Critical local smoothing for the three-dimensional wave equation

Proof. Failure of the first assertion provides s_k\to0 and \mathcal E_k\in\mathcal A_{s_k} with \mathcal B_{s_k}(\mathcal E_k)>s_k^{-\Gamma-\eps}. The resulting sequence is admissible in the fixed setup and has upper exponent at least \Gamma+\eps, contradicting (2.1). For the second assertion, take the maximum of the finitely many required constants at the remaining scales and 1.□

Proof. The adjoint V^* has norm one. Apply this fact to each P_jVg and then use orthogonality. The reconstruction assertion follows from V^*V=\Id. The assertion about copies follows by applying \norm{T_jg}\le\norm g separately to each j.□

Proof. The right side of the assumed inequality is at most N\max_jF(g_j)^{1/q}. Raising to the qth power proves the extraction estimate. Its logarithmic loss is q\log N_s/\log(1/s); finitely many such losses still tend to zero. For (2.4), F(g)^{1/3} is the norm of (w_\lambda^{-1/6}A_\lambda g)_\lambda in the corresponding \ell^3 sum of Hilbert spaces, and Minkowski’s inequality applies. For a countable label set, apply the inequality first to finite subsets and then use monotone convergence.□

Proof. A continuous coordinate function attains its extremum on a nonempty compact set. The set where that extremum is attained is closed in the same compact set and is nonempty. Induction proves the first assertion. For the alternative, pass first to a convergent subsequence of the bounded maximizing sequence. Closedness and continuity attain the supremum of g. At each subsequent step, pass to a convergent subsequence of the stipulated bounded minimizing sequence. Closedness preserves feasibility, and continuity of the coordinates preserves the preceding equalities and attains the next infimum. Each resulting slice is a nonempty intersection of closed sets.□

Proof. Pass to a convergent subsequence of (c_n). Its limit c is strictly less than c_0. Closedness places (z_0,c) in the minimizing slice, which is a contradiction.□

Proof. Choose one fixed m with r_m<\tau/2. The sequence in \mathscr S_m then contradicts the definition of \Gamma.□

Proof. For each conditioning value, sum the conditional probabilities of the at most N values below e^{-u}/N, and then integrate. This proves (2.6) and its normalized version. For K>C, the tail integral identity gives

For s\le1/2 the right side tends to zero uniformly as K\to\infty. This is uniform integrability; truncation at K proves the final assertion.□

Proof. For the marginal of any variable W, the likelihood ratio L_W=dQ_W/dP_W is P(E\mid W)/q, hence is at most 1/q. Moreover Q\{L_W<s^\tau\}\le s^\tau by integrating L_W against P_W. Apply these two facts to W=(X,Y) and W=Y. Except on the union of the two exceptional sets, both ratios belong to [s^\tau,s^{-\alpha}]. The conditional likelihood ratio is L_{(X,Y)}/L_Y, so it belongs to [s^{\alpha+\tau},s^{-\alpha-\tau}]. Taking logarithms proves (2.7). An old exceptional event of P-probability r has Q-probability at most r/q, proving the stated qualification and the consequences.□

Proof. Condition on Y and compare the joint law of (U,V) with the product of its conditional marginals. At actual samples their likelihood ratio is e^j, so integrating over the set where it is less than e^{-r} proves the tail bound. Therefore \E(j_-)\le1, while the definition of conditional mutual information gives \E j=I_P(U;V\mid Y). It follows that \E(j_+)\le I_P(U;V\mid Y)+1. Markov’s inequality and the negative tail bound prove convergence in probability after division by L.

Let B=\mathbf 1_E. The chain rule gives

The left side is the average, over the two values of B, of the corresponding conditional mutual informations. Its contribution from B=1 is P(E)I_Q(U;V\mid Y), proving (2.8).□

Proof. A base boost replaces X by X-x_0 and replaces the last base position by its appropriate linear tilt in B_t; Equation (3.1) then has the same form. A common normal shear subtracts a(F) from velocity and b(F)+ta(F) from position. On each hybrid fiber it is a Weyl translation, and the free propagator intertwines the two translations up to a scalar phase. The scalar phase can be included in the conjugated mask. If a time interval of length L is normalized to unit length, positions are divided by L, velocities are unchanged, and H becomes H/L. Dividing normal coordinates and velocities by F_0 changes H to H/F_0^2. Dividing base coordinates by (r_0,r_0,r_0^2) and normal coordinates by F_0 divides all weights by A=r_0^{3+\eta}F_0^{1-\eta} and multiplies the regularity constant by A. Energy is preserved by the corresponding unitary coordinate changes. Thus Equation (3.6) is unchanged.

The scalar transfer kernel is \ip{U_t^{H,\sigma}\phi_{D,N}^{\ell,\sigma}} {\phi_{D',N'}^{\ell',\sigma}}. Free propagation changes the first packet’s position width to at most Cq_\ell, since |t|H/q_\ell\le Cq_\ell, and leaves its Fourier width comparable to q_\ell^{-1}. Integrating two Gaussians, in physical space for the position difference and in Fourier space for the momentum difference, gives an upper bound of the form

where C_s and the Schur integration volumes have only fixed-power costs on the setup. Each of the two separate Gaussian bounds implies the displayed joint bound after reducing c. Schur’s test therefore makes the discarded kernel smaller than every power of s. A finite product and a polynomial sum preserve this conclusion. If no chain of the correspondences connects two row sets, the operator between those sets is negligible; this statement does not require a positive-kernel description of its main part.

At a repeated cut the absolute kernel is exactly

Its Schur norms are bounded independently of H,\ell. Cauchy–Schwarz against this kernel bounds output energy on a test by a Gaussian-weighted sum of input energies on its normal enlargements. Since f\ge d_\ell and \ell f\ge q_\ell, the enlargement in the jth Gaussian shell is obtained by multiplying the normal width by C(1+j). The resulting regularity bound is C\kappa w(Q)\sum_{j\ge0}e^{-cj^2}(1+j)^{1-\eta}. Exact base indices are never mixed. This proves the last assertion.□

Proof. Base indices move exactly and masks are contractions on each fiber. In the hybrid model, the pointwise packet bound gives a joint phase density at most C H^{-1} times the marginal base energy density. Lemma 3.4 propagates normal tails. The density and base-support bounds give, uniformly in shear parameters,

For f\le1 use joint density, and for f>1 use marginal base density. Dividing by r^{3+\eta}f^{1-\eta} shows that either sufficiently small width gives an arbitrarily small prescribed power: the factors are r^{1-\eta} and f^{1+\eta}, respectively. If neither width is small and one is sufficiently large, the total-energy bound suffices.

We record an elementary sublevel estimate to handle coefficients. For a nonzero real polynomial Q of degree at most two in d\le3 variables, on a fixed cube,

Here the coefficient norm is the maximum absolute coefficient. In one variable, three points in a set of measure m, separated by at least cm, and Lagrange interpolation show \norm Q_{\mathrm{coeff}}\le C\rho\norm Q_{\mathrm{coeff}}/m^2. For the induction, write Q as a polynomial in the last variable and choose a coefficient polynomial with coefficient norm equal to \norm Q_{\mathrm{coeff}}. Outside the set where this coefficient is smaller than \rho^{1/2}\norm Q_{\mathrm{coeff}}, the one-dimensional bound is C\rho^{1/4}; the exceptional base set has measure at most C\rho^{2^{-d}} by induction. This proves Equation (3.9), with harmless changes of constants.

For widths in a fixed polynomial range, discard negligible normal tails and clip base centers to a fixed-power box. The two normal shear polynomials have coefficient norm comparable, by a fixed-power change of coordinates, to the maximum of |c|,|d|,|\alpha|,|\beta(T)|,|K|. If that maximum is sufficiently large, at least one of their simultaneous sublevel conditions has arbitrarily small prescribed polynomial volume by Equation (3.9). The density bound handles its mass. Finally, m_\lambda^{3/2}/\sqrt{w_\lambda} =m_\lambda\sqrt{m_\lambda/w_\lambda}, so small ratios are summed against assigned mass, not against label cardinality.□

Proof. Subtract the test’s shear. Backward packet correspondences place its normal predecessors in root conditions of width s^{-o(1)}f, since every intervening d_{\ell'} is at most d_\ell\le f. Split its X-range into O(\ell^{-1/2}) intervals of width r\sqrt\ell. For each such interval, backflow of the two base position conditions is covered by a bounded number of root tests of base width r\sqrt\ell: the horizontal error is O(r\sqrt\ell) and the curvature error in the tilted coordinate is O(\ell r^2). Summing their weights gives

Contraction on this predecessor region and Lemma 3.4 prove the claim; polynomial test bounds absorb the negligible errors. Total output mass is at most s^{-1}e, so Equation (3.11) also gives \mathcal B_s\le s^{-1+\eta/4-o(1)}.□

Proof. Start above the fixed-power upper bound furnished by Lemma 3.5, and descend through dyadic thresholds. At threshold b, remove the entire remaining part of a test whenever its mass exceeds bw. Its mass is at most 2bw, by regularity of the preceding remainder. A polynomial lower bound on the allowed weights makes the removal procedure finite by total mass. The union removed at this threshold is dominated by that preceding remainder and is therefore 2b-regular. The unremoved measure is b-regular. Continue to the prescribed floor. The removed row sets are disjoint and their projections sum to the identity. Synthesis is a contraction, and Lemma 3.4 gives post-synthesis regularity.

For a fixed terminal assignment, its cubic readout to the power 1/3 is a weighted \ell^3 norm of Hilbert norms. Hence coherent decomposition costs at most a fixed power of the number of levels by the triangle inequality. That number is subpower, and at least one level preserves any extremal output exponent.□

Proof. A new terminal class has p'\ge p by minimality and d'=\Gamma+(p'-1)/2. Its assigned mass and the fixed weight count imply p'+d'\le p+d. Both inequalities force p'=p and d'=d.

For a non-remainder peeling level at \ell=s^{a+o(1)}, apply the prefix exponent to its raw heavy-test assignments. Lemma 3.8 gives

The tail exponent, applied separately to each interval and summed with its own post-energy, gives cubic readout at most

Combining this with Equation (3.15) recovers the full extremal upper exponent. Since the chosen level preserves that exponent, both estimates, and the synthesis energy bound, must be saturated in exponent. The floor remainder is negligible: its summed input energy is at most \ell^{-1}e, and a sufficiently small regularity floor makes the tail readout smaller by a fixed power.

Write b/\kappa=s^{P+o(1)}. The saturated prefix has mass-to-weight exponent P/a, so minimality implies P/a\ge p. For the tail first select a homogeneous dominant terminal mass class, then an interval whose normalized readout is near maximal. That tail realizes \Gamma in the closure and has mass-to-weight exponent (p-P)/(1-a); hence (p-P)/(1-a)\ge p. The one-time bound bounds P on the fixed gap, so these are legitimate finite closure data. Thus P=pa, and saturation in Equation (3.15) gives z=e s^{-da+o(1)}.

For a sublevel, use the unpartitioned level in the prefix argument. It bounds the sublevel’s raw energy from above. The preserving tail bounds its post-energy from below by the same exponent. At the root, the given input regularity and energy replace the prefix estimate. At the last cut a repeated-cut regularity estimate replaces the tail.□

Proof. Subtract the first shear and center its base chart. The intersection imposes, on bounded (X,B,Y),

where \max(1,|\alpha|,|\beta|,|k|)\asymp M. If \max(|\beta|,|k|)\gtrsim M, multiplying the first expression by \beta and subtracting k times the second gives

On a dyadic shell R\asymp\max(\sqrt M,|2t+\alpha|), its admissible t-set lies in at most two intervals of length O(M\delta^{-1}/R). In the lowest shell the same conclusion follows from proximity to the quadratic vertex. Since \max(|\beta|,|k|)\gtrsim M, the horizontal set is covered by O(\delta^{-1}R) squares of side 1/R.

In such a square use the coordinate Y-2x_{\mathrm{cell}}B. After the vertical terms are removed, the horizontal derivatives of the two expressions are O(R+M/R), so their oscillation across the square is O(1) because R\ge\sqrt M. At least one vertical coefficient has size comparable to M. Therefore O(\delta^{-1}R^2/M) vertical intervals of length R^{-2} suffice. The intersection is covered by O(\delta^{-2}R^3/M) Heisenberg base cells of width O(1/R), together with the first normal conditions enlarged by O(\delta^{-1}). Regularity bounds their total mass by

There are only logarithmically many shells on a polynomial setup.

Otherwise |\alpha|\asymp M and |\beta|,|k| are sufficiently small relative to M on the bounded base box. For fixed Y, the second expression has a B derivative comparable to M, and the first then has an X derivative comparable to M. The volume is O(\delta^{-2}M^{-2}). Its Heisenberg neighborhood of width 1/M obeys the same estimate, since the expressions change only by a constant there. It can consequently be covered by O(\delta^{-2}M^2) cells of weight O(M^{-3-\eta}), proving the same bound. Bounded M needs no subdivision.

For the partner count, every intersecting partner in the shell has its whole projected footprint in a bounded enlargement of the first base box, and in normal width O(\delta^{-1}M_0) about the first shear. This follows by extending the difference polynomials from an intersection point across bounded base coordinates. At each descendant time, contraction and starting regularity bound the total mass of these assigned rows by s^{-o(1)}\kappa_i w_j O(\delta^{-1}M_0)^{1-\eta}. Divide by the assumed mass per label and sum over at most O(\delta^{-1}) descendant times.□

Proof of Proposition 3.10. Read mass at a gate. First observe that the asserted mass lower bound follows whenever the gate and weight count have been constructed. Insert an ordinary split immediately after the gate in a thought experiment preserving the terminal exponent. Proposition 3.9 requires summed post-energy s^{o(1)}E_i, and contraction bounds it by the raw gate mass. At the final cut use the fixed terminal count and minimality of p. At the root, repeated-cut regularity is s^{-o(1)}\kappa, so reducing post-energy by a fixed power would lose the terminal exponent by the full tail bound.

Removing incompatible rows. Construct the gates backwards. Suppose the j-gate is fixed and set a=j-h, \delta=s^h. At a>0 make an ordinary preserving split; at a=0 use the root state. The resulting states g_a have total energy at most s^{-o(1)}E_a and regularity at most s^{-o(1)}\kappa_a. At j their row measure is at worst s^{-o(1)}\kappa_a-regular by Lemma 3.7. Delete labels whose newly read mass is less than \delta^D\kappa_aw_j. Their total mass is at most

After synthesis they remain s^{-o(1)}\kappa_a-regular. Their tail readout, relative to the required scale E_j\sqrt{\kappa_j} at j, therefore loses at least \delta^{D-3p/2-o(1)}. Since p<3, an absolute D makes the deletion harmless. At j=1, use the output ratio bound directly.

Every retained descendant test projects to a starting test Q^-_\lambda with base width O(r_j) in \ell_a position units and normal width O(\delta^{-1}f_j). Base backflow follows from the quadratic identity; normal backflow follows from Lemma 3.4, taking its truncation exponent smaller than h/10. Each such projected test contains every predecessor on the allowed correspondence chains. All label counts here are polynomial by their weight count and the polynomial lower bound for w_j.

Call a starting row ambiguous if it lies in two projected tests with M>\delta^{-D_0}. Lemma 3.11, its partner count, and the j-gate weight count bound its total mass by

Choose D_0=D_0(\eta) large enough that this is smaller than E_a by a fixed power of \delta. Restricted rows retain regularity after synthesis, so their tail is negligible by the model exponent. Remove them and also rows belonging to no projected retained test. The latter have negligible transfer to the j-gate. Both removals use one row projection followed by synthesis, not successive assumptions about disjointness of synthesized states.

Shear charts. Partition the remaining rows into charts G. First record a lattice Heisenberg base cell of width r_j. Choose one attached test in a fixed order and round its five shear data at that cell center: \beta(T_a), K, P, and its two normal values. Round in normal units

The five data determine the polynomial by a triangular change of coordinates. Nonambiguity makes every attached test’s data lie within a bounded number of these rounded values. Thus each j-label meets only boundedly many charts. In each chart, after subtracting the common rounded shear, base and normal support are bounded in (r_j,f^*,\ell_a) units, and every attached j-test has bounded coefficients in those units.

Full chart regularity. The charts now have compatible geometry. To apply the model exponent to their synthesized states, we also need full regularity. The next peel gives regularity on a bounded range of tests; the subsequent reanalysis, smoothing, and cutoff estimates extend it to the full test family.

Within each chart perform a truncated peel: normalized widths belong to [\delta^{D_1},1], with the Planck floor retained, and normalized shear coefficients have absolute value at most \delta^{-D_1}. Centers are unrestricted. The physical weight multiplier is w^*=r_j^{3+\eta}(f^*)^{1-\eta}. Keep one density and width class coherently across all charts; there are subpower many classes, so some class preserves the output. Let b_* be its density bound, or its floor if it is the remainder. Then b_*\le s^{-o(1)}\kappa_a, the raw mass in each chart is O(b_*w^*), and the synthesized chart energies satisfy \sum_G e_G\le s^{-o(1)}E_a.

We justify applying the full model exponent with regularity b_* to each chart. Same-cut Gaussian reanalysis preserves its truncated regularity: normal shell enlargements can be covered by translates at unchanged widths and coefficients. Smooth the input and the entire subexperiment by independent base boosts, base position translations, and the two normal translations, each of size \delta^l, where l>D_*+10. Retain the smoothing parameter as an exact auxiliary index. These are exact symmetries. The smallest attached output accuracy is \delta^{1+D_*}, so assigned masses persist on bounded enlarged tests. In normalized units the smoothed base and joint densities are at most

This follows by averaging translations, whose coordinate Jacobians are one, and using the raw chart mass bound. Base support is bounded; normal support outside s^{-o(1)} is negligible by Gaussian tails.

Here are explicit choices showing that the fixed density cost disappears from the regularity estimate. Choose D_2 with (1-\eta)D_2>5l+1. Equation (3.8) gives

Thus any width below \delta^{D_2} is harmless. Widths above one are handled by bounded base covering or, for the normal width, by dropping the normal conditions and covering the s^{-o(1)} normal support with unit translates. For widths in [\delta^{D_2},1], choose D_3 so that 2^{-3}D_3>5l+4D_2+1. Equation (3.9) makes tests with a coefficient above \delta^{-D_3} harmless, even after dividing by their minimum weight \delta^{4D_2}. All remaining tests pull back under each jitter to permitted truncated tests, provided D_1>D_2+D_3+10. For them the original truncated regularity bound is used directly and incurs no factor \delta^{-5l}. This proves full regularity s^{-o(1)}b_*w^* in normalized units. Tiny-energy chart pieces use Equation (3.10); the others satisfy the polynomial setup relative to their own energy.

Saturation and induction. The model exponent and the bounded chart overlap now bound the coherent cubic readout at j by

Indeed, for a label reached by at most C chart states, \norm{\sum_{G=1}^Cv_G}^3\le C^2\sum_G\norm{v_G}^3. The already established read-mass lower bound and weight count, by Hölder, give the lower bound

Consequently b_*=\kappa_a s^{o(1)}, and the floor remainder can be excluded. Use the heavy tests in the selected truncated level as the new documented labels. Their total weight is at most s^{-o(1)}E_a/\kappa_a. Their truncated widths and coefficients, together with chart compatibility, give the width bounds and Equation (M). The constants depend only on the choices above, hence only on \eta, not on the number of mesh steps. Any reachable j-test is attached to the actual supporting row of the new gate; no assertion about its mass in an earlier state is used.

This completes the backward induction. Finally insert ordinary regularization before the gates in forward order, preserving the terminal exponent at every step. Proposition 3.9 gives the desired benchmarks. At the final cut its prefix minimality and the repeated-cut regularity bound give the same conclusion; the root uses its given bounds. Additional finite row statistics may be flattened in this same forward order. Dyadic upper endpoints may be clipped at r_j, so repeated rounding does not create exponential drift in r_i/r_j\le1. Constants per step disappear before the mesh limit because every inner mesh is finite.□

Proof. The group energies sum to at most s^{-o(1)}E_i, and each group has regularity s^{-o(1)}\kappa_i after reanalysis. Hölder, gate count, and read mass give cubic output at least s^{o(1)}E_j\sqrt{\kappa_j}. The overlap assumption compares this to the sum of the group readouts with subpower loss. Each group readout is at most s^{(1/2-\Gamma)(j-i)-o(1)}e_G\sqrt{\kappa_i}. Thus near-extremal normalized groups must exist.

Summing reachable weights over groups costs at most s^{-o(1)}E_j/\kappa_j. Groups violating the displayed relative count with an extra factor s^{-\eps} have total energy at most

Their readouts are negligible by the group upper estimate. Delete them, then extract a near-maximal remaining group and let \eps\downarrow0. Its terminal homogeneous class has exponent p by the same minimality and count argument as Proposition 3.9. Finitely many intermediate count deletions are handled in the same way.

Only polynomially many groups are relevant: there are polynomially many labels and at most subpower many groups reaching each. Remove globally unreachable groups as one complementary row projection. Negligible transfers are then estimated with polynomial group and test counts. The fixed-threshold argument following Equation (3.10) discards tiny energies before the uniform exponent is applied. A loss s^{-\eps'} from a later outer limit is allowed only when \eps'/(j-i)\to0; choose larger vanishing slack in the count deletions. These observations ensure that extraction produces genuine polynomial subexperiments in the closure, not a formal limit object.□

Proof. Admissibility gives the upper bound one. Suppose the ratio is at most s^\xi for a fixed \xi>0. Set a=1-h, with h>0 sufficiently small relative to \xi,\eta, and \delta=s^h. Apply the deletion and chart partition in the proof of Proposition 3.10 on [a,1]. Each output has bounded chart overlap; summed chart energy is at most s^{-o(1)}E_a, raw chart mass is at most s^{-o(1)}\kappa_aw^*, and regularity down to the root Planck floor is at most s^{-o(1)}\kappa_aw^* in chart units. The normalized output normal width is \delta^{D_*}, whereas the final packet velocity width is smaller by s^{\xi/2}.

Jitter as in the preceding proof. The averaged initial row measure can now be regarded as classical input regular down to width zero. Above the hybrid floor use hybrid regularity. Below that floor use the density estimate s^{-o(1)}\kappa_aw^*\delta^{-5l} and Equation (3.8). The fixed positive gap \xi makes the tiny-width gain dominate \delta^{-5l} when h is sufficiently small. All input cutoffs then hold classically for non-tiny chart pieces.

We spell out why arbitrary hybrid masks inside the segment do not invalidate the classical upper comparison. In chart units use a normal phase lattice of side \zeta=\delta^{D_*+3}. At a final time t, bin analyzed rows by (N',D_t'-tN'). Along the fixed stack, all packet drifts in these units are much smaller than \zeta if h is small enough. Thus, at every fixed exact index, contraction and localization bound the output mass in one lattice cell by a constant times initial row mass in finitely many neighboring cells, plus negligible fiber energy. Only polynomially many cells matter.

Reduce the assigned mass in each cell by its common excess factor so that it fits this neighboring initial mass. The reduction loses only the negligible error after summing cells and integrating exact indices. Divide once more by the bounded neighbor-overlap constant and allocate the remaining assigned masses on those initial cells, separately for each final time. These are legitimate classical assignments: exact base indices have the same endpoint, and a neighbor cell arrives within O(\zeta) in normal phase, less than the output positional tolerance \delta^{1+D_*}. Hence it lies in a bounded enlargement of the original assigned test. Keep the jitter parameter among the exact indices throughout. This is an energy comparison, not a positive packet identity for a masked quantum evolution.

Applying the classical exponent chart by chart bounds the retained hybrid cubic readout by s^{-o(1)}\delta^{1/2-\Gamma_{\mathrm{cl}}}E_a\sqrt{\kappa_a}. Saturation requires s^{o(1)}\delta^{1/2-\Gamma_{\mathrm{hyb}}}E_a\sqrt{\kappa_a}, a contradiction. The prefix ending at any documented i>0 also realizes \Gamma,p, by its read mass and count, so the same argument applies there.□

Proof. Apply the one-step chart partition at the root and Lemma 3.12. Subtraction of the full chart shear bounds all reached leaf coefficients by fixed powers of s^{-1} in leaf units. This proves finite feasibility for c. When approaching finite c, the same chart partition can be used without subtracting its quadratic terms. After a base boost and translation, the absolute quadratic coefficients are still bounded by fixed powers in leaf units: conversion from s|K| to full-duration K costs at most s^{-1}. Subtracting only the linear part then gives finite feasibility for c_1 when needed. Once c_1 is minimized, that linear subtraction is unnecessary, since the leaf-center derivatives and the bounded quadratic coefficients already bound the linear term. Constant normal translations affect neither cost. Base boosts and translations preserve R_2 and the corresponding ray derivative P. Each output knows boundedly many charts, so all these extractions retain the relative count bound.

After r_1=f_1=1, Proposition 3.10 gives a uniform Lipschitz bound for the interpolated width exponents and their endpoint values. Arzelà–Ascoli on finer meshes gives Equation (3.19). Equation (Pl) at each positive depth, compared with its value at depth one, yields n(i)=(1-i)/2 in the strict hybrid case, and continuity handles zero. Telescoping Equation (M) proves Equation (A). For the first inequality evaluate each earlier affine coefficient at the later time T_j, which lies within its interval; the corresponding K error is multiplied by at most that interval’s length. A finite mesh contributes only O(L_*\,\mathrm{mesh})+o(1) to the exponent. Remove labels on no full geometric chain, whose transfer is negligible.

For Equation (B), work first on a fixed fine mesh, with a deletion tolerance larger than several mesh-error constants. Suppose the part of the i-gate violating the bound could preserve the terminal exponent. Ordinary-regularize just before that gate. Its total raw mass is at most s^{-o(1)}E_i. At the next depth j, keep only labels that can continue to a leaf in the graph imposed by the two curvature comparisons in Equation (M). Their total read mass is at least s^{o(1)}E_j.

For any predecessor on the high part, the leaf bound and telescoping force its affine trajectory to satisfy

at some terminal time T_1 within the reached j-interval. The exponent \mathrm{err} is smaller than the deletion tolerance. An affine function whose value or duration-scaled slope violates Equation (B) either never enters this range or enters it on an interval much shorter than \ell_j. Thus each high input row predicts only O(1) possible T_j values. For each T_j, restrict the input to the rows predicting it and apply contraction; all other transfers into the kept rows are negligible. Summing gives total read mass at most s^{-o(1)}E_i, contradicting E_j/E_i=s^{-d(j-i)} and d>0. Therefore the low part preserves the output. Make this deletion at every mesh depth, then pass to finer meshes. Strict tolerances and the ordered closure limits justify the same conclusion for limiting extremal configurations.□

Proof. Write b=b_2. At a positive depth i where b(i)=\min_{[i,1]}b, the prefix realizes \Gamma,p by read mass and count. Multiplying Equation (B) by r_i^2/f_i=s^{-b(i)+o(1)}, its normalized terminal curvature cost is at most (b(i)+c)_+/i. Minimality of c therefore implies

Since c>0 and i>0, this yields b(i)\ge c(i-1) at every positive future minimum. For any other depth, a later future minimum gives the same lower bound, because c(i-1) increases with i; continuity handles zero.

Comparison with root labels using Equations (A) and (B) gives

Suppose first 0<c<1 and at some interior i the shape inequality is strict. The lower comparison line for b(k)-k is strictly decreasing. By continuity, Equation (3.20) improves \ell_i|K_i|\le s^{c(i-1)-o(1)} by a fixed power at this i. The minimum of b on [i,1] likewise strictly exceeds c(i-1). Consequently, for some \eps>0, every connected leaf satisfies

At the regularized i-gate, bin \beta_i(T_i) at that precision and subtract the rounded constant quadratic shear. Every reached leaf knows only boundedly many bins. Group extraction gives an experiment on [i,1] at \Gamma,p whose terminal curvature cost, after dividing exponents by 1-i, is strictly smaller than c. This contradicts its minimality.

For c>1, a strict shape inequality at i instead gives

since its lower comparison line is strictly increasing. Equation (A) then compares the pair (\beta(T_i),K) for the i-label and a leaf to precisions s^{c(i-1)+\eps} and s^{c(i-1)-i+\eps}, respectively; extrapolating \beta to T_i costs at most \ell_i times the K error. Bin this jointly predictable pair and subtract its polynomial trajectory. Group extraction again lowers c, a contradiction.

Now let c=0<c_1. A future minimum of b_1 is also a future minimum of b_2=b_1-u, since u is nonincreasing. At such a depth, Equation (B) makes the normalized prefix curvature cost zero. On a connecting chain the center derivatives at depths i and 1 differ by at most s^{\underline b_{1,i}-o(1)}, where \underline b_{1,i}=\min_{[i,1]}b_1. To see this, first compare them at the actual ray by Equation (A). Moving to the center at i costs at most the curvature bound times r_i; its exponent is at least \underline b_{1,i} because

The leaf-center change has zero cost. Lexicographic minimality at positive future minima therefore gives b_1(i)\ge c_1(i-1), as in the curvature argument. At any strict interior point, bin and subtract the jointly predictable linear coefficient. The extracted tail keeps zero curvature cost while strictly lowering c_1. Thus equality holds everywhere.

Both b_1 and b_2=b_1-u are now nondecreasing. Equation (B) therefore bounds curvature at depth i by s^{b_2(i)-o(1)}. Taylor expansion at the base center leaves the coefficient P_i(x,b) times the increments in X and B_{T_i} in the respective normal conditions. Every remaining term is quadratic in horizontal increments or linear in the tilted vertical increment, so has size at most s^{-o(1)}f_i in velocity and s^{-o(1)}\ell_i f_i in position. The existing row supports can consequently be kept.□

Proof. Choose an interior common differentiability point of the Lipschitz profiles and set J=-u' and q=b_2' when relevant. If c=1 and the shape is not the equality line, there are points with q<1: otherwise absolute continuity and the endpoint value would contradict the strict gap somewhere above i-1. If the shape is the equality line, take q=1. Take shrinking segments [A,B] about this point, put h=B-A and \Delta=s^h, and normalize duration and terminal widths to (\ell_A,r_B,f_B). For each fixed h, take the mesh fine enough and all preceding closure errors small enough that their exponents are o(h). Differentiability then gives Equation (F) uniformly in the normalized depth. It remains to justify extracting actual experiments without losing the costs or compatibility.

At the ordinary-regularized A-gate partition rows by a containing lattice base chart in r_B,\ell_A units, and boost and translate to its center. Every later connected label at depth l\le B knows only s^{-o(h)} possible charts, since r_l\le s^{-o(h)}r_B and its base backflow is bounded in those units. Use the following additional bins, written temporarily in the old physical width exponents:

1. (i)

   If c>1, bin (\beta_A(T_A),K_A) at widths s^{b_2(A)}, s^{b_2(A)-A}, and subtract the rounded polynomial shear. Equation (A), extrapolated to T_A, makes these bins known to every connected later label up to s^{-o(h)} choices.

2. (ii)

   If 0<c<1, bin only \beta_A(T_A) at width s^{b_2(A)} and subtract that constant quadratic coefficient. The bounds for K in Proposition 3.15 make every later own-time coefficient predict the bin.

3. (iii)

   If c=1 and q\le1, bin the pair at widths s^{b_2(B)+A-B} and s^{b_2(B)-B}, and subtract the rounded trajectory. On this segment b_2(k)-k is affine nonincreasing up to o(h), so Equation (A) again gives the required prediction.

4. (iv)

   In the linear alternative subtract no curvature. Bin the A-label’s center derivative at width s^{b_1(A)} and subtract that linear shear; Proposition 3.15 supplies the joint prediction from every connected later label.

The subtractions are made after centering and include free constant terms. All residual curvature data have bounded-power size in \Delta units. For example, in case (iii), the uncertainty of K contributes at most s^{b_2(B)+A-B-o(h)} to the terminal quadratic coefficient, and at most s^{b_2(B)-o(h)} to its terminal duration-scaled K term. This gives terminal cost at most one, and when q<1 yields precisely s|K_l|\le s^{-o(1)} after normalization. Cases (i), (ii), and (iv) give the other displayed bounds directly. Their terminal cost is at most its prescribed minimum; minimality prevents a strict improvement.

When c>0, also bin and subtract a linear shear at a sufficiently large-power tolerance in \Delta units. This has no effect on the curvature costs. Indeed compare derivatives first on the connecting ray by Equation (A); \min b_1 differs from b_1(B) by O(h). The already bounded residual curvature permits recentering at the common base chart, still at a fixed-power cost. Every later label therefore predicts the additional bin up to s^{-o(h)} choices. In the linear alternative these derivatives are already bounded. Finally bin the actual residual normal phase at A at a sufficiently large-power tolerance and remove its constant velocity and position. Base backflow, bounded residual coefficients, and bounded packet drifts make each later reached label predict these bins with the same small multiplicity. The starting labels themselves also meet only that many groups.

Apply Lemma 3.12, including the relative count conditions at the finitely many intermediate depths. Transform the selected experiment by the indicated symmetries and remove labels unreachable from the selected group. Negligible errors are evaluated on each inner polynomial setup before its outer limits. Gate counts are now relative to the extracted experiment’s own energy and inherited regularity. Its raw starting mass is at most that regularity times a fixed power of \Delta^{-1}, by its bounded support; synthesized tails are negligible. Its energy is at least that regularity times an inverse fixed power for a near-extremal extracted group: otherwise contraction onto the normalized contributing output tests would make its normalized readout negligible by Equation (3.10). Thus all cutoff requirements hold after normalization.

If a documented root gate is needed, restore it on the extracted post-state. Classically keep the old row ownership. In the hybrid case reanalyze and retain rows having a nonzero original group row, in the same exact fiber, within s^{-\eps} packet widths. Use essential neighborhoods, defined by positive fiber mass in the relevant box, and assign one witnessing original label. Gaussian localization makes projection and synthesis approximate the identity on this input up to negligible error. Enlarge the root normal widths accordingly. Choose \eps=o(h) smaller than the truncation slack at the first mesh step. Every connection from the restored gate then extends geometrically to an original supporting row, so the original compatibility estimates remain valid with vanishing normalized errors.

The mesh in the normalized depth may now tend to zero, always after the inner scale limits. In relative count selections take vanishing \Delta-exponent slack larger than the total overlap and extremality errors. This proves realization of \Gamma,p and the same terminal cost minima in the closure. All arguments concerned finite masks and finite meshes; no infinite product of operators has been taken.□

Proof. Restrict synthesis at i to S. Its summed input energy is at most s^{-o(1)}\Prob^i(S)E_i, and its regularity remains at most s^{-o(1)}\kappa_i in each interval. By locality, its output on U agrees with the original output up to negligible error: input outside the active essential support has zero state. The tail exponent bounds its cubic readout by

Hölder and the j-gate weight count give

The scale powers cancel by 2d=2\Gamma+p-1. Divide by Equation (3.21). Superpolynomially small input pieces are treated by Equation (3.10); polynomial weights and counts absorb the negligible operator errors. At final readout the restriction to U means partial assignment, so the same argument applies.□

Proof. Compute the probabilities under the gate’s preselection law, and bin their nonnegative \log_s scores. Scores beyond a sufficiently large power occur on power-small mass, by the finite-name capacity bound, also after conditioning on y. Their contribution cannot preserve the terminal exponent: restrict to those rows and use the tail exponent, or Lemma 3.18. There remain subpower many bins for each fixed finite request. The coherent triangle inequality selects a class preserving the output. Such a class has probability s^{o(1)}, since a power-small fraction would again lose the output exponent. At final assignment one uses (\sum_{b=1}^k m_b)^{3/2}\le k^{1/2}\sum_{b=1}^k m_b^{3/2}.

Here is the required change-of-law calculation. If Q=P(\,\cdot\mid A) with P(A)=s^{o(1)}, then for any measurable name or conditioning field F,

The first inequality follows from Q(B)\le P(B)/P(A); the second follows by integrating r_F over the indicated event. Write D for one prescribed name and C for its finite-valued conditioning data. Apply these inequalities to (y,C,D) and (y,C). Their ratio is the likelihood ratio for the conditional probability of D given (y,C). Its \log_s is arbitrarily small outside a power-small event. Thus the scores selected before conditioning have the same limiting values under the retained law. This is also Lemma 2.11.

Make later gate choices only after earlier ones have been realized. Countably many eventual requests mean increasing finite lists followed by subsequences; they do not mean infinitely many projections on one state. Dense-query extension is legitimate only when the corresponding names have explicitly verified small-list relations as the query parameters approach one another.□

Proof. Fix \eps>0. At each (y,C_j) the set of D_j atoms of probability greater than s^{z-\eps} has cardinality at most s^{-z+\eps}. Pulling this list back through the conditioning and observed-name lists gives at most s^{-z+\eps-o(1)} possible D_i values per (y,C_i). Outside the exceptional predecessor set, each such value has conditional probability at most s^{z-\eps/4}. Their union therefore has power-small \Prob^i mass. Lemma 3.18 makes the corresponding large-atom event power-small at j.

For the other direction, the good D_i atoms of conditional probability at least s^{z+\eps/4} have at most s^{-z-\eps/4} values per (y,C_i). Forward their observed-name lists and use the backward conditioning lists. The resulting set has at most s^{-z-\eps/4-o(1)} D_j values per (y,C_j). Missing this set requires an exceptional predecessor and is power-small by Lemma 3.18. Within the set, atoms below s^{z+\eps} have total conditional probability at most s^{3\eps/4-o(1)} by counting. This proves the claim. The same proof within one law shows invariance under changes of discretization with the stated list bounds.□

Proof. Mark j-rows whose relay predicts a target datum whose proposed time list contains their T_j. An i-row can reach marked rows at no more than N=s^{-d(j-i)+\eps-o(1)} possible times. For each such time restrict the input to its possible predecessor rows, and use contraction and locality. Summing squared norms over times costs at most N per input row, so marked mass at j is at most

If l>j, apply Lemma 3.18 to transport this exceptional event; the final probability is at most s^{2\eps/3-o(1)}. For l=j divide directly by Equation (3.21). To obtain the conditional score claim, take the proposed list to be the time atoms larger than s^{d(j-i)-\eps}; there are at most the stated number. Finite grids and removal of the resulting exceptional events give the corresponding prefix mass bounds. Availability of the relay at the tested prefix is essential to the input multiplicity estimate.□

Proof. Write z=\log_{1/s}(E_i/(\kappa_iw)). A list of at most s^{-z+\eps} names has predecessor mass at most s^{\eps-o(1)}E_i by regularity and the covering hypothesis. Lemma 3.18 makes its probability power-small on the later law. Apply this to the list of atoms larger than s^{z-\eps} to obtain the lower score bound.

For the reverse bound, the i-gate weight count and the prediction hypothesis cover all reachable later rows by at most s^{-o(1)}E_i/(\kappa_iw_i)=s^{-z-o(1)} names. Any remaining rows have negligible probability by locality and Lemma 3.18. Within a list of that size, atoms smaller than s^{z+\eps} have total mass at most s^{\eps-o(1)}. This proves equality.□

Proof of Lemma 3.19. Only finitely many cuts and queries occur at this inner setup. Keep the exact-index spaces themselves in the row code, together with the Euclidean phase, time, and finite label coordinates; a standard Borel space admits an injective Borel realization in a Polish space. Thus equality of a preserved exact index is a Borel diagonal condition and distinct indices are never identified. Use the measurable separable-Hilbert coordinates of the direct integral to represent the finitely many operator masks. Their countably many matrix coefficients have Borel versions under the reference measure; on the common exceptional null set set the mask to zero. This leaves the pointwise contraction and the multiplication operator unchanged. At every fixed cut and interval, take a Borel measurable-field representative of the analyzed state and let H_i be its Borel positive-norm set. The full analyzed row-energy law is concentrated on H_i. Changing the representative on a reference-null set, or inserting \mathbf1_{H_i} before synthesis, leaves the physical state unchanged. In an i-to-j predecessor test use the current input state’s H_i and the marked end event as vertex conditions. At intermediate cuts use the Borel masks and geometric correspondences, not the original state’s support: a restriction at i can undo cancellation there. If a support condition for a particular propagated component is needed, recompute its Borel representative after that physical restriction. Thus ambient zero-state starting rows are not witnesses, without omitting possible intermediate transfer from the actual restricted input. Fractional assignment densities relative to the actual row measures likewise have Borel versions and give the same assigned measures. Finite time ownership, test inequalities, exact-index equality, and the truncated Gaussian correspondences in (3.7) are Borel in the code. Their finite intersection in the full path space is therefore Borel.

Projection of this Borel relation is analytic. Analytic and coanalytic sets are universally measurable, so completion of the active Borel law gives Borel inner and outer versions of equal mass. For finitely or countably many laws use a weighted sum of their normalizations to choose common versions. In particular, taking the inner version of the marked end rows preserves each retained row’s literal witness. Intersect this inner version with the current end-state H_j, which is Borel and has full end-row energy. Taking the outer version of their predecessor set preserves inclusion of all geometric predecessors. One may also choose a Borel inner predecessor version S^-\subset S of equal full analyzed input mass. The difference S^+\setminus S^- carries zero current pre-synthesis input state, so synthesis restricted to S^+ or S^- is identical. This last equality concerns the actual state, not an arbitrary probability law on completing paths; a witness may have zero conditional mass.

When finitely many witness bins are used, first form the analytic possible-end-row set for each bin. Lexicographically disjointize them, take Borel inner versions under the full analyzed end law, intersect each with the current end-state H_j, and then project the unrestricted starting-to-end geometric correspondence for each disjoint marked group. Take Borel inner and outer predecessor versions S_b^-\subset S_b\subset S_b^+ of equal full analyzed starting-row energy. If path concatenation places S_b in a Borel allowed starting-bin set P_b, intersect both versions with P_b; the original starting-bin partition and its overlap count are unchanged. Use S_b^+ for pointwise locality and S_b^- for the physical state and input-energy accounting. Every kept end row still has a witness in its assigned bin. A later physical restriction uses its Borel version in a newly formed finite-path relation. If an approximate inner version is convenient, losing row energy at most \xi^2 changes the restricted input state by at most \xi in norm; the following contractions preserve that bound. Choose \xi smaller than every required power of s before the exponent limits, so polynomially many descendants and tests add only negligible errors. Finally, choose joint versions under the side-row law and disintegrate. This proves almost-everywhere, not pointwise, conditional assertions.□

Proof. For the scalar case, apply the cited Theorem 1.3. Here the enlarged cap in the displayed formula accommodates the bounded overlap of the smooth partition. For completeness, the Hilbert-valued extension introduces no dimension-dependent loss. In a finite-dimensional subspace apply the scalar theorem to \ip{F}{G}, where G is a standard complex Gaussian vector. Its fourth moment is a fixed constant times \norm{F}^4. Moreover,

Average the scalar inequality and use this bound inside each double integral on the right. Orthogonal finite-dimensional projections and monotone convergence give the separable case. The weighted versions follow by summing translates with rapidly decreasing coefficients; changing an adapted smooth cutoff amounts to convolution on its dual scale and gives the same summable enlargement. For a scalar formulation localized by a decaying spatial weight, see also [13, Theorem 1.5].□

Proof. Set L=\delta^{-4} and form the L^2(\mu)-valued function

Choose Schwartz functions \chi,\phi with sufficiently small compact Fourier support. Require \chi to be bounded away from zero on the time interval under consideration and \phi to be bounded away from zero near zero. The three packet coordinates show that

Consequently \rho-\xi^2/(4\zeta)=O(1), so Theorem 4.1 applies in the scaled coordinates just described. Spatial integration in the two packet coordinates gives \norm{F}_{L^2}^2\lesssim E.

A fine angular projection has a uniformly smooth multiplier in these rescaled frequency coordinates. Thus its packet components vanish unless X belongs to a bounded enlargement of that fine angular sector, and they retain arbitrary rapid decrease in the same spatial packet coordinates. We claim that for an R-cap and any of its tiles,

Only X within O(R) of the cap center x_0 occurs. A ray meeting a tile near time t_0 backflows into horizontal width O((1+|t_0|)R) and tilted vertical width O((1+|t_0|)R^2). Indeed its tilted vertical velocity relative to the central ray is (X-x_0)^2=O(R^2). These enlarged boxes can be covered by a polynomial number in 1+|t_0| of the boxes in the hypothesis. The time decay of \chi absorbs that polynomial. For packets not meeting the tile, sum spatial shells in packet units: their energy decreases faster than any prescribed power, whereas the number of regularity boxes needed in the shell grows only polynomially. Integrate the normalized spatial density of each packet first, and then integrate against \mu. This proves the claim, including weighted tiles.

At a fixed cap scale the sum of these tile integrals is O(E) by overlap. Since |U|\asymp R^3, Equation (GWZ) gives

For a square counted at time t, throughout a sufficiently short interval of length comparable to \delta the packet cores of all counted rays lie in a fixed enlargement of the square. The integral of \norm{F}^2 over that enlargement is therefore at least a constant times the counted mass. Cauchy–Schwarz in a spatial region of area O(\delta^2), followed by integration over the short time interval, contributes at least c\delta^{-1}\mu\{Q_z(t)\in D\}^2. Bounded overlap in space and time proves Equation (kin).□

Proof. We first prove the free estimate. The density bound controls the mass of a small test by s^{-C}ef^2; for very large f use total mass. Consequently widths and reciprocal widths outside a sufficiently large polynomial range contribute an arbitrarily small power, by summing assigned mass. Split the remaining widths into O(1+\log(1/s)) dyadic classes. Normal dilation reduces each class to f=1; the change in the weight is canceled by the inverse change in the regularity constant. In the hybrid case admissibility now gives H\leq\tau. Each test is contained in boundedly many lattice cells of dimensions 1\times\tau. Splitting assignments between these cells and combining assignments in the same cell reduces their square sum to the square sum of the cell masses, up to an absolute constant.

For classical free evolution, express the latter square sum as a collision count against \mu_0\otimes\mu_0. Pairs with velocity separation comparable to r, \tau\leq r\lesssim1, can occupy a common cell at only O(r^{-1}) sample times. Such a collision also requires initial position separation O(r). For each first ray the regularity hypothesis bounds the mass of the second rays in question by Cb_0r^{1-\eta}. The contribution of this separation scale is at most Cb_0er^{-\eta}. Pairs separated by less than \tau use at most \tau^{-1} times and lie initially in a box of width O(\tau); their contribution is at most Cb_0e\tau^{-\eta}. Sum the logarithmically many scales. Velocities in a common unit cell differ by only a fixed constant, so there are no additional separation scales.

We give the hybrid proof in detail. If \tau=s^{o(1)}, packet locality and contraction bound the mass of each output cell at one time by s^{-o(1)}b_0. Its input predecessors lie in a root regularity box of width s^{-o(1)}, since d_\tau\leq1. Sum cell masses and the \tau^{-1}=s^{-o(1)} times. Hence suppose \tau has positive limiting depth, and use \tau as the inner scale. All cutoff powers remain finite. Fix an arbitrarily small \eps>0. Every loss below is \tau^{-C\eps} with a fixed C; tails outside \tau^{-\eps}-enlarged adapted neighborhoods are made negligible before \eps is sent to zero.

Partition the input by smooth compactly supported unit Fourier-velocity caps with bounded overlap. Their energies e_{\rm cap} sum to O(e). The Gaussian final analyzer sees, in any unit output velocity window, only caps within distance O(\tau^{-\eps}), modulo negligible operator norm. Thus Cauchy–Schwarz loses only \tau^{-O(\eps)} on reducing the square sum to the contributions of individual caps. Gaussian decay justifies summing this error over all relevant caps and cells; the polynomial support convention also permits restricting to polynomially many significant caps. Boost each cap to |N|\lesssim1, and call its input g and its free evolution u(t,x).

Plancherel in the analyzer’s velocity variable identifies its spatial marginal with a Gaussian average of \norm{u(T,\cdot)}^2 of width q_\tau\leq\tau. For any t\in[T,T+\tau], the band’s spatial mass on an interval transfers, in either time direction, into its O(\tau^{1-\eps})-neighborhood with negligible L^2 error. To verify this, insert a smooth Fourier plateau on the band. Away from velocities in its support, integration by parts in its propagation kernel gives a rapidly decreasing Schur tail. Its wavelength is O(H)\leq O(\tau), which is smaller than the allowed neighborhood. The retained operator is uniformly bounded on L^2.

At each t, enlarged spatial cells overlap O(\tau^{-\eps}) times. Average the preceding comparison in each terminal interval and expand the square of a cell integral. Up to negligible errors and \tau^{-O(\eps)}, the cell square sum is bounded by

Dropping the velocity cell after fixing a cap has only the already allowed multiplicity. A boost merely translates the spatial grids at each time. Errors can first be summed over polynomially many cells covering significant support; the remaining mass is negligible.

For the Fourier calculations, first fix two exact indices, with parameters \sigma,\sigma', and view the integrand as the squared norm of the tensor product of their fields. We integrate the resulting inequalities over this pair of indices before applying input regularity. In particular, all energies in the regularity bounds below include integration over their exact indices. Decompose Fourier-velocity pairs into Whitney scales

At scale r, use products of smooth multipliers a_I,b_J on intervals of width O(r), with center separation O(r) and boundedly many partners per interval. Require separation \gtrsim r, except for the nearest pairs at r_0. One explicit construction starts with smooth lattice partitions \psi_i^r, supported on |N-ir|\leq r, and the near-pair symbol \sum_{|i-j|\leq C}\psi_i^r(N_1)\psi_j^r(N_2). Telescope these symbols through dyadic scales. Expand a difference using the partitions at both adjacent scales in both variables. If C is a sufficiently large fixed constant, every surviving tensor term is separated by a constant times r and only boundedly many terms occur locally. The last near-pair symbol is the remainder at r_0.

Insert a Schwartz time cutoff with compact Fourier support and modulus bounded below on [0,1]. At fixed \Delta and fixed indices, the spacetime Fourier supports of products at any one Whitney scale have bounded overlap. After multiplying frequency coordinates by H, these supports satisfy

On fixing the first coordinate, the second is a positive quadratic in the transverse coordinate. At separation r\geq\sqrt H, its derivative has size comparable to r, and uncertainty O(H) permits only boundedly many interval pairs of width r. In the nearest part, the positive coefficients in the first coordinate already give bounded overlap. Plancherel at each scale and Cauchy–Schwarz over the logarithmically many scales reduce Equation (pair) to the sum of the individual spacetime product integrals with the squared time cutoff.

Localize each initial factor a_Ig,b_Jg further by G(x/r-k), k\in\Z. Here G is Schwartz, has Fourier support in a sufficiently small neighborhood of zero, and \sum_kG(v-k)=1. Such a function is obtained by choosing its Fourier transform supported near zero with the normalization prescribed by Poisson summation. The squared factors have bounded sum. Multiplication enlarges a velocity band by at most cH/r\leq cr, preserving the separation of separated pairs. Write e_{I,k} for the energy of a localized factor, including its exact indices. Then

For the second bound, use the original boosted input, before cap selection. The localized operator needs only its analyzed rows with velocity within O(\tau^{-\eps}r) of I and position within O(\tau^{-\eps}r) of rk. On those rows apply the boundedness of synthesis, the Fourier multiplier, and the position multiplier, followed by input regularity. The root packet widths are \sqrt H\leq r. Excluded velocities give Gaussian frequency tails; excluded positions give Gaussian spatial tails and the Schwartz kernel tail of a multiplier whose spatial width is O(H/r)\leq O(r). Applying Schur bounds after rescaling these kernels proves negligible norm for the discarded transfers. This proves Equation (LC) with constants independent of the exact-index spaces and Hilbert multiplicities.

Truncate the time integral to |t|\leq\tau^{-\eps}. The time cutoff’s Schwartz decrease, together with polynomial Bernstein bounds for the bandlimited fields, makes the error negligible. For a localized piece, discard the output outside

where C>2 is fixed large enough. To obtain the L^2 error bound, first discard the position multiplier’s tail beyond \tau^{-\eps}r, retaining a smooth propagation plateau equal to one on the original enlarged Fourier support. The nonstationary kernel tail then applies, since r^2\geq H and |t|\leq\tau^{-\eps}. Tensor-product errors are negligible by Bernstein on the untruncated factors. These errors sum over polynomially many significant position indices; the remaining indices can be discarded together by the original support and tail bounds. On retained outputs a product with the shifted factor requires

because |N_I-N_J|\lesssim r and |\Delta|\lesssim\tau^{1-\eps}\lesssim\tau^{-\eps}r. At each point only \tau^{-O(\eps)} such localized terms occur, so Cauchy–Schwarz separates them at this loss.

For a separated localized pair, extend its spacetime integral to the whole plane. Bilinear Plancherel gives

Indeed the map from momenta to their summed spatial and temporal frequencies has multiplicity at most two and Jacobian |Hp_1/\sigma-Hp_2/\sigma'|=|N_1-N_2|\gtrsim r. The shift introduces only a phase. This calculation is componentwise valid for Hilbert-valued fields and then integrates over the independent exact indices. The shift interval cancels the prefactor \tau^{-1} in Equation (pair), up to \tau^{-O(\eps)}.

There are two possibilities for the nearest scale. If r_0=\sqrt H, spatial Bernstein has bandwidth O(r_0/H)=O(r_0^{-1}); combine it with the other factor’s conserved L^2 norm over the retained times. If r_0=\tau, extend the shift integral to all of \R, after imposing the position-pair restriction just proved. The spatial and shift integrals then give the product of the two energies, and the surviving prefactor is \tau^{-1}=r_0^{-1}. Thus all cases are bounded by

Each localized piece has only \tau^{-O(\eps)} allowable partners. Use Equation (LC) for one energy and sum the other to obtain \tau^{-O(\eps)}r^{-\eta}b_0e_{\rm cap} at each scale. Since r\geq\tau, summing scales and caps proves the free estimate after letting \eps tend to zero. In particular this argument does not replace a finite-time spatial mass by an unrestricted spacetime L^4 norm when H is very small.

It remains to insert masks. Induct on their number. At the first cut of length \ell carrying a mask, peel the raw analyzed rows into dyadic scalar-regularity levels. At threshold b, use the tests with widths f,\ell f to remove disjoint heavy parts of mass greater than bf^{1-\eta}. The preceding threshold ensures that their union has regularity O(b). Include an arbitrarily low polynomial floor remainder. If M is the row contraction, write it as the coherent sum \sum_bM\mathbf1_{\mathrm{level}\ b} before synthesis. The free estimate to this cut shows that its level’s summed raw mass z_b satisfies

The post-mask energies sum to at most z_b, and same-frame Gaussian locality gives regularity O(b) after reanalysis. Apply the induction hypothesis separately on each descendant experiment, whose relative terminal time length is \tau/\ell, and sum its energies. The scale factors multiply as \ell^{-\eta}(\tau/\ell)^{-\eta}=\tau^{-\eta}. The floor has an arbitrarily small regularity constant; a trivial total energy bound makes its contribution negligible. Post-states too small to satisfy polynomial bounds relative to their own energy are instead estimated directly on the original polynomial terminal tests.

There are only logarithmically many levels. For fixed assignment multipliers the fourth root of the weighted mass-square sum is an \ell^4 norm of Hilbert norms and hence a seminorm in the input state. The triangle inequality therefore sums the coherent levels at a subpower loss. Repeating this a fixed number of times completes the induction. Repeated cuts and subpower time gaps use the same-frame bound and the elementary estimate already proved, with subpower loss.□

Proof. Partition the exact base variables at the initial time into lattice boxes M of widths (\delta^L,\delta^L,\delta^{2L}), using the tilt of the center of each X-bin for the last coordinate. These variables do not mix under propagation. For a fixed listed angle cell, a terminal label permits initial horizontal and tilted vertical positions in intervals of width O(\delta) each. To check this, backflow its terminal base footprint through a bounded time; variation within the angle bin contributes O(\delta^L) horizontally and O(\delta^{2L}) vertically in the bin’s tilt, while the terminal footprint has width O(\delta) in both relevant directions. A change from the label’s tilt to the bin’s tilt has bounded coefficients. Therefore at most

base boxes M are relevant to any ending label and its whole list.

On the initial gate within M, bin both curvature coefficients, at its starting time and in segment units, to width \delta^{-\max(1,Q)}. Equation (A) shows that each ending label knows only s^{-o(1)} possible bins, including the extrapolation of its quadratic coefficient back to the initial time. Subtract the shear of that rounded curvature trajectory. In the linear case omit this operation and use the enclosing linear tests supplied by the affine reduction. Residual curvatures of the two endpoint tests are now bounded by s^{-o(1)}\delta^{-\max(1,Q)} throughout the segment.

The endpoint derivatives at a true ray differ by at most s^{-o(1)}\delta^{-\max(0,C)}. Recenter at the ray F_M through the base-box center; this adds at most s^{-o(1)}\delta^{L-\max(1,Q)}. Both quantities are bounded by s^{-o(1)}\delta^{1-L}, by Equation (ML). Bin the residual initial derivative at that accuracy and subtract the resulting linear shear. The ending label, M, and the curvature bins determine only s^{-o(1)} possibilities for this last bin. The residual ending derivative at F_M is consequently at most s^{-o(1)}\delta^{1-L}.

For the prediction version, clip the lists to a polynomial parameter range large enough to contain all accurate guesses. Subtract the rounded curvature’s derivative at the true ray from each guess, and bin the result to width \delta^{1-L}. Make one copy of the classical ancestor row for each distinct resulting bin. This costs at most s^{-o(1)}\delta^{-Z} in total energy. On the event of an accurate guess, recentering at F_M has the same error as before. The ending label and M again know only s^{-o(1)} pertinent bins. If the guess uses the true derivative but the test is enclosed by a linear one, their coefficients differ by at most s^{-o(1)}, which fits the tolerance. Copying is used only in the classical model, where energy contributions add pointwise over exact indices.

After these subtractions, the part of each ending label over M lies in an ordinary scalar test of velocity width s^{-o(1)} and position width \delta s^{-o(1)}. To verify the tighter position statement, Taylor-expand the two shear polynomials about F_M. Their horizontal linear terms are bounded by \delta^{1-L}\delta^L=\delta. The remaining terms are bounded by \delta^{-\max(1,Q)}\delta^{2L}\leq\delta. The tilted vertical coordinate has precisely the latter order, also after base free evolution. The velocity estimate follows from the same expansion with its less restrictive unit tolerance.

For each starting time and bin tuple, ordinary regularity before the gate gives scalar input regularity constant

Indeed a scalar regularity box pulled back by the subtracted common shear, together with the base box M, is a full cylinder test of base width \delta^L. Restriction to the gate and same-frame reanalysis preserve this upper bound. Apply Proposition 4.3 to each such experiment, allowing all subsequent masks and retaining exact indices. The energies sum to at most s^{-o(1)}E_i, or to s^{-o(1)}E_i\delta^{-Z} in the prediction version. The scalar cutoff conditions follow from the original density and phase cutoff bounds. As throughout, extremely small-energy pieces are bounded directly on the polynomial output tests.

Let m_{\lambda,M} be the actual read mass of the tested event. Each pair (\lambda,M) sees only s^{-o(1)} bin tuples. Other tuples have no allowed packet chain and contribute negligible operator norm. Cauchy–Schwarz in the contributing fields and Equation (SC) imply

There is no coherent loss over M, which is an exact base partition. In the classical prediction version use only accurate copied parts; they majorize the tested event pointwise and have the same small-list property.

The documented count and the base-box count give at most

pairs. Apply Cauchy–Schwarz to their masses and divide by E_j^2. Since E_j/E_i=\delta^{-d+o(1)} and \kappa_j/\kappa_i=\delta^{p+o(1)}, the exponent is d-p+2+\eta(L-1)-R-Z. Finally d-p+2=1-d+2\Gamma, giving the assertion.□

Proof. In normalized tail units the leaf curvature is at most s^{-o(1)}\delta^{-1} throughout the segment. Its derivative on its own base footprint is at most s^{-o(1)}, by the two zero-cost bounds at the leaf. Recentring on a compatible base box of width \delta changes the derivative by only s^{-o(1)}. Thus the Taylor bounds in the preceding proof hold directly, without curvature or derivative bins. The scalar-channel and counting arguments are unchanged.□

Proof. Write \Gamma=\Gamma_{\rm hyb} and assume both costs vanish. Normalize the leaf widths to one. The bottom Planck assertion (Pl) gives H\leq s and H=s^{1+o(1)}. At each leaf,

Taylor expansion on its base footprint, whose horizontal and tilted vertical position widths are s, shows that its normal velocity and position lie in unsheared intervals of widths s^{-o(1)} and s^{1-o(1)} respectively.

Initial localization. Partition X into unit bins. Split the initial normal field by a smooth compactly supported unit Fourier-velocity partition of unity, and then by unit-width Schwartz position factors forming a partition of unity and having bounded Fourier support. Summed energies are O(e). Every leaf sees only s^{-o(1)} groups: its angle and normal velocity windows have subpower widths, and its position backflows into the corresponding subpower neighborhood of an initial position bin. This follows through the fixed stack of packet kernels since d_s\leq1. Gaussian and Schwartz tails justify deleting all other transfers with negligible norm; first truncate to polynomially many significant input bins. Equivalently use arbitrary fixed-power enlargements and then send their exponents to zero.

By the triangle inequality on each assignment, it suffices to estimate the groups separately and sum their cube sums at subpower loss. Tiny-energy groups admit direct polynomial-test bounds. On a remaining group boost and translate so that |X|\lesssim1, the normal Fourier velocities are bounded, and the initial normal position has negligible mass outside |D|\leq s^{-o(1)}. Denote its energy by e_g.

The group has input regularity O(\kappa). The same assertion holds after replacing it by the direct sum of a smooth Fourier-velocity partition at any width r\gg\sqrt H. Here is the operator justification needed later. The conjugated operators in the root packet frame have rapidly decreasing Schur kernels in its normal packet units. For a vector of Fourier multipliers the \ell^2 norm of its derivative of order k is O(r^{-k}). Rescale the Gaussian frequency-side inner product to root packet units and integrate by parts; since r\gg\sqrt H, this gives a bound for the whole vector kernel with summable tails. Unit position factors have the analogous space-side bound. At each exact base point these operators need only bounded enlargements, and then summable enlarged shells, of a normal test. Same-frame reanalysis therefore proves the claimed regularity for the sum measure. Integrating over the subpower normal support gives the base energy marginal regularity

We used R^{3+\eta}\leq R^3 and covered normal support by subpower many unit windows. All these weights are polynomial, so tails remain negligible.

After the boost, base leaf positions lie in squares of side s^{1-o(1)}. Pass from assignments to lattice position masses at subpower loss and drop angular restrictions. Let y denote all exact indices, Y_d(T) the indices whose base position at T belongs to a square d of side s, and u_y(T,x') the output field before final analysis. The unnormalized functional is bounded, up to subpower loss, by

Indeed the analyzer’s spatial marginal is convolution by a probability Gaussian. Jensen removes that convolution in this upper bound, and Hölder on a normal bin of length s cancels the functional’s factor s^{-1/2}. We may replace the fields by approximations having negligible L^2 error at every time: the original assignment functional absorbs these errors by its polynomial parameters. We may likewise restrict the output to a polynomial range of x' when necessary.

A stopping scale and its envelopes. Fix 0<\zeta<1/4, set r\asymp s^\zeta dyadically, and choose comparable dyadic scales

Then 1\gg\ell_0\gg B_*\gg s on sufficiently accurate outer approximations, and d_{\ell_0}\asymp r, q_{\ell_0}\asymp B_*. We will bound the expression in Equation (HL) by

where D_0 is absolute and independent of the number of masks. Separated velocity caps give the first term: a transverse crossing estimate will be combined with the kinetic bound. Nearby caps give the second term: we compare them with one experiment read at time length B_*, apply the exponent \Gamma to its prefixes, and use the kinetic bound on each remaining gap. Since B_*=s^{1-\zeta+o(1)}, both terms improve on the extremal exponent once D_0\zeta<\Gamma.

Split the group input coherently into O(r^{-1}) smooth velocity caps of width r centered at bounded lattice points \theta_k. Call their outputs u_{k,y}. Let e_y be the original group fiber energy, so \int e_y\,dy=e_g and each cap fiber has energy O(e_y).

Add the cut \ell_0, including any masks already at that length. For a fixed stack of n stages use slack s^{-\eps}, with \eps>0 arbitrarily small. At each mask restrict its rows to velocities within C_ns^{-\eps}\max(r,d_\ell) of the cap center, and post-compose with a smooth projection onto a slightly larger Fourier band. Choose successively increasing constants to include the preceding supports. The modified operators have bounded norm, and Gaussian frequency tails make their output differences negligible in L^2, uniformly per fiber and branch. At the stopping interval I, write their states as w_{I,k,y}; their bandwidths in velocity are O_n(s^{-\eps}r). The fixed factors s^{-C_n\eps} below become subpower losses by taking the inner limit before \eps\downarrow0, even if the mask counts grow in a subsequent outer approximation.

For T\in I the modified fields satisfy

To prove this, insert a smooth plateau on the stopping state’s band before its first propagation. For duration at most \ell_0, its kernel has size O(s^{-C_n\eps}/B_*) and rapid tails beyond the displayed displacement. After the stop, d_\ell\gtrsim r and q_\ell\lesssim B_*. The position kernel of a mask restricted to row velocities of width O_n(s^{-\eps}d_\ell) has absolute operator-valued bound O_n(s^{-\eps}/q_\ell) and Gaussian tails on scale q_\ell: insert the two normalized analyzing packets, bound the row contraction by one, and integrate their product against the frame measure. Post-projection and bandlimited free evolution for a duration at most \ell have Schwartz tails on the same scale, with slack. The effective uncertainty parameter is bounded in (q_\ell,d_\ell) units. Each kernel has integral bounded by s^{-C_n\eps}. Compose after the first convolution and apply Cauchy–Schwarz against its absolute kernel. The resulting rapid tails can be bounded by negligible times e_y. This proves Equation (env), also for repeated cuts.

Expand the squared norm of \sum_ku_{k,y} in Equation (HL). Pairs with centers within s^{-2\eps}r are bounded by s^{-O(\eps)}\sum_k\norm{u_{k,y}}^2. Call the remaining pairs high pairs, and write V=|\theta_k-\theta_{k'}|\geq s^{-2\eps}r for their separation. Only their sum will incur an ordinary power of r^{-1}.

The transverse crossing estimate. For normal fields f,g and a>0, define the crossing energy

For each high pair and exact index,

We prove the localization needed before the stopping cut, so that independent estimates on many short intervals do not introduce a scale loss. At each cut at or before the stop call the two modified band states g_{1,I},g_{2,I}, and put W_I=\ell_I V. For sufficiently large fixed D, their stopping crossing energy satisfies

We establish this by pulling back successively from each cut to its parent, increasing the slack constant a fixed amount at each step. At the root total energy gives the assertion.

Work in a common Galilean frame where both band velocities are O(V). They remain separated by a constant times V before the stop. For a parent I and its children J at the next cut write W=\ell_IV, w=\ell_JV, and

The required one-step estimate is

Thus the child crossings must be charged to nearby pairs in the parent, with no factor for the number of children. We first control the position spread of the cut operators, then sum the freely evolving separated pairs by bilinear Plancherel.

Decompose each parent state into

where P_\alpha is a smooth plateau of width comparable to V, equal to one on the input band and with the two enlarged supports still separated. At a child crossing, only pairs with |b-b'|\leq s^{-O_D(\eps)} contribute, up to negligible error. Indeed free band propagation has negligible tail beyond a slack multiple of W, and the child masks and projections have position width at most a slack multiple of w. The needed inequalities are

which follow from \ell_J\geq\ell_0 and V\gg r. Here is a quantitative tile argument for this step. For intervals E_m=[mw,(m+1)w), put

where \widetilde P_\alpha is a slightly larger plateau equal to one on the support of P_\alpha. Its two supports remain separated by a constant times V. Uniformly in J, t\in J, and the measurable row contractions, \norm{B_{\alpha,J,t}}_{2\to2}\leq C_n, and there is S_0=s^{-C_n\eps} such that, for every N and R\geq 2S_0,

We verify that this bound requires no derivatives of a mask. For the length-\ell frame, Plancherel in packet velocity gives, for any position set E and L>0,

Indeed its squared norm is bounded by the supremum, for x\in E, of the Gaussian integral over those packet centers D. To bound a mask between separated position sets, split packet centers into those within one third of their separation from the input set and the complement. On the complement use this analysis bound; on the first part use its adjoint for synthesis into the output set. The row contraction preserves both sets of centers. Thus the off-diagonal mask norm has Gaussian decrease on scale q_\ell, uniformly in the mask and Hilbert multiplicity.

The Fourier projections have Schwartz kernel tails on scale H/r. For U(T_J-t)\widetilde P_\alpha, velocities are O(V) and travel at most O(w), while the kernel’s wavelength is O(H/V)\leq O(w). Outside a sufficiently large multiple of w, its phase derivative is bounded below by a constant times position displacement. Integration by parts gives arbitrarily rapid integral tails in displacement divided by w. All three widths are at most w by the displayed inequalities. Products of the finite number of such operators have the matrix tail bound (4.1): at large total tile separation one factor has comparably large separation, and convolution of their summable tails preserves arbitrary decay. The slack enlargements only replace w by S_0w and insert a fixed power of S_0.

Choose R=s^{-C'_{D,n}\eps} with C'_{D,n} larger than the constant in S_0. Equation (4.1), with N arbitrarily large after fixing \eps, allows deletion of all matrix blocks with |m-n|>R in negligible operator norm. If B^R is the retained operator, Cauchy–Schwarz and the individual block bound K_{mn}\leq C_n give the explicit local inequality

For S\geq1, cover the crossing region by tile pairs |m-m'|\leq S+2 and apply the local inequality to both factors. Each input pair occurs at most (2R+1)^2 times. The triangle inequality between tile indices therefore gives

The error estimate follows by viewing \mathcal C_a^{1/2} as the L^2 norm of a restricted tensor product and using the negligible operator-norm difference between B and B^R.

Apply the same matrix argument at tile width W to A_{\alpha,J}U(T_J-T_I)P_\alpha. All its movement and kernel widths are at most a slack multiple of W. After negligible deletion, on each output W-tile there are only s^{-O_{D,n}(\eps)} contributing parent pieces. Pointwise Cauchy–Schwarz for their sums in both factors bounds the child crossing by the sum of the crossings of individual parent piece pairs, with a slack factor. A contributing pair must have |b-b'|\leq s^{-O_{D,n}(\eps)}: the output positions are within s^{-D\eps}w\leq s^{-D\eps}W and both input tile indices are within a slack distance of their output tile. Dropping the output tile restrictions on each retained pair only increases its crossing integral. This proves the earlier parent-pair restriction with its required restricted-region bookkeeping.

For a retained pair write f_{\alpha,b}=P_\alpha(\mathbf1_{[bW,(b+1)W)}g_{\alpha,I}) and v_{\alpha,b}(t)=U(t-T_I)f_{\alpha,b}. At every t\in J, B_{\alpha,J,t}v_{\alpha,b}(t) is exactly the cut image of this piece. Use Equation (4.2) with S=s^{-D\eps}, and write L=s^{-O_{D,n}(\eps)} for its enlarged radius and prefactor. All next-cut children have the same length \ell_J and are disjoint. Hence averaging over each child gives

The last line extends the time integral to \R and uses bilinear Plancherel for the separated plateau supports, independently for each shift. Thus the only scale factor is w/(\ell_JV)=1; there is no factor equal to the number of children.

Finally, \norm{f_{\alpha,b}}_2^2\leq C\norm{\mathbf1_{[bW,(b+1)W)}g_{\alpha,I}}_2^2. If |b-b'|\leq K_0=s^{-O_{D,n}(\eps)}, every pair in these original tiles satisfies |x_1-x_2|\leq(K_0+2)W. The tiles are disjoint, so

Combining this with Equation (4.3) proves the one-step pullback into the enlarged, restricted parent crossing energy. Finite induction proves the stopping estimate. Polynomially many intervals multiply only negligible errors, whose decay order was chosen after all cutoff powers were fixed.

From stopping energy to terminal crossings. Now insert Equation (env) for both outputs from a stopping interval. For fixed z_1,z_2, their kernel cores overlap in a position interval of length at most s^{-O_n(\eps)}B_*, and at only s^{-O_n(\eps)}B_*/(sV) terminal times. We used B_*\gg s to absorb the possible extra endpoint time. They overlap only if |z_1-z_2|\leq s^{-O_n(\eps)}\ell_0V. The two envelope prefactors, the position length, and s times the time count give s^{-O_n(\eps)}V^{-1}. The stopping crossing bound proves Equation (cross). A constant additive tail from one envelope is integrated using the other’s L^2 bound, or the spatial integral of its core, so unbounded position causes no extra error.

Summing the high pairs. At a given time put H_y(x')=\norm{u_{k,y}(T,x')}\norm{u_{k',y}(T,x')}, and define

with zero integrand when e_y=0. Contraction gives \int\int_{Y_d(T)}H_y\,dy\,dx'\lesssim M_{T,d}, while Cauchy–Schwarz in y bounds its squared L^2 norm by M_{T,d}N_{T,d}. Interpolation, or Cauchy–Schwarz between these two integrals, therefore gives

Lemma 4.2 applied to the original group’s base marginal gives \sum M_{T,d}^2\leq s^{1-o(1)}\kappa e_g. Equation (cross) gives \sum N_{T,d}\lesssim s^{-1-O_n(\eps)}V^{-1}e_g. Cauchy–Schwarz in (T,d) bounds one high pair by s^{-o(1)-O_n(\eps)}V^{-1/2}e_g\sqrt\kappa. There are O(r^{-2}) pairs, and the finite-sum inequality for the 3/2 power introduces only another fixed power of their number. Hence all high terms in Equation (HL) are bounded by

for an absolute D_0, independent of the number of masks.

Summing the low contribution. Partition each stopping interval into intervals J_* of length B_*, and let L_b=[bB_*,(b+1)B_*). Use one actual cylinder experiment whose root is the direct sum of the r-cap inputs, whose masks through the stopping cut act componentwise, and which thereafter evolves freely. For each J_*, let \nu_{J_*} be this experiment’s analyzed row measure at its left endpoint. All J_* and all position bins L_b thus belong to the same experiment; its root energy is O(e_g) once, not the sum of copied energies over J_*. The analytical approximation operators used to prove the envelopes are not included in this experiment. For T\in J_* and x'\in L_b,

To prove this, move the windows in Equation (env) from the stopping time to T_{J_*}. The inverse bandlimited transfer needs displacement uncertainty at most \ell_0O_n(s^{-\eps}r)=O_n(s^{-\eps}B_*), so local stopping energy is bounded by the freely continued energy on the displayed enlarged window. Pass to analyzed mass using its spatial Gaussian marginal, since q_{B_*}\lesssim B_*. Gaussian frequency tails allow restriction to |N|\leq C, because d_{B_*}\asymp\sqrt r and cap centers are bounded. Take \eps small relative to \zeta and the fixed number of cuts. The modified stopping states differ negligibly from the actual direct-sum states. Cover enlarged windows by s^{-O_n(\eps)} ordinary B_*-bins, with the same overlap bound.

Integrate in x'; a polynomial restriction is allowed as explained after Equation (HL). The low contribution is thus, up to slack and negligible errors, at most

Peel ordinary cylinder regularity levels of the positive measure \nu_{J_*}, and let z_\beta be summed mass at threshold \beta. Within each level and each interval the regularity is O(\beta). The heavy pieces are disjoint prefix assignments across the full J_* family, so the definition of \Gamma applied once to this actual direct-sum experiment gives

Its input energy is O(e_g) and its input regularity is O(\kappa), by the direct-sum bound proved above. The scale B_* has fixed positive depth and the cutoff powers and number of masks are fixed first. An arbitrarily low floor contributes negligibly by total mass.

Split Equation (low) by these logarithmically many levels. For fixed J_*,b,\beta, restrict to its position and velocity window and take the base marginal. In units of duration B_* it has regularity at most C\beta R^{3+\eta}\leq C\beta R^3 for R\leq1: apply full cylinder regularity with normal width O(1). This width is admissible because d_{B_*}\asymp\sqrt r. Apply Lemma 4.2 with \delta\asymp s/B_*. The sum of squares is at most s^{-o(1)}\delta\beta times the marginal’s mass, and the sum of masses is at most O(\delta^{-1}) times that mass. Cauchy–Schwarz bounds the corresponding 3/2 sum by s^{-o(1)}\sqrt\beta times the marginal’s mass. Summing the bins and levels and using the preceding prefix bound gives

for the low term. No ordinary power of the number of caps was used here.

Finally sum the original groups and let the inner slack losses vanish. Since B_*=s^{1-\zeta+o(1)}, the low term has exponent strictly below \Gamma. Choose \zeta>0 so small that D_0\zeta<\Gamma; the high term also has exponent strictly below \Gamma. The outer zero-cost and Planck defects only contribute s^{-o(1)}. Their use in passing from labels to grids and groups has bounded multiplicity independent of the number of cuts; choosing the stopping scale from the actual H avoids accumulating its defect along the masks. The resulting strict improvement contradicts extremality and proves the proposition.□

Proof. The separated-set version is the point–line dual of the discretized Furstenberg theorem of Ren and Wang [25, Theorem 4.1]. We explain the passage from probabilities, which is the only additional point needed here. Choose a'<a and b'<b sufficiently close to a,b for the prescribed final loss. Take the slack \sigma in the hypothesis smaller than both \min(a-a',b-b')/2 and one quarter of the slack allowed by the cited separated-set theorem at a',b'. Choose the additional sampling loss below another quarter of that slack. Replace line parameters by representatives of their \Delta-grid cells. Points incident to a line remain within O(\Delta) of its representative. On the finite grid sample \lceil\Delta^{-b'}\rceil line parameters independently from \nu. For a dyadic ball of radius r\geq\Delta, the expected number of samples is at most

For any prescribed small additional power loss, binomial tails bound all these counts simultaneously by that power times (r/\Delta)^{b'}. Indeed there are only polynomially many grid balls to consider, whereas the probability of exceeding a sufficiently large small-power multiple of the displayed expectation and 1 decays faster than every fixed power of \Delta. This also bounds multiplicities in a single cell. The probability of one line grid cell is O(\Delta^{b-\sigma}), so the expected fraction of samples repeating an earlier cell is O(\Delta^{b-b'-\sigma}). Discard these repetitions at a small-power cardinality cost. Perform the same grid sampling on each retained line with exponent a'<a; its point-cell probabilities are O(\Delta^{a-\sigma}). The expected fraction of point repetitions is O(\Delta^{a-a'-\sigma}). The number of lines is polynomial, so the binomial count bounds hold simultaneously; by averaging, discard any small fraction of lines with excessive point repetitions. Equalize cardinalities at a further small-power cost. The resulting sets satisfy the nonconcentration and cardinality hypotheses of the separated-set theorem. Its conclusion, continuity of F, and choices of a',b' sufficiently close to a,b give (RW). Small changes of width and projection of the point supports onto representative lines change covering numbers only by bounded factors. All small losses are chosen before \Delta tends to zero.□

Proof. Write p(u\mid D) and q(u\mid D,D') for the two conditional masses. Conditioning on (D,D') and summing over u gives

This is (LP). If Z=\log(q(U\mid D,D')/p(U\mid D)), the same argument gives \mathbb P(Z<-t)\leq e^{-t} and hence \mathbb E Z_-\leq1. Since \mathbb E Z=I(U;D'\mid D), Markov’s inequality applied to Z_+ proves the mutual-information assertion.

For lists, conditional on the known data, the total probability of names in a list of size L that each have conditional probability less than \rho^\epsilon/L is at most \rho^\epsilon. Combine this observation with the chain rule for conditional masses and the fact that \mathbb P(U\mid D)\geq\mathbb P(U,V\mid D) at the sampled names. The argument still applies if only values attained on the specified event are listed.

Let E be the restricting event, with \mathbb P(E)\geq c>0. The new conditional mass is

Both factors in the ratio are at most 1. Under the restricted law, the probability that either is smaller than \rho^\epsilon is at most 2\rho^\epsilon/c, by integrating the corresponding conditional probability of E. Thus the normalized logarithm of the ratio tends to zero in probability. This also proves the assertion with nonatomic side information.□

Proof. We first justify one bootstrap step. Suppose the unconditional line scores, conditional only on D, are at least yz'-\chi at a sufficiently fine finite grid of depths z' between 0 and a fixed z>0. We claim that the point score at depth z is at least (F(a,y)-\omega)z in probability, when the input tolerance and mesh are sufficiently small in terms of \omega and z.

Otherwise pass to a subsequence on which violation has probability at least a fixed c>0. A violating point belongs to a cell of conditional mass greater than \rho^{(F(a,y)-\omega)z}; for each D there are fewer than \rho^{-(F(a,y)-\omega)z} such cells. Intersect this event with all the good input-score events. Choose a value of D for which this intersection has conditional probability bounded below, say by c/2. Retain fine line cells on which the conditional probability of the intersection is at least c/4. Their total retained witness mass is bounded below in terms of c. Restrict the original line probability to these witnesses and normalize. Any occupied coarser line cell contains a good witness, so its original mass satisfies the required bound \rho^{yz'-\chi}; restriction and normalization cost only a constant depending on c.

In each retained fine line cell, restrict its conditional point probability to witnesses and normalize. Each occupied coarser point cell contains a witness to the first inequality of (cond), so its mass is bounded by a constant times \rho^{az'-\chi}. Cover Euclidean balls by boundedly many grid cells. Between successive queried scales use the next larger queried cells; a depth mesh of size \vartheta costs at most \rho^{-2\vartheta} in the ball estimates. At the coarsest end the trivial probability bound gives the same small-power loss. Thus, in units \Delta=\rho^z, the retained line and point measures meet Theorem 4.7 with exponents y,a and arbitrarily small nonconcentration slack. Replace each fine line cell by a representative line; its witnesses move by O(\Delta). Equation (RW), with a loss less than \omega z, contradicts the upper bound on the number of popular point cells. This proves the step. Point–line duality proves the symmetric step.

By (LP), the starting marginal bounds are x=a and y=g, up to arbitrarily small losses. The two steps iterate

Use the preceding values on both right-hand sides, so the sequences are increasing and bounded. At their fixed point, y>a and x>g. For example, y\leq a would give x=a+y\geq2y and then F(g,x)>y, a contradiction. Consequently the fixed-point equations reduce to

This map is a contraction in the maximum norm. Its fixed point is (2a+g,a+2g) when a+g\leq1, and (1+a,1+g) when a+g\geq1. Thus finitely many iterations suffice to reach the strict target in (inc). Choose the finite grids and tolerances backwards from the final iteration. The argument for output depth z uses \rho^z as basic scale and is identical after this change of units.□

Proof. Zero shear cost and Taylor expansion on a terminal test imply unsheared velocity widths s^{-o(1)} and spatial widths s^{1-o(1)} after a bounded velocity boost. The possible unit boost bins therefore cost only s^{-o(1)} per label. This proves the first assertion about C(1). Conversely, for each fixed value c_0 of C(0), take the union of the root predecessors of all terminal samples with C(0)=c_0. This whole union lies in only boundedly many root time/test pairs of bounded weight: A and terminal time are bounded, so exact backflow places the initial boosted positions in a bounded box. Consequently a list of at most s^\tau e/\kappa values of C(0) has predecessor mass at most Cs^\tau e, for every fixed \tau>0. Rarity transfer (Sp) makes the corresponding terminal event power-small. Apply this to the list of atoms with probability greater than s^{-\tau}\kappa/e. This is the popular-list argument of Lemma 3.23, and gives the typical lower score \log_{1/s}(e/\kappa)-o(1) for C(0) without counting terminal times separately. The documented count and the label-to-name list give the typical upper score \log_{1/s}(E_1/\kappa_1)+o(1) for C(1). Their difference is p+d. Apply forward deterministic flattening at final readout on successively finer finite lists. The relative branching bound and Lemma 4.8 imply that all limiting increments lie between 0 and 4(b-a). Dense-list extension gives the asserted profile and endpoint bound.

For the angular assertion, ordinary-regularize at j=1-v. Fix a terminal label and an angular s^v-cell. In duration units of the j-interval, terminal spatial uncertainty is s^{v-o(1)} in each of the three boosted spatial coordinates. Backflow through a duration at most 1 adds uncertainty O(s^v) horizontally and in the normal position. After tilting by the angular-cell center, the velocity curvature contributes O(s^{2v}) in the last base coordinate, while terminal position uncertainty there is still s^{v-o(1)}. Consequently s^{-v-o(1)} full tests with r=f=s^{v-o(1)} cover all predecessors. Each has weight s^{4v-o(1)}, so their total mass is at most

Classical masks are pointwise contractions and do not enlarge this mass. Labels of read mass below s^\tau\kappa_1 have total probability at most s^{\tau-o(1)} by the count. On the remaining labels the conditional angular atom bound follows, with arbitrarily small \tau. A label predicts C(a') with a subpower list. Testing popular angular atoms given C(a'), and pulling these lists back to labels, gives the same bound by Lemma 4.8.

For the time assertion apply Lemma 3.22, with the exact particle as common side information and C(a) as target. At each of the cuts a and a+zh, use the current time and boosted-position bins at the common width s^a as relay names. Bounded boosted velocity moves position by only O(s^a) throughout the depth-a interval. Thus the earlier row predicts the later relay with bounded ambiguity, and that relay predicts C(a) with bounded ambiguity. The lemma and (Sp) exclude lists of fewer than s^{-dzh+\tau} fine time names for any fixed \tau>0. Testing the atoms that would violate the stated score gives the assertion. Approximating depths on the finite cut mesh costs only the ordered vanishing exponent errors.

It remains to prove plane avoidance. Write the plane condition as

Choose a fixed h'>0 sufficiently small compared with \epsilon h, with i=1-h'>a+Kh, and put \Delta=s^{h'}. Fix a small \lambda>0. When |\gamma|\leq\Delta^\lambda, the boundedness of A_2 reduces the condition to a quadratic sublevel set in A_1 of thickness O(\Delta^\lambda). At least one of |\alpha|,|\beta| is bounded below. Such a sublevel set is covered by boundedly many intervals of length O(\Delta^{\lambda/2}): when the quadratic coefficient is bounded below this follows by factoring at the vertex, and otherwise the derivative is bounded below on the bounded interval, after a bounded partition. A terminal label knows only subpower many choices of the conditioning chart. Apply the zero-cost case of (mem) with L=1 on a final depth gap \lambda h'/3. Its cell width is larger than these intervals, so a bounded list suffices for each chart. The exponent 1-d+2\Gamma is positive. This part has power-small probability.

Suppose now |\gamma|>\Delta^\lambda. Ordinary-regularize at i. A row at this cut knows only subpower many compatible boost and C(a+Kh) charts, since its exact particle and current time start predict the coarser name. Make this subpower number of copies and restrict each to the selected plane slab. In its segment duration units subtract the velocity shear

The corresponding spatial shear is one of the allowed cylinder shears. Its coefficients are O(\Delta^{-\lambda}) on any nonempty bounded chart. Choose h' small enough that s^{\epsilon h}\Delta^{-\lambda}\leq\Delta. The residual velocity therefore has magnitude O(\Delta). Partition the residual initial normal position into intervals of length \Delta. On each such slab the base marginal satisfies

Indeed the specified set is contained in a full input test with normal width comparable to \Delta. Since R^{3+\eta}\leq R^3, (kin) applies with constant s^{-o(1)}\kappa_i\Delta^{1-\eta}.

For each chart and initial normal slab, let e_* be its input mass and m_{t,b} its base position-cell masses on the \Delta time grid. Equation (kin) and the trivial time-copy bound give

Cauchy–Schwarz therefore bounds their 3/2-sum by s^{-o(1)}\sqrt{\kappa_i}\Delta^{(1-\eta)/2}e_*. Subsequent classical contractions only decrease these masses. A terminal label meets at most s^{-o(1)}\Delta^{-O(\lambda)} base position cells and initial normal slabs: its spatial diameter after the shear is s^{-o(1)}\Delta^{1-O(\lambda)}, and residual velocity is O(\Delta) over the unit duration. Each label also sees only subpower many charts. Summing the preceding estimate and using the finite-list convexity inequality for each label yields

where m_\nu are terminal masses restricted to the plane event. Their total is at least s^{o(1)}\mathbb P^1(\text{event})E_1. The terminal count and Hölder’s inequality yield the reverse estimate

Thus the event probability to the power 3/2 is at most s^{-o(1)}\Delta^{\Gamma-\eta/2-O(\lambda)}. Choose \lambda sufficiently small relative to \Gamma-\eta/2. This proves the fourth assertion, uniformly in the plane choice. Maximizing conditionally over a countable sufficiently fine parameter grid, and then enlarging the slab by a bounded factor, also shows that the conditional supremum of slab probabilities tends to zero for typical C(a+Kh) names. Measurable approximate maximizers suffice.□

Proof. Suppose such a configuration exists and use Lemma 4.10. Put m=3-p. The relation between p,d,\Gamma gives m=2-2d+2\Gamma>0, while p\geq1 gives m\leq2. The profile supplied by that lemma satisfies D(1)\leq p+d. We will prove D'(a)\geq1+3d at almost every interior depth, contradicting p+d=1+3d-2\Gamma. At a short gap, conditional sampling and planar incidences will force information in a two-dimensional projection of the spacetime position. Recovering the two omitted coordinates from time uncertainty gives the required derivative bound.

We first identify depth parameters at which conditioning changes angular information negligibly on a short depth interval. For each dyadic h>0, take subsequential limits, simultaneously on dense depth lists, of

These functions have values in [0,2Kh], are nonincreasing, and are Lipschitz with constant 4, by relative branching of C. The limiting normalized information equals (f_h(a)-f_h(a+Kh))/h. Integrating it over 0<a<1-Kh gives at most 2K^2h. Summing over dyadic h and applying Tonelli’s theorem shows that, for almost every interior a,

in the ordered limiting sense. Fix such an a at which D is also differentiable.

Write \rho=s^h, C=C(a), and C'=C(a+Kh). Subtract the lower endpoints of the time and position bins of C and divide by their common width. The finer name C' determines a representative O\in\mathbb R^4 with error O(\rho^K) for

Here W is determined by the exact particle and C, by evaluating position at the start of the time bin; all displayed variables are bounded. At each fixed finite collection of queried z>0, Lemma 4.10 gives the lower scores

in probability.

Draw M rows independently conditional on C', with M any fixed integer, and put L_i=A_i^{(Kh)}, using bin representatives when evaluating e_i=e(L_i). The rows share O and C. The time bound survives conditioning on all pins. Indeed L_i is known from particle i, and conditional independence gives

For completeness, the second inequality follows by expanding conditional entropy: the pins have additive entropy given C', whereas their entropy given C is at most the sum of their individual entropies. The first inequality is conditional data processing. The information bound also controls the information about \theta_i conditional on its particle, by the mutual-information chain rule. Apply Lemma 4.8. Independently, the angular bounds conditional on C' survive conditioning on the other pins by the product law.

For every fixed small \epsilon>0, with probability tending to one, all four-element subsets satisfy

To see this, sample the vectors successively. Uniform conditional plane avoidance from Lemma 4.10 excludes a slab about any plane containing the affine span of previously sampled V’s. The resulting lower bounds for successive distances multiply to a determinant bound. For fewer than three predecessors extend their span to a plane. Vertical planes give the second inequality. A union bound suffices because M is fixed. Rounding errors O(\rho^K) are smaller than all these tolerances. The implied power constants are absolute.

For i<j, let \pi_{ij} be orthogonal projection to the quotient plane with kernel \operatorname{span}(e_i,e_j), using measurable orthonormal coordinates. Write g_{ij}(A_k) for the projective direction of \pi_{ij}e(A_k). Choose the finite query depths that will be needed in Lemma 4.9. Color each increasing triple i<j<k by the vector of \epsilon-quantized scores

at these depths. There are finitely many colors: scores outside [0,2] can be assigned one overflow color, which has probability tending to zero because a direction has O(\rho^{-z}) cells and z\leq1. Take M sufficiently large that the finite Ramsey theorem [24, Theorem B, p. 267] provides four indices all of whose triples have the same color. Only this monochromatic four-set conclusion for a finite coloring of triples is used here.

We verify the recovery estimate used with this monochromatic set. For its largest index k, choose the smallest index i and the other two indices j,\ell. A direction for \pi_{ij}e(A_k) specifies the three-dimensional linear space \operatorname{span}(e_i,e_j,e(A_k)); the other direction specifies \operatorname{span}(e_i,e_\ell,e(A_k)). The determinant bound implies that these spaces meet transversely in \operatorname{span}(e_i,e(A_k)), with inverse norms at most \rho^{-O(\epsilon)}. Thus two direction cells of width \rho^z constrain any bounded candidate with first coordinate 1 to within O(\rho^{z-O(\epsilon)}) of the affine line through e_i,e(A_k). On this line, parameterized by (1-t)e_i+te(A_k), the equation “third coordinate equals the square of the second coordinate” has residual

The roots 0,1 are simple with derivative bounded below by a power \rho^{O(\epsilon)}. Consequently the candidate angular parameter lies in a bounded number of balls of radius \rho^{z-O(\epsilon)}. This is a list of at most \rho^{-O(\epsilon)} cells at the queried width.

It follows from the angular lower bound and list comparison that the joint score of these two directions is at least mz-O(\epsilon)-o(1), on transverse samples outside a vanishing exceptional set. The sum of their individual scores is at least the joint score minus o(1) by (LP) and the chain rule. Their colors agree, so each individual score is at least (m/2)z-O(\epsilon)-o(1), simultaneously at all query depths. Since there are only finitely many index choices, one fixed triple i<j<k has these properties on an event of probability bounded below independently of \rho. Restrict to that event. By Lemma 4.8 all previously obtained finite score bounds remain valid in probability, with vanishing additional losses. One can further restrict to a fixed bounded slope chart after one of finitely many fixed rotations, retaining positive probability.

Set D_*=(C,\mathbf L_{\ne k}) and define

These form a point–line incidence to accuracy O(\rho^K) in bounded charts. We check (cond) with a=d and g=m/2. After adjoining particle k, the line is known. Its projected speed is at least \rho^{O(\epsilon)}, by transversality, so a point cell of width \rho^{z'} permits at most \rho^{-O(\epsilon)} time cells of that width. The conditional time bound and (LP) prove the first inequality of (cond). For the second, adjoin C'. The point name is then known, and a line cell specifies its direction to boundedly many direction cells. The score bound obtained from the monochromatic triple and (LP) prove the required line score. All tolerances can be made smaller than the tolerance in Lemma 4.9. Hence

with an arbitrarily small prescribed strict loss.

We must also recover the two coordinates removed by \pi_{ij}. Complete the quotient coordinates to an orthonormal basis using first the unit vector along the part of e_j perpendicular to e_i, and then the unit vector along e_i. Given particle j and D_*, the quotient point Q_1 has only boundedly many possibilities: its velocity component is annihilated up to O(\rho^K), because L_j is known. The next coordinate has speed at least \rho^{O(\epsilon)} and therefore reveals \theta_j to a list of \rho^{-O(\epsilon)} cells. Its score conditional on Q_1,D_* is at least d-O(\epsilon) by the time bound, the bounded-list version of the chain rule, and (LP). For the last coordinate adjoin particle i. Both preceding coordinates are predictable with bounded ambiguity, whereas the last coordinate reveals \theta_i with the same accuracy. This contributes another d-O(\epsilon). The chain rule, bounded changes between orthonormal coordinate grids, and (LP) to remove the pins give

The order of choices is as follows. First choose the target strict loss and the finite incidence iterations and query grids. Next choose the color accuracy, transversality tolerance, and input score tolerances sufficiently small. Then fix a finite Ramsey number M. Only after these choices let dyadic h\downarrow0 at the selected a, taking all preceding scale and mesh limits first for each h. The stationary information loss then tends to zero, and every event restriction after this information comparison has probability bounded below by a constant depending on M, not on the inner scale.

Given C, the name O_1 and the refinement C(a+h) determine each other with bounded ambiguity. Thus its limiting score in \log(1/\rho) units is (D(a+h)-D(a))/h, which tends to D'(a). Since d+m/2=1+\Gamma>1, the preceding inequality gives D'(a)\geq1+3d for almost every a. Integrating and using Lemma 4.10 gives

a contradiction. This proves the proposition.□

Proof. The time cells are nested. For base coordinates, backflow uses B_t=B_{t'}+(t-t')X, Y_t=Y_{t'}+(t-t')X^2. After subtracting the tilt 2[X]_vB_t, the unresolved contribution of the time displacement is quadratic in the unresolved angular increment. Its sizes are therefore s^{a+v} and s^{a+2v}. Changing the angular cell center produces its known linear correction and an error of the latter size. These facts give both the base lists and their relative branching bounds.

Compare two curvature displays in the coarse time units. Their differences have sizes O(s^{b(a)}) and O(s^{b(a)-a}). For a nonconstant display, extrapolation backward from the finer time preserves these bounds because b'-1\geq0. Constant displays require no slope comparison. After both displays have been guessed, the unresolved part of their derivative difference has size at most s^{b(a)+v}, which is no larger than s^l. Thus the derivative cells have bounded comparison lists.

For completeness, the normal estimate uses the difference of the two shears, rather than either shear separately. Its derivative on the actual base ray is O(s^l), since both displayed derivatives approximate P. Its curvature difference has size O(s^{b(a)}) in the coarse time units. On the unresolved base cell, Taylor’s identity bounds its velocity uncertainty by

and its position uncertainty by s^a times this quantity. These estimates are stronger when the normal widths are enlarged. Shear subtraction commutes with backflow. This proves the lists for the residual coordinates even when the absolute shear or absolute position is large. The same calculations with unequal resolutions give at most s^{-C(|\Delta a|+|\Delta v|)-o(1)} choices. Comparison through a common coarser pair proves the final assertion.□

Proof. Fix a canonical name at depth a. On its ancestor rows the base coordinates lie in the displayed base cell; after the displayed shear the two normal coordinates lie in cells of widths s^{w-g},s^{a+w-g}. Conversion between t_a and the actual gate time has the same precision because the residual velocity is known to that precision. Thus those rows are covered by a small list of regularity tests of weight s^W, in a small list of time intervals. Their benchmark mass is at most s^{pa+W-o(1)}\kappa. Division by the assigned-law benchmark s^{-da}e, and the sparse-list consequence of (Sp), exclude cells whose typical probability is larger than (\kappa/e)s^{(p+d)a+W-o(1)}. This is (canon). In the hybrid comparison the artificial ray differs from the ancestor row by at most s^{-o(1)} in normal velocity and s^{a-o(1)} in position, by (Pl). Since the base is identical, all shears preserve this additive comparison. It fits the required widths when w<0. The classical virtual endpoint uses the same sparse-list argument on the final law.

For the upper bounds at the stated equality points, the reached label determines the name to a small list. Curvature comparisons are those of (F). If c\leq1, omitting the slope of the curvature trajectory costs at most the own curvature width in its own duration units. The derivative discrepancy along an actual ray is bounded by s^{\min b_1-o(1)} on the connecting interval. Once the curvature cuts are guessed, recentering over the label’s base cell adds at most s^{b_2(a)+u(a)-o(1)} to that derivative. The Taylor calculation in Lemma 5.3 then gives the normal widths. In a strict family the residual own slope is bounded by s^{b_2(D)-D-o(1)}, so its omission is harmless at D. The documented label count supplies the reverse entropy inequality.

Assertion (ii) follows directly from (A). The slope comparison on [i,D] predicts the global bin and gives the first bound. Multiplying this residual slope bound by the earlier duration gives the artificial depth b(i). The coefficient comparison itself is finer, since \min_{[i,D]}b_2>b(i), and gives the second bound.

For (iii) and (iv), first guess the earlier curvature cuts. Backflow of the later base test, with X^{(v)} when required, gives the earlier base widths: its uncertainties s^{j+u(j)} and s^{j+2u(j)} fit when the right excess v-u(j) is small compared with j-a. At zero excess the inequalities are weak and the allowed small lists suffice. At the later time the unresolved angular part of the relative derivative has size at most s^{b(a)+v-o(1)}; the unresolved position part of a curvature slope has the additional later-time factor. The remaining derivative discrepancy fits by the stated condition on b_1. Taylor’s identity now gives the normal comparison, using n(j)\geq w(a,v), with strict slack for a small positive excess.

When a curvature slope was omitted, perform this comparison at T_j, not across the whole earlier duration. The omitted slope has size s^{b_2(j)-j-o(1)}; its contribution from unresolved own positions to the derivative is s^{b_2(j)+u(j)-o(1)}. It fits the later normal widths. The unresolved angular increment multiplies only the curvature difference, of size s^{b(a)-o(1)}. Having compared velocity and position at T_j, backflow gives the earlier position assertion. These bounds depend only on the later label and the allowed common base region, so the lists are uniform over all completing paths of that label.□

Proof. At a fixed a, take derivatives in x for a countable dense set of conditioning depths. Their values lie in [0,1], and are nonincreasing as the conditioning depth increases. Monotone envelopes therefore define the trace from conditioning depths below x. For x\in[v,v+h], its derivative at conditioning depth v lies between that trace and the derivative at depth x-\eps, provided h<\eps. At common Lebesgue points, integration over [v,v+h], followed by h\downarrow0 and then \eps\downarrow0, makes the two bounds agree. This proves (ang). Countable envelopes and continuity recover the original profiles. Time monotonicity follows from the corresponding monotonicity of every difference quotient.

For fixed dyadic h, the function a\mapsto U(v+Gh;a,v) is nonincreasing and bounded by Gh. The integral of its decrement over a translation of length Gh is consequently O_G(h^2) on a compact interval. Divide by h and sum over dyadic h. The resulting summable nonnegative functions tend to zero almost everywhere. Nesting bounds an arbitrary smaller gap by a comparable dyadic gap. The same proof can be integrated in v, giving the interior statement. For a fraction-s^{o(h)} final restriction, first transfer the deterministic scores. Chain rule and bounded relative name capacities then recover the normalized entropy difference; direct conditioning of the old information budget is not required.□

Proof. Use monotone envelopes of the derivatives for countably many time depths. At a point of the graph, squeeze the integrand on each side between its trace and its value with time moved by a fixed \eps to that side. Lebesgue differentiation in x, the positive affine change of variable x=B(z), and then \eps\downarrow0 prove the assertion. Joint and conditional scores agree with these deterministic entropy increments by chain rule and flattening. This argument uses no regularity of a coefficient as a function of the exact particle.□

Proof. Pull a proposed angular list back to \lambda_j using its canonical prediction list. Apply (mem) with time gap h=z/L_{\rm raw}, then transfer the resulting rarity to the final law by (Sp). This proves the prefix statements and their derivative consequences. Conditioning additionally on \lambda_j,X^{(v)} proves the right-excess inequality.

For the prediction variant of (mem) on [j-h,j], only a list for the slope at depth b_1(j) is needed, given y_0,t_{j-h}. If strict extra prediction persists, this list has arbitrarily small exponent in s^h. In case (ii), its cost is at most C_{\rm rate}\nu(j)+o(1): the score at B(j-h) vanishes, and the remaining interval is above the graph at the earlier conditioning time. Apply Lemma 5.6. Restrict the final event to successful slope predictions, whose failure is negligible at a fixed strict tolerance. Their lists are then available on every tested completing path. The prediction memory bound and (Sp) prove both enhancements.□

Proof. For an exact side, the phase coordinates are fixed by the particle. In the quadratic case x_*>l also resolves the residual derivative to finer precision than the root unit.

For a coarse side first subtract the S-shear. Its base uncertainties are s^V,s^{A+V},s^{A+2V}, and its normal uncertainties are s^{w(A,V)},s^{A+w(A,V)}; the same bounds hold after flow to t_a. At a representative base point the root and side derivatives differ by O(s^{\min\{l(a,v),l(A,V)\}}). Their curvature difference has size O(s^{b(A)}) in earlier-duration units. Changing to the fixed representative frame therefore adds velocity error at most

with an additional factor s^A for position; omit the second term in the pure linear case. The inequalities in Definition 5.8 put all these errors strictly below the root widths. The quadratic derivative itself varies by at most O(s^{l(A,V)}). The change of base tilt fits because A+v+V>a+2v.

An actual cell may differ from the representative cell at a grid boundary. Guess its boundedly many neighboring linear-cut and angular-center cells first. A linear-cut difference multiplies only unresolved base coordinates at their root widths when converting residual names; the resulting errors are root normal widths. The tilt conversion has the same property. This gives the two-directional lists without comparing normal coordinates in different frames directly. All fine phase comparisons above are in the single representative frame.

A measurable representative can be obtained by realizing the conditional kernel with a uniform seed and fixing a seed that is valid almost everywhere, by Fubini. The same choice can be made with the patch parameter under Lebesgue measure, and kept fixed as v varies. Only its real coordinates and cell data are used. Conditional observations still use the original kernel; no extra shared mark is conditioned upon.□

Proof. At the gate i=a+p'h, every predecessor on a tested completing path predicts the static side and the other root data to small lists. For a coarse side this uses exact base flow, the strict normal slack for hybrid backflow, and (A). For an exact predicted coefficient, use its arbitrarily small-cost list and discard its negligible final failure event. The row predicts \theta_{p'}, and it predicts r to depth p' if c'\geq1, or depth c'p' otherwise. In a linear branch the latter accuracy always holds. When c>1, the predicted curvature trajectory is evaluated at the endpoint; its bounded normalized slope makes replacing that time by the gate time cost only \delta^{p'-o(1)}.

Here is the causal counting step. Use Lemma 3.19: take a Borel inner version of the prediction-success event, mark rows at a later gate j' that have a continuation to it, and retain a Borel inner version of that analytic mark under the full analyzed j'-row energy law, inside the current end-state support. For each of their times, restrict the i-inputs to a Borel outer version of the listed predecessor set. It includes all predecessors under the unrestricted truncated i-to-j' geometric correspondence, not just predecessors on successful completions, and has the same full analyzed input energy. A path from a complementary input to a marked row would concatenate with its completing path, and so would contradict the definition of that predecessor list. Correspondence locality therefore makes complementary transfer negligible. Ordinary contraction, summed over the listed next time cells, costs their multiplicity times the raw mass at i. Applying (Sp) onward to the readout prohibits lists of exponent strictly below d(j'-i), including the small predictor multiplicities. Testing popular cells proves the stated time lower surprises. The same proof starts at p'=0.

For c'<1, use Lemma 5.6 with V_*=\beta^\diamond,B=b for constant quadratic cuts, or with the raw slope and B=b_1 in a linear branch. The side is the specified static side. Its additional root data have zero exponent beyond S,t_a,[V_*]_{B(a)}. At the later time, p'>c'(p'+t) for sufficiently small t/p', so the r-increment lies above the graph and has rate \nu. Adding the coefficient bins between B(i) and the tested depth to the causal time bound loses at most c'(\nu-\nu_-)t: this is precisely the mixed coefficient information exposed by the new time bits. The r-increment before those bits has rate at least \nu. Chain rule gives joint rate at least

and also the separately stated time bound. Known shifts of bin origins and virtual raw coefficient comparisons cost small lists only.□

Proof. For the clipped assertion, root additions to y_0,t_a have zero cost and the offset in (xi) is then known. Every strictly finer raw slope bin has positive deterministic score by the definition of p_*. After trimming exceptional heavy bins, conditional independent sampling gives the separation. In a virtual use keep the query below b_1(D).

We prove the power statements by contradiction. Failure gives a predicted tube of width s^{(1-o(1))z}, where z=\lambda h>0, carrying mass s^{o(z)}; all previous errors are taken to zero in z-units first. A nearly vertical line confines the time coordinate to a power-small interval and is excluded by Proposition 5.10. Hence, after an arbitrarily small loss, the line is the graph of an affine function with slope s^{-o(z)} in mark coordinates. In physical units it predicts an affine trajectory \beta_* with

Exact extra-prediction sides are restricted to their successful lists, taking their costs and failure tolerances sufficiently small in z-units.

We shall extract a segment [i,j]. By Lemma 3.19, take a Borel inner version of the retained final event, mark j-rows having a completion to it, and then take a Borel inner version of this analytic mark under the full analyzed j-row energy law, inside the current end-state support. Every retained marked row still has a completion. Their mass is at least s^{o(z)}E_j by (Sp); the count bound gives a cube-sum lower bound s^{o(z)}E_j\sqrt{\kappa_j}. Group starting rows by the binned polynomial subtraction and their starting interval. Disjointize the finite possible-bin relations and retain their Borel inner versions, so each marked end row has a bin witnessed by a completion. Use the Borel outer predecessor set for that bin, formed from the unrestricted truncated i-to-j geometric correspondence. It includes all its correspondence predecessors without changing full analyzed input energy; its Borel inner partner carries the same full analyzed state inside the original starting-bin partition. Each end label has s^{-o(z)} chosen bins, as checked in the cases below. Path concatenation and locality then justify restricting synthesis to those starting sets. Input energy summed over groups is at most s^{-o(z)}E_i, each group’s regularity is at most s^{-o(z)}\kappa_i, and the end counts gain only the same multiplicity. Hölder preserves the cube-sum lower bound over groups. After rescaling, these grouped experiments realize the preceding extremal parameters in the closure. Thus a fixed improvement in the minimal cost is impossible.

Suppose first 0<c<1. Set i=a+z/2, j=a+3z/4, and choose 0<\omega<\min\{(1-c)/2,c/4\}. Subtract the prediction at the i-time start, rounded at depth b(i)+\omega z. Starting rows know it from the side. Moving the prediction to the endpoint time costs at most s^{b(a)+z/2-o(z)}, and comparing the end label with the final coefficient costs s^{b(j)-o(z)}. Both are finer than the chosen bin, so the end label knows few bins. The residual end coefficient is at most s^{b(i)+\omega z-o(z)}, while the duration-scaled own slope already has size s^{b(j)-o(z)}. In segment units the curvature cost is at most c-4\omega+o(1)<c.

For c>1, set i=a, j=a+\gamma z, with 1/c<\gamma<1, and choose \gamma<\tau<1. On the part where s^j|K_1-\beta_*'|>s^{b(a)+\tau z}, the end trajectory has the same slope discrepancy to within a factor two, since c\gamma>1>\tau. Its meeting the predicted trajectory within the tube confines the terminal time, per end row and predicted line, to length s^{j+(1-\tau-o(1))z}. The causal time count excludes this part by a power. On the remainder, extrapolation to t_a gives coefficient and slope errors bounded by

Subtract the predicted pair at those base depths improved by \omega z, with 0<\omega<\tau-\gamma. It is jointly list-recoverable, and both residual curvature terms at the end have cost strictly below c.

For c=1, use i=a, j=a+2z. The additional left slack makes \lambda_j predict the coarse side: its base width has u(j)>V, its derivative comparison fits l(A,V), and its normal comparison fits by the same Taylor and hybrid-drift bounds as in Proposition 5.4. It also predicts the root curvature bins. Subtract the predicted pair at depths b(a)+\omega z,b(a)-a+\omega z, where 0<\omega<1/2. The residual end coefficient is bounded by s^{b(a)+\omega z-o(z)}; the end-duration times the residual slope is bounded by s^{b(j)-o(z)}. Hence the end curvature cost is strictly below one. This case does not require the slope of the prediction to be inferred from the end curvature alone: it is predicted by the recovered side.

Finally, in the pure linear case choose i=a, j=a+\gamma z with c_1\gamma>1. A value prediction for r predicts the terminal linear coefficient at depth b_1(a)+z-o(z). Subtract its bin at b_1(a)+\omega z, 0<\omega<1. The end label knows the bin by its finer coefficient comparison. The curvature cost remains zero and the linear cost becomes at most c_1-\omega/\gamma+o(1), a contradiction.

For clarity, the recovery margins behind the group count are as follows. When 0<c<1, the prediction-transport and end-label errors have depths b(a)+z/2 and b(j)=b(a)+3cz/4, both finer than the subtraction depth b(i)+\omega z=b(a)+(c/2+\omega)z by the two restrictions on \omega. When c>1, both errors in the displayed value–slope pair exceed their subtraction depths by (\tau-\gamma-\omega)z>0. At c=1, the recovered coarse side supplies the slope bin to a small list, and the end coefficient has depth b(j)=b(a)+2z>b(a)+\omega z. In the pure linear case the terminal coefficient has depth b_1(j)=b_1(a)+c_1\gamma z>b_1(a)+\omega z. Thus the pre-existing side and rounding lists leave only s^{-o(z)} bins per fixed end label in each case.□

Proof. The marks and their law before evaluation of the representative phase are independent of v on the patch. Their total entropy is O(h\log(1/s)). For sufficiently large fixed K, the pair C_*(v),C_*(v+Kh) determines F_{\leq p'} up to bounded lists. In a quadratic template the new derivative bin reads P to depth K. The new residual phase can be converted to the old shear frame using the known shear difference and finer base bins; every old coordinate is then known to depth at least K. In a linear template the linear cut is unchanged. Consequently telescoping the decreasing function H_s(\boldsymbol\Xi\mid C_*(v)) over a translation of size Kh bounds the integral of I_s(F_{\leq p'};\boldsymbol\Xi\mid C_*) by O(h^2)+o(1). Here I_s=hI_\delta, since \log(1/\delta)=h\log(1/s).

Integrate also in a on compact patches when needed. For fixed dyadic h, take lower limits of the sum of the finitely many nonnegative errors through the earlier approximations. Fatou preserves the integral bound. After division by h, the dyadic integrals are summable, so almost every root admits accurate experiments with all the required normalized errors tending to zero. Countably many finite cutoffs and tuple sizes are handled by a diagonal. This establishes existence of the stated tangent laws; it does not assert stationarity for arbitrary sampling schemes that change with v.

Consider next observation i and its actual child name \mathcal C(a+Kh,v), with K larger than all required query depths, also after multiplication by c' and, for nonconstant quadratic cuts, by c'-1. It predicts C_* and reads \Xi_i,Q_i to bounded lists. Here are the details for the last assertion. The child time \theta' approximates \theta_i arbitrarily finely. Write W'=B+\theta'X, V'=Y+\theta'X^2. The difference between child and root shears, in the common frame, is

All coefficients and offsets are determined by the two names. In the quadratic case \mathcal R,\mathfrak b approximate r_i,\mathfrak k_i, and \mathcal L+2\mathcal RX-2\mathfrak bW' approximates P. Thus \mathcal L-2\mathfrak bW' reads P-2r_iX; adding back the known \mathfrak bV' to the residual velocity reads N-PX+r_iX^2. The base reads supply W',V', and the residual position supplies D_0'+\theta' N. For a linear branch the same derivative expression reads r_i, while \mathcal R,\mathfrak b are negligible because b+v>l, or identically zero. It therefore reads N-r_iX. Changing the curvature basis in a clipped child costs s^{b+v}, smaller than every required s^l\delta^{M'} by the fixed branch slack. Nonconstant quadratic cuts read r_i with error O(\delta^K+\delta^{c'K}) and the slope to depth (c'-1)K. Large constant offsets are converted at the exact child-bin time before comparison. Finally the common phase error is negligible by Lemma 5.9.

Equation (ng) bounds the information that the child name gives about the fine angle, conditionally on the root. Thus adding (Q_i,\Xi_i) costs o(1). To add the other marks, first include a sufficiently fine phase bin. Given it and \Xi_i, the fiber read has only bounded ambiguity. The remaining conditional phase information is bounded by (vb), using chain rule. This proves the first line of (fib). The second line is the deterministic angular profile and the root comparison from Lemma 5.9.□

Proof. Conditional resampling preserves the conditioning data and their joint marginal. Given those data it is independent of the rest of the history. Apply (fib) to this unchanged marginal; chain rule shows that retaining an angular prefix only decreases the information budget for the remaining angle bits. Summing the finite number of budgets gives the joint assertion. The first three identities in (leg) follow from equality of the first three fiber reads. The derivative read gives \Delta P=2r_iA. Substitution in the remaining quadratic read gives \Delta N=PA+r_iA^2; the linear read gives the last identity. Increase the finite cutoff to make all rounding errors smaller than the queried scales.□

Proof. If a conditional distribution is supported on a list of size N, then the total probability of values of conditional probability less than \rho^\epsilon/N is at most \rho^\epsilon. Apply this observation to each refinement and each list comparison. It bounds the refinement score between zero and K(q'-q)+\epsilon+o(1), except on a set of probability tending to zero. Bounded total range gives the same control on whole scores; values exceeding the range exponent by \epsilon have probability at most \rho^\epsilon, also conditionally on D. Thus every finite vector of scores is tight. A diagonal subsequence gives a consistent joint limit on the countable query set.

For exactly nested names the log-probability chain rule is an identity. For list nesting, insert the coarser name into the finer name. Its additional conditional score tends to zero by the preceding list bound, so the same identities survive in the limit. Equation (LP) gives the conditioning inequality at each fixed query. Apply it to the increment name with the coarse name already conditioned upon to obtain Equation (5.1). A countable union of null sets gives all inequalities simultaneously. The loss from refining a conditioning name F_r to F_{r'} is nonnegative and at most i(F_{r'}\mid F_r,D), by the chain rule and nonnegativity of conditional scores. It is therefore at most K(r'-r) with the appropriate branching constant. Thus the required mixed score functions are Lipschitz also in their conditioning depths. The Lipschitz estimates extend the limiting scores continuously from rational depths. Comparisons with neighboring rational depths also give the claimed limit at any additional fixed real query. The same reasoning applies when several arrays are refined together.□

Proof. For rational r, choose the derivative f_r(q)=\partial_qH(q,r) outside one common null set. Increment monotonicity gives 0\le f_{r'}(q)\le f_r(q)\le K for rational r<r'<q. Put

For any fixed real u, differentiate the increment inequalities comparing H(\cdot,u) with all rational conditioning depths. For almost every q>u and each rational r<u, they imply

For the first inequality, compare with a rational depth between u and q. Absolute continuity therefore gives, for every u<v and rational r<u,

Choose p to be a Lebesgue point of g and of every f_r defined in a neighborhood of p, and outside the common derivative null set. These conditions exclude only a null set. For fixed rational r<p, the lower and upper errors in Equation (5.2) are bounded, uniformly over the indicated triangle, by

respectively. The first is o(t). The limsup of the second divided by t is at most f_r(p)-g(p). Taking the infimum over rational r<p makes it zero and proves the assertion.□

Proof. For A=U,V,(U,V), apply Lemma 5.15 pathwise to H_A(q,r)=i(A_q\mid F_r,D). Its diagonal vanishes by the list hypothesis, and its increment inequality is Equation (5.1). Put u=p+\lambda t and v=p+t. The chain rule first gives

Write i(A;B\mid C)=i(B\mid C)-i(B\mid A,C) for limiting conditional information. These quantities are nonnegative by Equation (LP). Since U_u is determined by F_u,D, the additional loss from the future part of V satisfies

The second inequality follows by expanding the information involving (U_u,F_u) in both orders and using nonnegativity. The last equality uses nesting and determination of F_p,V_u from F_u,D; its o(t) bound follows from the uniform triangular trace. This proves Equation (5.4). Finally, Fubini’s theorem converts the pathwise almost-everywhere statement into an almost-sure statement at almost every fixed p.□

Proof. Let R=dP_\rho/dQ_\rho. The elementary estimate P_\rho(\log R<-u)\le e^{-u} implies \E_{P_\rho}(\log R)_-\le1, and hence

Markov’s inequality proves \log R=o(L_\rho) in probability. Relative entropy decreases under measurable maps, so this also applies to the likelihood ratios for each joint name and its conditioning field. Subtracting those two log ratios proves the conditional-score assertion. Applying the log-sum inequality to E,E^c gives

which proves the event assertion.

For restriction to E, use \operatorname{KL}(P_\rho(\cdot\mid E)\Vert P_\rho)=\log(1/P_\rho(E)). To verify the information assertion, let X,W,D denote the variables in question and set Q=P_D P_{X\mid D}P_{W\mid D}. With c_\rho=P(E), the mixture identity for relative entropy gives

where h(c)=-c\log c-(1-c)\log(1-c)\le\log2. This identity remains valid if E also depends on auxiliary variables: apply the mixture identity to the marginal laws of (X,W,D). The relative entropy on the left is at least I_{P(\cdot\mid E)}(X;W\mid D), by decomposition against the product of the restricted conditional marginals. Division by c_\rho\ge c proves the result.□

Proof. The case S=0 follows from the nonempty output list on G, so assume S>0. A finite choice of rotated projective charts allows restriction to a positive-probability part of G where projections have the form

with bounded u,z. Multiplying a projection by a bounded nonzero factor only changes interval constants. First restrict to G. Lemma 5.17 preserves the vanishing normalized information budget. If c_D=\Prob(G\mid D) and \Prob(G)\ge c, then under this restricted law the parents with c_D<c/2 have total probability at most 1/2. On the other parents, normalization of Equation (5.6) costs at most 2/c. The conditional information is nonnegative and has vanishing average after normalization by \log(1/\Delta). We can therefore choose a parent with these mass bounds and with conditional information o(\log(1/\Delta)).

Fix that parent. Write P for the restricted joint law and \mu,\nu for its source and record marginals. Let R be the relation that the projected source is in the record’s output list. Since P(R)=1, the log-sum inequality gives

Thus product sampling still hits the lists, although a fixed positive product probability is not asserted.

Fix a small \epsilon>0. For sufficiently advanced experiments, the marginal bounds have constants at most \Delta^{-\epsilon} and \eta\ge\Delta^\epsilon. Draw M=\lceil\Delta^{-b}\rceil independent full records W_1,\ldots,W_M from \nu. With probability tending to one faster than any fixed power of \Delta, their empirical direction law \nu_M obeys

Indeed, for each dyadic interval of radius r, the count is binomial with mean at most M\Delta^{-\epsilon}r^b; the asserted threshold is larger by a factor comparable to \Delta^{-\epsilon} and is at least a constant multiple of \Delta^{-2\epsilon}. The binomial upper-tail bound is \Prob(N\ge k)\le(eMq/k)^k for a count with M trials and success probability q: sum over the k successful indices and use \binom{M}{k}\le(eM/k)^k. This bound, followed by a union bound over O(\Delta^{-1}) dyadic intervals down to scale \Delta, proves the claim. Every other interval is covered by a bounded number of these intervals at comparable radius.

The expected average relation density of this sample is \eta. Since the probability that Equation (5.8) fails is o(\eta), there is a sample satisfying that equation and

Let E=\{z:\theta(z)\ge\eta/4\}. The bounds 0\le\theta\le1 imply \mu(E)\ge\eta/4. The probability \mu_E=\mu|_E/\mu(E) has planar ball masses at most C\Delta^{-2\epsilon}r^S. For each z\in E, put the uniform probability on its successful sampled records. Its direction masses are at most C\Delta^{-3\epsilon}r^b, by Equation (5.8) and \theta(z)\ge\eta/4.

Dualize: the source point z parametrizes the line

The map from z to the line parameters is bi-Lipschitz on the bounded chart, so this family supports an S-Frostman probability with the preceding small power loss. On each \ell_z, the successful records give a b-Frostman probability: the direction parameter and distance along the line are uniformly comparable. Yet all these points lie within O(\Delta) of a union of at most

grid cells, using the output lists separately for every sampled record. Repeated records or repeated directions cause no problem. Apply Equation (RW) with its point exponent b and line-family exponent S. For any fixed \omega>0, choosing \epsilon small enough gives the lower covering bound

Comparison with the upper count excludes every strict deficit in Equation (5.7).

For completeness, suppose mass bounds are supplied only at radii \Delta^{\sigma_j} on a grid with successive gaps at most \gamma, including the two endpoints. Enclose a ball of arbitrary radius by a bounded number of cells at the next coarser tested radius. The additional loss is at most \Delta^{-2\gamma}, since S\le2 and b\le1. Choose \gamma before the other small losses to fit the strict deficit. Finally, errors of size \Delta^{1-\tau} are handled by rerunning the proof at that effective final resolution. The output count exponent becomes x/(1-\tau), and the small power constants are rescaled by the same factor; the ball powers S,b are unchanged. The conclusion converges to the displayed one as \tau\downarrow0.

For the last assertion, the restriction estimate in Lemma 5.17 has upper bound I(Z;W\mid D)/c_\Delta+h(c_\Delta)/c_\Delta. This is o(\log(1/\Delta)) under the stated assumptions, since h(c)/c\le\log(1/c)+1. Selecting parents with c_D\ge c_\Delta/2 loses at most half the restricted probability and adds only an exponent-zero normalization factor to the mass bounds. The rest of the proof is unchanged.□

Proof. Put

Choose \kappa,\epsilon_0>0 so small that \kappa+\epsilon_0<g/4 and \kappa<(1+A/d)/2. We choose a separation parameter \lambda>0 in the clustered case below. We prove the assertion with \eta_0=\eta, where \eta is chosen last, sufficiently small relative to these parameters. The exact requirements on \eta are collected at the end of the proof.

Cover the bounded space of lines meeting the support by finitely many slope–intercept charts and partition each chart into \Delta-squares. Assign each witness line to one such square c, and let T_c be a fixed C\Delta-tube containing the \Delta-neighborhoods of every line assigned to c. Define

A pair at separation r\geq\Delta^\kappa belongs to at most C(1+r^{-1})\leq C\Delta^{-\kappa} such pairs of footprints. Indeed its allowable line directions occupy intervals of total length O(\Delta/r), while the intercept has only O(1) choices at width \Delta for each direction cell. Consequently

Call a cell spread if J_c\geq\Delta^{2A+\epsilon_0}. There are at most

spread cells.

For spread cells, this upper count will contradict the Furstenberg lower count \Delta^{-F(A,d)+g/4}. In a nonspread cell, the small mass of separated pairs instead forces most of the tube’s mass into one ball of radius \Delta^\kappa. We treat these two cases in that order.

Suppose first that spread cells account for at least half of the mass in (5.10). Pass to one line chart, losing a fixed factor, and discard pins whose partner mass in this relation is less than \Delta^{2\eta}. The retained pins have mass at least \Delta^{2\eta}. Their normalized distribution has ball bounds \Delta^{-3\eta}R^d, with a further fixed constant harmless. For each retained pin x, push its restricted partner measure to the line-parameter points \mathfrak l(x,y) and normalize. This probability has ball bounds

To check this, choose one witness line in a parameter ball which meets the support of the pushforward. Every partner belonging to that ball lies in a CR-neighborhood of this witness line, so (rad-assume) applies. Values CR>1 are covered by the mass bound and a fixed constant.

Under planar duality the parameter points for a fixed pin lie on its dual line. In the chosen bounded chart the map from pins to dual-line parameters is bi-Lipschitz. Thus (RW) applies with point exponent A and line exponent d, giving at least \Delta^{-F(A,d)+g/4} parameter cells. This contradicts (5.11), since \epsilon_0+\kappa<g/4 and \Delta is sufficiently small.

It remains to treat witness pairs assigned to nonspread cells. For any such occupied cell put m_c=\mu(T_c); then m_c\geq\Delta^{A+\eta}. Averaging over z\in T_c gives a point z_c\in T_c such that

Hence the cluster

has mass at least \tfrac12\Delta^{A+\eta}, and therefore at least \Delta^{A+2\eta} when \Delta is sufficiently small.

Choose

Discard witness pairs at distance less than \Delta^\lambda. Their total mass is at most \Delta^{d\lambda-\eta}, which is negligible compared with \Delta^\eta. Since \lambda<\kappa, at least one endpoint of every remaining pair has distance at least \tfrac14\Delta^\lambda from its cluster. Symmetry permits orienting a relation of mass at least a fixed multiple of \Delta^\eta so that its first endpoint x has this property. Retain pins whose partner mass is at least \Delta^{2\eta}; the set of retained pins again has mass at least \Delta^{2\eta}.

For each retained pin take a maximal separated collection of its witness directions, with angular separation

A ball of directions of radius CH accounts for partner mass at most C\Delta^{-\eta}H^A\leq\Delta^{A(1-2\lambda)-2\eta}, by the tube upper bound centered at one witnessing line. It follows that the collection has at least

members, with harmless rounding of this lower bound. Assign to each member the cluster of its witness cell. These clusters are disjoint: every point of such a cluster has distance \gtrsim\Delta^\lambda from x, so its direction from x differs from the witness direction by O(\Delta^{1-\lambda}), which is much smaller than H.

Let E' consist of the new pairs (x,y) with y in one of these chosen clusters. The choices can be made measurably by a fixed finite grid and a deterministic greedy ordering. Using (5.12) and the cluster mass gives

Retain with each new pair its original witness line \ell_0 and cluster K. We have

Here x\in\ell_0, y\in N_{C\Delta}(\ell_0), and |x-y|\gtrsim\Delta^\lambda. The angle of the two lines is O(\Delta^{1-\lambda}). On the cluster, whose diameter is O(\Delta^\kappa), rotation about y costs at most O(\Delta^{1+\kappa-\lambda})=O(\Delta). This proves (5.13) without thickening its final tube by a power of \Delta.

Set q=(1+A/d)/2. By (5.9),

Pigeonholing the dyadic annuli between radii \Delta^q and C\Delta^\kappa therefore assigns to every pair of E' a radius D in this range for which its cluster contributes at least \Delta^{A+3\eta} mass in the annulus D\leq|z-y|\leq2D. There are O(\log(1/\Delta)) choices of D. For one common D, the corresponding relation has mass at least \Delta^{4\lambda}. Choose y so that its set of pins has mass at least \Delta^{5\lambda}, and put

This pin submeasure has angular ball bounds

whenever the angular ball meets the retained pin set; increasing the exponent loss by \eta absorbs any additional fixed constant. In fact choose one pin in that ball and use its original witness line \ell_0. The new line differs from \ell_0 by O(\Delta^{1-\lambda}), whereas R\gtrsim\Delta^{1-\kappa} and \lambda<\kappa. Every pin in the angular ball is consequently in N_{C'R}(\ell_0). The original upper bound in (rad-assume), not an upper bound asserted for a newly conditioned measure, proves (5.14).

A maximal collection of pin directions separated by a sufficiently large constant times R has at least \Delta^{5\lambda+3\eta}R^{-A} members. Their C\Delta-tubes in the annulus of radius D about y are disjoint. Each contains mass at least \Delta^{A+3\eta} from its assigned cluster. Consequently

The ball upper bound, with its fixed-radius constant absorbed, is at most \Delta^{-2\eta}D^d. These inequalities imply

contrary to the choice of \lambda,\eta for small \Delta.

To meet all the losses used above, after choosing \kappa,\epsilon_0,\lambda in that order choose \eta>0 with

and smaller than the loss allowed by (RW) with conclusion loss g/4. These are finitely many compatible requirements. Finally choose \Delta_0 small enough to absorb the fixed constants and logarithmic factors in both cases.□

Proof. Discard at each tested depth the time bins which violate the asserted mass bound. Their average mass tends to zero; finitely many such trimmings have the same property. Each time interval is covered by a bounded number of comparable bins, so the trimmed time marginal, and hence every planar ball, has the required unnormalized bound. Increasing finite grids can be chosen diagonally. Only grids fixed before the \rho-limit are used.

The displayed tube masses give legitimate limiting random variables. Indeed an endpoint ball of a sufficiently smaller constant radius lies in the joining tube. Among O(\rho^{-2t}) planar cells, the total trimmed mass of those having mass less than \rho^{2t+\alpha} is O(\rho^\alpha). The probability that either endpoint was trimmed away tends to zero. Thus the exponents can be clipped on exceptional sets and jointly subsequenced at countably many rational queries. Nearby depth queries handle constant changes of width.

If 0<e_*<d on a set of positive probability, choose an essential value A\in(0,d). Given the tolerance in Lemma 5.21, restrict to |e_*-A|<\epsilon, with \epsilon sufficiently small. At each such sample the lower inequalities e(u)\geq(A-2\epsilon)u hold for every sufficiently small rational u, while e(t)\leq(A+2\epsilon)t holds at arbitrarily small rational t. A countable union therefore supplies a fixed small t and a positive-probability event on which these inequalities hold at t and at every point of a prescribed finite grid \sigma t, 0<\sigma\leq1.

Choose that grid with mesh small relative to the tolerance of Lemma 5.21, including a first point so small that the trivial mass bound fills the gap to radius one. Monotonicity fills all intervening radii. Transfer the strict inequalities to the approximating laws and put \Delta=\rho^t. The time bounds give (5.9); the tube bounds give (rad-assume) with any prescribed sufficiently small loss. The event has fixed positive probability at this stage. Since both marks were sampled independently given Z, its surviving pair mass is the average of the corresponding \mu_Z\times\mu_Z relation masses. One can consequently choose Z with pair mass bounded below by a fixed positive constant, in particular by \Delta^{\eta_0} for small \Delta. The event and its transpose may be combined to make the relation symmetric. This contradicts Lemma 5.21.

A uniform positive tube-avoidance exponent gives e_*>0. The dichotomy then gives the last assertion.□

Proof. Since S is known from Z, conditioning on A does not change the full record kernel under R. Conditional iid sampling also allows the other full records to be held. On parents of positive probability, the exact conditional pair law is consequently

Its moving planar marginal is \mu_z^K=(\pi_z)_\#\nu_z^K=\mu_z|_K/\mu_z(K). This factorization holds before selecting G. Take the jointly Borel inner version of G from Lemma 3.19 under the sum of the P- and R-laws on the full record tuple. It preserves their masses and the pointwise witness bounds; all conditional assertions here are for almost every common datum. Its direction marginal on G is

where \mathcal W_{D,J} is the set of planar projections of surviving pin records in that direction interval having some full-record witness V_y with projection in K. The inner probability is at most one. In particular the bound does not contain a factor \mu_z(K)^{-1}.

If \mathcal W_{D,J} is nonempty, choose one full-record witness and denote its planar projections by (x_0,y_0). Since y_0 is within O(\delta^b) of c_K and the pair is separated, the directions of \mathfrak l(x_0,y_0) and \mathfrak l(x_0,c_K) differ by O(\delta^{b-\chi})=o(r). Every participating pin lies in a fixed bounded region and makes angle O(r) with that witness direction about c_K. It is therefore in N_{Cr}(\mathfrak l(x_0,y_0)). The original trimmed tube bound proves Equation (5.15). Notice that the source law is still \mu_z^K; it has not been replaced by the conditional law of the trimmed measure.

For the change of law use the full finite tuple carrying the event, the common data, and all tested source and pin names. Its relative entropy is at most the stated budget by data processing. Write f=dP/dR on this tuple and f_D=dP_D/dR_D. The likelihood estimates in Lemma 5.17 show, for fixed \eps,t>0, that outside a set of vanishing P-probability,

The conditional likelihood on the retained event is at most \delta^{-2\eps t}. Integrating it over either the source or the pin proves the claimed unnormalized marginal bounds under P. Intersecting further with a popular-output event only decreases these masses.

Finally, decomposition of relative entropy conditional on D, followed by its decomposition against the product kernel \nu_z\times\nu_z^K, bounds I_P(V_y;V_x\mid D) by \operatorname{KL}(P\Vert R). The name W_y is a function of V_y,D, so data processing gives the asserted bound on I_P(W_y;V_x\mid D), also with the full pin record retained in the projection lists. The scale conversion is valid because t is fixed. If a strict violation is retained on an event of probability c(t)>0, take the earlier information error negligible compared with c(t)t\log(1/\delta) before invoking the positive-event restriction rule. This uses no uniform assertion as t\downarrow0.□

Proof. It suffices to treat 3\epsilon/d<1: by increasing C_d, the conclusion is immediate from the total mass bound for larger \epsilon. The time bound also bounds planar balls. Put a=\Delta^{3\epsilon/d}. A ball of radius a about x has mass at most C\Delta^{2\epsilon}, so at least half the mass of each witness tube is outside that ball, for small \Delta. Choose a maximal family of witness directions separated by C\Delta/a. Outside B(x,a) their \Delta-tubes are disjoint when C is sufficiently large. The number of chosen directions is at most C\Delta^{-\epsilon}. Maximality places all witness points within O(\Delta/a) of one of the representative lines.

Each representative is a witness, so its slope m satisfies c\Delta^\epsilon\leq|m|\leq C\Delta^{-\epsilon}. On its enlarged tube,

For the points under consideration, |\theta_y-\theta_x|\geq\Delta^\epsilon. On each of the two possible signs of this difference, the inverse of the quadratic has derivative at most C\Delta^{-2\epsilon}. Thus f_x(y)\in I restricts \theta_y to at most two intervals of total length C\Delta^{\sigma-C_d''\epsilon}. Apply the time-marginal bound and sum over at most C\Delta^{-\epsilon} representative lines. This gives (5.16); intervals longer than one are handled by the total mass bound. Rounding the queried lengths to a finite grid merely adds its mesh to the loss.□

Proof. We first record precisely how a high radial bound produces a direction bound. On any finite grid of sufficiently small depths u=\sigma t, a high witness satisfies

with a further arbitrarily small absolute loss when transferring to the approximating laws. For fixed Z,\xi_i, an angular bin which meets the witness set has all its witnessing partners inside a constant enlargement of one witness tube. This gives the same bound for the unnormalized partner-direction sublaw. One may instead fix Z and a j-parent bin at depth p\gg t. Use its representative as the center. After discarding pairs closer than a sufficiently small fixed power, the change of center is negligible at every queried angular width. An angular bin with a witness again places all its participating pins inside an enlargement of that witness tube. Thus the pin-direction sublaw, unnormalized and with this parent held, has the same exponent d. These are bounds on a restricted measure, not claims that the two marks remain independent after the witness restriction.

The cumulative score of f given Z,\xi_i is Lipschitz, starts at zero, and is differentiable at almost every depth. If (5.17) fails on a set of positive probability, Fubini’s theorem and the fundamental theorem of calculus provide a fixed differentiability depth p>0 and a positive-probability high set where the incremental rate is strictly less than d. Choose t\ll p small enough to test this strict deficit, together with a finite prefix grid on which the preceding high tube bounds hold.

Adding \xi_{j,p} to the conditioning can only lower the tested limiting incremental score, by the score calculus. The coarse product is known to bounded lists from this parent and the pin. After translating and dilating the j-parent, the product increment is, up to O(\rho^p) in dilated units, the linear form with coefficient vector

Its norm is bounded below by an arbitrarily small power with probability tending to one. Since t\ll p, both its quadratic remainder and the finite-record errors fit the tested resolution, with arbitrarily small exponent loss. The source increment has prefix exponent at least d by hypothesis. The gradient directions, given Z,\xi_{j,p}, have unnormalized exponent at least d on the high relation: they are the joining directions just estimated, with the two coordinates interchanged. Before restriction, the pin is independent of the j-increment given this parent, since the two marks were independent given Z. The conditional projection rule applies and gives projected rate at least \min\{d,(d+d)/2,1\}=d, contradicting the strict deficit. Integrating the almost-everywhere rate bound proves (5.17). Popular-cell trimming converts it to the corresponding unnormalized interval-mass bounds whenever needed.

On \mathcal G_0, the definition of a pointwise liminf supplies arbitrarily small rational t with e(t)<\epsilon t at each sample. A strict violation of a differentiable limiting rate persists at every sufficiently small layer at that sample. A countable union therefore selects one fixed such t on a positive-probability violating set. Impose the two scalar separations at threshold \rho^{\epsilon t}; their exceptional probability tends to zero in the earlier limits. The trimmed time marginal has the required bounds at the finite queried depths. At \Delta=\rho^t, the condition on e(t) gives the tube lower bound of Lemma 5.24, with an arbitrarily small additional loss. That lemma proves the asserted interval bounds at all the necessary prefixes.

Finally, these conditional bounds can be averaged over Z and independently sampled auxiliary records when the parent conditioning keeps the other pin fixed and permits that mixture. A different conditioning requires its own justification. In particular the selected low-set event may have a small positive probability depending on t. One first fixes this t and its finite queries, and only then makes the earlier information errors negligible relative to that probability. Nothing here permits transferring an information budget through a restriction whose probability is merely a power of the fine scale.□

Proof. Let O be the actual child name at (a+\lambda h,v). Given C_*, its limiting entropy divided by \lambda is \partial_a\mathcal H. Even after giving a sufficiently fine phase name, the time bits cost at least d\lambda: reveal Z and apply the conditional time-mark bound. The remaining entropy is bounded below by a sum of

over consecutive layers. The chain rule makes this a telescoping sum of information about phase.

We will lower-bound the phase information by integrating the expected values of the following local rates over the indicated intervals:

The middle interval is absent when c=1. The first rate comes from recovering the phase after paying for its angle; the other rates come from readings recoverable from the output.

For p+t<\min(1,c)\lambda, the output together with C_* and X_{p+t} reads F_{p+t} up to negligible lists. Its positions give the two base coordinates at the child time; undoing its displayed shear gives the common-frame coordinates and, in the quadratic case, the residual derivative. The extra angular increment has conditional cost at most mt by (ang). Consequently Equation (6.4) pays the expected w-rate on this interval.

For the remaining intervals, we must identify the projected readings and show that their coefficient information is paid for. Let

The child-time error is O(\delta^\lambda). The converted base positions read W',V' to depth \lambda. In common parent units, the child shear difference has known constant offsets and coefficients \mathcal L,\mathcal R,\mathfrak b at time \theta'. Its residual velocity and position are displayed to depths c\lambda and (1+c)\lambda. The derivative

approximates P in the quadratic case, or r_i in the linear case, to depth c\lambda.

In a linear branch the curvature coefficients vanish, or are negligible at every tested fixed power by the branch slack. For constant quadratic cuts, \mathcal R=r_i+O(\delta^{c\lambda}) and \mathfrak b=0. For the quadratic equality branch with c>1,

This is the own-time coefficient comparison, extrapolated back to normalized time zero after subtraction of the root trajectory. All its nonconstant normalized coefficients are bounded; constant offsets are retained exactly in the translations.

In the quadratic case the own-time algebra can be checked without an implicit change of origin. Put

The derivative comparison gives |\epsilon_0|\lesssim\delta^{c\lambda}. With actual displayed coefficients held fixed, the residual velocity has gradient

and the residual position has gradient

Subtracting \theta' times the former from the latter gives K-\mathfrak aH+\epsilon_0\,dB. These identities are evaluated at the actual phase and follow directly by differentiating the shear polynomials; the known constant offsets have zero variation. At a parent representative the derivative error is O(\delta^p+\delta^{c\lambda}). Multiplication by the parent width gives O(\delta^{2p})+O(\delta^{p+c\lambda}). The second term is negligible on a microlayer with t<c\lambda, and fits a binary layer whenever p\leq c\lambda. The derivative display itself has gradient dP-2\mathfrak a\,dX+2\mathfrak b\,dB. Equation (6.5) therefore gives the three claimed G_i readings with the stated precision.

In the linear case the shear is linear, with |r_i-\mathcal L|\lesssim\delta^{c\lambda}. The displayed residual velocity and the backflowed residual position have respective gradients

They differ from the two components of G_i by (r_i-\mathcal L)dX and (r_i-\mathcal L)dB. Before backflow the position gradient is dD+\theta'\,dN-\mathcal L(dB+\theta'\,dX), which gives x_i with the additional O(\delta^\lambda) time-coefficient error.

Center each displayed quantity by its value at F_p, evaluated with its actual displayed coefficients and offsets. Taylor’s formula has error O(\delta^{2p}). The residual-position gradient converges, with a positive power error on an interior layer, to x_i. Below depth \lambda, the base-position gradients additionally give W_i,H. If c>1, subtract \theta' times the displayed velocity from the position. In the quadratic case the two resulting gradients give K-\beta_iH and J-\mathfrak k_iH, and the derivative equation gives L_i. In the linear case they give the two components of G_i. Thus the intermediate interval gives \pi_i when c<1 and G_i when c>1, while the final interval always gives x_i.

For completeness, the projected entropy is indeed paid by Equation (6.4), even though its coefficients may depend on phase. Freeze the needed angular and curvature coefficients into a name \Omega at a depth q with

Shrink the microlayer first so that such a fixed q exists. On a compact interior interval of depths, the preceding output comparisons recover \Omega from O,C_*,F_p up to negligible lists. The centered reading U is also determined, up to such lists, by both (F_{p+t},C_*,F_p,\Omega) and (O,C_*,F_p,\Omega). For finite tested names A,B,\Omega,U and arbitrary conditioning data C, the chain rule and the two reconstruction bounds give

To obtain this inequality, first adjoin \Omega to B, at cost at most H_\delta(\Omega\mid B,C), retain the conditional information given \Omega, and apply the chain rule to the two reconstructions of U. Use A=F_{p+t}, B=O, and C=(C_*,F_p). All three error terms tend to zero at the fixed layer, before division by t. Conditioning further on the stored marks lowers this entropy. Translation at the parent and the coefficient-freezing comparison then identify the relevant contact-reading rate.

Tile compact subintervals of the three depth ranges by smaller layers. Bounded convergence for the score derivatives and nonnegativity for the information sums permit passage to their integrals. Truncate near the endpoints, then remove the truncations using the uniform relative capacities. Finally average in \lambda logarithmically. Dilation invariance of this mean gives Equation (time-rate).

For width variation, increasing v by \lambda h in a linear branch reads at most F_\lambda and one additional \lambda-depth scalar in vertical position. This proves the linear upper bound. In a quadratic branch the same change refines X,B,P by \lambda and the three recentered coordinates by 2\lambda. In changing the linear shear, the new binned coefficient differs from the true normalized P by O(\delta^\lambda). After translation, its effect on unresolved horizontal widths is O(\delta^{2\lambda}). Hence the new width name and F_{\lambda,2\lambda} determine one another up to bounded lists, given C_*. Equation (vb) allows marked conditioning in these finite-depth entropy comparisons. Since F_0 has zero normalized entropy, logarithmic averaging of Equation (6.3) proves Equation (width-rate).□

Proof. Adjoin Q_i,X_p to the conditioning. Along this fiber the phase is determined by X to every fixed tested accuracy. Differentiating the fiber equations gives Equation (6.6). Their quadratic remainder on an angular parent cell is O(\delta^{2p}). Thus F_p and the annihilating child readings have negligible list cost with the enlarged conditioning. A nondegenerate component recovers X_{p+t} up to negligible lists. For a subpower lower bound on its derivative, fix the separation exponent much smaller than the strict loss in the test, apply one-dimensional inversion, and then let that exponent decrease. Equation (fib) gives the residual rate m. The reverse upper bound for angular rates follows from (ang) and conditioning.

For the center mode, walk successively along fibers i,j, with endpoints F_0,F_1,F_2 and angular coordinates X_0,X_1,X_2. Adjoin the starting side, very fine X_2 bits, and X_{1,p}. The finite-walk likelihood comparison and (fib) leave rate m for X_{1,p+t}. The separation |X_1-X_2|\geq\delta^\chi holds with probability tending to one for every fixed \chi>0: conditional on the previous history, a new angular prefix has positive score at each positive depth, so it cannot lie in a bounded list of prescribed bins with nonvanishing probability.

Apply (leg) on the two legs and differentiate with X_2 fixed. The resulting center derivative at F_2 is

In particular its last component is 2(X_2-X_1)(\theta_ir_j-\theta_jr_i)dX_1. The same parent-cell Taylor and inversion argument applies. The endpoint has the original marked marginal, so the result is a statement about its original conditional rates. Substitution in the formula for x_i or x_j verifies the last assertion.□

Proof. The chain rule gives S\geq x+M. The projection rule gives x\geq\min(1,S,(S+d)/2). If x<1, then S>x, so the only remaining possibility is 2x\geq S+d\geq x+M+d. This gives x\geq M+d.

For random rates these deductions are applied to a positive-probability set with a strict deficit. Bin the rates to sufficiently short fixed intervals, use mixed-prefix linearity to obtain the finite-prefix source bounds, and use popular projection cells to obtain the output lists. The information error vanishes at this fixed layer. The projection rule excludes the deficit. This procedure also allows the high-set radial witnesses and the low-set pin-product witnesses of the preceding section; it never conditions on an unspecified subpower-probability event.□

Proof. In all uses of Lemma 6.4, the planar source and any common child conditioning depend only on phase and the held records. Equation (vb), followed by the conditional chain rule, permits removal of the varying record. Its directional sublaw bounds, initially conditional on Z and the held records, integrate to the tested conditioning. This verifies the information requirement without asserting independence after a witness restriction.

For W_i, use the source (X,B), conditional on the center in the quadratic case and on H in the linear case. The direction parameter is \theta_i, with exponent d, and \ell_i gives residual m.

For the quadratic center inequality, condition on x_i to child depth and use (H,J). It determines all center coordinates with x_i. Vary j and project with kernel the image of b_{ij}. The direction is a nonsingular linear transform of the joining-line direction in mark space. Radial spreading and Lemma 6.3 give a projected rate at least l_* and residual m. Their sum gives |C_z|-|x_i|\geq m+l_*.

For the remaining horizontal rate, condition on the whole center and quotient (dX,dB,dP) by \ell_i. One may use the quotient coordinates (W_i,dP-2r_i\,dX). The missing coordinate has rate at least m. On the quotient, vary j and project with kernel the image of \ell_j. Its direction is another nonsingular transform of the joining line, and its residual is m. Thus the quotient rate is at least m+l_*, proving the third bound in (modes). Time separation suffices for the nondegenerate coordinate comparisons.

Next, x_i is a combination of the two center components of quadratic G_i. Given x_i, the form J-\mathfrak k_iH sees b_{ij}. Given the center, L_i sees \ell_j. Both tests are nondegenerate because

has subpower separation by line avoidance. They contribute two residual copies of m. The pair G_i,G_j determines all three center coordinates and two independent horizontal combinations. Indeed the two coefficient pairs (\beta_i,\mathfrak k_i) and (\beta_j,\mathfrak k_j) have subpower separation by the same line test. One scalar coordinate remains, with rate at most one. Subadditivity and exchangeability prove the second bound in (Gq).

For the linear x_i bound, use the planar source (dX,x_i) and put f=(\theta_i-\theta_j)(r_j-r_i). Both projections x_i and x_i-f\,dX have residual rate exactly m: \ell_i and \ell_j, respectively, give the lower bound, and the angular upper bound gives equality. Consequently their projected rates agree. Vary j in the second projection and use the pin-product direction alternatives with Lemma 6.4. This proves |x_i|\geq l_*. For G_i, its image of \ell_j is nondegenerate by crossing and has direction \theta_j. The same argument gives a projected rate l_* and residual m. Two G readings determine every coordinate except Y, so the last expected inequality follows.

Finally (\pi_i,X) misses only one additional scalar coordinate by rank. In this comparison all inversions have bounded coefficients. Its missing angular rate is at most m, and the other scalar costs at most one. Thus \alpha\leq|\pi_i|+m+1, which is (miss).□

Proof. Walk 2n distinct legs. Adjoin the starting side, very fine even angular coordinates, including X_{2n}, and the odd angular parents at depth p. These data predict the endpoint parent to negligible lists. Summing the B,Y formulas in (leg), and translating at that parent, makes the endpoint H reading equivalent to

The coefficient of H may first be frozen at the known fine endpoint or at the prescribed fixed positive prefix; the increment comparison gives the same rate.

Let s_k be the rate of the partial sum with these data, and set s_0=0. The source consisting of S_{k-1} and the new odd quadratic scalar has joint prefix rate at least s_{k-1}+m. To see the second contribution, reveal the other fine endpoint bits, which predict the previous sum. The derivative of the new scalar is 2(X_{2k-1}-X_{2n}), with subpower separation, and (fib) gives its remaining angular rate m.

For the projection, omit the current pair of records. The difference of their time marks has exponent d. Here is the required conditional comparison. If X_j^* are fixed finite-depth angular names fine enough for all these tests, the walk rule gives

This follows by the conditional chain rule in chronological order: the next endpoint is sampled from its conditional fiber kernel, the marked marginal is stationary, and its angular information cost is (fib). The initial angular coordinate is included in Z_0 and need not be independent of it.

The reference law in Equation (6.8) factorizes the omitted record and the source angular bits conditionally on the remaining data. The source parents are predicted to negligible lists by the odd parents and the retained data. Conditional chain rules give the same negligible information budget after these parents are revealed. Conditional likelihoods at actual samples agree up to vanishing exponents. Trim excessive likelihood ratios and time-ball failures with any fixed small power slack. The unchanged start-side marked marginal supplies the unnormalized direction bounds, which integrate to the source conditioning. This verifies the projection-rule hypotheses at the actual law; no assertion merely typical for the reference law is transferred without a likelihood bound.

The projection rule therefore gives

Iterating this monotone map from zero converges to \min(1,m+d): below that value each of its three entries exceeds the current value, and its least positive fixed point is \min(1,m+d). Adding the auxiliary data can only lower the original endpoint score, whose parent was already predicted. Given any strict desired error, choose a finite n making the iterate sufficiently close to the limit, and only then choose the common replica experiment and cutoffs.□

Proof. The target and the resampled sides are exchangeable conditionally on the shared fiber data. Their angular information bounds give the stated separations. Holding R_s and X_F fixes A. In the quadratic case the leg formulas give

Differentiating and substituting X_F=X_R+A and P_F=P_R+2r_jA into the contact forms gives Equation (trans). In the linear case use N_F=N_R+r_jA and D_F=D_R-\theta_jr_jA. Direct substitution proves the assertion about x_j.

The conditioning order is essential. First freeze every coefficient involving the moving mark at a depth below p and above t. With the full record known, translate at F_p to compare the original and frozen readings. Equation (vb) then removes the remaining fine record from the list test. Only after this removal do we adjoin Z_{R_s} and the motion data. At a mark parent of depth p, these data predict F_p and every held annihilating child reading up to negligible lists. An informative component recovers the scalar innovation by inversion, with arbitrarily small power loss. Its Taylor error is O(\delta^{2p}). Holding the other mark scalar to depth p+t, when permitted by its innovation law, adds only an error of that order of accuracy.

Auxiliary angular bits do not invalidate removal of the moving record. For example, for two draws and fixed sufficiently fine names,

Indeed let \mathcal B=(Q_j,\mathcal D) and Y_s=X_{R_s}^*. The variables Y_1,Y_2 are conditionally independent given \mathcal B, and their joint conditional law depends on (Z_F,\boldsymbol\Xi) only through \mathcal B. Since \mathcal B is determined by these latter data, the conditional chain rule gives

The inequality subtracts the nonnegative term I_\delta(Y_1;Y_2\mid C_*); the last equality is (fib) for the stationary marginals. The target bits X_F^* may be included in the finite phase budget in (vb). At R_s, its own angular bits are known, and the other angular bits satisfy the corresponding version of Equation (6.9). Finally, (Z_{R_s},\boldsymbol\Xi) has the original marked marginal, so the replicas are conditionally independent there and the stated mark innovation law applies. Conditional chain rules permit the specified mark parents and child scalar bins. This proves the score comparison without asserting independence conditional on all the exact motion data simultaneously.□

Proof. For the quadratic x_i bound, move a record j\ne i. Its gradient in mark space is

Keep Z_R, the moving mark parent \xi_{j,p}, the needed target angular bits, and other records except i,j. Use i as the varying projection record. The input mark increment has dimension at least d. By the parent-representative version of radial spreading, the gradient directions over i have unnormalized Frostman exponent d on the high witness relation. Swapping and signing coordinates does not change that assertion. Pair separation makes the gradient magnitude nondegenerate up to subpower loss.

To verify the output and independence hypotheses, start from a proposed low-score list for x_i conditional on F_p,\mathcal D_{-j}. The removal of record j is allowed by (vb). Given its parent and the motion data, F_p has negligible lists. Taylor expansion at the phase predicted by the mark parent, followed by dilation by \delta^{-p}, turns this into a list for the stated linear projection. The law of the angular bits has negligible likelihood cost relative to retaining (Z_R,\boldsymbol\Xi) and sampling those bits independently given C_*. Under this comparison kernel the planar source and the projection record are independent given the test data. Trim likelihood and direction failures with fixed small power slack; the marked marginal is unchanged. Thus the source and direction dimensions are both at least d, and the projection rule gives |x_i|\geq d. Precisely, Lemma 5.23 applies with side Z_R, auxiliary name X_F^*, S=C_*, moving mark j at parent depth b=p, and pin record i. It supplies both the unnormalized parent-direction bound and its conditional likelihood transfer, without dividing the pin bound by the moving parent’s probability.

For the mixed quadratic bounds, move i itself. Hold r_i to child depth. The H component reads the time innovation, of rate at least a_*, and x_i annihilates it. Next hold \theta_i to child depth. The J component reads the r_i innovation, of rate \nu, while both H and x_i may be held. The chain rule gives the two center bounds. If \nu>0, use the planar pair (J,K) conditional on H and the projection K+\theta_iJ. Holding \theta_i to child depth, this projection and H annihilate the r_i innovation, while J reads it. Remove the full i record from the planar source by (vb), then use the time direction and Lemma 6.4 with residual \nu. The resulting projected rate is n_*. Restoring the record and H identifies it with x_i. When \nu=0 use (cross-q). Finally W_i, even conditional on the whole center, costs at least l_* by (modes).

In the linear case condition on (X,B,H) to child depth. The planar source is (dN,dD) and the projection is dD+\theta_i\,dN. With \theta_i held to child depth, the r_i innovation has rate \nu, is annihilated by this projection, and is read by dN. Lemma 6.4 gives rate n_*, also for x_i with this finer conditioning. Add |W_i\mid H|\geq l_* and |H|\geq l_* from (H-walk). This proves the first two claims. Given \pi_i, the angular mode gives rate m for X; after those readings are held, dN still reads the r_i innovation with residual \nu. Consequently \alpha\geq|\pi_i|+m+\nu.□

Proof. In the first case move j\ne i, holding r_j to child depth. The time innovation still has rate at least d. The reading

annihilates that motion. It is a projection of G_j with direction \theta_i. The image of \ell_i in G_j is nondegenerate by crossing and is annihilated by V. Lemma 6.4 therefore gives |V|\geq l_*.

For c=1 move i and set V=x_i. Do not hold the refined r_i; the joint-parent time innovation still has rate d, and x_i annihilates the full motion. For c>1, move i, set V=G_i, and hold r_i to child depth. The allowed depth condition p+t<cp gives the time rate d, and G_i annihilates this remaining motion.

In all cases use two conditional draws R_1,R_2 as in Lemma 6.7. The motion of (W_i,H) has components (-A_s,A_s^2)d\theta. After adding the auxiliary angular bits, first test H+A_1W_i with the second draw. Its coefficient is A_2(A_2-A_1), with subpower separation, and V is held. Next hold both V and H+A_1W_i, and test W_i with the first draw. The held combination annihilates this motion and the coefficient of W_i is -A_1. The chain rule gives |(W_i,H)\mid V|\geq2d. The conditioning comparison in Lemma 6.7 justifies each test before dropping the auxiliary bits.

In the first case (W_i,H,V) and \pi_i are equivalent with the marked data, since their last components differ by (r_i-r_j)W_i. In the other two cases the held readings also annihilate \ell_i, leaving angular rate m. These observations prove all three inequalities.□

Proof. Fix p<z<2p and refine the horizontal depth from p to p+t, leaving z fixed. Given F_{p,z}, this increment is equivalent to the new horizontal bits: recentering contributes only O(\delta^{2p}). For every angular mode, Q_i,X_p,\mathcal D predict F_{p,z} up to negligible lists. Indeed the contact-center derivative on the fiber is O(\delta^p) on the parent, and its variation is O(\delta^{2p}).

Two time-separated angular modes are linearly independent. First test a horizontal scalar annihilating \ell_i and seeing \ell_j; then test a scalar seeing \ell_i. These tests remain valid after holding the scalar annihilating both modes. Lemma 6.3 therefore gives horizontal increment rate at least 2m. Integrating in p from \lambda to 2\lambda with z=2\lambda fixed yields e(\lambda)\geq2m in the averaged sense. Boundary endpoints are handled by the Lipschitz continuity of the array on the wedge.

For the additional rate, a horizontal scalar annihilating both modes is

Set q_0=z-p and u=p-q_0>0. Resample R on the i fiber while sharing X_{q_0}. Then A=X_F-X_R is O(\delta^{q_0}) and, for every fixed \chi>0, at least \delta^{q_0+\chi} in magnitude with probability tending to one. This follows from the conditional version of (fib). The marked marginal is stationary, and the target angular bits beyond the common prefix retain negligible information with Z_R,\boldsymbol\Xi given C_*,X_{q_0}. To see the exact comparison, set \mathcal S=(C_*,X_{q_0}) and \mathcal B=(Q_i,\mathcal D,X_{q_0}). The resampling kernel is the conditional side law given \mathcal B. The conditional Markov property and (fib) give

The second inequality uses that X_{q_0} is a prefix of X_F^*. Thus the comparison law keeps (Z_R,\boldsymbol\Xi) unchanged and samples X_F^* independently with its conditional law given \mathcal S. The common prefix is already determined by Z_R.

Start the moving mark at depth u. Given Z_R, fine X_F bits, and \xi_{i,u}, the name F_{p,z} has negligible lists. Horizontal uncertainty is O(A\delta^u)=O(\delta^p). Guess the boundedly many corresponding horizontal bins. With these bins fixed, the recentered uncertainty is

The mixed quadratic mark remainder is O(A^2\delta^{2u}), which is smaller.

Freeze the i coefficients in Equation (6.10) at their u-parent representatives. Indeed its expanded form is

Each coefficient changes by O(\delta^u). On a horizontal parent its centered increment has size O(\delta^p), so the change is O(\delta^{p+u}), negligible at depth p+t when t<u. Absolute shifts at the parent are retained during this comparison with the full record. Remove the remaining fine i mark by (vb) before adjoining motion data. With X_F fixed, the resulting mark-plane projection is

After dilation of the mark parent, a low-rate list at depth p+t becomes a projection list at depth t, up to arbitrarily small power losses. Its source has joint dimension at least d. More explicitly, the common test data are Z_R,X_F^*,\xi_{i,u} and the held records other than i,j. The source is the child increment of \xi_i and the projection record is j. Under the comparison law they are independent given these data: the marks are independent given Z_R, and X_F^* is independent of them given \mathcal S. Conditional chain rules retain a vanishing information error at each fixed u,t. The original low-score list can be unioned over the bounded parent possibilities described above, giving a list for each j record. The mark innovation bounds apply at Z_R,\xi_{i,u}. Radial spreading at this same parent gives the unnormalized direction bound of exponent d over j on the high witness relation. Finite-prefix likelihood trimming, with fixed small power slack, verifies the projection hypotheses at the actual samples. This is Lemma 5.23 with side Z_R, auxiliary name X_F^*, S=(C_*,X_{q_0}), moving mark i at parent depth b=u, and pin record j. The required separation exponent can be chosen so that u-\chi>t, since t\ll u. Thus U has incremental rate at least d in the strict-error sense.

This last argument does not assume homogeneous horizontal increments under the finer center conditioning. More explicitly, an expected increment rate strictly below 2m+d along a sequence t\downarrow0 would, by bounded relative capacities and the two residual angular rates, give a positive-probability strict deficit for the U list. Radial witnesses provide the required finite-prefix bounds with probability tending to one as t decreases. Select one sufficiently small fixed t, take the tangent information errors to zero first, and apply the projection rule to exclude this deficit. Integration over the open wedge, followed by Lipschitz endpoint control, proves (e-q).□

Proof. The center name and two independent horizontal forms leave only one scalar coordinate; line avoidance supplies subpower singular-value bounds. Relative capacities therefore give 0\leq b_{\rm miss}\leq1. The joint increment of L_i,L_j given F_{p,2p} costs at least 2mp. Indeed L_i annihilates \ell_i exactly and L_j sees it, and conversely. Given Q_i,X_p,\mathcal D, the center bins to depth 2p have bounded lists by the fiber Taylor estimate. The horizontal invariance is linear, so these two comparisons apply on the whole binary layer. The conditional chain rule yields e(p)\geq b_{\rm miss}(p)+2m.

On (\lambda,r_0\lambda) use the microlayer estimate and 2\E|G_i|\geq\E\alpha-1. Tile [r_0\lambda,c\lambda] by binary intervals [p,2p], where p=c2^{-j}\lambda. On each such interval,

The output comparison in Proposition 6.2 then reads G_i to depth 2p in the contact basis at F_p. The Taylor remainder is O(\delta^{2p}), and the coefficient error in (6.5) times the parent width is no larger. Freeze the true (\beta_i,\mathfrak k_i) at depth p as output-predicted auxiliary data before adding marked conditioning. Subtract the velocity using the actual child bin time \theta'; this avoids a \lambda-depth error from an incorrect terminal-time backflow.

The pair G_i,G_j at this accuracy recovers the three recentered center readings and L_i,L_j, up to subpower losses. Hence symmetry and subadditivity lower-bound the information from a single output by half the isotropic binary increment minus pb_{\rm miss}(p)/2. Logarithmic averaging of that isotropic increment divided by p gives \bar\alpha: integrate its expected derivative on [p,2p] and use dilation invariance. The binary interval lengths sum to (c-r_0)\lambda, and the initial interval has length (r_0-1)\lambda. Their sum proves (bin).□

Proof. Write x,\pi,G,w,\alpha,e for the expected barred rates. Equations (time-rate), (width-rate), and (canon) reduce the claim to S\geq0, where

For c>3 in the quadratic case we may replace the last term in T by (bin). The omitted favorable \eta term is \eta c in the linear case and \eta(c-k) in the quadratic case. It is nonnegative for the stated speeds.

Quadratic freezing. Suppose first c<1. Equations (modes) and (mix-q) give

If \nu\leq k_0, then n_*=d+\nu, a_*\geq d-c\nu, and the last maximum is at least m+l_*. Thus

The desired bound follows from 2cm\geq(1+c)(1-l_*). If m\geq k_0 this is immediate. Otherwise it is equivalent to (1+3c)m\geq(1+c)k_0, which follows from m\geq k_0/(1+c) and 1+3c\geq(1+c)^2 for c\leq1. If \nu\geq k_0, then n_*=1 and the maximum is at least a_*+\nu. Equation (6.12) gives

because l_*\geq d and a_*+c\nu\geq d.

For c\geq1, the center and horizontal bounds give w\geq2m+2l_*+x and G\geq2m+x. Hence

If m\leq k_0, subtracting 1+2d+c gives (1+c)(2m-k_0)\geq0. If m\geq k_0, substitute l_*=1 and use monotonicity in m.

Quadratic motion. For 1\leq c\leq3, use 2G\geq\alpha-1, x\geq d, e\geq2m+d, and k=(c+1)/4. Cancellation of the \alpha terms gives

For c>3, set b=\bar b_{\rm miss}\in[0,1]. The two width bounds imply e\geq2m+\max(b,d). Substitution of (bin) gives

For b\leq d the right side decreases in b, so its minimum occurs at d. For b\geq d it is affine, so only b=d,1 need be checked. The respective values are

both nonnegative since r_0<2.

At c=1, using only e\geq2m and x\geq0 gives

This proves the third assertion, including m=0.

Linear freezing. If c<1 and \nu>0, Equation (mix-l) gives T\geq2l_*+(1+c)n_*+c\nu. Its excess over 1+2d+c is

For \nu\geq k_0 this is nonnegative. For \nu\leq k_0, if m\geq k_0 it is also immediate. Otherwise m\geq k_0-c\nu lower-bounds it by (1-c)k_0+\nu\geq0. If \nu=0, angular invariance gives w\geq\pi, so T\geq\pi+cx\geq2d+(1+c)l_*. The threshold gives l_*=1. For c=1 use w\geq x+2d and x\geq l_*=1. For c>1 use w\geq2d+G, G\geq m+l_*, and x\geq l_*. These yield T\geq2d+c(m+1)+1\geq1+2d+c.

Linear motion. For c\geq1, put k=(1+c)/2 and use 2G\geq\alpha-1. Cancellation gives S\geq1+x-2d-m. Since m\leq k_0, l_*=d+m; the bound x\geq l_* leaves S\geq k_0. For c<1, put k=c. The \alpha terms cancel directly, leaving

Use x\geq l_* and \pi\geq2d+l_*. The latter follows from (double) if \nu=0; if \nu>0, (mix-l) implies it because l_*+n_*\geq2d. Thus

Finally take k=1 and 2d\leq c<1. Equation (miss) gives

When m>0, crossing and \ell_j give x\geq m; when m=0 the same inequality is trivial. Since m\leq1, this lower bound is at least c-2d\geq0.

Every calculation involves finitely many linear inequalities with bounded coefficients on a compact parameter range. Strict-error versions therefore give the claimed stability; changing the speed costs the corresponding Lipschitz error.□

Proof. The angular capacities make M Lipschitz. Put m(p)=M'(p) at its Lebesgue points. Fix the tested observation i, its posterior draw R, and the other target observation j. Write A=X_F-X_R and, for a small fixed \eps>0, condition on

Its normalized entropy given C is O(\eps). In the contact frame of the current prefix define

Unless specified otherwise, rates are conditional on (C,\Omega,F_p); an additional held reading is held to child depth p+t. First restrict p to compact subintervals of (0,1) and (1,2). After \eps and a positive contact-prefix depth are fixed, take t smaller than these depths and than p.

We first account for the cost of \Omega. For each of the nested refinement families

where \Omega_0 is any fixed subtuple needed below, the integrated expected information loss on adding \Omega is at most H_\delta(\Omega\mid C)=O(\eps). This is the conditional chain rule applied along the refinement family. The cumulative losses have nonnegative Lipschitz limits, so their derivatives have the same integral bound.

Before this addition the first family has rate m(p) by stationarity. For the second, the posterior draw has the original side-and-observation marginal. Conditional independence given (C,S_i) and the first stationarity hypothesis give

Its time innovations therefore retain the lower rate d, by (LP). The same information comparison and M(z)\ge g_*z show that \abs A\ge\delta^\chi with probability tending to one for each fixed \chi>0: the contrary event puts X_F in a bounded list of depth-\chi bins predicted by X_R.

These budgets justify all subsequent rate errors as follows. Fix a strict proposed deficit and a positive lower bound for its probability. Let \eps\downarrow0, discard layer points where one of the finitely many loss densities is too large, and then choose typical remaining points for the pathwise and mixed-prefix differentiations. At each such point let t\downarrow0. The expected loss divided by t tends to zero along this procedure; the likelihood inequalities (LP) turn it into a vanishing score error on all but a vanishing probability. The same argument gives a vanishing product-likelihood cost, measured relative to t\log(1/\delta), for each planar projection test. Conditioning additionally on a child reading which does not use the removed subtuple costs no more: the chain rule bounds the remaining information by that of the full source, with the negligible list entropy of that reading added. All rates have bounded capacities, so discarded probabilities and layer sets contribute vanishing errors to the integrals. This procedure requires no uniform differentiability in \eps.

The output chain rule now gives

where \alpha_0 is the expected isotropic phase rate before adding \Omega. Indeed, conditional on fine phase, the time contribution is at least d after conditioning further on \mathscr Z. Below depth 1, adding the angular bits recovers the phase bits and costs at most their rate m. Between depths 1 and 2, first add the coarse mark coefficients predicted by the output and then add \Omega. Translate the displayed reading at F_p and use (s-line) to obtain x. Tiling compact interior intervals and taking limits proves (L-chain); bounded capacities allow the omitted endpoints to tend to 0,1,2.

Both angular fiber modes \ell_i,\ell_j annihilate G to the required accuracy. They give

For completeness, add S_i,X_p for the first mode, or S_j,X_p for the second. The fiber equations predict the parent and all annihilating readings; replacing line coefficients by their depth-\eps representatives introduces error O(\delta^{p+\eps-o(1)}). First use dB+\theta_{i,\eps}dX, which sees \ell_j, and then dX, which sees \ell_i. The two times are separated by every fixed power threshold, by conditional independent sampling and their time-score bound. Each inversion thus costs an arbitrarily small power, and the first budgets in (7.1) supply the two residual rates m(p).

Next vary \widetilde\theta at R, holding \mathscr Z_R and X_{F,6} and starting inside a depth-p time parent. The shared fiber label identifies the target mark with (\widetilde\theta,\beta_R+\kappa_R\widetilde\theta) to the fine prescribed accuracy. Equations (leg) and (trans) give the response in G as

up to the coarse-coefficient error. Here \beta_R-\beta_F=-\theta_i(\kappa_R-\kappa_F) to the same accuracy. This response annihilates x,y, whereas its J' and L components are separated from zero by arbitrary power slack, using (mis) and the separation of A. The parent is predicted by the fiber formulas. The remaining Taylor error is O(\delta^{2p-o(1)}), which is finer than \delta^{p+t}. The contact forms are frozen; no derivative of the line field is taken. Consequently the time innovation, with the second budget in (7.1), gives

Apply the conditional projection rule of Proposition 5.19 twice. For the source (J',K'), initially retain only the line coefficients from \Omega, and vary the time slope \theta_{i,\eps}. Conditional on \mathscr Z, the trimmed direction sublaw has exponent d; integrating this unnormalized bound supplies the required direction bound at the source parent. The last budgets in (7.1) justify reinstating the remaining records for the score and list tests. If S is the joint source rate, (L-res) gives S\ge|x|+d-o(1), and the projection bound |x|\ge\min(1,S,(S+d)/2)-o(1) therefore gives |x|\ge\min(1,2d)-o(1).

For the second application condition on x to child accuracy, retain the target entries of \Omega, and vary A_\eps in the projection y of (L,J'). Here is the exact information comparison for this conditioning. Write T for the target subtuple of \Omega, B_*=A_\eps, and D_*=(C,F_p,T). Let W_* be the child bin of the frozen reading x, and let Z_* be the planar child source (L,J'). Both are determined to negligible lists by a sufficiently fine relative child phase name V_* and D_*, without using B_*. The name V_* may be taken at depth p+(1+\xi)t with arbitrarily small fixed \xi>0; the extra capacity is O(\xi t). The conditional chain rule gives

For the first inequality, adjoin V_* to the source, and use I(V_*;B_*\mid D_*,W_*) \le I(V_*;B_*\mid D_*)+H(W_*\mid D_*,V_*). The second equality uses nested phase bins, with the negligible list error if their grids are only list-compatible. The cumulative mutual information on the right is nondecreasing and bounded by H_\delta(B_*\mid C,T)=O(\eps). Consequently its loss density is one of the integrated budgets already treated above. Divide (7.2) by t, use that budget at the retained layer points, and then let the strict resolution slack \xi tend to zero. This proves the needed vanishing source-record information after the child conditioning. In particular, W_* uses the frozen target coefficients and no A; coefficient replacement is by a known translation at F_p and an error O(\delta^{p+\eps-o(1)}), not by an A-dependent reading.

For the direction bound, let the full target data include its side and its two observations. The names D_*,W_* are functions of these data. Conditional on the full target data, R still has the posterior kernel given (C,S_i). Stationarity and M(z)\ge g_*z trim its angular law to unnormalized ball bounds of exponent g_*. More explicitly, for each required fixed depth and strict tolerance retain only posterior angular cells whose conditional mass given (C,S_i) is at most the corresponding \delta^{g_*z-\text{tolerance}}. Their union has probability tending to one by the stationary scores, and their unnormalized mass in any constant-multiple depth-z ball has the same bound up to a fixed neighbor factor, for every conditioning value. Translation and reflection give the same bound for A for every full-target value. Integrating these restricted kernels over the full target conditional on (D_*,W_*) preserves the bound. No normalization of the retained sublaw is used in this step. The projection uses layers t\ll\eps, so rounding the direction is harmless. The last budget, now with mixed-prefix linearity for the source conditioned on x, again permits the full record in the test. Its residual beyond y is at least d-o(1). Thus

Each assertion means that a fixed strict deficit of positive probability contradicts the corresponding finite projection test with the loss order specified above.

Finally \alpha_0\ge\E|F|, and \int_0^1m(p)\,dp=M(1)\ge g_*. Substitute (L-mode) and (7.4) into (L-chain), and then send all the budget and endpoint errors to zero. This gives (line-time). To check the last claim, if d\le1/2, use g_*\ge1-2d and \min(1,d+g_*)\ge1-d; if d\ge1/2, use g_*>0 and \min(1,d+g_*)\ge d. In either case the stated lower bound is at least 2+3d.□

Proof. Here u=0 and b_1=b_2=a-1. Use the unclipped quadratic template and request the canonical profiles also on the fixed boundary v=0, where E(a,0)=0. Countably many depth queries suffice. Choose a typical strong a at which the prediction slack persists immediately to the left and (ng) holds on that boundary for all fixed accuracy demands. The reached label \lambda_a knows the boundary name. The strengthened version of (mem) with L=1, ending at a, together with (Sp), therefore gives angular prefix scores with lower slope k_0+2\Gamma, to arbitrary strict slack. The comparisons can use nearby actual gates with their earlier errors already tending to zero. If this slope exceeds the angular capacity there is already a contradiction; otherwise it supplies a positive g_* with g_*\ge1-2d.

Fix the exact side Z=(y_0,[P]_{x_*},t_a,R_a). It is the same side, with the same marks, at v=0 and on the tested interior patch. Suppose the required uniform power avoidance fails. Then, along z=\lambda h\downarrow0, a line predicted from Z has tube mass s^{o(z)} at normalized width s^z. This formulation allows \lambda subsequently to decrease, as required in the high-set radial estimate (rad). A vertical line, or a slope of size at least s^{-\chi z}, restricts \theta on the bounded mark support to an interval of positive power width; the conditional time-list bound excludes such mass. Hence, with subpower normalized coefficients, the remaining line event is

Include successful lists of size s^{-o(z)} for [P]_{x_*} given (y_0,t_a), using strong prediction and the deterministic profiles. Put h'=z/400, \delta=s^{h'}, and restrict the single final law to this event. All fixed strict time-score failures were power-small in these units before restriction, by the causal time-list count; likelihood comparison therefore preserves the time-score hypotheses.

Use \mathscr Z=Z, C=C_*(a,0), and the representative phase at v=0. The predicted line satisfies (s-line). Let S_i contain depth-12 marks and representative quadratic fiber entries. They are read to negligible lists from (C,\mathcal C(a+40h',0)). Actual and representative roots are mutually predicted to negligible lists: guess neighboring derivative bins if necessary and transport normal centers with the base-cell accuracies; the true derivative varies by only O(s^{x_*}) on this side. The deterministic scores persist under the s^{o(h')} restriction. Transferring those scores first, and then using the entropy chain rule, transfers (ng) to I_\delta(X_6;S_i\mid C)=o(1) with the preceding angular lower profile. This does not transfer an arbitrary small mutual information bound directly through a rare-event restriction. Further subsequences supply M. The output O_i=\mathcal C(a+h',0) has the recovery properties of Lemma 7.1, by the displayed-shear estimates underlying (time-rate). Boundary equality in (canon) gives

All profile errors are taken to be o(h') before z decreases.

It remains to verify (mis). Its failure at a fixed small \chi>0 gives, by independent posterior sampling and selection of a most popular interval, a prediction of a \delta^\chi interval for \kappa_F from (C,S_i) with nonvanishing success. Mark an end gate near j=a+60h' using the Borel inner completion mark of Lemma 3.19, inside the current end-state support, and apply group extraction on [a,j]. The successful completions have mass s^{o(h')}. The label \lambda_j knows the canonical names through a+40h' at width 0=u(j), and hence C,S_i, to negligible lists. Starting rows know the lists for Z along these paths, including the successful derivative lists.

Choose 0<\omega<\chi and bin the value and slope of the predicted trajectory at t_a at widths

Include the queried t_a in the list. The slope bin is predicted at the end from (C,S_i). Once it is chosen, the value bin is read to small lists from \beta_j(T_j): comparison with \beta_1(T_1) and transport from T_j to T_1 cost s^{b(j)-o(h')}, and the line-tube error is still smaller. Extrapolation of the binned slope back to t_a costs only its chosen value width. Disjointize the finite possible-bin relations as in Lemma 3.19. Each retained marked end row has a witnessing completion in its chosen bin; path concatenation puts all its predecessors in the corresponding Borel outer starting list. This predecessor projection uses the unrestricted truncated [a,j] geometric correspondence, not only successful completions. Its Borel inner partner carries the same full analyzed pre-synthesis input state in the original bin partition. After the common subtraction, the end curvature value is bounded by s^{b(a)+\omega h'-o(h')}, and its duration times the residual slope is at most s^{b(j)-o(h')}. As b(j)-b(a)=60h', the curvature cost on this segment is strictly less than 1. Group extraction contradicts its minimality. Fixed list slacks are chosen first to dominate mesh and closure errors and then sent to zero in h' units. Only the curvature cost is improved.

Thus (mis) holds. Lemma 7.1 contradicts the displayed boundary upper bound. This proves uniform line avoidance and hence the stated radial spreading. The restriction was made on the deterministic-profile single law before the new line experiment, so no stationarity claim on a subpower part of an already constructed many-replica law is needed.□

Proof. The boundary is an open quadratic branch strictly before D:

In hybrid applications this also supplies strict normal slack. The derivative to be proved is at fixed v_0, so interior stationarity in width is unnecessary.

We first specify the forward profiles needed in the hybrid case. At each fixed rational 0<j<1, preflatten the actual read rows \mathbf P^j in chronological order, before the final deterministic profiles. Make finitely many requests and then diagonalize. On these rows use a version \mathcal C^{[j]}(a,v), a<j, with the same exponents and endpoint constant as the artificial template, but with the j-label and its coordinates as the actual data. Namely, set \zeta=b_2(D)-D, bin \bar k_j=[K_j]_\zeta, subtract that trajectory, and use the residual \beta'_j(T_j),P_j for the curvature and derivative inputs. Here the prime denotes residual coordinates. The queries include the roots at v_0=0 and their angular refinements. In the classical case any fixed a<j<D is allowed: evaluate these names via the encountered label \lambda_j on the final read law, with no change to an earlier gate.

On connections from j to D, the names \mathcal C^{[j]} and \mathcal C have bidirectional lists of vanishing exponent, also at nearby times below j. To verify this, their global slope bins differ by at most s^{\zeta-o(1)}, by (A). Compare the total displayed curvature trajectories, including the global terms, at t_a. The discrepancy at T_D has accuracy \min_{[j,D]}b_2>b(a); the difference of global bins is multiplied by O(s^a), and the residual j-slope by O(s^j). The resulting error is s^{b(a)-o(1)}, and the slope error is s^{\zeta-o(1)}. The raw derivative discrepancy fits the displayed derivative accuracy by (A). On the shared base cell, the shear-difference Taylor formulas then give the normal widths. Hybrid fake drift fits by the strict slack and (Pl). These comparisons do not estimate the large global terms separately at absolute time. In hybrid, the forward inequality (Sp) transfers the already flattened j-row scores to the final law; classically the comparisons are pathwise. Thus the needed scores on the two systems agree, with power-small strict-tolerance exceptions and persistence under restrictions of vanishing exponent. No earlier gate is modified after computing later profiles.

This comparison concerns the canonical names and their angular refinements. It does not identify the new fine line-coefficient side data at j with analogous data at D. Those finer data are introduced on the j-law below, and are predicted from earlier gates ending at j. In particular the coarse bound K_j-K_D=O(s^{\zeta-o(1)}) is not used as a depth-30 prediction of their normalized slopes.

Fix a typical a<D for the boundary memory, differentiability, and (ng). Choose a<j<D, rational in hybrid, and then \delta=s^h with h small compared with these fixed gaps. For the latent side choose A_0<a, V>v_0 with the patch slack of the coarse-side construction, and include

Add depth-30 bins, in \delta units, of the bounded quantities

Use their representatives for (\beta_F,\kappa_F) and choose the representative root and phase on this refined side. With

Equation (s-line) follows because b(a)=a+\zeta.

The conditional time-score bound follows from the causal time-list count. Indeed a row at any nearby prefix gate between a and a+O(h)<j, on a path to the tested event, predicts the side to small lists. Put \Delta_K=b_2(j)-j-\zeta=(1-q)(D-j)>0 and choose 100h<\Delta_K, with h also smaller by a fixed large factor than all static-cell, derivative, and normal-drift slacks. On a connection from a gate i\in[a,a+O(h)] to j, Equation (A) gives

For the second bound use \min_{[i,j]}b_2\ge a+b_2(j)-j and multiply the slope discrepancy by O(s^a) when extrapolating to t_a. Thus both normalized errors are finer than the new depth-30 bins. The inequality

controls both K_j-K_i and the extrapolated curvature discrepancy at t_a more accurately than every new tangent bin, by (A). First subtract the guessed global bin. The residual j-slope has size s^\zeta, so the T_j-based root curvature cut also has a small list. Earlier static phase and derivative data are predictable using l(A_0,V)<\min b_1, exact base flow, and the hybrid normal slack. The expanded lists at the prefix gate therefore exclude too-popular time bins, with power-small exceptions at each fixed strict tolerance. In classical final-path tests the exact particle index is retained throughout.

Let S_i consist of depth-12 marks and quadratic fiber entries, and put O_i=\mathcal C^{[j]}(a+h,v_0). The pair (C,\mathcal C^{[j]}(a+40h,v_0)) reads S_i to negligible lists. Actual-root comparisons and the constant-cut displayed-shear calculations give the output recoveries in Lemma 7.1. The deterministic profile transfer just proved transfers (ng) to its stationarity hypothesis, with the prefix lower bound from final memory; moreover

These statements persist under a vanishing-exponent fraction restriction of the j-row law.

We prove the remaining mismatch property while keeping a,j,D,h fixed. Suppose a \delta^\chi interval for \kappa_F is predicted from (C,S_i), where 0<\chi<1 is fixed and small. Choose 0<\omega<\chi and group on [j,D] by bins of K_j and \beta_j(T_j) of widths

Starting rows know these bins. The end label knows the final canonical names at the used times below j at width v_0, hence their j-versions, C,S_i, and \bar k_j, to small lists. It therefore predicts the slope bins on success paths. Given one such slope, it guesses the value bin at T_j from \beta_D(T_D): Equation (A) and time transport apply because \min_{[j,D]}b_2>b(j). Include the ordinary time and start-rounding choices. After subtraction the residual end value is bounded by s^{b(j)+\omega h-o(1)}, and the residual slope times the terminal duration by s^{b_2(D)-o(1)}. Since b_2(D)-b(j)=D-j, the curvature cost on [j,D] is at most 1-\omega h/(D-j), a fixed strict improvement.

Classically, failure of (mis) yields a prediction with nonvanishing final mass. Use the Borel inner success and completion marks of Lemma 3.19 at D, inside the current end-state support; Equation (Sp) and the count bounds supply the group lower bound, using outer hulls of unrestricted geometric predecessors. Exact particle and time starts fix the earlier gate ownership, so these are actual path groups. Group extraction contradicts minimality of the curvature cost.

In hybrid the working law is \mathbf P^j, and a prediction-success event there might have little terminal mass. To address this, discard before j-synthesis all rows whose depth-\chi slope bin has conditional mass at least s^\tau given (C,S_i), with \tau>0 sufficiently small. These maps use the single observation on that row and their Borel versions under the full analyzed j-row energy law as in Lemma 3.19. For each conditioning value there are at most s^{-\tau} discarded bins. We use a closure argument because \tau is not yet a vanishing exponent. Hold a,j,D,h,\chi,\omega fixed, and put \gamma_0=\omega h/(D-j)>0. If arbitrarily small \tau allowed the discarded synthesis to approach terminal saturation, choose \tau_n\downarrow0 and genuine approximating experiments whose terminal deficit and prior mesh, profile, and regularization losses tend to zero faster than \tau_n. For each n group on the same fixed interval [j,D] by the preceding construction. Each reachable end label has the expanded list of groups. Finite-overlap inequalities have normalized loss O(\tau_n/(D-j)), while other list and approximation losses can be made o(\tau_n)/(D-j). Regularity is retained before propagation, grouped energies sum to the j-energy budget, and counts have the same overlap loss. The quantitative loss form of group extraction selects genuine experiments whose extremality and count losses tend to zero in the [j,D] units. Their curvature costs are at most 1-\gamma_0+o(1), since h,j,D,\omega remain fixed. They realize the original extremal exponents in the closure with cost at most 1-\gamma_0/2 eventually, contradicting minimality. This applies the vanishing-overlap clause along \tau_n; it does not apply a zero-loss extraction statement at fixed \tau. Consequently some sufficiently small fixed \tau>0 leaves a strict positive terminal-exponent deficit on the discarded part. The triangle inequality for the terminal functional shows that the complementary synthesis preserves the terminal exponent. Its pre-synthesis row fraction is consequently s^{o(1)}.

Restrict only the j-row probability law to this complement. On retained rows the original conditional slope-bin mass is below s^\tau. Except on a vanishing restricted fraction, conditioning normalization given (C,S_i) multiplies mass by at most s^{-\tau/2}: the excluded conditioning cells have original retained mass at most s^{\tau/2}, whereas the total retained mass is s^{o(1)}. Neighboring-bin counting therefore makes posterior collisions at the tested precision vanish. The same fixed representative maps, time bounds, and deterministic scores are preserved by this restriction. For finitely many separation queries repeat the exclusion with previous errors sufficiently small; diagonal limits give (mis) for every fixed threshold. This is a restriction of the j-law for the entropy test, not a claim about the final law of a modified stack. The strict curvature improvement is needed only for fixed h.

Lemma 7.1 now gives \partial_a\mathcal H(a,v_0)\ge2+3d. On this fixed-width quadratic branch, \partial_a W=1-\eta and p+d=1+3d-2\Gamma in (canon); hence \partial_a E\ge2\Gamma+\eta\ge2\Gamma. Finally, the inequalities used above hold for every q<1: b_2(j)-j-\zeta=(q-1)(j-D)>0, and \min_{[j,D]}b_2>b_2(D)+j-D whether q is positive or negative.□

Proof. Use the actual C-shear at t_a. Center the residual normal velocity and starting position in their C-bins, and divide by s^{w-g},s^{a+w-g}, respectively. The bounded pair V=(V_n,V_d) is a function of (y_0,C). The output and root know \theta'=(t_{a+h}-t_a)/s^a. Given C, let S_k contain

1. (i)

   X^{v+K_*h};

2. (ii)

   B_{t_a} to depth a+v+k_Rh and Y_{t_a}-2[X]_{v+K_*h}B_{t_a} to depth a+2v+2k_Rh;

3. (iii)

   in the quadratic case, P-2R_aX+2k_aB_{t_a} to depth l(a,v)+k_Rh, in the root curvature cuts.

These data refine with k, up to zero-cost lists. Request the finite deterministic profiles of (S_k,V_z), z,k\in\mathcal K, and their needed joints, without altering the earlier profiles. Write D(z\mid k)=\lim H_\delta(V_z\mid C,S_k).

Two geometric comparisons will be used:

Here and below all prediction lists have vanishing exponent at fixed h. To verify the first line, the child basis minus the root basis on the actual ray has derivative discrepancy O(s^l) and curvature discrepancy O(s^b) in s^a duration units. In curvature cases b+v\ge l. Evaluating this difference at the known root base-bin center instead costs only O(s^w) in normal velocity and O(s^{a+w}) in position, including the vertical base tilt. Since (c+1)h\ll g, these errors fit both child accuracies in the enlarged normal units. Evaluate all known offsets at the exact bin times. Thus the predictions are single centers with the required radii, not merely unspecified lists. For C^k, the S_k data supply its refined base and derivative bins. The same shear-difference comparison bounds the normal error within its requested flat widths.

For the entropy bound in (7.5), first give X^{v+K_*h}. Backflow from (O,C) reads both the horizontal and tilted vertical positions at normalized depth at least 1. In the tilted coordinate the velocity is X^2-2[X]_{v+K_*h}X, whose error about the refined center is quadratic. The two extra position costs are therefore (k_R-1)h and (2k_R-1)h. In the quadratic case the output displays derivative refinement ch, which suffices because k_R\le c. Changing back to the root curvature basis costs O(s^b) in earlier-time units; multiplying by the unresolved base errors in the guessed position bins costs at most O(s^{l+k_Rh}). No further derivative entropy is required.

Conditionally on (y_0,C,S_k), time bins at \delta-depth z\in[0,1] have typical score at least dz. Indeed (C,S_k) is predicted to small lists from (y_0,t_a) on all paths outside the profile exceptions. Root curvature predictions follow from (A), (F), and cut ownership. The quadratic derivative query is strictly below p_*(a); in a clipped linear branch it uses only l=b_1(a), and in pure linear form it uses the center slope comparison. Exact classical flow predicts the phase entries. Expanding any popular time list by these predicted lists and applying the causal time count through a+zh gives power-small failures at each fixed strict tolerance. Quantizing the child bin-start time changes only bounded factors. Likewise, deterministic profiles and the neighboring-bin likelihood bound give conditional V-ball mass at most \delta^{D(z\mid k)-o(1)} about typical samples, for z\in\mathcal K. These are bounds for the unrestricted conditional mass of those balls. The trimmed time law has unnormalized interval bounds of exponent d.

We claim

The first term is supplied by H_\delta(O\mid y_0,C,S_k). For the information with y_0, draw an independent reference y'_0 given (C,S_k), and let V' be its normal pair. If it is admissible for (O,C), the first comparison in (7.5) forces agreement in V_n to O(\delta^c) and in V_d+\theta'V_n to O(\delta^{c+1}). Hence \abs{V'-V}\lesssim\delta^c. If its distance lies between consecutive grid widths \delta^{z+\mu} and O(\delta^z), where \mu=\operatorname{mesh}\mathcal K, the velocity difference is comparable to the full difference unless the latter is already at final width. Consequently the time projection confines \theta' to an interval of length O(\delta^{c+1-z-\mu}). At the final width no time gain is charged.

Average reference admissibility over actual samples satisfying both typicality bounds. Given the actual (y_0,C,S_k), use the conditional ball bound for V' and, for each fixed V', the trimmed time interval bound. Summing the finitely many annuli bounds this average by the power in (sh) minus its first d, with strict tolerance and the O(\mu) loss. Markov’s inequality makes the reference admissibility probability at most that power with slightly more tolerance for most actual (O,C,S_k). The actual posterior given (O,C,S_k) is supported on the admissible set. The relative entropy bound for a measure supported on a reference set of mass q, namely \log(1/q), therefore gives the information term in (sh). All trimming was on the actual sample in this averaging argument; no exceptional-set transfer to the independent product law is used.

For z_2>z_1, the increment D(z_2\mid k)-D(z_1\mid k) is nonincreasing in k, because S_k refines. Thus the largest minimizer z(k) in (sh) is nondecreasing. A nondecreasing self-map of a finite ordered set has a fixed point: iterate from its least element. At such a point z(k)=k, use (7.5) and refinement of C by O to obtain

The weight exponent difference, later minus moved, divided by h, is -(3+\eta)k_R+(1-\eta)(c-k). Subtracting this and p+d as in (canon) gives

Both extra displayed terms are nonnegative: k\in[c,c+1], d\le1, and k_R+c-k\ge0 in the stated branches. Choose \mu small enough to retain \Gamma. The finite refinement proof can be applied at any macroscopic path node to the same profiles E^g,M_X; it does not select microscopic configurations adaptively. This proves (step).□

Proof. Fix compact domains and all Lipschitz and speed bounds before choosing density scales. Let C_0 denote constants depending only on these bounds and K_*. Outside a null set of widths set

with the terminal endpoint convention if the set is empty. Take rational queries in a containing interval and truncate this threshold to its closure if necessary. Monotonicity implies positivity at almost every relevant point above \sigma, whereas m_{t'}(v)=0 at each rational t'<\sigma(v) outside its null set of exceptional widths.

We first justify integration on a prescribed positive segment. Fix a slab [s,T] and a measurable width set P where m_t(v)>0 for almost every t in that slab. Consider a proposed segment of speed k_+, duration \ell, and width interval W. At its initial point one has either

To see this without assuming the proposed line is generic, perturb its initial width. For almost every perturbation, Fubini’s theorem and the Lipschitz chain rule give the moving estimate almost everywhere and valid vertical estimates at almost every encountered width. Time spent outside P is bounded by |W\setminus P|/k_+, with an error tending to zero under the perturbation. If freezing is allowed and the high-rate set m>k_0 on that line has positive time measure, choose a regular point arbitrarily close to its first essential occurrence. The measure of earlier high-rate times can be made arbitrarily small. Integrate the moving estimate to that point, using the Lipschitz bound on those earlier high-rate times, and then freeze to T. Monotonicity retains m>k_0 on the vertical continuation, and endpoint nonnegativity proves (7.8). If there is no positive-measure high-rate set, integrate to the moving endpoint to get (7.7). Let the earlier-high-time tolerance and then the perturbation tend to zero. A subsequence either always freezes or always reaches the proposed endpoint, and Lipschitz continuity gives the corresponding alternative at the original starting point.

Suppose J>0 and put T(v)=v/J on an interior seam-width interval. It is covered by

One of these sets has a density point in that interval. If v_*=Jt_* is a density point of P=\{\sigma(v)\le T(v)-\Delta\}, \Delta>0, fix a small \eps>0 and start at (t_*,v_*+\eps r). Propose a positive segment of duration r. For r small, every visited width in P has threshold below t_*, while density gives |W\setminus P|=o(r). The segment stays to the right of the seam. Either (7.7) or (7.8), with T=t_*+r, implies

Seam equality bounds the left side by C_0\eps r, a contradiction after first choosing \eps small and then r small.

Now let v_*=Jt_* be a density point of Z_0. Start at the same width and make steps of duration h until less than h remains in [t_*,t_*+r]. At a step starting at (t_i,v_i), the speed bound gives v_i-Jt_i\ge\eps r. For h small compared with \eps r, all y\in I_i=[v_i-h,v_i] satisfy

These intervals have disjoint interiors, since every step increases width by at least h. Choose a fixed finite rational grid meeting every interval (t+b_0/4,t+b_0/2) for t\in[t_*,t_*+r]. Choose g with A_0g<b_0/4. For each step take a grid point t'_i in this interval for t=t_i+h. It is admissible in (shift) for the entire I_i. On I_i\cap Z_0 it satisfies t'_i<T(y)\le\sigma(y), and hence m_{t'_i}(y)=0. Integrating (shift) divided by h over I_i gives

On the good portions this integrand is dominated by

which tends to zero almost everywhere and is bounded. Notice that only grid times below the threshold are included. All visited widths lie in a window W_r of length O(r) based at v_*. Disjointness and bounded convergence give

First choose r using density so that the first term is small compared with \Gamma r, then g\ll\eps r, and only then let h\to0. Telescoping (step), using (enl) and endpoint nonnegativity, gives

which is impossible with these choices.

Suppose finally J=0. Fix \eps>0 and a compact positive-length interval I_0 of starting times. Let L_0 exceed the speed bound by a fixed margin. Fubini’s theorem gives

For r sufficiently small, there is therefore t_*\in I_0 with transition-set length at most \eps r. In (0,L_0r) put

The remaining set has length at most \eps r. Widths in P have positive trace throughout the slab [t_*,t_*+r]. Choose a rational t'\in(t_*+5r/4,t_*+7r/4); the trace vanishes at that time for almost every width in Z_0. Approximate P to symmetric-difference error \eps r by a finite union P' of intervals, with N endpoints. Now choose g with A_0g<r/8 and C_0(N+1)g\le\eps r, and subsequently let h\to0 with h\ll g and h<\eps r/2.

Start at (t_*,\eps r). Outside P' make an off-gap step; inside P' apply the positive-segment comparison up to its next right endpoint or the final time. Stop immediately if the terminal alternative (7.8) occurs. Consistent half-open endpoint conventions prevent zero-length transitions. Width increases before any terminal freeze, so there are at most N+1 positive segments. Changes between E and E^g cost at most C_0(N+1)g. Their positive-segment width intervals are disjoint and lie in P', so the total bad-width charge is at most C_0|P'\setminus P|\le C_0\eps r.

For off-gap steps the intervals I_i=[v_i-h,v_i] again have disjoint interiors. Since their right endpoints are outside P', their intersections with P' lie within distance h of its endpoints. Consequently

The one fixed t' is admissible in (shift) for every step. On Z_0, Equation (7.6) and bounded convergence give zero limiting integral; on the transition set and the parts of P just estimated use F_h\le K_*+1. Hence

Each off-gap step pays \Gamma times its duration and each positive segment pays 2\Gamma, subject to the listed errors. A terminal freeze pays to the final time; otherwise less than h remains unused. Initial seam equality, initial displacement, and final nonnegativity yield

Let h\to0 with earlier choices fixed. The choice of g gives a contradiction for \eps sufficiently small, completing the proof.□

Proof of Theorem 3.6. First suppose a classical extremal configuration has \Gamma>\eta/2. After it is excluded, suppose a hybrid configuration has \Gamma>\eta/2. It then has the strict gap \Gamma>\max(\Gamma_{\mathrm{cl}},\eta/2) required for (Pl). The affine reduction (F) and the zero-cost arguments leave precisely the positive-cost cases treated below. All uses of (surplus) are on valid branches with the indicated crossing or spreading hypotheses.

We describe once the integration conventions. Almost-everywhere estimates on affine rays are integrated on generic arbitrarily nearby rays and passed to the desired one by Lipschitz continuity. A speed with surplus 2\Gamma may be decreased slightly while retaining surplus at least \Gamma. Rays from the terminal corner can start at a small positive backward time, with initial E tending to zero. If a high-rate linear portion is encountered, the freezing alternative in the proof of Lemma 7.5 applies.

We also require a short-slab observation. Suppose b_1'>0 and choose a typical j with E(j,u(j))=0. Work in a short slab ending at j, omitting a small fixed fraction next to its final edge. For v>u(j) sufficiently close to u(j), the pair (\lambda_j,X^v) recovers \mathcal C(a,v) to small lists throughout the shortened slab. In the unclipped quadratic case, choose it so that b(a)+v<b_1(j). If R_j is the angular profile given \lambda_j, conditioning gives

An integrated lower slope for R_j through u(j) supplies positive-measure sets of arbitrarily close widths with derivative at least that slope minus any prescribed strict error. Otherwise integration would contradict the lower slope. A bound above capacity is already impossible. Choose these widths generically for all almost-everywhere assertions. If the supplied derivative exceeds a freezing threshold, (surplus) holds throughout the slab, except for any stipulated small time-density error. Its positive integral contradicts nonnegativity: the terminal width perturbation and the omitted edge cost arbitrarily little by Lipschitz continuity and E(j,u(j))=0. When a threshold depends on the mixed trace \nu, choose j to be a Lebesgue point of that trace. The strict margin at j then survives off a time set of arbitrarily small relative length. All extra j-profile queries are readout refinements after j has been selected.

Finally, if p_*>b_1 on a positive-measure set of interior times, monotonicity of p_* and continuity of b_1 give a smaller open interval with uniform strict prediction slack. Otherwise p_*=b_1 at almost every time under consideration. We call these the strong and weak alternatives, respectively.

Equality curvature: q=c_2>0. Put c=c_2 and C_1=c-J=b_1'. If J>0, consider left quadratic points 0<v<u(a) with l<\min_{[a,1]}b_1 and, in hybrid, w<0. Write v=u(j) for a later j<1. Left memory gives

The coarse-side construction and avoidance supply radial spreading, including the extra left-slack version when c=1. If 0<c\le1, then L_{\mathrm{raw}}\le1+c, so (7.10) exceeds the quadratic freezing threshold k_0/(1+c). On every compact interior time slab all sufficiently small v>0 meet the hypotheses. Integrate at those widths, let v\downarrow0, and then let the trimmed endpoint tend to 1. The equality E(1,0)=0 contradicts the resulting positive drop. The same argument applies for c>1 when L_{\mathrm{raw}}\le2, using the threshold k_0/2.

If c>1 and J\ge(c+1)/4, use a ray from the terminal corner at a speed just below (c+1)/4. It stays in the left region, has 2k<c, and satisfies the derivative prediction condition; if b_1 decreases forward, l<0 suffices. The positive quadratic motion in (surplus) is contradictory. This covers hybrid equality configurations because (Pl) gives J=(c+1/2)/2 there.

The remaining configurations are classical and have b_1'>0: either J=0, or c>1 and 0<J<(c+1)/4, with C_1>1. Indeed if C_1\le1 in the latter range, then L_{\mathrm{raw}}\le2, a case already excluded. In a strong interval use the unclipped quadratic template near the seam with exact side and prediction slack. Avoidance supplies radial spreading; at c=1,J=0 this is supplied by Lemma 7.2. If c\le3, the strengthened memory bound (mr) uses L=\max\{1,(1+c)/2\}. For c<1 it exceeds k_0/(1+c), and for 1\le c\le3 it exceeds k_0/2. Equation (7.9) therefore gives the short-slab freezing contradiction. Shrink the slab to retain l<p_* throughout. If c>3, apply Lemma 7.5: its positive speed is k_+=(c+1)/4>J, and the finite-step speeds have minimum c/2>J. They are all at least 1, the normal-width exponent increases forward, and no high-rate switch is needed.

In the weak alternative use the clipped template. Its right branch v>u(a) is linear of rate C_1, with exact construction and crossing at almost every weak time. If c<1, necessarily J=0 here and C_1=c. At a typical mixed-trace time j, use both memory inequalities (mr). Together with (7.9), their strict margin gives positive m above k_0-c\nu(a) outside an arbitrarily small relative set of times. The low-rate linear freezing bound contradicts the short-slab observation. In all other weak cases C_1\ge1. For 0<m\le k_0 use k_+=(1+C_1)/2, and finite-step speeds have minimum C_1; both are strictly greater than J in the remaining range. For m>k_0, freeze on the right by the linear bound. A continued freeze crossing left uses the quadratic freezing bound, the same monotone clipped-template trace, coarse side, and line avoidance. The forward width rates are C_1 on the right and c on the left, both positive. Lemma 7.5 applies and gives a contradiction.

Pure linear cost: c_2=0<c_1. Write c=c_1. The linear construction has crossing. If classical and c<1, use both inequalities (mr) at a typical mixed-trace time and the short-slab argument just given; there is now no side-of-seam restriction. In hybrid, (Pl) gives J=c+1/2. Use a ray from the corner at a speed slightly below c for c<1, and slightly below (1+c)/2 for c\ge1. It lies in v<u(a), w<0, and left memory gives

For c<1 this exceeds the threshold k_0/(1+c) for linear motion at speed c; for c\ge1 use the stated positive linear motion while m\le k_0. If m>k_0 on a positive time set, switch at a regular point to freezing. Its hypotheses remain valid backwards. These motions contradict nonnegativity and terminal equality. The same argument, with exact side, works classically for c\ge1 and J\ge(1+c)/2. For the remaining classical case c\ge1, J<(1+c)/2, use Lemma 7.5 on the pure linear template: k_+=(1+c)/2>J, finite-step speeds are at least c>J, and any high-rate freeze stays in a linear branch.

Strict curvature: q<1=c_2. Put C_1=q-J<1 and use the artificial template ending at D. At v_0=u(D) memory supplies the lower slope

If C_1\le0, then L_{\mathrm{raw}}=1, so this is positive and at least 1-2d. This explicitly includes every negative q allowed by the affine reduction. If C_1>0, the same conclusion follows whenever d\ge C_1/(1+2C_1), since (1-d)/(1+C_1)\ge1-2d is exactly that inequality. Take D=1, integrate Proposition 7.3, and contradict E(D,v_0)=0.

It remains that C_1>0 and d<C_1/(1+2C_1). If C_1\le1/2, then d<1/4. From the terminal corner D=1 take a quadratic ray of speed slightly below 1/2. It satisfies l<\min_{[a,D]}b_1 and w<0 strictly in the interior: along a speed k<1/2 ray these are -(1-k)(1-a)<-C_1(1-a) and -(1-2k)(1-a)<0. The coarse side is valid. The unit-cost quadratic motion bound without spreading, which also permits m=0, gives the contradiction. All hybrid strict configurations have been covered: (Pl) gives J=(q+1/2)/2, so C_1=q/2-1/4<1/4.

Finally suppose 1/2<C_1<1. The configuration is classical, and 2d<C_1. If the raw final derivative has a strong interval, choose a typical D in it, retaining prediction slack immediately to its left. The strengthened memory estimate (mr) has L=1 for this boundary comparison via the encountered \lambda_D on final paths. Proposition 7.3 again contradicts E(D,u(D))=0. Otherwise take the clipped artificial template with D=1 and move from the corner at speed 1, using generic small perturbations. This is strictly right of the clipping switch v=b_1(a)-b(a)=(1-C_1)(1-a). The classical linear branch has crossing at almost every weak time. Its arbitrary-m motion bound for 2d<C_1<1, including m=0, gives the contradiction without requiring a positive angular trace.

Every affine positive-cost configuration has now been excluded. Together with the zero-cost arguments this proves the classical bound, and subsequently the hybrid bound with its required strict separation, as claimed.□

Proof. The two angular symbols have disjoint supports unless the angular assertion holds. On their overlap use frequency coordinates q_\ell(\xi-2\rho A) and H\rho. The amplitude is compactly supported and smooth with uniformly bounded derivatives. The quadratic part of the propagation phase has the same property because uH/q_\ell^2=O(1). The remaining linear phases are precisely the two position differences in Equation (cone-kernel). Integration by parts gives that bound, including its normalization 1/(q_\ell^2H). The row measures and the inverses of the packet scales have finite-power size. Schur’s test therefore makes the omitted operator norm smaller than any prescribed power by choosing N last.

For a path through a finite sequence of cuts, an angular change at length \ell_m acts for at most O(\ell_m) additional time. It contributes O(q_{\ell_m}) in transverse position and O(\ell_m d_{\ell_m}^2)=O(H) to the tilt-corrected vertical position. Replacing an intermediate tilt by the original tilt costs at most O(d_{\ell_m}q_{\ell_m})=O(H), with the same dilation factors. Summing finitely many such bounds proves Equation (acc). Successive truncation and contraction also show that an output region with no retained predecessor path has negligible energy. This last assertion uses operator norms and does not posit a positive transition kernel for the wave.□

Proof. If f exceeds a sufficiently small inverse power of P, cover by round tests; their number and the reciprocal of f^{1-\eta} are polynomial in P. Otherwise Equation (8.1) gives |g|\asymp1 on the set. Subdivide its normalized phase box into parabolic boxes of transverse width a=P^{-D}, centered on actual witnesses. After centering at a witness, boosting to its ray, applying the parabolic change of scale, and dividing the polynomial by a, its transverse linear coefficient is g at that witness. Its vertical and quadratic coefficients are O(aP^C); its constant and central-ray derivative are O(P^Cf/a). Choose D first and the smallness threshold for f second. All coefficients then satisfy the bounds in Definition 8.2, after bounded further subdivisions. The discriminant remains one. The new relative thickness can be taken as P^{C''}f/a; thus the absolute uncertainty width is at least rf. There are polynomially many boxes, and their total test cost has the asserted bound.