QAOA attains the SK ground-state energy in the thermodynamic-first limit

Proof. This is Theorem 1 of [4], together with its normalizations in Equations (2.3), (2.6), and (6.1)–(6.2). Its Equation (3.11) gives the fixed-depth tree value, and Equation (6.3) identifies that value with the SK limit. That paper uses the negative of our tree cost. Conjugation by X on one bipartition of the tree changes the sign of every Z_iZ_j, preserves every mixer, and fixes the initial vector. It therefore changes the sign of the edge correlation as well, giving precisely (2.1). The SK convention already agrees with the positive Gaussian Hamiltonian in its Equation (6.1). Consecutive gates of the same kind and inverses are included by inserting zero-angle gates.□

Proof. We give the exact finite-degree decomposition. Put M=D+1, let E be the pointed child space, and write N_M for the number of nonvacuum children in \mathsf k\otimes\operatorname{Sym}^M(E). Let R_M contract a distinguished child against its vacuum, and let

The latter is an isometry from E^\circ. Under occupation embeddings, J_M^*=\cL^* and

Indeed, in the m-particle sector the fraction of placements surviving deletion is \binom{M-1}m/\binom Mm=1-m/M.

Delete the distinguished edge and let P_A,P_B denote the vacuum projections on its two half-trees. Decompose f_{D,i} with P_B+(1-P_B) and f_{D,j} with P_A+(1-P_A). The term with both vacuum projections is zero: the surviving half-tree vectors are centered. If just one side is excited, the other side being vacuum forces the root and all the other branches of the first vector to be vacuum. With s_D=J_M^*f_D and r_D=R_Mf_D, the two such terms are \braket{s_D}{r_D}/\sqrt M and its conjugate. Heights in this pairing are matched by adding empty descendants. Each remaining excited-side vector has squared norm \bra{f_D}N_M\ket{f_D}/M, by symmetry among branches. Thus

After multiplication by \sqrt D/2 the error tends to zero. Equation (2.8) and strong convergence show that the first term tends to \re\braket{\cL^*v}{v}=\re\braket{v}{\cL v}. This proves (2.6).

Finally, (2.7) follows by adding and subtracting \braket{w}{\cL v} and using \|\cL\|=1.□

Proof. On finite-particle vectors, \|a(d)\xi\|\leq\|d\|\|N^{1/2}\xi\| and \|a^*(d)\xi\|\leq\|d\|\|(N+1)^{1/2}\xi\|. The fundamental theorem of calculus applied to e^{\ii t q(d)}\xi proves (2.16). Conjugation gives

Commuting N through a field changes that field to another field with the same norm of its argument. Expanding the mth power and applying the creation and annihilation bounds therefore proves (2.15). For the weighted version use

and [\dd\Gamma(\Lambda),a^*(d)]=a^*(\Lambda d). Every expansion uses only finitely many weighted norms of d. Approximation in graph norms proves the identities and estimates on the stated domains. A spin-controlled displacement is the sum of two ordinary displacements on orthogonal spin subspaces.□

Proof. Put g=QT_D\vac, h=QT_D^*\vac, and A=QT_DQ. For centered T_D its normalized sum has the exact occupation representation

where factors beyond the truncation are zero. On the m-particle sector the two square roots express, respectively, the numbers D-m of vacant and D-m+1 of potentially filled slots. The formula first holds on algebraic tensors and then on smooth vectors. Creation and annihilation have uniform weighted bounds by the proof of Lemma 2.5. For the remaining term, a one-copy bound between powers of \Lambda implies a bound for its sum between powers of 1+\sum\Lambda_j: apply the one-copy bound to each occupied slot, use \Lambda_j\leq\sum\Lambda_j, and pay at most one additional power for the number of occupied slots. This argument works at any desired output power. It proves the required bounds for (2.18). On a fixed-particle smooth tensor the last term tends to zero and the square roots tend to one; density and (2.14) finish the proof. For an average the same decomposition has off-diagonal factor D^{-1/2} and complementary-block factor D^{-1}, while its vacuum block tends to the stated scalar.□

Proof. We first identify the candidate limit from exponential vectors; then we prove the bounds needed for smooth convergence. If x,z\perp\vac are smooth, the vectors (\vac+x/\sqrt D)^{\otimes D} converge to \varepsilon(x)=\bigoplus_{m\geq0}x^{\otimes m}/\sqrt{m!}, and similarly for z. The coefficient of D^{-1} in the one-copy matrix element is

Taylor’s formula makes the remaining error o(D^{-1}), also for converging triangular arrays of smooth x,z. The full pairing consequently tends to e^c. For t=\sum_\alpha a_\alpha T_\alpha\vac, the exponential-vector pairing of e^{\ii q(t)} has an additional term -\|t\|^2/2. Since \|t\|^2=\sum_{\alpha,\lambda}a_\alpha a_\lambda \langle\{T_\alpha,T_\lambda\}/2\rangle_0, the correction in (2.19) is exactly the required one. Twice differentiating K(x)^*K(x)=1 at zero shows that its real part is zero. The displayed limit is thus unitary.

We next prove uniform bounds for the product U_D(a)=K_D(a/\sqrt D)^{\otimes D}. For one child let A=Q\Lambda Q and B_D(a)=K_D(a/\sqrt D)^*A K_D(a/\sqrt D). Taylor’s formula at zero, with the assumed bounds, gives

and the complementary block has polynomial smooth bounds. Here and below P_m denotes some fixed nonnegative polynomial in 1+|a|, whose degree is allowed to depend on m. The same vacuum-row decomposition as in (2.18), now applied to the unnormalized sum of the B_D(a), shows that

has polynomial bounds from any sufficiently high input weight to any prescribed output weight, uniformly in D. Composing this estimate a fixed number of times bounds its powers. Since W^mU_D=U_D(U_D^*WU_D)^m, this proves the assertion for U_D itself; the adjoint is treated identically. All identities may first be made on smooth tensors, which the product preserves, and then closed using these very estimates.

For derivatives,

The one-copy vacuum mean is O(D^{-1/2}P(a)), since it vanishes at zero. Lemma 2.6, with this scalar part retained, gives a uniform bound. Further differentiation produces products of finitely many such sums. A sum carrying k\geq2 derivative-scaling factors is bounded by the average estimate times D^{1-k/2}\leq1. Induction proves all jet bounds.

Weak convergence on the total set of exponential vectors, together with equality of the limiting norms, proves strong convergence. Higher-weight convergence follows from (2.14); derivative convergence follows from finite differences and the uniform bounds on one further derivative. This also gives local uniformity in a. Since the strong limit is unitary, the adjoints converge strongly as well: for unitaries U_D,U, one has \|(U_D^*-U^*)\xi\|=\|\xi-U_DU^*\xi\|. The adjoint jet bounds already proved, followed by the same finite-difference argument, give convergence of adjoint jets.

To verify the unscaled-parameter assertion, the only additional estimate concerns a logarithmic derivative A_\nu(x;\theta)=K(x;\theta)^*\partial_{\theta_\nu}K(x;\theta). It satisfies A_\nu(0;\theta)=0. Moreover

Its vacuum entry at a/\sqrt D is therefore O(D^{-1}P(a)), its vacuum row and column are O(D^{-1/2}P(a)), and its complementary block is O(D^{-1/2}P(a)) with smooth bounds. The unnormalized sum of its D copies consequently has uniform smooth bounds by the decomposition used above. Every further \theta-derivative has the same two vanishings, and mixed a-derivatives are treated by the normalized-sum or average estimates. This proves all joint jet bounds. Pointwise convergence for fixed \theta, finite differences, and the adjoint argument just given prove the parametric conclusion.□

Proof. At finite D, factoring out the new child costs gives the exact left update

The one-copy observable is centered, because Z is odd under the joint spin/seed symmetry and the preceding word preserves that symmetry. Its vacuum column is v_Z. On each root Z-eigenspace, the update is a product of small one-child unitaries. Lemma 2.7 gives precisely the cost in (2.20); the scalar correction is zero for an exponential of a fixed observable. The smooth estimates follow inductively from the same lemma and Lemma 2.5. Mixers are local, and a seed gate is multiplication by a two-by-two unitary; its input derivatives have polynomial seed bounds. Both therefore preserve the smooth domain. The update commutes with the root Z, proving the assertion about v_Z; conjugation proves the rotated-axis statement.

For the full-root assertion use M=D+1 in the occupation decomposition while retaining a/\sqrt D in each one-copy factor. In the proof of Lemma 2.7, every limiting coefficient is multiplied by M/D\to1, and every bound is unchanged up to a constant. Induction on the fixed word proves that full-root and cavity-root maps have the same limit. Padding the subtree by the finite light-cone radius justifies all factorizations before limits are taken.

The finite-degree ordinary insertion vectors meet the hypotheses of Proposition 2.2: parity makes them centered, the construction preserves rooted symmetry, and the smooth weight dominates 1+N. Their edge pairing is the expectation of Z_iZ_j. The root-shift formula and Proposition 2.1 therefore give v_W=\cE(v_Z).□

Proof. First identify the limiting signal operators. Write U(t)=\lim_D e^{\ii t\cdot S_D} for the strong operator limit. Strong limits of products of uniformly bounded operators preserve the group law, and the limits at t and -t preserve inverses; thus each U(t) is unitary. For smooth \xi,

The uniform polynomial bounds give strong continuity on this dense domain, hence on the full space. The strongly continuous joint unitary group therefore determines strongly commuting self-adjoint generators S_1,\ldots,S_b. Passing to the limit in the integrated one-parameter group identities identifies each generator on smooth vectors with the corresponding polynomial limit. The smooth bounds make this domain invariant, so iteration identifies all polynomial limits. This is the realization used in F(S).

We give the estimates, which also specify the substitution on the smooth domain when F_D is unbounded. For such coefficients, commutation with the spectral measures is understood in the closed-operator (or decomposable-operator) sense. On the invariant smooth domain we use the equivalent identities with the joint unitary group. Sharp spectral projections are not required to preserve the smooth domain. These identities pass to the limit because the moving inputs e^{\ii t\cdot S_D}\xi converge in every required graph norm, as do the coefficients acting on them. Thus the limiting coefficients retain the commutation needed for the Fourier products below. Suppress D. Fix an output weight m, and take r_m,k_m from (2.21). Choose an integer q>k_m+b. For a sufficiently large integer N, the family

and its first q derivatives are integrable in x as operators from some fixed input graph norm to the r_mth output graph norm. Such an N exists because only finitely many polynomial growth exponents are involved. With \widehat G(t)=(2\pi)^{-b}\int e^{-\ii t\cdot x}G(x)\,\dd x, integration by parts gives

Therefore

converges absolutely in the mth graph norm and has a bound of the form (2.13). The polynomial on the right acts first; its bound can be absorbed by increasing the input weight last. This order of choices avoids any circular dependence of exponents.

Integration by parts in t, with the same integrable graph-norm bounds, shows that the result does not depend on N: multiplying a symbol by 1+|x|^2 corresponds to applying 1-\Delta_t to its Fourier transform and hence to multiplication by 1+|S|^2 after integration. The common smooth domain is preserved by all factors. Fubini’s theorem and the convolution formula show on that domain that

In this identity the order of F and G is retained; one only commutes the Fourier factors and polynomials in S past the operator coefficients, as permitted by the hypothesis. It can first be checked for rapidly decreasing smooth symbols and then for the divided symbols above, moving their polynomial factors to the input. Taking adjoints in pairings of smooth vectors likewise gives F(S)^*=F^*(S) on the common domain. For scalar symbols this construction is the usual joint spectral functional calculus by Fourier inversion. In particular the constant symbol 1 substitutes to the identity. Thus a pointwise unitary symbol and its adjoint substitute to mutually inverse isometries on a dense invariant domain. Their extensions are unitary on the full Hilbert space. These facts also identify the construction with the finite-tree substitution where the controls already have a simultaneous multiplication representation.

For convergence, first restrict x to a compact set. Convergence of the integrands on smooth inputs and the integrable bound just proved give convergence of the Fourier integrals. The polynomial decay supplied by (1+|x|^2)^{-N} controls the discarded tails uniformly. The same proof applies to every scalar-parameter derivative and to adjoints. Uniform higher output bounds then give smooth convergence by (2.14). For coefficients merely measurable in s, use dominated convergence in s; for polynomial seed bounds increase the seed-moment input weight.

Finally, every x-derivative of e^{\ii t f(s,x)} is bounded by a polynomial in |t| times a fixed polynomial in |x| (and |s| in the moment-weight case). Applying the same argument proves the last assertion. The proof used a joint spectral representation and Fourier inversion, without choosing independent field coordinates.□

Proof. We give both the finite-tree recursion and the analytic induction. This is necessary because a global operator-norm error proportional to the number of vertices would not imply the assertion.

Boundary inputs. Unroll the fixed expression and index its normalized-sum occurrences by their named roles, retaining repeated occurrences separately. These indices form the finite boundary-channel list \alpha=1,\ldots,b. Cut an edge immediately above a branch root. For an occurrence A_{I_\alpha J_\alpha}f_\alpha, replace the parent’s contribution to the normalized sum by a formal real coordinate y_\alpha, retaining the receiving mask I_\alpha. Write y=(y_1,\ldots,y_b); these are substitution variables, not additional random seeds. A sender supplies the coordinate m_\alpha(y)=\1_{J_\alpha}(s)f_\alpha(y), evaluated at that sender. The receiving child uses m_\alpha(y)/\sqrt D on its role mask and ignores the coordinate when that role is inactive. All outgoing messages that are requested by the parent are unchanged by the cut: their formulas exclude the parent’s receiving type. The parameters of the gate are computed from the finite list of zero-boundary child signals and these added inputs. At the branch root this gives a target matrix V(y), equal to identity on nontargets, and the outgoing message list m(y). The same list is used for each child with its receiving masks imposed there. Indeed, a component sent to a child excludes that child’s type from its own computation, so it contains no contribution from that child. The mask exclusions are needed for exactly this assertion.

We first derive the factorization in the physical coordinates, before conjugating by the preceding circuit. Retain the target operations at distance at most d from the branch root, but compute all their parameter formulas on the padded tree. Let G^{[d]}(y) be this truncated operation, and temporarily write V_{\rm phys}(y) and m_{\rm phys}(y) for the root matrix and outgoing messages just defined. For d=0 the operation is only V_{\rm phys}(y). For d\geq1, partitioning the targets among the root and its children gives

The same formula list is evaluated in each child with its receiving masks imposed there. The exclusion rule ensures that the input sent to a child does not feed back through that child’s receiving type.

The zero-boundary child operation is the baseline to factor out. Put

where U is the preceding subtree circuit and U_j its child circuits. For d=0 take B^{[0]}=1. The residual operation in a child, expressed in its preceding frame, is

At the current branch use the same definition,

Thus K^{[d]}(0)=1. This compares two operations on the same branch; it does not discard the zero-input operation.

Conjugate the physical signals by P, writing V(y)=P^*V_{\rm phys}(y)P and m(y)=P^*m_{\rm phys}(y)P. Equivalently, conjugate each zero-boundary child signal by its U_j, while leaving the root spin in its present frame. Define E^{[d]}(y)=P^*(B^{[d]})^*G^{[d]}(y)P. Factoring the baseline from the physical identity gives the exact finite-degree recursion

For d=0 the first line is E^{[0]}(y)=V(y). For d=1 it multiplies the root matrix by the ratios of the child target matrices. Notice that m(0) need not vanish: removing the parent input leaves the root’s own signals. Hence E^{[d]}(0) is generally nontrivial, although every ratio K^{[d]} equals one at zero.

The second line follows by canceling B^{[d]}(B^{[d]})^* between the two factors. At zero external input the same factorization gives

The new child circuits are G_j^{[d-1]}(0)U_j, so the new root map is

Only targets within the finite influence distance can respond to a cut input. Taking d above that distance gives the actual root update; all more distant zero-input operations are already factored into the child circuits.

The factors and their parameter arguments commute before the preceding circuit is applied: targets are disjoint from controls, and each site supplies only its prescribed axis. More explicitly, let \mathcal A_j be the abelian von Neumann algebra generated by all seeds and prescribed control axes in child branch j, excluding target spin axes. For every scalar a, each truncated gate G_j^{[d]}(a) commutes with \mathcal A_j, since it changes only target spins and has coefficients in that algebra. Consequently

Before the final root conjugation by u, every component of m(y) and every zero-boundary parameter sum is affiliated with the abelian algebra generated by the root controls and the child algebras U_j^*\mathcal A_jU_j. Thus K_j^{[d]}(a) commutes with every spectral projection of every such component, even when that component uses signals from branch j itself. The unused components of a common message list cause no exception. This proves the exact commutation required for substituting the operator m(y) into the scalar family. All these algebras also commute with root-seed multiplication, as do the preceding root maps, so the seed-dependent version of commuting substitution has its additional commutation hypothesis. This also proves that any two descriptions giving the same finite-tree operation, including its cut versions, have the same root map.

Zero-boundary signals. Assume by induction that the preceding maps u,u^* have the asserted smooth bounds and limits. Compute a signal in its physical control algebra from the bottom roles up. At a sum, its zero-boundary child columns have already been computed; after conjugation by the preceding child maps, they still commute, are still centered by odd symmetry, and have the required smooth bounds. A signal at the new root is calculated pointwise in the resulting child sums and the root’s chosen axis, and is then conjugated by the preceding u when its one-copy Heisenberg version is needed.

Here is the induction that also controls exponentials. Local commuting seed/axis variables plainly have joint exponentials with polynomial smooth bounds. Suppose this is true for the child signal list. Lemma 2.7 gives (2.23), including the polynomial frequency bounds on joint exponentials, and Lemma 2.6 gives polynomial convergence. Lemma 2.10 then computes each smooth pointwise function and gives polynomial bounds for the joint exponentials of every new finite signal list. Conjugating by u and u^* preserves those bounds and convergence. This proves the signal assertions in order of calculation depth, without any assumption on an unknown outgoing exponential.

Propagation of a gate across a cut. Proceed next in d in (2.26)–(2.27). For d=0, V(y) is a smooth two-by-two matrix function of the signal lists just treated; commuting substitution gives all assertions. Suppose they hold for the child family K^{[d-1]}(a), including all input jets and adjoints. It equals identity at a=0. Treating the inputs as odd under the symmetry gives \mathscr P K^{[d-1]}(a)\mathscr P=K^{[d-1]}(-a), where \mathscr P\vac=\vac. Therefore each first derivative has vacuum mean zero.

For scalar a, apply Lemma 2.7 to the child product. With \partial_\alpha K^{[d-1]}(0)=\ii T_\alpha, its limiting value is

The second term is scalar for scalar a. After substitution it can be an operator in the commuting control algebra; it must therefore be retained. The product lemma supplies polynomial bounds for every scalar a-jet. The components of m(y) are smooth functions of a finite list of commuting normalized signals and a root control axis. The finite-tree commutations already established imply the hypotheses of Lemma 2.10, which substitutes a=m(y). Use the spectral projections of the fixed root axis if needed. Multiplication by V(y) completes the induction for E^{[d]}(y), and conjugation in (2.27) completes it for K^{[d]}(y). The argument includes all boundary-input derivatives by the ordinary chain rule.

This proves (2.28) with smooth bounds and limits. It adds one gate to the preceding sequence. Ordinary cost and mixer updates may be inserted between these gates by Lemma 2.9; its proof requires only smooth bounds on the preceding maps and preservation of symmetry. Local seed gates are handled as in that lemma. Induction on the finite sequence proves the first assertion. The same proof with D+1 top factors instead of D gives the same limit, since (D+1)/D\to1. Smoothness supplies the bounded occupation required for Proposition 2.2.

External parameters can be carried through this induction without being scaled. At the child-product step use the unscaled-parameter assertion of Lemma 2.7; at the substitution step use the mixed-jet version of Lemma 2.10. Each unitary strong limit is unitary, so adjoint convergence follows from the identity in the proof of Lemma 2.7; its jet version follows by finite differences with the adjoint jet bounds. For real commuting signals the adjoint is the same signal, and compositions use the reversed products of the already convergent adjoint families. Thus adjoint convergence is established at each induction step, rather than inferred from bounds alone.

Continuity inside a sequence. Repeat these two inductions with a further approximation index. Compact convergence of a fixed number of matrix derivatives and uniform polynomial bounds are precisely the hypotheses of commuting substitution. The normalized sum and product lemmas allow the child maps themselves to vary with this index. Hence every new root map and its adjoint converge on smooth vectors with uniform smooth bounds. A fixed finite surrounding sequence is handled by induction and the product rule. The assertion does not make a bound uniform in the length of that sequence. For a pointwise product formula on a single target and a common commuting control algebra, the entire product is first regarded as one matrix-valued function; the theorem then applies to its smooth matrix limit. This is how arbitrarily long local commutator or sum formulas can be used without an estimate proportional to graph size. For measurable seed limits, dominated convergence in the root seed and then the same child induction give the stated extension. Norm continuity (2.7) proves energy convergence.

Conditional bounds and masks. Take W_0=1. Every root map commutes with root-seed multiplication, by inspection of the finite-tree gates and their factorizations. Disintegrate the root Hilbert space over that seed. All preceding estimates remain valid in each fiber with the same constants: coefficients are uniformly bounded there, and the normalized child columns are averaged over independent child seeds. Moreover the weight commutes with root-seed multiplication. Integrating the squared conditional estimate over J proves (2.25). With polynomial seed coefficients use instead the seed-moment weight and integrate the polynomial majorants. This completes the proof.□

Proof. Differentiating (2.30) at zero gives \mathscr P H_\alpha\mathscr P=-H_\alpha, proving centering. To justify all Hamiltonian-jet bounds without an estimate on an extensive Hamiltonian, introduce an unscaled real time t and use the unitary cut ratio

All current Hamiltonian terms commute before conjugation, and simultaneous conjugation preserves this property. In particular H_D(a) and H_D(0) commute. Thus the display is exactly the prior-frame ratio of the zero- boundary gate and the gate with boundary input a. A local matrix e^{\ii t h(x)} and all its mixed t,x derivatives have polynomial bounds from the assumed bounds on h (use Duhamel’s formula for matrix derivatives). The external-parameter version of Theorem 2.12, which also applies to its truncated cut recursions, therefore gives uniform smooth bounds and convergence of K_D(a;t) with all time and boundary-input jets. In particular

These are the required bounds for the response jets; no bound for the possibly extensive H_D(0) is asserted or used.

For the child-product limit compute the first two boundary jets at zero:

In Lemma 2.7 one has T_\alpha=tH_\alpha; the two anticommutator terms cancel exactly. The product of the D child ratios therefore tends, with all time jets, to the exponential of \ii t times (2.31). Differentiating at t=0 establishes the asserted smooth Hamiltonian limit itself. The finite-degree child products are unitary groups in t. Their strong limits at positive and negative times retain the group law and inverses, while convergence of time jets gives strong continuity. Their self-adjoint generators consequently act on smooth vectors by (2.31); no separate choice of closure of the displayed expression is used. The exclusions and Lemma 2.10 allow operator inputs and preserve the group law and this generator identification. Repeating this argument at the finitely many affected levels proves the recursive assertion.

For R_j with this self-adjoint realization, the spectral inequality |e^{\ii tx}-1|\leq|t x| proves (2.32). Uniform smooth bounds and (2.14) then give higher-norm convergence; applying the same reasoning to the adjoints and using finite composition proves the final statement.□

Proof. Let S be a standard Gaussian in the local root-seed factor and start with g_0=S. Recursively set

and use (3.3)–(3.4) to form Y_j,g_j as soon as Z_j is available. The constants c_j here are those defined under the independent Gaussian law above. We will verify that the recursively constructed variables have precisely that law.

At each step identify all earlier vectors by the compatible empty-descendant embeddings of Section 2. These embeddings preserve the earlier Gaussian positions: lifting an embedded child vector gives the embedded copy of its previous lift. Thus the recursion uses the same earlier variables at the next height. Inductively g_j has height at most j and Z_{j+1} has height at most j+1. All vectors can be compared in the same finite-height space whenever an inner product is taken.

We show inductively that g_0,\ldots,g_j are real centered orthonormal vectors. The claim holds for g_0. If it holds through g_{j-1}, the positions Z_1,\ldots,Z_j are jointly centered Gaussian with identity covariance, hence independent standard normals. For j\ge1, the coefficient c_j a(t_{j-1},Y_{j-1}) is measurable in Z_1,\ldots,Z_{j-1}. Conditional mean zero of Z_j proves that g_j is centered and orthogonal to all earlier g_i with i\ge1. The normalization in (3.4) gives \norm{g_j}=1. The receiving-root seed S belongs to a different tensor factor from every child-Fock position, so it is independent of all these positions. In particular \braket{g_0}{g_j}=0. The child directions themselves may contain seeds at their own roots; this does not change the independence at the receiving root. This closes the induction and also proves that Z_1,\ldots,Z_{N+1} have the required joint law. The coefficients are bounded at the fixed grid times, so every constructed column has moments of every finite order.

The map u(t,\cdot) is odd and a(t,\cdot) is even. Consequently each Y_j and g_j is odd under simultaneous seed reversal. Since \Phi_{\rm G}(-z)=1-\Phi_{\rm G}(z), the same is true of F_N outside its null zero set. Hence F_N is centered.

It remains to compute its energy, not just its distribution. The vector F_N is constant in the root-seed factor and depends only on the finite orthonormal family of Gaussian modes \mathsf G(g_0),\ldots,\mathsf G(g_N). The adjoint root shift extracts precisely its constant-root-seed one-particle component. Since q(g)=a^*(g)+a(g) has variance \norm g^2, this component is

There are no other first-chaos coefficients: F_N is independent of Gaussian modes orthogonal to the displayed family. Moreover F_N is independent of the receiving-root seed, so \braket{F_N}{g_0}=0=A_0. Inner products with the other g_j are exactly the scalar coefficients A_j. All vectors are real, and therefore

This proves the claimed identity, with the Gaussian variance and root-shift normalizations used throughout the paper.□

Proof. Replace a Gaussian local seed s by a smooth bounded odd truncation s_R satisfying |s_R|\le|s| and s_R\to s pointwise. It converges in every finite moment. At a fixed finite depth this changes the original formula by an amount tending to zero in every required finite moment: use (2.10) at each aggregation, Gaussian moment bounds for its finite list of fields, and the polynomial bounds for the derivatives of the pointwise maps.

Fix 0<\kappa<1. The desired final angle

is bounded and odd, and is a Lipschitz function of the clipped output. It satisfies \sin(2\theta_\kappa)=(1-\kappa)m^*. We approximate this angle and all its subformulas beginning at the output and working toward the leaves. For the finite-dimensional ideal input law at a pointwise operation, a continuous polynomial-growth function has an arbitrarily accurate L^p approximation, for any fixed finite p, by a bounded smooth function with bounded derivatives. This follows by a smooth cutoff outside a large box and mollification; the uniform moments control the cutoff error. Symmetrizing by F(x)\mapsto(F(x)-F(-x))/2 preserves oddness and does not increase its error under the symmetric input law.

Once an outer map has been replaced by a fixed bounded Lipschitz map, perturbations of its inputs propagate continuously in L^2. For its aggregate inputs this is exactly (2.10), jointly for the finite list of child columns. We can therefore choose the approximations at the next lower level sufficiently accurately to preserve the already chosen output accuracy. There are finitely many operations, so induction supplies a formula in which every transmitted scalar function is bounded, smooth, and odd. No uniform bound on Lipschitz constants over the sequence of approximation choices is needed; each outer choice is fixed before its inputs are approximated.

Finally a bounded smooth cutoff supported strictly inside a box can be extended periodically to a larger box. Fourier approximation of this smooth periodic function gives finite trigonometric sums converging uniformly, with any prescribed finite number of derivatives, on that box. To control the complement, first fix the box so that its probability is small and choose the Fourier approximation uniformly on the entire period; both the cutoff and its approximant then have a common finite sup norm. The same output-to-leaf continuity argument applies. Odd symmetrization leaves only sine terms \sin(\lambda\cdot x) for an odd scalar output. This constructs the asserted trigonometric formula. Choose its angle error smaller than \eta/4 and \kappa<\eta/2. Since x\mapsto\sin(2x) is 2-Lipschitz, the resulting output satisfies (3.11).□

Proof. Unroll the fixed expression into a finite expression tree. For each occurrence, each possible current nonhelper color c, and each possible set A of ancestor colors, make a copy of the lower formula. Its aggregates retain only children whose colors avoid A\cup\{c\} and all helpers. At a retained child of color d, the recursive copy uses ancestor set A\cup\{c\} and current color d. Thus its own aggregates exclude A\cup\{c,d\}. All sets encountered have size at most h+1. There are finitely many copies because the original depth and the color partition are finite. The aggregation at color c is the sum of its source-color aggregations. Linearity of \mathsf G identifies this finite sum with the lift of the corresponding masked child column.

These copies also define an orientation-free all-neighbor rule. When a recursive call crosses from a vertex i of color c to a vertex j of color d, the ancestor list at j contains c. Every aggregate at j therefore gives the neighbor i a zero mask. Its sum over all neighbors equals its sum over neighbors other than i, with the same normalization D^{-1/2}. Induction over the expression depth identifies the all-neighbor calculation with the descendant calculation on each active branch. The outer root has no parent and uses all its neighbors. In particular the finite rule uses the aggregations A_{RS} of the synthesis interface without requiring a parent orientation. A full root has D+1 neighbors; its limiting formula agrees with the D-child convention by the full-root assertion of Theorem 2.12.

Here is a uniform error estimate. Give the original formula the extra independent colors but ignore them in its computation. At one aggregate, let g be its ideal bounded child column, with |g|\le K, and let g^{A,d} be the recursively modified column on a retained color d. Couple the two fields using (2.10). The modified aggregate has child column

The color masks are orthogonal, and g is independent of the colors at its root. Hence

Every masked column is centered: color masks are even and the conditional formula is odd in the Gaussian seeds. Thus the Gaussian-isometry identity applies without a mean correction. For a list of k aggregates the squared Euclidean input error is the sum of their squared errors. A pointwise map with Lipschitz constant K' multiplies that sum by at most (K')^2. Inducting over the finite expression depth, with the unchanged bounded local seed as the base case, proves a conditional squared error bounded by C(\alpha+p_H) uniformly over current and ancestor colors. Setting the helper-root output to zero adds at most \norm f_\infty^2p_H.

This estimate has the fan-in of the original formula, not the number of color copies: disjoint child masks were combined by squared norms before applying a Lipschitz bound. Each nonlinear pointwise map still receives the original finite list of grouped fields: if it originally had k field arguments, its colored version is F(S,Y_1^{\mathrm{col}},\ldots,Y_k^{\mathrm{col}}), where

The conditional derivative bounds in these k arguments are unchanged. The finite sums defining grouped fields are additional linear operations and can be unbounded; boundedness is required of the transmitted columns and final angles, not of Gaussian fields. For a trigonometric formula, substitute each grouped linear sum into the next pointwise map. A trigonometric term in linear combinations of the source fields is still a trigonometric term, so this substitution preserves the finite trigonometric form without retaining separate linear pointwise maps. For the synthesis interface, a rotation by a grouped field is exactly the product of the source-field rotations on the same axis, since these classical fields commute. If a compilation instead regards all the source fields as separate coordinates, its derivative constants may depend on the fixed number M. This is harmless: M and all these finite lists are fixed before any helper refinement or quantum approximation is taken. Only the classical error constant in (3.13) and the bounds in the original grouped coordinates are asserted to be independent of M.□

Proof. Use (3.10) with error \varepsilon/3 and write m^*=z. Apply Lemma 3.2 to obtain an angle \theta whose output m=\sin(2\theta) has \norm{m-m^*}<\varepsilon/12. Both outputs are bounded by one, so (3.9) loses less than \varepsilon/6. Fix this entire bounded trigonometric formula, including its finite depth and Lipschitz constants.

Reserve a distinct helper type for each propagation depth. Choose their positive probabilities with total p_H<\eta_H sufficiently small, and partition the remaining probability into finitely many colors of maximum probability \alpha sufficiently small. Apply Lemma 3.4 to the angle calculation and use the 2-Lipschitz property of sine. Taking \alpha,p_H smaller if necessary gives \norm{m^{\mathrm{col}}-m}<\varepsilon/6. Another application of (3.9) loses less than \varepsilon/3. The total loss is less than \varepsilon. Oddness gives centering, and the remaining claims follow directly from the construction. Every choice has been made for a fixed finite calculation in the limiting rooted space; no uniform rate in the number of colors, circuit depth, or helper probabilities has been used.□

Proof. The Weyl estimates in Lemma 2.5, applied to \sum_j\xi_jd_j, give, for every a,

Only upper bounds for weighted column norms enter this inequality. For an integer N, put g_{s,t}(x)=e^{\ii tF_s(x)}(1+|x|^2)^{-N}. Fix the desired output power. First choose enough derivatives that integration by parts gives an integrable Fourier bound with weight (1+|\xi|)^{r_a}, and then choose N large enough that those derivatives are integrable, uniformly in s,t. The polynomial derivative assumptions permit this order of choices. Functional calculus gives, on smooth vectors,

with the chosen Fourier normalization absorbed in \widehat g. The polynomial on the right has a fixed smooth power bound by the creation and annihilation estimates. Fourier integration therefore proves the assertion. Differentiating in any additional parameters also differentiates the fields and the polynomial factor. If v(y)=\sum_j\xi_jd_j(y), the central commutator relation gives

Repeated derivatives are the same Weyl operator times finite ordered products of derivative fields and scalar Gram pairings. Each field insertion has a smooth bound polynomial in |\xi|, and every scalar pairing is quadratic in \xi, by the weighted column-jet hypotheses. The Leibniz rule gives the same finite-power bounds for derivatives of (1+|Q|^2)^N; its differentiated fields need not commute with the original list. Increasing the Fourier decay and input weight powers just chosen proves the jet assertions. No change of field coordinates, and hence no inverse Gram matrix, occurs.□

Proof. Normal ordering gives a pair-creation kernel, its adjoint, and a one-particle kernel:

where the first kernel is understood symmetrically. Orthogonality of different masks gives

The same inequalities hold with d,e interchanged. Pair creation by a Hilbert–Schmidt kernel has norm at most \sqrt{(n+1)(n+2)} times its kernel norm on the n-particle sector; the corresponding estimates for annihilation and a one-particle kernel are bounded by a constant times n+1. Commuting powers of W through these operators inserts powers of \Lambda on their kernel indices. This proves the asserted smooth smallness.

For completeness, this smallness of the generator alone is not used to assert an exponential power bound. Every iterated commutator \ad_W^j(Q) is another quadratic with uniformly bounded Hilbert–Schmidt kernels of the displayed form. Consequently

Writing [W^a,Q] with these commutators on the left yields \norm{[W^a,Q]\zeta}\le C_a\norm{W^a\zeta} for positive integer a. On smooth vectors, differentiation of \norm{W^ae^{\ii tQ}\zeta}^2 and Gronwall’s inequality now give a bound independent of the column count. At a fixed finite column count these differentiations are justified by Lemma 4.3; its constants are not used in Gronwall’s bound. Equivalently one may first truncate the weights and pass to the limit using these same inequalities.

Finally, spectral calculus gives

The uniform higher-power bounds and interpolation prove the last assertion. The same conclusions hold for U^*QU when U is unitary and both U and U^* have uniform smooth bounds. They are uniform for families of unitaries with common such bounds: indeed, e^{\ii tU^*QU}=U^*e^{\ii tQ}U, and the smooth power losses compose.□

Proof of Proposition 4.2. We prove a single replacement with arbitrary preceding root maps from the gate calculus: they preserve the joint parity, commute with root seed multiplication, and have uniform conditional smooth bounds. These hypotheses are preserved by all the gates used here. For the sender partition put

We will estimate the Hamiltonian correction at each root cut, prove uniform smooth bounds for its exponential, and then iterate the replacement. At helper roots a scalar phase remains; we will check that it cancels from the child signals before the next replacement. The partition changes only the terminal coefficients of the lower calculation: the formula f and its finite channel list stay fixed.

On the range of (4.10), and hence in its recursive occupation realization, we may set X_k=1 at every k\in K. The preservation facts just established apply to the exact replacement and all its surrounding gates.

The relative root operation. Fix the partition and introduce an unscaled real time t, keeping the preceding circuit fixed as t varies. Write H_D for the desired Hamiltonian and \widetilde H_D for the complete contraction Hamiltonian. On a padded rooted tree T, retain target terms through a depth d larger than the influence distance. On each child component T_a, use the same formula with zero parent input and target depth d-1. All parameter formulas are computed on the padded tree. For either Hamiltonian F_D, put

All terms in the two Hamiltonians and their branch baselines commute: they are diagonal in the same physical site axes. Factor the preceding circuit as U_D=\mathcal U_Du_D, where \mathcal U_D=\bigotimes_aU_{a,D}. The new child circuits are e^{\ii tF_D(T_a)}U_{a,D}. Factoring their product from e^{\ii tF_D(T)}U_D, as in (2.28), gives exactly

Consequently the relative root maps satisfy

This is a unitary group in t, whose exponent is a root cut difference. The zero-input branch Hamiltonians have been subtracted; no limit of the extensive Hamiltonian difference is being asserted. The identity holds on the compressed space as well.

At this fixed partition, Theorem 2.12 applies to both complete operations with t as an external parameter. The complete contraction is the gate at the K pivot, with coefficient p_K^{-1}\sum_l B_{k,l}F_{k,l}. Their local matrices have polynomial bounds for all mixed time and input derivatives, locally uniformly in t. The root maps, their adjoints, and their time derivatives therefore have smooth degree limits. Passing to the limit in (4.12) gives a unitary group \mathcal T_{\mathcal P}(t): its group law and inverses follow from the finite-degree identities, and time-derivative convergence gives strong continuity. Its self-adjoint generator A_{\mathcal P} is defined by this group. Differentiation at zero shows that every smooth vector is in its domain and that, under the occupation identifications,

We now compute this derivative by separating diagonal paths, simple paths, and lower-formula responses. This is an algebraic decomposition of the finite-degree cut difference, not an application of the gate calculus to each separated Hamiltonian. All contributions to (4.13) use the preceding full frame. Thus top-path expressions first computed in the child frame receive one final conjugation by u; the lower-response formulas already display it. The smooth bounds are preserved by this conjugation. The diagonal average and its helper-root scalar are included before any phase is removed.

The diagonal paths. The terms j=j' in (4.9) equal

Away from the incidence lost at a K-root cut, the last factor tends to one, with all smooth bounds required in the gate calculus: it is a normalized average of bounded local seed masks. This remains true after conjugation by preceding maps with uniform smooth bounds. One can see the rate on the symmetric embedding by decomposing that average into its vacuum scalar, vacuum row and column, and occupied block. The centered row and column are O(D^{-1/2}), and the occupied block is O(D^{-1}) with a loss of one number power.

If the removed root lies in K, the lost incidence contributes (Dp_K)^{-1}\sum_{j\sim o}\1_J(j)b_jf_j. The average has a scalar vacuum limit. Thus it contributes only a phase conditional on the root’s membership in K. Below we also recover this scalar as the same-neighbor contraction of the full K-root expression. No estimate discards an uncentered normalized sum.

The nonbacktracking paths at top masks. Remove the terms j=j'. A remaining path i,j,k,j' is simple on a tree. At a terminal j'\in J_l, looking away from its parent, its extra coefficient is

Here all inner fields are computed in descendant trees. Let d_l be the child vacuum column of \1_{J_l}Xb, in its preceding Heisenberg frame. It is centered, since it is odd, and (4.4) gives \norm{d_l}_a\le C_a\sqrt{p_l}. The first normalized lift is q(d_l). Masking its next root on K, conjugating by the preceding map, and lifting again gives

The constants include p_K^{-1}, which is fixed. The reversed coefficient, with f in place of b, has the same bound. This reasoning uses centered lifts at both steps; the parity of the transmitted fields supplies their centering.

At a root in I, the final incoming column is the combination over J_l of the small coefficients just estimated. Its squared weighted norm is bounded by C\sum_l p_l^2, since conditioning on J_l contributes one factor p_l and (4.16) contributes another. The extra root Hamiltonian is its normalized lift times the root axis P, and is O_{\mathrm{sm}}(\rho).

At a root in J_l, the two possible occurrences of that root, as j or j', give a bounded smooth factor f or the field b times one of the small coefficients in (4.16). The two factors use distinct first-step masks. Thus there is no further same-neighbor term to subtract. The root correction is O_{\mathrm{sm}}(\sqrt{p_l}), uniformly conditionally on the root seed in this part. It is a smooth function of a fixed number of commuting fields with uniformly bounded weighted columns.

At a root in K, let d_l,e_l be the incoming child columns of \1_{J_l}Xb and \1_{J_l}Xf. All their fields commute. The root contribution of the full top-path sum is p_K^{-1}\sum_l q(d_l)q(e_l). The terms using the same neighbor are the limit of the average

In particular, this is an average on one branch, not the product of two independent lifts. After subtracting it, the correction is

The weighted columns belonging to different parts are orthogonal. Lemma 4.4 gives Q_K=O_{\mathrm{sm}}(\rho) and uniform exponential bounds independent of the number of parts. The scalar \kappa is the harmless local phase identified from the diagonal paths.

This list accounts for all top-path cuts. If the root belongs to I,J,K, a top path not containing it cannot change after the cut: the dependencies of f exclude all three masks. At a J root, its own f uses only descendants. A root outside both the top masks and the lower calculation has no correction. It remains to check cuts inside the lower computation of f.

Responses through the lower formula. We have accounted for cuts through the four-vertex helper paths. A cut can also change the value of f at their terminal vertices. The following recursion keeps those changes, including mixed responses between channels, in the estimate.

First work in the physical frame. Write H_{\rm off}(T)=\sum_{l,j'\in J_l}\omega_{j',l}f_{j'}^T, where \omega_{j',l}=X_{j'}R_{j',l} initially denotes its full top-path coefficient. If a cut root v has a lower mask, it cannot lie on any of the top paths. Each coefficient is consequently unchanged and lies wholly in one cut component. For the child components T_a, the exact finite-tree identity is

Only terminals within the fixed influence radius contribute. At an active terminal j'\in J_l, a boundary input into its lower computation therefore changes the Hamiltonian by

Whenever this difference is active, its parent lies in a lower mask, outside I,J,K. Consequently every first step of R_{j',l} goes into a descendant branch. That coefficient has no boundary response of its own on this active path. If the parent is in a top mask, the boundary input into f is absent, and (4.19) is zero. This observation ensures that using the downward expression (4.15) loses no term.

Here is the precise difference recursion. At a root v of the lower calculation let m(y) be its fixed finite list of outgoing channel values. Let \mathcal H_v^{[d]}(y) be the sum of the weighted terminal terms with terminal distance at most d, evaluated with boundary input y, and put \Delta_v^{[d]}(y)=\mathcal H_v^{[d]}(y)-\mathcal H_v^{[d]}(0). The coefficients are computed downward, as just justified; the terminal formulas themselves use all required sites. In the physical frame the recursion is a local term (4.19), when the root is terminal, plus

Only channels of the fixed f-calculation are included. At the actual output root, where there is no parent input, the root Hamiltonian correction is obtained from the local term and the child sum evaluated at D^{-1/2}m(0), without this subtraction. The definition counts distance to the terminal vertex; the top coefficient beyond that vertex has already been computed downward. Taking d beyond the finite influence range gives exactly the Hamiltonian difference across the cut. Indeed the identical recursion with \mathcal H_a first holds by partitioning terminals among children, and replacing \mathcal H_a(z)=\Delta_a(z)+\mathcal H_a(0) cancels the constant terms. This also explains the argument m(0) in the second summand.

Now factor the preceding circuit as U_v=(\bigotimes_aU_a)u_v. These preceding maps are fixed in the boundary variable of the new Hamiltonian. Simultaneous conjugation of the physical identity gives

where \widehat\Delta_a=U_a^*\Delta_aU_a, and T_v(y) is the local terminal term (4.19), zero at a nonterminal root. The direct term and messages are interpreted in the same child frames. There is no derivative of U_a or u_v. Although a preceding Heisenberg image of the top coefficient may appear to have larger support, its boundary independence follows from the physical identity before this fixed conjugation. Repeated lower masks in different expression branches cause no additional terms: these are identities for the whole nonlinear function of all incoming channels, so their mixed responses are retained.

The required parity covariance is explicit: if \Pi is the joint parity, then \Pi\omega\Pi=-\omega and \Pi f(y)\Pi=-f(-y). Thus \Pi\{\omega[f(y)-f(0)]\}\Pi=\omega[f(-y)-f(0)], and the recursion preserves this covariance. We pass the algebraic recursion (4.20) to the limit by the centered normalized-sum calculation used in the proof of Corollary 2.13. At a fixed partition, the terminal coefficients and their jets are finite products and smooth compositions of centered odd normalized sums, together with the separately estimated bounded-mask averages normalized by D^{-1}. The normalized-sum bounds and the conditional bounds for preceding maps give uniform degree bounds in every required higher smooth norm; these constants may depend on the fixed partition. They propagate through the fixed-depth difference recursion. Taylor expansion in the boundary inputs therefore retains the centered first response and the scalar second response, while its third-order remainder carries D\cdot D^{-3/2}. Commuting substitution handles the outgoing message arguments. These are termwise limits in the derivative of the complete relative family, not a unitary calculus for the separated Hamiltonians. If D_\alpha,D_{\alpha\beta} are the first two input derivatives of a child response at zero, set

Its contribution is

followed by the preceding root conjugation. At a nonterminal lower root with no parent input, the Hamiltonian correction instead uses the unsubtracted child contribution

This is the root correction specified after (4.20), rather than the value at y=0 of the boundary difference, which is zero. Both expressions use the same response columns and scalar second responses. First derivatives are odd and hence centered. The same calculation applies to every fixed input jet. The partition estimates below are made after this degree limit.

At depth zero, (4.16) and the derivative bounds of f make every fixed response jet O_{\mathrm{sm}}(\varepsilon), conditionally in the local seed. More explicitly, the sum of terminal jets over l is a direct integral on the disjoint events J_l, so its conditional bound is the maximum of their C\sqrt{p_l} bounds, not their sum. Suppose the induction holds at depth d-1. All weighted norms of r_\alpha and all |c_{\alpha\beta}| are then bounded by C\varepsilon. The functions m and their derivatives have fixed polynomial bounds. Multiplication, differentiation, and the preceding root conjugation in (4.22) therefore give the same smallness at depth d, with a possible finite loss of powers. This induction has fixed finite depth and a fixed number of channels, both independent of the number of parts J_l. It requires no independence between different lower channels. In particular it applies when two expression branches reuse some lower masks. At a nonterminal lower root, if there are k outgoing channels and s ordinary Gaussian inputs used to compute m, the root correction uses at most s+k field directions. Each r_\alpha is a single child vector containing the whole lower calculation; its Hilbert-space support need not be finite dimensional. The scalar second responses contribute no new directions.

The diagonal degree-factor errors have the same response recursion with terminal coefficient

in place of X_{j'}R_{j',l}. On an active lower response path the parent is not in K or I, so this is again a downward coefficient. Its limiting smooth bounds vanish by the average estimate already proved. Thus all its lower responses vanish as well.

Uniform topology and iteration. After the degree limit, the diagonal degree-factor errors vanish. The preceding computations identify the total action in (4.13). At a helper root its scalar part is \sigma_{\mathcal P}(s)=\kappa\1_K(s), with

This bounded root-seed multiplication commutes with every root map. Hence

is a strongly continuous unitary group. Its self-adjoint generator R_{\mathcal P} acts on smooth vectors by the total remaining correction. The four cut cases, in the common preceding frame, give

Since the parts partition the fixed sender mask,

Both quantities tend to zero under the refinement hypothesis, with f and p_K>0 fixed.

To obtain uniform smooth bounds for this actual group, consider the full field expression for the remaining correction at each root. At an I-root it uses a bounded number of combined columns. At a J_l-root include the fields for both possible sender occurrences. At a lower root use all ordinary channel fields and response columns in the unsubtracted root formula. These current-generator fields commute: before preceding conjugation they belong to the same physical seed and fixed-axis algebra. Thus the total expression is a smooth function of a fixed number of commuting fields and, when present, the fixed root axis, with the uniform polynomial derivative bounds already proved. At a K-root it is the complete centered quadratic Q_K. Define V_{\mathcal P}(t) by the joint spectral calculus of these expressions, as a direct integral over the root seed, with the final conjugation by u wherever it is not already displayed. Lemma 4.3 gives its uniform conditional smooth bounds in the fixed-field cases; Lemma 4.4 gives the count-uniform bound at K. These bounds concern the total expression at each root.

We verify that V_{\mathcal P} is the actual relative group \mathcal S_{\mathcal P}. At fixed partition the field-calculus group preserves the smooth domain and has smooth time derivatives there, with \partial_tV_{\mathcal P}(t)\zeta =\ii R_{\mathcal P}V_{\mathcal P}(t)\zeta by the action identified in (4.13). For a smooth vector \zeta, therefore, V_{\mathcal P}(s)\zeta remains in the domain of R_{\mathcal P}, and its image under that generator varies continuously by the smooth bounds. Differentiating gives

Evaluation at s=0,t proves \mathcal S_{\mathcal P}(t)\zeta=V_{\mathcal P}(t)\zeta, and density extends the identity to the Hilbert space. Thus the uniform smooth bounds just proved apply to the relative root group itself; no separate closure of the formal sum of corrections is chosen.

Spectral calculus for its generator now yields

Uniform higher smooth bounds upgrade this to convergence in every fixed lower graph norm; the same holds for adjoints. The relative factor multiplies the desired root map on the right, so

This proves convergence of the actual root maps up to their root-seed phase, with uniform conditional smooth bounds. The same argument applies on cut subtrees, and the fixed lower response jets retain the estimates proved above.

A root-seed phase cancels in each descendant Heisenberg conjugation before its vacuum column is lifted; helper-labeled descendant spaces need not be removed. Starting from the fixed initial preceding maps, apply this single-replacement argument successively through the fixed finite sequence. The continuity part of Theorem 2.12 applies at each step with the conditional smooth bounds just established. It gives insertion convergence off K and proves the simultaneous contraction statement.□

Proof of Theorem 4.1. The finite nested construction proves (4.2). To identify the target value, the ideal seed-only rotations give

where both coefficients are classical functions of seeds; all descendant spins remain in their pointed spin vectors. In \braket{v_{\mathrm{cl}}}{\cL v_{\mathrm{cl}}}, the second summand on the left is orthogonal to the range of \cL. Its shifted version on the right has a child spin excitation and is orthogonal to the first, purely classical summand on the left. Consequently \cE(v_{\mathrm{cl}})=\cE(m). The same identity holds after omitting helper insertions, because m=0 there.

For any circuit preserving root seed labels, \norm{\1_Hv}=\sqrt{p_H}: the mask commutes with the circuit and with Z, and Z is unitary. Since \cL is an isometry, Proposition 2.2 gives

Both projected insertion vectors have norm at most one, so (4.2) changes their energy by at most 2\eta. This proves (4.3). All approximation limits above compare already-defined fixed-word tree values. After choosing finite approximants, the resulting seeded word and all its parameters are independent of the degree, system size, and disorder.□

Proof. The first assertion follows from the creation–annihilation estimates in Lemma 2.5, or by differentiating a Weyl displacement on \mathcal S. In the Taylor remainder one uses \norm{q(d)^2\eta}\leq C\norm d^2\norm\eta_1. The Weyl product identity then gives the assertion for two nearby directions. Resolving P into its two spectral projections proves the spin-controlled statements.

For the second assertion, the Weyl product reduces the question to a matrix coefficient of W(f_r/r), where \norm{f_r}\geq\kappa. Choose the unit quadrature Q_r=q(\ii f_r/\norm{f_r}). Equation (5.1) translates Q_r by 2\norm{f_r}/r. Split each vector using the spectral projection of Q_r onto [-\norm{f_r}/(2r),\norm{f_r}/(2r)]. The two translated central intervals are disjoint. The complementary pieces have norm bounded by C_Mr^M\norm\eta_{k_M}, by the spectral theorem and the uniform estimate \norm{Q_r^M\eta}\leq C_M\norm\eta_{k_M}. This proves (5.2). Root spin matrices commute with these projections. Finally, (5.3) is exactly the second Weyl identity; its right-hand side is a spin rotation followed by a displacement of size O(r). Both preserve the required tail bounds. Notice that this last argument precedes, and does not use, any claim that the probe acts almost trivially on an ordinary packet.□

Proof. Centering follows from the global spin-flip symmetry; every initialization gate is even and the inserted Y or Z is odd. The norm statements follow from unitarity. Because R_\star^*ZR_\star=Y and YX=-\ii Z, expansion of (5.6) gives

where the omitted special–special term has norm r, and the remaining scalar phase tends to one. The directions d_{0,r} are bounded. Resolving the root spin shows that both displayed high vectors are sums of coherent packets whose displacements have norm tending to infinity. Their projections onto a fixed number sector therefore vanish, directly from the coherent-state expansion. At a fixed output Z-spin the centers in \ell_r and in the displayed expression for w_r differ by 4\gamma_re+O(1); their overlap tends to zero by Lemma 5.2 (or the vacuum Weyl formula). Finally, split a bounded family at N\leq M. Its complementary norm is uniformly O(M^{-k}), while its fixed-sector pairing tends to zero. First take r\downarrow0 and then M\to\infty.□

Proof. Choose the axes so that every c_{j,a}\neq0, and so that the pairs (c_{j,1}^2,c_{j,2}^2) are distinct unless the original vectors are equal or opposite. Such axes exist: each prohibited equality or zero is a nontrivial trigonometric equation in the axis angle, apart from the stated equal-or-opposite exception. Delete duplicate square pairs; on each deleted pair the product c_{j,1}c_{j,2} is the same.

In G=SU(2)^m\times\R^2, take the two generators having spin components -\ii c_{j,a}P_a and clock component the a-th coordinate vector. Their brackets have zero clocks. A first bracket has \ii X-coefficient equal, up to a common nonzero real constant, to c_{j,1}c_{j,2}. Repeated double brackets with the first or second generator multiply this coefficient by c_{j,1}^2 or c_{j,2}^2, again up to nonzero common constants. Thus the generated Lie algebra contains the clock-zero vector with coefficients

for every real polynomial F. Polynomial interpolation on the finite set of square pairs makes all these coefficients equal to one.

For completeness, exponentials of this Lie-algebra vector are reachable by finite products of the two original one-parameter groups. Let \mathcal R denote the subgroup of such finite products, and let \mathfrak g_0 be the Lie algebra generated by the two generators. The span of their conjugates by elements of \mathcal R is invariant under each of the two adjoint one-parameter groups. Differentiation shows that it contains all iterated brackets and hence equals \mathfrak g_0. Choose finitely many of these conjugates forming a basis. The product map of their one-parameter groups has surjective differential at zero, so its image contains a neighborhood of the identity in the connected analytic subgroup with Lie algebra \mathfrak g_0. Each factor is itself a reachable conjugate. Subdivide any one-parameter path into finitely many pieces in this neighborhood. The whole exponential is therefore in \mathcal R. Applying this to the common X-generator with zero clocks gives (5.7).□

Proof. Let V_j be a prefix of the short probes, and write v_{P,r}^{(j)}=(V_ju_r)^*P(V_ju_r)\vac. Center preservation and all centered-tail bounds follow first from Lemma 5.2, without estimating the spin phases. At a standard stage every ordinary probe has zero center phase.

At a huge-cost stage there is the exact identity

Group the ordinary packets by their original sign y. The diagonal term of sign y on the right-hand side is y\braket{PV_j\psi_y}{V_j\psi_y}, which is real. This remains true although the probes may rotate the spin within that group. The two ordinary centers are separated by 2|t|, since \norm{v_r}=1. An ordinary center and a special center are also separated: their imaginary projections against v_r differ by a quantity bounded away from zero. Lemma 5.2 makes all these cross terms smaller than any fixed power of r. Terms containing only special packets are O(r^2), and the remainder terms are o(r). Consequently, at every intermediate probe prefix,

More quantitatively, let \varepsilon_r=o(r) bound the norm of the discarded vacuum remainder at the start of the probe sequence. For every fixed M, the preceding argument gives

The constant depends on the finite pulse list, a positive lower bound on center separations, and finitely many designated vacuum-tail seminorms. The exact probe prefixes preserve the norm of the discarded remainder; no number-operator estimate is imposed on it. This derivation used only center preservation, so there is no circular small-phase assumption.

By (5.3), an ordinary center -tyv_r therefore has spin phase o(r) in every probe. In either geometry a probe acts as identity plus O(r) on the ordinary part and on that part with any spin matrix inserted. On the special part the norm is O(r), so unitarity suffices. Using

for the next probe V, or equivalently estimating (PV-VP)(V_ju_r)\vac, gives

for the fixed finite list.

After removing an ordinary center, expand the displacement in each probe to first order. Equation (5.9) supplies only an o(r) additional phase in the huge-cost geometry. Replacing v_{P,r}^{(j)} by its starting value in the first-order displacement costs O(r^2), by (5.11). The first-order sum is therefore

Here is a uniform remainder bound for this cancellation. For a fixed ordinary center d_r, let \mathcal B_r(d_r) be the net probe action conjugated by W(d_r/r), and put

Then for every designated centered tail \eta,

Indeed, remove each spin phase by its operator-norm bound and replace each displacement direction by its starting value. The latter replacement costs at most C r\Delta_r\norm\eta_1 through the finite product. In the product with starting directions, hold those directions fixed at their given r-values and introduce an auxiliary pulse strength \lambda. Its first derivative at \lambda=0 is zero, by the clocks. Its second derivative is a finite sum controlled by \norm{q(v)q(w)\eta}\leq C\norm v\norm w\norm\eta_1; the intervening displacements preserve this seminorm uniformly. Evaluating the Taylor remainder at \lambda=r gives C_Kr^2\norm\eta_1; no differentiability of the directions in r is assumed. The constant C_K depends only on the fixed pulse list; all actual insertion directions have norm one. At a standard center \alpha_r=0. At a huge-cost center (5.10) gives \alpha_r\leq C(r^2+\varepsilon_r+r^M)=o(r). Together with \Delta_r=O(r), this proves the claimed uniform o(r) estimate using the single tail seminorm \norm\eta_1. It applies unchanged to a root-spin-inserted tail. If the input also contains a discarded norm remainder \rho_r, its contribution to the net identity error is at most 2\norm{\rho_r}, by unitarity. No field operator is applied to such a remainder in this argument.

On a special packet the change of its phase coefficient during the sequence is O(r), by (5.11) and the boundedness of its scaled center. The small displacements contribute O(r) on its tail, and its starting phase coefficients converge to the prescribed c_{j,a}. The resulting finite spin product thus converges to (5.7). Multiplication by the packet’s amplitude O(r) makes its error o(r). All quantities defining the probes came from the vacuum. The estimates after removing a center, however, use only the tail seminorms. They therefore apply to every bounded input family with the same center list.□

Proof. Put \widehat v_{P,r}=A_r^*PA_r\vac. This is a bounded smooth family. Unitarity and A_r\widehat v_{P,r}=PA_r\vac give

Each term is O(r) by the exact special norm and the ordinary approximation. The possibly nonconvergent scalar phase cancels.□

Proof. Write u_{\mathrm{in},r} for the exact prefix before the echo, and V_r for its selective flip. The next two prefixes are

Their actual Z-insertions satisfy

The first equality holds because the forward cost commutes with the root Z. It splits the ordinary part into spins y with centers -tyv_r, and changes each special center to d_{zy,r}-tyv_r. Centered tails remain bounded. The standard-stage imaginary projection against v_r is unchanged by adding -tyv_r. Thus the assumed special visibility separates special and ordinary centers at this huge-cost stage.

Denote the exact ordinary and special components after that forward cost by \Psi^{\mathrm{cost}}_{O,r} and \psi^{\mathrm{cost}}_{S,r}, respectively. Lemma 5.5 gives net identity to o(r) on the ordinary part, including its Z-inserted version. On the special part it gives -\ii X to error o(r), again including the Z-inserted version. Since XZX=-Z, it follows that

The last equality uses transport, which the forward cost preserves. The same calculation shows that transport changes sign at this flip.

On an ordinary input the reverse cost combines with the forward cost as

The Weyl phase \xi_r is independent of the root spin because the two spin factors multiply to Z^2=\1. It may fail to converge. Equation (5.20) makes the residual direction bounded and proves (5.18). The ordinary o(r) probe error is carried through the reverse cost by unitarity. No insertion approximation has been divided by r: the cancellation in (5.21) used the two actual directions, and only then was (5.20) invoked.

A special packet has spin -y between the flips. Its two cost displacements therefore add as -tyv_r-tyv'_r, giving (5.19). Here y remains its label from the standard stage before the echo; it is not relabeled when the spin flips. Its scalar phases are harmless. The second selective flip is applied at a standard ordinary geometry, since (5.21) has already removed the huge ordinary centers. It acts trivially to o(r) on ordinary tails, restores special spin y, and restores the plus sign in transport. At this second-flip probe start the exact special norm is still r, and the ordinary low action already has the updated unitary W_Z(g_r)A_r. Lemma 5.6 consequently approximates the actual insertion to O(r) by this updated low insertion, irrespective of the temporarily negative transport sign.

The special insertion remains transported because each cost commutes with Z, while each selective flip anticommutes with Z on the special component to error o(r). In particular, this proof needs only forward estimates on that component. Boundedness of the residual direction implies uniform moment bounds for the new A_r,A_r^*. Lemma 5.6 supplies bounded approximants to the next actual insertion. Finally, v'_r-v_r=O(r), so the updated D_r retains the asserted bounded-approximant property.□

Proof. Every A_r in the invariant is a finite sum of root spin matrices times Weyl displacements. Its directions are finite linear combinations of earlier actual insertion directions, d_{0,r}, and the bounded echo directions g_r. Its adjoint and its insertion vectors have the same finite-Weyl-sum form. The number of summands is finite for the fixed circuit; no bound uniform in circuit length is required.

For such sums all vacuum inner products reduce to

Thus only finitely many direction inner products are needed at a step. The initial directions satisfy d_{0,r}=2b\ell_r+o(1), and the echo directions satisfy g_r=-2tw_r+o(1). Norm-small errors in these bounded directions may be removed using Lemma 5.2. Every actual standard insertion differs by O(r) from its bounded smooth low insertion, by Lemma 5.6. Thus, when computing a new Gram entry, an earlier actual insertion may first be replaced by its low insertion at error O(r). Lemma 5.3 therefore makes its pairing with \ell_r,w_r vanish. Together with \braket{\ell_r}{w_r}\to0, this is exactly the orthogonal-sum rule for external and endogenous directions. Formula (5.23) now proves the Gram assertion inductively.

Here is the additional calculation for the root-shift energy, so that ordinary Gram convergence is not silently substituted for convergence of a different operator. A low Y- or Z-insertion is centered: the circuit preserves the joint spin-flip parity and these matrices are odd. Write such an insertion as

In the unseeded representation, \cL h=\ket+\otimes a^*(h)\Omega. The one-particle coefficient of W(f)\Omega is \ii e^{-\norm f^2/2}f. Consequently

Each pairing \braket{f_j}h is one of the same finite linear combinations of insertion and external-direction Grams. In the formal seeded representation the same formula holds, with h inserted in the endogenous summand; the external components of f_j then pair to zero. More explicitly, the external two-mode Fock factor represents the Gaussian variables at the root, while the child Fock factor is built over endogenous directions. The root shift puts the root-seed factor in its pointed vacuum and creates h only in that child factor. The two-factor decomposition is identified with Fock space over their orthogonal direct sum. The high-direction argument gives exactly this zero pairing in the actual limit. Finally, the actual insertion differs in norm from h by O(r). Since \cL is an isometry,

which proves energy convergence. A finite scalar bulk phase cancels from every insertion. The same argument applies just before an echo’s second flip, because its ordinary centers have already canceled and its low update has already been made.□

Proof. For a bounded input \eta_r, use the standard-stage uniform forward approximation on both \vac and \eta_r. After the huge cost their ordinary pieces have centers -tyv_r and root spins y. In the pairing defining \braket{u_r^*Yu_r\vac}{\eta_r}, the inserted Y reverses the spin. Nonzero spin pairings therefore require the two opposite ordinary centers. Their distance is 2|t|, so their contribution vanishes by (5.2). Every contribution with a special component is O(r); the ordinary remainders are o(r). This proves the first assertion. Before the huge cost, the invariant approximates every special center by a bounded smooth vector to O(r). In particular the intermediate directions v'_r in (5.19) differ by O(r) from the corresponding earlier standard insertions, by (5.20). Adding the forward displacement -tyv_r preserves this property. The first assertion and Cauchy–Schwarz prove the second.□

Proof of Theorem 5.1. Start with the requested finite seeded word and a tolerance. For a small fixed b>0, prepend R_\star e^{2\ii b Zq(\ell)} in the formal seeded calculus, with \ell an additional independent Gaussian coordinate at every site. As b\downarrow0, its energy tends to that of the original seeded word. Indeed, finite bounded-direction Weyl words are strongly continuous in this initial displacement, as are their insertion and energy Grams. At b=0 the initial global mixer acts on the original \ket+ product state by a scalar, so it changes no expectation; the extra Gaussian coordinate is then unused.

Choose b sufficiently small for this error, and perturb the target angles sufficiently little that all the visibility conditions hold, with another arbitrarily small energy error. Fix the resulting finite probe lists by the successive construction above. With these lists fixed, take r>0 sufficiently small. The packet induction and Lemma 5.8 show that the ordinary word consisting of the initialization, unscaled costs, compensated mixers, and seed echoes has energy arbitrarily close to that of the perturbed formal seeded word. Summing the three errors proves the assertion.

Every gate used is a cost or mixer pulse, possibly with negative angle. Toggled probes and all compensations are finite such words; consecutive gates of the same kind can be combined, or zero-angle gates inserted to obtain the stipulated alternating QAOA format. After b, the perturbation, the finite probe lists, and r have been chosen, the entire word and its angles are fixed. Proposition 2.1 therefore applies before any approximation parameter is changed. This proves the claimed order of limits and parameter independence.□

Proof of Theorem 1.1. Fix \varepsilon>0. Apply Proposition 3.5 with error \varepsilon/4 and helper bound \eta_H=(\varepsilon/8)^2. It gives a finite bounded smooth classical calculation with output m and helper probability p_H satisfying

Apply Theorem 4.1 with \eta=\varepsilon/8. The resulting finite seeded circuit has insertion v with

By Theorem 5.1, a finite ordinary word approximates this seeded value to error less than \varepsilon/4. Its value is therefore greater than \Pstar-\varepsilon.

All choices just made concern limiting tree calculations and deterministic approximation parameters. After these choices there is a single finite word with real angles independent of D, n, and J. By Proposition 2.1, its tree value equals its fixed-parameter SK value. Write the word in the form (1.3); it gives v_p(\gamma,\beta)\ge\Pstar-\varepsilon for some finite p. The upper bound and monotonicity established above finish the proof.□

Proof of Corollary 1.2. The normalized negative cost used by Basso et al. [4] is

Thus its cost gate with angle \gamma_j is, up to a scalar phase, the usual MaxCut cost gate with angle 2\gamma_j/\sqrt D, with the same sign and unchanged mixer angle. Let r_p^+(D,\gamma,\beta) be the distinguished-edge correlation for the positive tree cost in Proposition 2.1, and put

That proposition gives \nu_p(D,\gamma,\beta)\to v_p(\gamma,\beta) for each fixed word.

For the observable at an edge, backward cancellation of commuting cost gates leaves only cost edges having an endpoint at distance at most p-1 from the distinguished edge’s endpoint set. An extra edge in this relevant cone would close a cycle of length at most 2p+1. Under the stated girth condition the cone is therefore the regular tree cone. Conjugating only this cone by X on one bipartition class reverses the cost sign and the distinguished Z_uZ_v, while preserving the mixers and the initial plus state. No bipartition of G is used. Every edge therefore has negative-cost correlation -r_p^+(D,\gamma,\beta), and

This is the finite-degree edge formula in the full version of [4, Equations (2.3)–(2.6) and (3.1)], in the convention of Proposition 2.1. Choose the word by Theorem 1.1 with v_p(\gamma,\beta)\ge\Pstar-\eta/4. For all sufficiently large D, fixed-word convergence gives \nu_p(D,\gamma,\beta)\ge\Pstar-\eta/2, proving (1.6).

For fixed D,p, let b_{p,D}(G) be the fraction of edges whose relevant cone is not a regular tree cone. A conservative radius-(p+1) neighborhood test bounds the number of such edges by a constant depending on D,p times the number of cycles of length at most 2p+3. Those fixed-length cycle counts are tight in the uniform simple regular model by McKay, Wormald, and Wysocka [17, Theorem 1]. Consequently b_{p,D}(G_{n,D+1})\to0 in graph probability and in mean as n\to\infty. Since each edge contributes a cut probability in [0,1], the exact value on the other edges gives

This proves convergence of the conditional quantum expectation to the tree value at fixed D,p; the whole random graph need not have large girth.

Dembo, Montanari, and Sen [7, Theorem 1.6(b), with the convention after Equation (1.7)] identify the optimal leading correction for the same simple regular model. After dividing their cut size by nd/2 and putting d=D+1, their theorem says that \sqrt D(\operatorname{OPT}(G_{n,D+1})-1/2) converges to \Pstar in the same successive graph-probability sense; here \sqrt{D/(D+1)}\to1. Their \Pstar is ours: pairing their ordered off-diagonal Gaussians and using sign symmetry gives (1.1) in law, while the diagonal is a mean-zero configuration-independent shift. For large D the deterministic quantity \Pstar-\nu_p(D,\gamma,\beta) is at most \eta/2. If \sqrt D\,\Delta_D>\eta, the absolute centered-optimum error or the absolute conditional-expectation error multiplied by \sqrt D must therefore exceed \eta/4. The first probability vanishes in the stated successive limit by the cited optimum theorem, and the second vanishes in the inner n limit by the preceding bound. This proves (1.7).□

Proof. Symmetry gives m(0,0)=0. The martingale identity and Itô isometry therefore give

The left-hand side equals t on [0,q]. Continuity and boundedness of a, together with continuity of X, prove the first assertion of (A.7), including both endpoints.

Since \gamma=\beta_0 on [q,1], the Cole–Hopf formula gives

Thus, by Cauchy–Schwarz and the identity just proved,

Itô’s formula applied to X_tm(t,X_t) and \phi(t,X_t) yields

The stochastic integrals have mean zero because a,m,b are bounded and X has finite moments. The elementary inequality

follows, for example, by differentiating the difference for x\ge0 and using evenness. Substitution of (A.8) consequently gives

Finally 0\le\E a(t,X_t)\le(\E a(t,X_t)^2)^{1/2}=1 on [0,q], so removing [T,q] costs at most q-T.□

Proof. We use the bounded-coefficient CLT and edge calculation in [9, Assumptions 2–3, Theorem 3.1, Proposition 2.1, and Lemmas 4.2–4.3]. The verification below has two parts: first the degree-dependent normalizers and moment bounds, then the edge value with its finite-step index. The functions a(t_r,\cdot) are bounded and Lipschitz, the drift is Lipschitz, and only finitely many r occur. Initially c_{D,0}=c_{\delta,0}, and x^r uses normalizers only through index r-2, so the normalization induction is not circular. Inductively, the Gaussian limit of the preceding messages and bounded continuity of a^2 give

Thus c_{D,r} converges, and the normalized coefficients and their Lipschitz constants are bounded uniformly for all sufficiently large D. This verifies the boundedness and normalization hypotheses used in the finite-step CLT, including when its coefficients depend on D. One may equivalently run its induction with this triangular sequence: convergent deterministic coefficients do not affect any induction step. All fixed moments are uniformly bounded. Indeed, an update sums the columns A^{r-1}_{v\to i}u^r_{v\to i} from disjoint child cavity trees. They are independent across children and centered by (A.14); the bounded multiplier is measurable before the innovation it multiplies. In this application their marginal centering also follows from seed parity, since a is even and b is odd. The moment inequality for sums of independent centered variables gives, for each integer k\ge2,

Induction over the finite number of updates proves uniform bounds of every order. They imply uniform integrability and hence the asserted moment convergence. The x and z conclusions follow from their continuous, polynomial-growth expressions in the messages.

For the energy, put

Proposition 2.1 of [9], with its degree k=D+1, is the exact identity

The second line is O(D^{-1/2}) by the fixed moment bounds. To compare node and cavity terms, split off the missing neighbor:

Its squared difference from u_{i\to j}^r is O(D^{-1}). The two Euler recursions and the spatial Lipschitz bounds therefore give, by induction over this fixed finite number of steps,

These are the comparisons in Lemma 4.2 of [9]; in particular, the use of the spatial Lipschitz bound for a is explicit.

Here is the reduction in Lemma 4.3 of that paper in our indices. Set s=\ell-1 and H=A_{v\to o}^{s-1}B_v^{s-1}B_o^s. This product depends only on the depth-(s-1) cavity labels on the v side and the depth-s cavity labels on the o side. Conditioning on these labels in (A.14) therefore gives

The variance estimate in [9, Lemmas 3.2–3.4] is uniform for the present normalizers. Indeed, if K\ge1 bounds the coefficients at the fixed horizon and \Delta_r=\E[((\tau^r_{i\to j})^2-1)^2], its proof uses only the variance recursion, independence of child cavities, and the exact normalization \E(A^r_{i\to j})^2=1 to give

Thus \Delta_s=O(D^{-1}), and the preceding conditional expectation is O(D^{-1/2}) by Cauchy–Schwarz. Replacing the two node coefficients by their cavity counterparts using (A.21) now yields

The middle equality uses the exact independence of the two cavity trees, and the last uses their coefficient normalization and one more node-to-cavity comparison. This proves the quantitative replacement without a convergence rate for c_{D,r}. All constants here may depend on \delta,L,\beta_0,T, which are fixed. Taking the degree limit and setting r=\ell-1 proves (A.19).¹ Finally, the coefficients multiplying U_r in (A.16) are measurable with respect to U_0,\ldots,U_{r-1}. Orthogonality of martingale increments and the definition of c_{\delta,r-1} give \E(Z^\delta_L)^2=\delta\sum_{r=1}^L1=L\delta.□

Proof. Take U_0 independent of W and \sqrt\delta\,U_r=W_{r\delta}-W_{(r-1)\delta} for r\ge1. Let \tau_\delta(t)=\delta\lfloor t/\delta\rfloor and use the continuous Euler interpolation

The drift is bounded and Lipschitz in space and time on [0,T]. Hence \E|\overline X^\delta_s-X^\delta_{\lfloor s/\delta\rfloor}|^2 \le C\delta. Subtracting the integral equations for X and \overline X^\delta, using Cauchy–Schwarz, and then Gronwall gives

The initial contribution is \E|\sqrt\delta U_0|^2=\delta. This proves the first bound of (A.22).

Because a is bounded and spatially Lipschitz, and \E a(t_r,X_{t_r})^2=1,

For small \delta all these expectations lie in [1/2,3/2]. The map x\mapsto x^{-1/2} is Lipschitz on that interval, proving the second bound of (A.22).

Equation (A.16) is the stochastic integral

The integrand differs from a(t,X_t) in squared mean by at most C\delta, uniformly in t\le L\delta. This follows from (A.22), the spatial and temporal Lipschitz bounds for a, and \E|X_t-X_{\tau_\delta(t)}|^2\le C\delta. Itô isometry, including the last interval of length at most \delta, gives (A.23), since m(T,X_T)=\int_0^Ta(t,X_t)\,\dd W_t. The same estimates in first mean compare the sum in (A.24) with its Riemann integral. The omitted first and last intervals have total length at most 3\delta and bounded integrands. This proves the final estimate.□

Proof. The proof first treats characteristic vectors, then arbitrary pointwise maps, and finally the occupation bound needed for energy. We use induction on formula depth. All intermediate moments are uniformly bounded: normalized centered sums satisfy the moment bound used in Lemma A.2, and polynomial-growth functions preserve the existence of sufficiently high uniform moments.

Suppose the real centered child columns f_{D,1},\ldots,f_{D,k} already converge in norm to f_1,\ldots,f_k in the common child Hilbert space. Fix t\in\R^k and put F_D=\sum_jt_jf_{D,j}, F=\sum_jt_jf_j. The limits are centered because the embeddings preserve the vacuum and the convergence is in norm. A joint characteristic vector for the parent sums is the symmetric tensor power of

Taylor expansion in L^2, justified by the uniform fourth moment, and centering give

The j-particle component of the embedded tensor power is

These are the components of e^{\ii q(F)}\vac. The expected occupation of the left-hand vector is D\norm{\zeta_D}^2=O(1), so its occupation tails are uniformly small. Thus convergence of each fixed grade implies norm convergence of the complete vector. A root with M=D+1 children has the same proof: use M factors and the normalization M^{-1/2}, with the same convergent child columns. Alternatively, retaining D^{-1/2} at that root gives the same limit because M/D\to1. The occupation bound in either convention is M\norm{\zeta_D}^2=O(1).

Finite linear combinations of these exponentials, multiplied by bounded measurable functions of the independent root seed, are dense in the limiting L^2 space. Indeed the root seed and aggregate fields have a product law. Products of dense families in the two factors are dense, and exponentials are dense in the field factor: an orthogonal L^2 vector would give a finite signed measure with zero Fourier transform and would therefore vanish. There is a useful precision concerning measurable seed factors. Let S denote the root seed, Y_D the finite list of aggregate fields, and Y its limit. The law of S is unchanged with D and is independent of Y_D and Y. For P(s,y)=\sum_jg_j(s)e^{\ii t_j\cdot y} with bounded measurable g_j, and a bounded target F(s,y) continuous in y for every fixed s, the function |F(s,y)-P(s,y)|^2 is bounded continuous in y for each fixed s. Weak convergence of Y_D therefore gives convergence of its field integral pointwise in s. The bound is uniform in s, so dominated convergence in the unchanged root-seed law proves

Thus measurability of g_j does not require a discontinuous test function in an appeal to joint weak convergence. For a polynomial-growth target, first multiply by smooth cutoffs outside large boxes in the seed and field variables. The cutoffs preserve continuity in the field variables, and uniform higher moments make the discarded squared norms uniformly small. This proves the induction step. It also proves continuity in deterministic coefficients, by applying the same argument to a convergent sequence of them.

For completeness, the occupation bound is not a consequence of the CLT alone. At a parent, let P_j average over the jth child, and write its output as H_D(s,S_{D,1},\ldots,S_{D,k}), where S_{D,l}=D^{-1/2}\sum_j f_{D,l}^{(j)}. The occupation operator in the symmetric embedding is N_D=\sum_{j=1}^D(I-P_j). With an independent replacement of child j, conditional variance and the mean-value theorem give

Here the derivative of H_D has polynomial growth and the sums, the replaced child columns, and their products have the uniform moments established above; Hölder’s inequality therefore gives a constant independent of j,D. Summing proves \braket{H_D}{N_DH_D}\le C. If needed, the same argument with distinct child replacements and mixed derivatives bounds the factorial moments of N_D of every fixed order.

The energy identity is now Proposition 2.2 applied to these classical vectors. More explicitly, with M=D+1 root children, the projection onto an excitation in a distinguished child has norm at most C/\sqrt M by symmetry and the occupation estimate. The term with excitations across the distinguished edge on both sides is therefore O(M^{-1}). The two terms with only one crossing select a single child; after multiplication by \sqrt M they converge to \braket{f}{\cL f} and its conjugate. The term with neither crossing is zero by centering. The factor M/(2\sqrt D) in (A.25) thus gives \re\braket{f}{\cL f}, since M/D\to1.□

Proof. Use (A.27) with error \varepsilon/3. Its underlying formula z^\delta has odd transmitted columns and jointly odd smooth local maps with polynomial-growth derivatives, as checked after Lemma A.4. It therefore meets the hypotheses of Lemma 3.2. Apply that lemma and then Lemma 3.4 with the choices in the proof of Proposition 3.5: the first output error is less than \varepsilon/12 in norm, the second less than \varepsilon/6, and one positive-probability helper is reserved per propagation depth with total probability below \eta_H. The two further energy losses are less than \varepsilon/6 and \varepsilon/3, respectively, by (3.9). Their sum with the initial loss is less than \varepsilon. Those two shared lemmas also give the boundedness, trigonometric form, exclusions, and parity asserted here. The entire formula and all helper probabilities are fixed after the degree limit of the finite calculation.□