The Binary Totient Series

Will Cook

Long paper

The Binary Totient Series

Erdős #249 By Will Cook July 2026; revised 1 October 2026 Authorship, AI use and citation

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Summary. The dyadic totient sections through level $e \geq 1$ have an explicit rational basis of size $2^{e} + 1$ . Truncating a scaled tail difference produces an integer whose residue certifies nonintegrality when its distances from both endpoints of the residue interval exceed an explicit error bound. The record develops the resulting finite denominator exclusions, equivalent residue criteria, and sufficient exponential-sum estimates. Rational comparison sequences identify the limitations of particular size, parity, rank, and finite-prefix arguments. The short paper supplies the all-base basis theorem; here each conditional result carries the arithmetic hypothesis still needed to apply it. Those hypotheses are not established in the form required to prove irrationality of S=∑ $φ (n) 2^{- n}$ .

This paper covers a claim-bounded problem-specific reasoning surface for Erdős #249: selected checked premises, finite certificates, closed routes, surviving obligations, and their exact evidence bands.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #249, which remains open.

Start here, by the editorial route

In this paper

Parameter ranges and quantifiers

The distinction between a finite calculation and an unbounded family is in the quantifiers. A statement such as $\forall a \geq 8$ is uniform in $a$ , whereas a list of computed scales is finite and a condition $\forall N_{0} \exists N \geq N_{0}$ asks for witnesses arbitrarily far out. The table uses these distinctions even when a uniform statement is conditional on an unproved hypothesis. A bound on a secondary variable, such as $J + (a + 6) < 2 \cdot 2^{a}$ in the positivity row, does not bound $a$ .

Some identities have no distinguished scale parameter. They are listed with the general identities. An equivalence can hold for all parameters without establishing either of the equivalent arithmetic assertions.

The LCM formulation of Problem 249 requires arbitrarily large scales: irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply

Irrational (\sum_{n \geq 1} φ (n) / 2^{n}) ⟺ \forall a_{0} \exists a \geq a_{0}, Ω_{a} \notin Z,

a cofinal statement in the exponent $a$ indexing $H = H (2^{a})$ . A finite calculation alone does not establish it; a further theorem could propagate finite information to arbitrarily large scales.

The table groups statements by their parameter ranges, not by proof status. In particular, some assertions with cofinal quantifiers are already proved, such as the prime constructions and rational comparison sequences. Others are implications whose quantified hypotheses remain open. The horizontal rule separates the cofinal group for ease of reference; it does not mark a boundary between proved and unproved mathematics. The descriptions identify this distinction in that group. The prefixes summarise the quantifiers; complete hypotheses are given in the cited statements. A definition records an object or condition; it does not by itself prove the associated soundness or existence assertion.

In the periodic-weight rows, write $w : N \to Z$ and $c_{w} (n) = \sum_{d ∣ n} w (d)$ for $n \geq 1$ . Positivity and nonvanishing refer to these divisor coefficients, not to the weight $w$ itself. Here “nonzero arbitrarily far out” means that for every $N$ there is $n > N$ with $c_{w} (n) \neq 0$ ; it does not mean nonvanishing on every arithmetic progression. The nonprincipal character $χ_{4}$ modulo four is $0$ on even integers and $(- 1)^{(n - 1) / 2}$ on odd integers.

Results and unproved conditions, grouped by their quantifiers. The dividing rule separates the cofinal group, not proof status. Complete hypotheses are given in the cited statements.
Result	Source	Quantifier prefix	Representation
Result	Source	Quantifier prefix	Representation


scaled totient tail (definition)	`totientTail`	$\forall N$	binary-weighted series
integer prefix and real tail	`two_pow_mul_totient_series_eq`	$\forall N$	binary-weighted series
$D (h, N, L)$ (definition)	`windowDiscrepancy`	$\forall h \forall N \forall L$	finite binary sums
$C (h, N, L)$ (definition)	`certifiedKill`	$\forall h \forall N \forall L$	finite binary sums
necessary depth inequality	`certifiedKill_depth_floor`	$\forall h \forall N \forall L$ , given $C$ $h N L$	finite binary sums
residue test soundness	`tail_diff_notMem_int_of_certifiedKill`	$\forall h \forall N \forall L$ , given $C$	finite binary sums
completeness of the residue test	`exists_certifiedKill_iff_tail_diff_notMem_int`	$\forall h \forall N$	finite binary sums
integrality from denominator divisibility	`tail_diff_int_of_den_dvd`	$\forall N \forall h \forall r : Q$ , given $S = r$ , $r . den ∣ 2^{N} (2^{h} - 1)$	binary-weighted series
tail-period law	`eventual_period_of_not_irrational`	$\neg Irrational S \to \exists h > 0 \exists N_{0} \forall N \geq N_{0}$	binary-weighted series
least common multiple $H (t)$	`periodLcm`	$\forall t$	LCM periods
a nondivisor below $2 t$ is a prime power	`eq_prime_pow_of_not_dvd_periodLcm`	$\forall t \forall j (0 < j < 2 t)$ , given $j ∤ H t$	LCM offsets
totient factorisation at divisor offsets	`totient_periodLcm_ray_split`	$j ∣ H (t)$ , with the prime-divisor hypotheses of Proposition 6.51	totient factorisation
integrality on the LCM grid under rationality	`rational_totient_series_forces_lcm_cone_flatness`	$\neg Irrational S \to \exists t_{1} \forall t \geq t_{1} \forall q > 0 \forall m$	LCM grids
second-difference residue test	`second_diff_notMem_int_of_certifiedRank2Kill`	$\forall h \forall N \forall L$ , given $second-difference residue condition$	finite binary sums
a nonintegral pair on a finite grid	`exists_nonintegral_pair_of_coneNonflatCert`	finite $Q \neq \emptyset$ ; all size and residue hypotheses of Theorem 6.56	finite LCM grids
finite exclusion of integer carries	`tail_diff_notMem_int_of_survivorKill`	$\forall h \forall N \forall K$ , given $finite carry exclusion$	integer recurrences
Farey neighbour denominator law	`farey_gap`	$a, b, c, d, r, s \in Z$ ; $b, d > 0$ , $b c - a d = 1$ , $a s < b r$ , $d r < c s$	rational intervals
rational-approximation criterion	`irrational_of_den_mul_abs_sub_tendsto_zero`	$u_{j} \in Q$ , $x \in R$ ; eventually $u_{j} \neq x$ , and $den (u_{j}) \| u_{j} - x \| \to 0$	general criterion
near-integer irrationality criterion	`irrational_of_int_mul_near_int`	$\forall ξ : R$ , $\forall q > 0 \exists m z 0 < \| m ξ - z \| < 1 / q$	general criterion
independence from a nonsingular evaluation matrix	`linearIndependent_of_separatedMinorCertificate`	a finite family with a square evaluation matrix of nonzero determinant	general criterion
dyadic totient-kernel rank $= 2^{e} + 1$	`linearIndependent_canonicalTotientKernelFamily`	$\forall e \geq 1$	dyadic sections
rationality forces unbounded carry-kernel rank	`not_irrational_totientSeries_implies_unbounded_carryRank_unconditional`	$\neg Irrational S \to \exists v > 0 \exists u \forall e$	carry sections
no fixed $D$ clears all $A (n) / n$ for $A = φ * μ$	`no_fixed_integral_primitive_euler_index`	$\forall D > 0$	Möbius sums
Lambert-series value identities	`tsum_totient_div_pow_two_eq_pnat_half_pow`	value identities, $L (μ) = \frac{1}{2}$ , $L (φ) = 2$ , $L (1) =$ Erdős–Borwein	Möbius sums
prime-power unit-gap criterion	`reducedDenominatorUnitGapCert_prime_pow_iff`	$\forall p prime, e > 0, N, K \geq 0$	rational intervals
positive rational-difference lower bound	`positive_rational_difference_lower_bound`	$\forall whole pfx : Q, pfx < whole$	general criterion
$L_{2} (μ) = S - \frac{1}{2}$	`tsum_totient_half_pow_eq_half_add_moebius_sq`	unconditional	Möbius sums
weighted squared-denominator identity	`tsum_lambert_linear_weight_sq_pure`	$\forall w : N \to R (\| w (d) \| \leq d) \forall r \in [0, 1)$	Möbius sums
$L_{2} (μ) = S - \frac{1}{2}$	Proposition 5.14	unconditional	Möbius sums
$L_{2} (1) = ζ_{q} (2) - ζ_{q} (1)$ identity	`tsum_one_div_mersenne_sq_eq_sigma_sub_tau_series`	unconditional	Möbius sums
$L_{2} (φ)$ = gcd moment	`tsum_totient_div_mersenne_sq_eq_gcd_moment_series`	unconditional	gcd moments
comparison of the weights	Propositions 5.14–5.18	unconditional	Möbius sums
gcd-divisibility factorizes	`tsum_pos_pair_both_dvd_half_eq_inv_mersenne_sq`	$\forall d : N, d > 0$	geometric weights
sum over positive coprime pairs equals $1$	`tsum_pos_coprime_inv_mersenne_eq_one`	unconditional (sum over all coprime pairs)	geometric weights
Stern–Brocot recursion with stopping	`cylinderMass_split`	positive integers $a, b$	Stern–Brocot tree
positive numerator coefficients	`mobiusNumeratorPolynomial_eq_gcdWord`	$\forall r (Squarefree r) \forall k < r$	divisor and cyclotomic sums
evaluation at $2$ gives the numerator	`mobiusNumeratorPolynomial_eval_two`	$\forall r (Squarefree r)$	divisor and cyclotomic sums
reduction to the radical of $H$	`scaledMobiusShadow_eq_radicalBase`	$\forall H > 0 \forall r > 0$	Möbius sums
numerator congruence modulo $Φ_{m}$	`cyclotomic_dvd_mobiusNumeratorPolynomial_sub`	$\forall r (Squarefree r) \forall m (m ∣ r)$	divisor and cyclotomic sums
gcd of the numerator and $Φ_{r} (2)$	`mobiusNumerator_gcd_cyclotomicValue`	$\forall r (Squarefree r)$	divisor and cyclotomic sums
Mersenne factors retained in the denominator	`upperHalfChannel_product_dvd_den_of_scale_primeFactors_le`	$t \geq 5$ , squarefree $r$ ; finite $P$ of primes dividing $r$ in $(t / 2, t]$ ; prime factors of $r$ and the multiplier are at most $t$	divisor and cyclotomic sums
denominator lower bound / exact value	`lcmHeight_scaledMobiusShadow_den_exact`	$\forall t$ (exact-value); $t \geq 5$ (lower bound)	Möbius sums
numerator congruence after adjoining a prime	`cyclotomic_dvd_primeJump_new_fibre`	$r \geq 1$ , $p$ prime, $p ∤ r$ , $m ∣ r$	divisor and cyclotomic sums
squared-Mersenne series: partial sum plus tail	`mobius_square_series_eq_partial_add_tail`	$\forall D : N$	Möbius sums
cancellation of the affine divisor terms	`joint35_oldChannel_zero`	$\forall d > 0 (d ∣ H)$ ; general: finite affine annihilator	LCM diagonal
new-divisor counting identity	`supportCoeff_mul_eq_add_defect`	$\forall a \in A (a > 0) \forall x > 0$	divisor counts
bound on zero blocks in divisor counts	`supportCoeffZeroWindow_length_le_eps_logb_add`	$\forall ε > 0 \exists B$ , given rationality of the base-2 support series	divisor counts
equivalent shift condition for the target $1 / 2$ (Problem 257)	`shiftWindowZero_iff_half_mem_mersenneAchievementSet`	iff, no extra parameter	Mersenne subsums
totient multiplication with common prime factors	`totient_mul_eq_overlapFactor_mul`	$\forall x > 0$ ( $j$ fixed)	totient identities
totient relative-Euler-product split	`totient_mul_eq_totient_mul_relativeEulerProduct`	$\forall j x > 0$	totient identities
formula for $φ (2 H (t) + j) - φ (H (t) + j)$	`lcmRayArithmeticLetter_eq_deltaTotient`	$\forall t \forall j$	Möbius sums
the finite diagonal sum equals $D (H (t), H (t), L)$	`lcmDiagonalArithmeticWord_eq_windowDiscrepancy`	$\forall t \forall L$	Möbius sums
#249 $⟺$ cofinal actual-orbit nonintegrality	`irrational_totientSeries_iff_actualLcmOrbitNonintegralitySupply`	iff	Möbius sums
the LCM tail difference is an integer translate of a multiple of $S$	`actualLcmTailOrbit_eq_scaled_totientSeries_sub_prefix`	$\forall a$	Möbius sums
explicit finite-block approximation	`abs_actualLcmTailOrbit_sub_rawApprox_lt`	$\forall a \forall q$	Möbius sums
an equivalent residue-interval condition	`exists_certifiedKill_iff_guardCylinderWitness`	$\forall h N, iff$	integer recurrences
sign of the three-term totient difference under the stated middle-value bound	`fixedRankSecondDifference`	$\forall H j$ , given MiddleRankTotientExtremal	totient differences
exact value on a specified dyadic family	`fixedRankSecondDifference_dyadic_fixture_exact`	$\forall n$ (closed-form)	totient differences
the rational parity comparison (definition)	`parityCoboundaryWeight`	$\forall n$ (definition)	binary-weighted series
$R_{N}$ as a sum over positive divisor indices	`totientTail_eq_tsum_positiveForeignResidueKernel`	$\forall N$	divisor and cyclotomic sums
$R_{N}$ as a shifted Möbius sum	`totientTail_eq_tsum_shiftedMobiusPulse`	$\forall N$	divisor and cyclotomic sums
absolute convergence and regrouping of Lambert sums	`tsum_lambert_pair_regroup_if`	$\forall w v : N \to R (\| w (d) \| \leq d, \| v (m) \| \leq m)$	divisor and cyclotomic sums
carry divisibility and integral tail differences	`carryShift_dvd_iff_tailDiff_mem_int`	$\forall N k$ , given the carry recurrence and limit	integer recurrences
rationality $\Rightarrow$ periodicity & unbounded rank	`not_irrational_totientSeries_implies_mod_period_and_unbounded_rank`	$\neg Irrational S \to \exists v u \forall e$	integer recurrences
the stated reconstruction cost is at least $3$	`three_le_totientAdjugateTailCost`	$\forall finite w : ι \to Q x : ι \to N$ , given $\sum w_{i} φ (x_{i}) = 1$	weighted tail bounds
directed residue interval (definition)	`directedCertifiedKill`	$h, N, L \geq 0$ ; soundness and the existential-depth equivalence are proved in Proposition 6.69	integer recurrences
rationality and an integer carry with vanishing boundary term	`binaryCoeffSeries_rational_iff_exists_temperedBinaryOrbit`	$\forall c : N \to N (c (n) \leq n)$ , iff	binary-weighted series
doubling-orbit rigidity	`doublingOrbit_eq_zero_of_tempered`	$\forall d : N \to R$ , doubling and $d (n) / 2^{n} \to 0$ imply $d \equiv 0$	integer recurrences
number of labels needed for exact tail recovery	`balancedPulse_no_autonomous_decoder`	$\forall m \forall State$ , collapsing-state $\to$ no decoder	binary-weighted series
endpoint residues do not determine the initial carry	`affineBinaryOrbit_mod_twoPow_eq`	$\forall a : N \to Z \forall u_{0} v_{0} \forall L$	binary-weighted series
integer-coefficient block-certificate dichotomy	`int_coeff_series_irrational_or_bpow_mul_eq_intCast_of_full_block_certificates`	$\forall b \geq 2 \forall c \forall q$ , given full-block certificate at $q$	binary-weighted series
divisibility under iterated forward differences	`dvd_iteratedForwardDifference`	$d ∣ f (n)$ for every $n$ ; every finite list of shifts	integer sequences
totient series as a coprime-pair sum	`tsum_coprime_pair_pow_eq_tsum_totient_mul_pow`	$\forall n$ ; $\forall r \in [0, 1)$	geometric weights
factorisation through a rational-valued linear functional	`linearDescender_eq_smul_eval`	rational vector spaces $V, W$ ; surjective linear $ev : V \to Q$ , linear $L : V \to W$ , $\ker (ev) \subseteq \ker L$	divisibility and linear algebra
divisibility of a reduced denominator	`divisor_dvd_divInt_den`	$\forall D > 0 \forall m ∣ D \forall a$	divisibility and linear algebra
the complementary denominator divides the integer multiplier	`scalarLocalization_complement_dvd`	$\forall x : Q \forall H (H ∣ x . den) \forall c ((c x) . den ∣ H)$	divisibility and linear algebra
largest-exponent nonvanishing criterion	`scaled_dyadic_sum_ne_zero`	$\exists m \in s$ (strictly maximal, odd coeff), $\forall i \neq m e (i) < e (m)$	divisibility and linear algebra
column rescaling and determinant nonvanishing	`det_residualMonomialMatrix`	$\forall d \forall e : Fin d \to N \forall z : Fin d \to C$ ( $z$ nonzero)	complex matrices
determinant $1$ for the divisor-incidence matrix	`incidenceMobius_det_eq_one`	$\forall N : N$	Möbius sums
error bound from convergence	`controlledForeignProjection_of_tail_limit`	$2 H \leq D$ ; convergence is supplied for $H > 0$ in Section 6.6.8	Möbius sums
carries realising fixed-precision unit residues	`fixedPrecisionTropicalNoGo`	$\forall u > 0$ (fixed precision), $\forall$ finite carry word	divisibility and linear algebra
vanishing of iterated prime dilation differences	`iteratedDilationDifference_eq_zero_of_cardFactors_le`	distinct primes $p_{1}, \dots, p_{r}$ , $r > K$ , $gcd (n, \prod p_{i}) = 1$ ; $Ω (a) \leq K$ on $A$	divisor counts
$φ (n) > \frac{3}{4} n$ for the stated rough integers	`three_quarters_mul_lt_totient_of_rough_lt_two_pow`	$\forall a \geq 8 \forall n > 0$ ( $n$ is $t$ -rough, $t = 2^{a}$ , $n < 2^{2 \cdot 2^{a}}$ )	Möbius sums
positivity of short-window totient differences	`lcmRayArithmeticLetter_pos_of_lt_two_mul`	$\forall a \geq 8 \forall j (0 < j < 2 \cdot 2^{a})$	Möbius sums
the negative representative of the true endpoint	`actualLcm_trueEndpointSurvivor_neg`	$\forall a \geq 8 \forall J K$ (size bound)	Möbius sums
integrality forces upper-endpoint residue	`actualLcm_integral_forces_topEdgeResidue`	$\forall a \geq 8 \forall J K$ (size condition, $2 H + J + K + 2 < 2^{K}$ )	Möbius sums
terminal dyadic staircase impossible under the size condition	`not_actualLcmTerminalDyadicStaircase_of_room`	$\forall a \geq 8 \forall J K m$ (size condition, modulus-wide)	Möbius sums
penultimate difference in a partial divisibility pattern	`puncturedDyadicStaircase_penultimate_eq_half`	$\forall a \geq 8 \forall J K m$ (hypotheses of Proposition 6.86)	Möbius sums
upper-endpoint residue-gap $\Rightarrow$ nonintegral	`actualLcmTailDiff_notMem_int_of_topEdgeResidueGap`	$\forall a \geq 8 \forall J K m$ (size condition, one-sided)	Möbius sums
relation between the centred residue and the actual carry	`two_mul_actualOddHalfCenteredLift_eq_terminal_sub_trueCarry`	$\forall a \geq 8 \forall q$ (size condition, fit bound $2 (H + q + 2) \leq 4^{q}$ )	Möbius sums
divisibility of the three-term totient difference	`two_mul_totient_dvd_fixedRankSecondDifference`	$\forall a \geq 4 \forall j > 0 (j^{2} \leq 2^{a})$	totient differences
equality of two finite rational approximations	`renormalizedResidueAgreement_of_twice_le`	$H > 0$ , $D \geq 2 H$ ; no separation hypothesis for this equality	Möbius sums
a Mersenne divisor need not divide a tail difference	`idCoeff_mersenneModulus_does_not_annihilate_tailShift`	$\forall K, q (K > 0, K < q, q ∣ 2^{K} - 1)$	Möbius sums
finite numerator separation implies nonintegrality	`scaleFullTarget_miss_of_lambert_projected_num_gap`	$\forall H, D$ fixed, given uniform numerator-gap bound	Möbius sums
periodic-weight Lambert dichotomy	`irrational_or_bpow_mul_eq_intCast_intWeightedErdosSeries_periodic`	$\forall b \geq 2 \forall m > 0 \forall w$ ( $w (n + m) = w (n)$ )	Möbius sums
periodic weight with nonnegative divisor coefficients	`irrational_intWeightedErdosSeries_periodic_of_coeff_nonneg_of_frequently_ne_zero`	$b \geq 2$ , $m > 0$ , $w (n + m) = w (n)$ ; $c_{w} (n) \geq 0$ and $c_{w} (n) \neq 0$ arbitrarily far out	Möbius sums
zeros on one progression for the character modulo four	`intWeightedCoeff_periodFourSignWeight_eq_zero_of_mod_four_eq_three`	$c_{χ_{4}} (n) = 0$ for every $n \equiv 3 (\mod 4)$ ; the weight is fixed	Möbius sums
positivity of the translated LCM tail difference	`actualLcmTailDiff_shift_pos`	$\forall a \geq 8 \forall J (J + (a + 6) < 2 \cdot 2^{a})$ , no rationality hypothesis	Möbius sums

Farey window $K = 240$ denominator bound	`gap_check_window_1_240_le_79639646646701375323355774875831053`	$\forall q : N, 0 < q \leq$ $79,639,646,646,701,$ $375,323,355,774,875,831,053$	rational intervals
denominator lower bound $S \neq p / q$ , $q \leq Q_{0}$	`tsum_totient_div_pow_two_ne_ratCast_of_den_le_79639646646701375323355774875831053`	$\forall p : Q, p . den \leq Q_{0}$	rational intervals
finite LCM certificates covering thresholds $a_{0} \leq 6$	`powerTwoActualLcmShortArithmeticKillSupply_through_six`	$\forall a_{0} \leq 6 \exists a L$	Möbius sums
contiguous diagonal certificate band through $t \leq 82$	`ErdosProblems.Skip.LadderT67.exists_diagonalKill_le_82`	$\forall t \leq 82 \exists L$ ; no $t = 83$ certificate claimed	LCM diagonal

residue certificates for $1 \leq h \leq 8$ , $N = 12$ , $L = 16$	`certifiedKill_all_small`	$\forall h \in [1, 8]$ , $C$ $h$ 12 16 (direct evaluation)	finite binary sums
denominator exclusions for $1 \leq h \leq 16$ , binary exponent $14$	`totient_series_ne_rat_of_den_dvd_pow_two_mul_mersenne_upto_sixteen`	$\forall h \in [1, 16]$ , denominator exponent 14 (direct evaluation)	finite binary sums
the second-difference test uses more depth at $(h, N) = (1, 8)$	`totient_tail_rank_two_kill_sound_but_not_shallower_cell`	first-difference certificate at $L = 8$ ; no second-difference certificate for $1 \leq L \leq 8$ , and one at $L = 9$	finite binary sums
listed diagonal certificates with $t \leq 64$	`certifiedKill_diagonal_all_imported`	$t \in {1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, \dots}$ (28 explicit values)	LCM diagonal
LCM certificates at $a = 4$ and $a = 6$	`lcmDiagonalArithmeticKill_two_pow_four`	$a = 4$ , $a = 6$ (concrete, formally checked)	Möbius sums
square-CRT coprime-factor examples: zero / nonzero differences	`squareCRT_clean_block_can_vanish`	witness $n = 52$ (vanishes); witness $n = 27$ (nonzero)	totient factorisation


certificate criterion (proved implication; unproved premise)	`irrational_totient_series_of_certificate_supply`	$\forall h \geq 1 \forall N_{0} \geq 0 \exists N \geq N_{0} \exists L, C h N L$	finite binary sums
period-multiple criterion (equivalent; unproved condition)	`irrational_totient_series_of_multiple_certificate_supply`	$\forall h_{0} > 0 \forall N_{0} \exists m > 0 \exists N \geq N_{0} \exists L$	multiples of periods
LCM-diagonal criterion (unproved condition)	`irrational_totient_series_of_lcm_diagonal_certificate_supply`	$\forall t_{0} \exists t \geq t_{0} \exists L$	LCM diagonal
LCM-grid certificate criterion (unproved premise)	`irrational_totient_series_of_lcm_cone_window_kill_supply`	$\forall t_{0} \exists t \geq t_{0} \exists q > 0 \exists m \exists L$	LCM grids
LCM nonintegrality criterion (unproved condition)	`irrational_totient_series_of_lcm_diagonal_nonintegrality_supply`	$\forall t_{0} \exists t \geq t_{0}$ , $R (2 H_{t}) - R (H_{t}) \notin Z$	real tail differences
finite-grid residue criterion (unproved premise)	`irrational_totient_series_of_lcm_cone_nonflat_supply`	for every $t_{0}$ , data $t \geq t_{0}, L, Q$ satisfying Theorem 6.57	finite LCM grids
first-harmonic norm criterion (unproved premise)	`irrational_totient_series_of_first_harmonic_norm_gap`	$\forall h > 0 \forall X_{0} \exists X \geq max (X_{0}, 1) \exists L$ , size condition $16 (2 X + h + L + 2) \leq 2^{L}$	binary-weighted series
four-tail residue criterion (unproved premise)	`irrational_totient_series_of_primeJumpSharpKill_supply`	$\forall t_{0} \exists t \geq t_{0} \exists p, L > 0$	binary-weighted series
short LCM-window criterion (unproved premise)	`PowerTwoActualLcmShortArithmeticKillSupply`	$\forall a_{0} \exists a L, a_{0} \leq a \land L < 2 \cdot 2^{a}$	Möbius sums
LCM-tail separation criterion (unproved premise)	`irrational_totientSeries_of_actualLcmOrbitSeparationSupply`	cofinal $1 / 32 +$ error separation at prescribed odd depths	Möbius sums
five endpoint criteria (unproved premises)	`irrational_totientSeries_of_flexibleActualTerminalDominanceSupply`	one of the five quantified conditions in Proposition 9.14	Möbius sums
proved rational counterexample to bound, parity and aperiodicity	`exists_totientParity_arbitrarilyManySeparatedCarry_rational_countermodel`	$\exists c : N \to N$ , $c \leq 6$ , $c \leq n$ , $c \equiv φ (\mod 2)$ , arbitrarily-separated blocks	binary-weighted series
proved mod-4 prime construction (CRT and Dirichlet)	`exists_prime_deltaTotient_four_mul_mod_four_two`	$\forall h > 0 \forall B \exists prime p > B$	Möbius sums
sufficient mod-4 condition (unproved premise)	`irrational_totientSeries_of_cofinal_modFourPulseSurvivorKill`	the displayed mod-four residue condition at arbitrarily large scales	Möbius sums
proved two-adic prime construction	`exists_prime_totient_twoAdic_pulse_divisors`	$\forall K \geq 2 \forall H > K \forall B \exists cofinally many primes p > B$	divisibility and linear algebra
proved implication from eventual integrality	`eventual_integral_tailDiff_has_cofinal_twoAdic_half_pulse`	$\forall K \geq 2$ , given the stated two-adic prime construction and eventual integrality	divisibility and linear algebra

Finite extension and cofinal estimates

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