Claims · 68 dated claims on Franel on a digit design, each with its tag and its witness; newest 2026-09-23.
Proved At frequency 1 the exponential sum of the denominator-restricted Farey set IS the design's Mertens meter: with S_F the whole numbers whose every digit lies in a digit set, sum of e(r) over r in {a/b reduced, b in S_F, b <= Q, 1 <= a <= b} equals M_F(Q) = sum of mu(b) over b in S_F, b <= Q, since sum over a mod b coprime to b of e(a/b) = mu(b) by Mobius inversion against the complete sums; every denominator in S_F up to Q = 10^5 has its literal sum of phi(b) roots of unity equal to mu(b), worst deviation 1.09e-11 at b = 86293 (base 3 digits {0,1}) and 1.36e-12 at b = 7247 (base 10 without 9), 0 wrong roundings. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelladder_check)
Proved At frequency m the same sum is sum over d dividing m of d M_F(Q/d; d), where M_F(x; d) = sum of mu(c) over c <= x with dc in S_F is the Mertens function of the DILATED design d^{-1} S_F; the classical divisor-shifted Mertens sums are the case S_F = Z, where every dilate is Z and all of them collapse to M, while for a digit design d^{-1} S_F is not S_F, is not a digit design and carries no digit test, so each d > 1 brings a new function; checked at m = 1, 2, 3, 4, 5, 6, 12 on base 3 {0,1} at Q = 2187, base 10 without 9 at Q = 1000 and the full-set control at Q = 300, exact integer against literal sum at every cell. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_denominator)
Proved The digit-restricted Franel identity, Fourier form: sum over k nonzero of abs(S_F(k,Q))^2 / k^2 = (pi^2/3) G_F(Q) with G_F(Q) = sum over d, e of (gcd(d,e)^2/(d e)) M_F(Q/d; d) M_F(Q/e; e), a finite sum of exact rationals; the kernel is the Smith gcd matrix that already carries the moire correlation law and the Gaussian identity, so digit restriction moves the entries and never the kernel; checked against the literal Fourier side truncated at abs(k) <= 200000 with the printed tail bound 2 m^2/K, gap inside bound at base 3 Q = 81 and 243, base 10 Q = 40 and the control Q = 40. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelkernel_sum and fourier_side)
Proved The digit-restricted Franel identity, rank form: if the node set has top node 1 and mean value sum of rho_r = (m_F(Q)+1)/2, then G_F(Q) - 1 = 12 m_F(Q) sum_j delta_j^2 as exact rationals, with m_F(Q) = sum of phi(b) over b in S_F, b <= Q and delta_j = rho_j - j/m_F(Q); closure under r -> 1 - r away from the node 1 is one sufficient condition for that mean value, holding for every denominator-restricted set and failing for every proper strict set; the proof is Parseval plus piecewise integration of (A(v) - m v)^2; True at base 3 Q = 81, 243, base 10 Q = 40 and the control Q = 40, which regenerates Edwards section 12.2. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelfarey_delta_square)
Proved The denominator lane's CONJECTURED shape implies the square-root ceiling for the design's Mertens meter: dropping every term but k = 1 and k = -1 from a sum of nonnegative terms gives 2 M_F(Q)^2 <= (pi^2/3) G_F(Q), which by the rank form is 4 pi^2 m_F(Q) sum_j delta_j^2 + pi^2/3; with A_F(Q) << Q^alpha (the block count) and m_F(Q) <= Q A_F(Q) << Q^(1+alpha), the conjecture sum_j delta_j^2 = O(Q^(-1+eps)) forces abs(M_F(Q)) = O(Q^(alpha/2+eps)), the constant absorbed and no unproved input entering; the measured exponent is -0.959 and -0.899, not -1, so only the conjecture yields the ceiling. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelscan_backward)
Proved The strict set's frequency-1 sum has an exact divisor form: it equals sum over b in S_F, b <= Q of sum over d dividing b of mu(d) times sum of e(a/(b/d)) over a <= b/d with da in S_F, by Mobius inversion of the coprimality condition followed by a -> da; the inner sum is a digit-restricted exponential sum over an arithmetic progression, the Type II object with no bound on the tree, so the identity is exact and inert; literal summation against the divisor route agrees to 6.28e-15 over the 64 denominators of base 3 {0,1} below 729 and to 1.95e-14 over the 162 of base 10 without 9 below 200. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelstrict_ramanujan_divisor)
Verified Literal enumeration of the strict digit-restricted Farey set reaches the counts the sieve prints without enumerating a fraction: 278, 4286, 67561, 1080458 at Q = 3^5, 3^7, 3^9, 3^11 on base 3 {0,1} and 1830, 147096, 11890654 at Q = 10^2, 10^3, 10^4 on base 10 without 9, 7 rungs and no disagreement. (witness: lab/py/restricted-franelstrict_literal against lab/rs/farey-discrepancydesign)
Proved For a design carrying the digit 0 the dilate d^(-1) S_F = {c : dc in S_F} is a regular language recognised least-significant-digit-first by a deterministic automaton whose d states are the carries of long multiplication by d: reading digit e from carry r writes the output digit (de + r) mod base, which must lie in F, and moves to carry floor((de + r)/base), which stays below d by induction, and after level digits dc is the level output digits with the terminal carry r_level written above them, so acceptance is exactly that r_level has all its digits in F; the accepting set is Acc_d = {0} union (S_F intersect [1, d)), of size A_F(d-1) + 1. The hypothesis 0 in F is load-bearing and not cosmetic: without it the run tests all level PADDED output digits, a leading output digit 0 is not a digit of dc, and the automaton recognises the padded set of the Mobius page instead, reading 8 at base 3 with F = {1,2}, d = 1 and level = 3 where the true count is 14. Regularity of the dilate itself survives without the hypothesis; the count identity does not. So the dilated Mertens sums of the restricted Franel identity run over regular sets, not over digit designs. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franeldilate_matrix, verb_converse0 mismatches of 840)
Proved The dilate's transfer matrix T_d(r, r') = #{e < base : (de + r) mod base in F, floor((de + r)/base) = r'} counts it, #{c < base^level : dc in S_F} = e_0 T_d^level 1_(Acc_d) for a design carrying 0, and EVERY column of T_d sums to exactly #F, in every base, at every digit set and every d, with no hypothesis at all: the pairs (e, r) in [0,base) x [0,d) are in bijection with v = de + r in [0, dq) by the division algorithm, the column at r' counts the v with v - base r' in F, and the window [base r', base r' + base) lies inside [0, dq) for every r' < d, so exactly #F of them qualify. Hence the all-ones vector is a positive left eigenvector and the spectral radius of T_d is #F for every d: the dilate carries the design's own mass exponent as its Perron root. The rows sum to g times #(F intersect (r + gZ)) with g = gcd(d, base), so they equal #F whenever gcd(d, base) = 1, giving #{c < base^level : dc in S_F} <= (#F)^level at those d with constant 1, again only for a design carrying 0: base 3 with F = {1,2} and d = 1 reads 14 at level = 3 against (#F)^level = 8. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_dilate, verb_converse)
Verified The dilate automaton and its transfer matrix are checked against brute-force enumeration: over d <= 64 at base 3 {0,1} and base 10 without 9 no column of T_d is off #F, while rows are off #F at 21 and 38 of the 64 respectively, every one of them at a d sharing a factor with the base; at base 3 {0,1} with d = 2 the transfer matrix [[1,1],[1,1]] with both carries accepting counts 2^level - 1 at every level <= 12, agreeing with literal enumeration of {c : 2c in S_F} at every rung and reading 4095 at x = 3^12 = 531441; and the count identity's scope is exact on the sweep over base = 3, 4, 5, every F, every d <= 6 and every level <= 5, with 0 mismatches in the 840 cases carrying the digit 0 and 399 in the 750 without it. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_dilate, verb_converse)
Proved The dilated meter is blind to the base's own powers: if 0 in F then M_F(x; base^j d) = M_F(x; d) for every j >= 0 and every d, since appending j zero digits neither leaves nor enters S_F, so the dilates repeat along every base-power ladder and only the base-prime part of d can move them. At base 3 {0,1} the d = 3 column reproduces the d = 1 column exactly, M_F(3^12; 3) = M_F(3^12) = 56 with both peaks 61. It is the lever that fixes the rate in the converse's hypothesis and that refutes the mass saving uniformly in d. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_dilate)
Proved The digit transform of a dilate is the transfer matrix in place of the digit symbol: for a design carrying 0, sum of e(ct) over c < base^level with dc in S_F equals e_0 M(t) M(qt) ... M(base^(level-1) t) 1_(Acc_d) with M(t)(r, r') = sum of e(et) over the digits e carrying r to r', and M(0) = T_d, by decomposing over automaton paths. That is the ladder of the Mobius page with the scalar symbol g_F(base^j t) replaced by a matrix, and the replacement is exactly what the route costs: the product no longer factors, so the sup-over-shift l^1 exponent that carries a Type I estimate for a digit design has no scalar analogue here. The matrix form gives the exact count at t = 0 and exact evaluation at any t, and gives no cancellation in mu; the Type II wall stands where it stands at d = 1. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_dilate)
Proved The converse of the restricted Franel identity, from (U') and with the dependence on d explicit. (U') gives G_F(Q) = O_eps(Q^(alpha + eps)) and hence sum_j delta_j^2 = O_eps(Q^(-1+eps)) on the denominator-restricted set, which is the denominator lane's conjecture. Write d = a d_base and e = b e_base with a and b supported on the primes dividing base; the two parts have disjoint prime support, so gcd(d,e) = gcd(a,b) gcd(d_base,e_base) and the kernel sum FACTORS. Each term is at most gcd(d,e)^2 (de)^(-1-alpha/2-eps) (d_base e_base)^((alpha-1)/2) Q^(alpha+2eps); the coprime factor carries exponent -3/2-eps and, writing d_base = g u and e_base = g v with gcd(u,v) = 1, is at most zeta(1 + 2eps) zeta(3/2 + eps)^2; the base factor is the product over p dividing base of sum over i, j >= 0 of p^(2 min(i,j) - (i+j)s) with s = 1 + alpha/2 + eps, which sums in closed form to the product of (1 + p^(-s))/((1 - p^(-s))(1 - p^(-alpha-2eps))) and is FINITE because alpha > 0. Then m_F(Q) >> Q^(1+alpha)/log log Q, the >> Q^alpha members of S_F in the top block below Q each exceeding Q/base with phi(b) >> b/log log b, so the rank form divides the bound down to Q^(-1+3eps). The exponent (alpha-1)/2 on d_base is critical, not chosen: at (alpha-1)/2 + delta on d_base the same argument gives only G_F(Q) = O(Q^(alpha + 2delta + eps)) and no threshold, the coprime g sum becoming sum of g^(-1+2delta) of size Q^(2delta), and at delta = 0 it is the harmonic sum, convergent only through the eps; the base factor never sees the exponent. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_dilate)
Verified The raw exponent readings on the dilates separate nothing and are not exponents: over base 3 {0,1} to x = 3^12 = 531441 the reading log max abs M_F(x;d) over log x is 0.311823 at d = 1 and at most 0.292046 over d = 2, 4, 5, 7, 8, 11, 13, 16, 22, 31, the d = 3 row being the d = 1 row by the free base powers rather than an independent reading; over base 10 without 9 to x = 10^7 the reading is 0.484570 at d = 1 against 0.489199 at d = 7 and 0.472377 at d = 2, and the crossing seen at x = 10^6, 0.495982 at d = 2 against 0.444731 at d = 1, reverses by x = 10^7, peaks 2026 against 2466. The local exponents between consecutive rungs swing over 0.24 to 0.845, so none of these readings is an exponent and none of them tests the converse's hypothesis, which is a statement about the ratio to d_base^((alpha-1)/2) x^(alpha/2) and is metered separately. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_dilate)
Conjecture The denominator-restricted set's Franel analogue is sum_j delta_j^2 = O(Q^(-1+eps)), equivalently G_F(Q) = O(Q^(alpha+eps)); the forward half of an equivalence with the square-root conjecture for M_F is open and needs the dilated sums M_F(x; d) for d > 1, for which no bound is known here, so only the implication above is proved and no exponent is claimed here. (witness: farey.md, The restricted Franel identity)
Conjecture The strict set's frequency-1 sum divided by its node count converges to the first Fourier coefficient of a limit measure of the strict set, nonzero; the readings are 0.335693, 0.343837, 0.345905, 0.346338 at Q = 3^5, 3^7, 3^9, 3^11 and 0.015138, 0.012250, 0.011561 at Q = 10^2, 10^3, 10^4, four and three nested rungs and no exponent claimed; that limit measure has no definition on the tree, and until it is named there is no Franel-type equivalence to state on the strict set. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_strict)
Conjecture (U'), the surviving hypothesis with the dilate's true mass: abs M_F(x; d) = O_eps(d_base^((alpha-1)/2) x^(alpha/2 + eps)) uniform in d >= 1 and x >= 1, with d_base the part of d coprime to the base. It is square-root cancellation in each dilate's own mass read correctly, since d_base^((alpha-1)/2) is the square root of the accepting-set constant at d_base and the base-smooth inflation is bounded; it is consistent with the free base powers by construction because (base^j d)_base = d_base; and its d = 1 case is exactly the square-root ceiling the forward implication already delivers. Metered as a ratio it does not fire: max abs M_F(y;d) over y <= x divided by d_base^((alpha-1)/2) x^(alpha/2) reads 0.9531, 0.7991, 0.9531, 0.6054, 0.9883, 0.3580, 0.5504, 0.8269, 1.0535, 0.4431, 0.5528, 0.3828 at d = 1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 22, 31 on base 3 {0,1} at x = 3^12, and 1.1276, 0.9264, 0.5523, 0.6173, 0.8583, 1.2702, 0.4594 at d = 1, 2, 3, 4, 5, 7, 11 on base 10 without 9 at x = 10^7, while the refuted exponent puts 1.1673 at d = 3 against 1.0535 as the coprime maximum. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_dilate)
Conjecture The dilate's mass constant is read off the automaton's accepting set under a SECOND coprimality: writing Delta_F for the gcd of the differences of the digits in F, for gcd(d, base Delta_F) = 1 the matrix T_d over #F is doubly stochastic, the carry chain is irreducible, its stationary law is uniform, and A_d(base^level)/(#F)^level converges to #Acc_d/d = (A_F(d-1) + 1)/d, which is O(d^(alpha-1)) and is exactly the saving a level of distribution for S_F at the modulus d would give. Verified to three decimals at level = 24 at every printed d coprime to the base, both metered designs having Delta_F = 1: base 3 {0,1} reads 1.0000, 1.0000, 0.7501, 0.8000, 0.5714, 0.5001, 0.5455, 0.5394, 0.5001, 0.3636, 0.3548 at d = 1, 2, 4, 5, 7, 8, 11, 13, 16, 22, 31 against 1, 1, 0.75, 0.8, 0.571429, 0.5, 0.545455, 0.538462, 0.5, 0.363636, 0.354839, and base 10 without 9 reads 1.0000, 1.0000, 1.0000, 0.9091, 0.9231, 0.8264, 0.8272, 0.8148 at d = 1, 3, 7, 11, 13, 121, 243, 729 against 1, 1, 1, 0.909091, 0.923077, 0.826446, 0.827160, 0.814815. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_dilate)
Conjecture The converse's hypothesis has a Mobius-free surrogate the lab can meter: square-root cancellation in each dilate's own mass is abs M_F(Q/d; d) <= N_F(Q; d)^(1/2+eps) with N_F(Q; d) = #{m in S_F : m <= Q, d divides m}, since the sum for M_F(Q/d; d) runs over exactly those m, so under that hypothesis the converse reduces to the divisor statement that B(Q) = sum over d, e of gcd(d,e)^2/(de) times sqrt(N_F(Q;d) N_F(Q;e)) is O(Q^(alpha+eps)), which mentions no Mobius function at all. This form stays consistent where the d-uniform bound above does not, reading abs M_F(Q/base^j) <= A_F(Q/base^j)^(1/2+eps) at d = base^j, which is the d = 1 ceiling again. Measured at base 3 {0,1}: B(Q)/Q^alpha reads 12.5146, 17.8640, 24.7369, 31.5935, 39.0671 at Q = 3^4 to 3^8 with local exponents 0.955, 0.927, 0.854, 0.824 falling toward alpha = 0.630930, and B(Q) over Q^alpha (ln Q)^2 falls 0.6480, 0.5920, 0.5693, 0.5342, 0.5058, so the range is consistent with Q^alpha times a power of a logarithm and no exponent is claimed. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelsmith_bilinear)
Refuted No Mertens-type sum over S_F equals the strict set's frequency-1 sum, because that sum is not real: at base 3 {0,1} and Q = 3 it is 1 + e(1/3) = 0.5 + (sqrt 3/2) i and at base 10 without 9 and Q = 10 it is -1.809016994 + 0.587785252 i, both exact algebraic sums evaluated past 1e-9, while M_F(Q), the count-weighted sum of mu(b) phi_F(b) and the normalised sum of mu(b) phi_F(b)/phi(b) are all real; the two witnesses carry the refutation alone; beside them sits an observation and not a mechanism, that the strict set also fails the pairing a -> b - a, failure of which is not shown to force a non-real sum. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_strict)
Refuted (U), the d-uniform dilated bound abs M_F(x; d) = O_eps(d^((alpha-1)/2) x^(alpha/2 + eps)), holds for NO design carrying both 0 and 1, so it cannot be the hypothesis of the converse. Base powers being free gives M_F(x; base^j) = M_F(x), so (U) at d = base^j demands abs M_F(x) <= C_eps base^(j(alpha-1)/2) x^(alpha/2+eps) at every j >= 0, and alpha < 1 drives the right side to 0 at fixed x, forcing M_F identically zero against M_F(1) = mu(1) = 1. Base 3 {0,1} at x = 3^12: the left side is 56 at every j = 0 to 12 while d^((alpha-1)/2) x^(alpha/2) falls 64.0000, 52.2558, 42.6667, 34.8372, 28.4444, 23.2248, 18.9630, 15.4832, 12.6420, 10.3221, 8.4280, 6.8814, 5.6187 over d = 3^0 to 3^12 and the ratio climbs 0.875, 1.072, 1.313, 1.607, 1.969, 2.411, 2.953, 3.617, 4.430, 5.425, 6.645, 8.138, 9.967, unbounded in j. The cause is that (alpha-1)/2 is the square root of the dilate's mass constant only where that constant is d^(alpha-1), and on the base-power ladder the constant is 1. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_converse)
Refuted The converse is one implication and not an equivalence: the threshold does not give (U') by the natural route. From G_F(Q) = O(Q^(alpha+eps)) the Fourier form gives termwise abs S_F(fill,Q) <= fill (pi^2 G_F(Q)/6)^(1/2), and Mobius inversion of S_F(m,Q) = sum over d dividing m of d M_F(Q/d;d) gives d M_F(Q/d;d) = sum over c dividing d of mu(d/c) S_F(c,Q), hence only abs M_F(Q/d;d) <= (sigma(d)/d)(pi^2 G_F(Q)/6)^(1/2), which is << log log d times Q^(alpha/2+eps) and GROWS in d where (U') needs d_base^(-1/2-eps) decay. So no biconditional is available, and none is claimed. (witness: farey.md, The restricted Franel identity)
Refuted Coprimality to the base alone does NOT give the accepting-set constant. At base 3 with F = {0,2}, where Delta_F = 2, the dilate d = 2 is coprime to the base and carries T_2 = [[2,0],[0,2]], so carry 1 is unreachable from carry 0, the closed class is {0} and the uniform stationary law is read on the wrong class: exhaustive counts are 2, 4, 8, 16, 32, 64, 128, 256 at level = 1 to 8, exactly (#F)^level, so the constant is 1 against #Acc_2/2 = 1/2. The split is exact where it is swept, over every base <= 7, every F carrying 0 and every 2 <= d <= 24 coprime to base, read at level = 400: 1747 agreements and 0 failures at gcd(d, Delta_F) = 1, 0 agreements and 148 failures at gcd(d, Delta_F) > 1. That same constant 1 sits at a d coprime to the base, so it also kills O(d^(alpha-1)) there, 1 against 2^(alpha-1) = 0.6444, and the base-smooth mechanism is therefore one cause and not the only one. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_converse)
Refuted The mass saving A_d(x) = O(d^(alpha-1) x^alpha) does not hold uniformly in d, and the base-power ladder is what refutes it: with 0 in F the dilate at d = base^j is S_F itself, so K_d = 1 exactly at every j while the ceiling base^(j(alpha-1)) tends to 0, and A_d(x)/(d^(alpha-1) x^alpha) is at least base^(j(1-alpha)), UNBOUNDED. Off the ladder the base-smooth dilates are denser than the design in the same way: at base 10 without 9 K_d = A_d(base^level)/(#F)^level reads 1.1111, 1.1358, 1.1111, 1.1413, 1.0700, 1.0343 at d = 2, 4, 5, 8, 16, 32 against the claimed ceilings 0.968781, 0.938537, 0.929003, 0.909237, 0.880851, 0.853352, and the accepting-set law fails there too, those same d carrying #Acc_d/d = 1, 1, 1, 1, 0.9375, 0.90625. The dilate is denser because the last digit of an element of S_F is uniform on F and F is not balanced modulo a prime dividing the base, so no equidistribution of S_F modulo d is available at base-smooth d. What it costs is the converse's first hypothesis and not its conclusion, the repaired hypothesis asking the rate on the coprime part only: over the 29 base-smooth d <= 1000 at base 10 without 9 the constant lies in [0.9273 at d = 512, 1.1637 at d = 625] and over every d <= 200 its inflation over the value at the coprime part of d lies in [0.9375 at d = 112, 1.1413 at d = 88], bounded on the metered range and unmeasured past it. (witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_dilate, verb_converse)
Proved The Jordan form of the restricted Franel kernel: with x_d = M_F(Q/d; d) for d <= Q and y_f = sum over m <= Q/f of x_(fm)/m, gcd(d,e)^2 = sum over f dividing gcd(d,e) of J_2(f) gives G_F(Q) = sum over f <= Q of (J_2(f)/f^2) y_f^2 with every weight J_2(f)/f^2 = prod over p dividing f of (1 - p^(-2)) in [6/pi^2, 1], hence (6/pi^2) sum_f y_f^2 <= G_F(Q) <= sum_f y_f^2, exact, for every real vector x, so for every design and every Q. Witness: farey.md, The restricted Franel identity; lab/py/restricted-franeljordan_form
Proved The vector y is a family of dilated sums: y_f = sum over n <= Q/f with fn in S_F of w(n) where w = mu * (1/id), w(n) = (1/n) sum over c dividing n of mu(c) c = prod over p dividing n of (1 - p)/n, abs w(n) = phi(rad n)/n <= 1, w(1) = 1; so y_f is the Mertens sum of the dilate f^(-1) S_F with mu replaced by w. Witness: farey.md, The restricted Franel identity; lab/py/restricted-franeljordan_form
Proved The sandwich with explicit logarithms: x -> y inverts by Mobius, x_f = sum over m <= Q/f of mu(m) y_(fm)/m; on vectors indexed by d <= Q both maps are upper triangular with entries 1/m and mu(m)/m at (f, fm), Schur's test with row sums at most H_Q and column sums sigma(d)/d and prod over p dividing d of (1 + 1/p) <= sigma(d)/d <= H_d bounds both squared operator norms by N_Q = H_Q max over d <= Q of sigma(d)/d <= H_Q^2 <= (1 + ln Q)^2, and so (6/pi^2) (1 + ln Q)^(-2) sum over d <= Q of M_F(Q/d; d)^2 <= G_F(Q) <= (1 + ln Q)^2 sum over d <= Q of M_F(Q/d; d)^2 for every design and every Q >= 1; Robin's unconditional sigma(d)/d < e^gamma ln ln d + 0.6483/ln ln d sharpens N_Q to O(ln Q ln ln Q). Witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_sandwich
Proved The ratio sum_f y_f^2 / sum_d x_d^2 has no constant bound over all vectors, and the witness proves exactly that and not the sharpness of N_Q: for x_d = 1 at the divisors of the primorial N = prod over p <= z of p <= Q and 0 elsewhere, y_f = sigma(N/f)/(N/f) at f dividing N and 0 elsewhere, so sum_f y_f^2 / sum_d x_d^2 = prod over p <= z of (1 + 1/p + 1/(2p^2)) > sum over p <= z of 1/p, unbounded as Q grows, of order ln ln Q; at z = 5, 7, 11, 13 the ratio reads 2.753472, 3.174922, 3.476671, 3.754393, equal to the product. Witness: farey.md, The restricted Franel identity; lab/py/restricted-franeljordan_form
Proved The mean-square Franel equivalence on a digit design: for every digit design with at least two digits, sum_j delta_j^2 = O_eps(Q^(-1+eps)) on the denominator-restricted Farey set iff sum over d <= Q of M_F(Q/d; d)^2 = O_eps(Q^(alpha+eps)) iff sum over f <= Q of y_f^2 = O_eps(Q^(alpha+eps)); the rank form, G_F(Q) = 12 m_F(Q) sum_j delta_j^2 + 1 when 1 in F and - 2 otherwise (the node 1 is present exactly when 1 in F, so c = 1/2 or c = 0 in the Parseval computation; the constant reads 1 at every jump Q <= 40 on base 3 {0,1} and base 10 without 9 and -2 on base 3 {0,2}, base 4 {0,2,3}, base 5 {0,2,4} and base 10 {0,2,5,7}, base 3 {0,2} at Q = 26 reading G_F = 5.043162 against 12 m S2 = 7.043162), with Q^(1+alpha)/ln ln Q << m_F(Q) << Q^(1+alpha) moves the threshold to G_F(Q) = O(Q^(alpha+eps)), the weights move it to sum_f y_f^2 at no cost and the sandwich to sum_d x_d^2 at the cost of (1 + ln Q)^2, absorbed by the eps. On the full set every dilate is M, the d = 1 term is M(Q)^2 and M(x) << x^(1/2+eps) bounds the sum by zeta(1+2eps) Q^(1+2eps), so the statement collapses to Franel's theorem with M(x) = O(x^(1/2+eps)) standing for RH. (U') implies the mean square since sum over d <= Q of d_co^(alpha-1) (Q/d)^(alpha+2eps) << Q^(alpha+2eps); the mean square does not return (U'), which is pointwise in d. Witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_sandwich
Verified The sandwich at base 3 {0,1}, Q = 3^4 to 3^8, and on the control at Q = 40, 81, 243: the Jordan form equals the gcd double sum as exact rationals to Q = 729 and to 2.8e-14 and 5.7e-14 in floating point at Q = 2187 and 6561, the control reading G_F(40) = 62.310829 again; G_F(Q)/sum_f y_f^2 reads 0.842880, 0.829928, 0.818649, 0.842080, 0.831385 on the design and 0.854165, 0.869367, 0.850790 on the control, inside [0.607927, 1]; G_F(Q)/sum_d x_d^2 reads 0.661794, 0.510529, 0.583192, 0.585221, 0.593465 on the design with sum_d x_d^2 = 19, 63, 113, 261, 673, and 1.093172, 1.337459, 1.582101 on the control, inside [(6/pi^2)/N_Q, N_Q] which reads [0.0436, 13.94] at Q = 81 and [0.0169, 35.95] at Q = 6561; the d = 1 share M_F(Q)^2/sum_d x_d^2 reads 0.210526, 0.253968, 0.035398, 0.187739, 0.005944; the power-iteration norms, lower bounds after 3000 steps, read 2.258774, 2.490385, 2.693333, 2.874809, 3.040083 for x -> y and 1.940642, 2.080130, 2.196254, 2.299018, 2.391003 for y -> x against N_Q^(1/2) = 3.733351, 4.338690, 4.906916, 5.412152, 5.995693; no constant and no exponent claimed for any ratio. Witness: lab/py/restricted-franelverb_sandwich
Proved Cross-reference to the row above: the termwise route to (U') stays refuted, and the biconditional it declines is available with the mean square over d in place of (U'): the Franel threshold on the denominator set is equivalent to sum over d <= Q of M_F(Q/d; d)^2 = O_eps(Q^(alpha+eps)), by the Jordan sandwich of the same section. Witness: farey.md, The restricted Franel identity; lab/py/restricted-franelverb_sandwich
Proved For every digit design and every Q >= 1, sum_{m in S_F, m <= Q} tau(m) <= B(Q) <= (1 + ln Q) sum_{d <= Q} (sigma(d)/d) N_F(Q; d) <= (1 + ln Q)^2 sum_{m in S_F, m <= Q} tau(m), hence A_F(Q) <= B(Q) <= C_eps (1 + ln Q)^2 Q^eps A_F(Q) and B(Q) = O_eps(Q^(alpha+eps)): the Mobius-free surrogate of the converse is a theorem, unconditionally. Witness: AM-GM on the symmetric kernel, sum_d N_F(Q; d) = sum_m tau(m), sum_{d dividing m} sigma(d)/d <= tau(m)(1 + ln m) and tau(m) <= C_eps m^eps, farey.md THE RESTRICTED FRANEL IDENTITY.
Verified At base 3 {0,1}, Q = 3^4 to 3^12, B(Q) reads 200.233, 571.647, 1583.160, 4043.971, 10001.178, 23553.362, 54650.025, 125375.511, 285237.429, the Jordan form agreeing with the gcd double sum to 2.4e-11 at Q <= 3^8; B(Q)/sum tau(m) climbs 2.9019 to 4.4174, the AM-GM bound over B(Q) climbs 2.728 to 5.402, and B(Q)/(Q^alpha (ln Q)^2) falls 0.6480 to 0.4007. Witness: lab/py/restricted-franel verb accepting.
Proved Under (SR), abs(M_F(Q/d; d)) <= C_eps N_F(Q; d)^(1/2+eps) for every 1 <= d <= Q and Q >= 1, the denominator lane's conjecture sum_j delta_j^2 = O_eps(Q^(-1+eps)) holds on every digit design with at least two digits, and no statement about S_F in residue classes enters. Witness: sum_{d <= Q} M_F(Q/d; d)^2 <= C_eps^2 A_F(Q)^(2eps) sum_m tau(m) and the mean-square equivalence, farey.md THE RESTRICTED FRANEL IDENTITY.
Proved For a design carrying 0, the accepting-set law's main term is A_F(Q)/d and not (#Acc_d/d) A_F(Q): A_F(d base^level) = #Acc_d fill^level - 1 + [d in S_F], false without 0 (base 3 {1,2}: A_F(9) = 6 against 4 at d = 1), so A_d(base^level)/fill^level -> #Acc_d/d is N_F(Q; d) ~ A_F(Q)/d read at Q = d base^level; at base 3 {0,1}N_F(3^level; 4) = 2^(level-2) + 2^(level/2-1) - 1 at even level, while A_4(Q/4)/A_F(Q/4) reads 0.5078 against 3/4 at Q = 3^14. Witness: counting the top digits of n <= d base^level; lab/py/restricted-franel verb accepting reads 71, 271, 1055, 4159 at level = 8, 10, 12, 14 and 4159/8191.
Proved At base 3 {0,1} and the repunit R_t = (3^t - 1)/2, t >= 2, an element of S_F coprime to 3: N_F(3^(2t); R_t) = 2^t + 1, N_F(3^(3t); R_t) = 2 3^t + 1, and N_F(3^(2rt); R_t) >= K_r(t), the pair count of 2r-tuples of t-strings with u_1 + .. + u_r = v_1 + .. + v_r, with K_1(t) = 2^t and K_2(t) = 6^t. Witness: R_t divides m iff (3^t - 1) divides twice the block sum, and complementing the v strings; lab/py/restricted-franel verb accepting reads 5, 9, 17, 33, 65, 129, 257 at k = 2, 19, 55, 163, 487 at k = 3, 71, 369, 2003 = 6^t + 5^t + 3^t + 1 at k = 4, and K_r(2) = 4, 36, 430, 5796, 82404 at r = 1 to 5 against brute force.
ProvedK_r(t) >= c_r rho_r^t with rho_r the Perron root of the carry class of 0, aperiodic by its positive diagonal, 3 rho_r = Lambda(2r) of mobius.md, and rho_r > 4^r/3 at r = 1 to 5, certified exactly by a positive rational vector v with min_c (Mv)_c/v_c > 4^r/3. Witness: lab/py/restricted-franel verb accepting prints Lambda(2r) = 6, 18, 65.664583, 257.340480, 1025.101288, ratios Lambda(2r)/4^r = 1.5, 1.125, 1.026009, 1.005236, 1.001075, the certificate True at every r, and the characteristic polynomials, x^3 - 32 x^2 + 246 x - 540 at r = 3 and x^5 - 512 x^4 + 65400 x^3 - 2541780 x^2 + 27112239 x - 85739148 at r = 5.
Refuted The accepting-set law as an upper bound N_F(Q; d) << A_F(Q)/d uniform in coprime d <= Q^theta, at theta = 1/2, 1/3, 1/4, 1/6, 1/8, 1/10: N_F(Q; R_t) R_t/A_F(Q) is unbounded like (3/2)^t at Q = 3^(2t), like (9/8)^t at 3^(3t) and 3^(4t), and like (Lambda(2r)/4^r)^t at 3^(2rt); the count attains the column-sum lemma up to 2^(-1-alpha) at Q^(1/2). Witness: the two rows above, lab/py/restricted-franel verb accepting.
ConjectureLambda(2r) > fill^(2r) at every r, which would carry the refutation to every fixed level theta > 0: the pointwise level of distribution of S_F at residue 0 is then no fixed power of Q. Witness: lab/py/restricted-franel verb accepting prints Lambda(2r)/4^r above 1 at r = 1 to 5 only; mobius.md proves Lambda(2r) >= fill^(2r) and not the strict inequality.
Verified The meter R(Q, d) = N_F(Q; d) d_co/A_F(Q) over every d <= Q^(1/2) at base 3 {0,1}, Q = 3^8 to 3^16, peaks at 2.6562, 3.7812, 3.8994, 5.6875, 5.7764, 8.5391, 8.6058, 12.8125, 12.8625 at d = 40, 121, 121, 364, 364, 1093, 1093, 3280, 3280, the repunit of length ceil(level/2) every time, no other cell within 0.02, the maximum over the coprime d the same cell; R(3^level, 4) = 1 + 2^(1-level/2) - 2^(2-level) at even level. Witness: lab/py/restricted-franel verb accepting.
Verified (U') at the repunits does not fire: at d = R_t, x = floor(3^(2t)/R_t) = 2 3^t + 2, t = 2 to 8, the 2^t + 1 Mobius values peak at 1, 2, 2, 5, 5, 6, 6 against the yardstick d^((alpha-1)/2) x^(alpha/2) reading 1.992 to 4.470, ratio at most 1.719 at t = 5, and peak/N^(1/2) reads 0.447, 0.667, 0.485, 0.871, 0.620, 0.528, 0.374, largest at t = 5. Witness: lab/py/restricted-franel verb accepting.
Provedsum_{a mod d} abs(hat F_level(a/d)) <= 2 e^(2 pi fill) base^(a_1) fill^level d^(a_1) with a_1 = log_base(B_base(F)/fill) the certified bound for the l^1 exponent, alpha_1 <= a_1, hence N_F(Q; d) << Q^alpha d^(a_1 - 1) for every d and every design, the root-of-unity identity behind it needing 0 in F and the bound not; the floor a_1 >= 1 - alpha puts it above the column-sum lemma fill (Q/d)^alpha at every d coprime to the base; at base 3 {0,1} it is Q^alpha d^(alpha - 1) and feeds B(Q) as Q^(2 alpha). Witness: the shifted-grid recursion of mobius.md step 2 with the supremum over a window inside the sum, Lipschitz constant 2 pi (base-1) fill of g_F, farey.md.
Proved At d = base^t - 1 and level = 2t, hat F_(2t)(a/d) = hat F_t(a/d)^2 and sum_{a mod d} abs(hat F_(2t)(a/d)) = d (fill^t + 2w) exactly, w = [0 in F][base - 1 in F], so the nonzero roots carry Q^((1-alpha)/2) times the main term while N_F(3^(2t); 3^t - 1) = 1 at base 3 {0,1}: the l^1 route to a pointwise Type I bound is dead at Q^(1/2) by the triangle inequality alone. Witness: Parseval mod d on t-strings; lab/py/restricted-franel verb accepting reads E_d = 10.375 = 728/64 - 1 at d = 728, level = 12.
VerifiedE_d, the nonzero roots' l^1 over fill^level, over d <= 3^(level/2) at base 3 {0,1}, even level = 8 to 16, peaks at the base power d = 3^(level/2) reading 13.6785, 27.5490, 54.5261, 106.9952, 209.0447, and reads 4, 6.5625, 10.375, 16.0781, 24.625 at d = 3^t - 1, equal to (3^t - 1)/2^t - 1. Witness: lab/py/restricted-franel verb accepting.
Verified Cross-reference to the sandwich row: the figure binary paper-franel-converse recomputes G_F(81) = 12.574087 and G(81) = 159.157609 from the dilated Mertens sums and asserts the rank form G = 12 m sum delta^2 + 1 on both sets to 1e-8, m_F(81) = 241, m(81) = 2020, sum delta^2 = 0.004002105 and 0.006524654, sum_d x_d^2 = 19. Witness: figures/src/bin/paper-franel-converse.rs.
Verified Cross-reference to the row on (U') at the repunits: the printed ratio peak/N^(1/2) at t = 5 is 0.870 (5/sqrt(33) = 0.8704), not 0.871; the note's digit is corrected, the paper's Fact 6.2 reads 0.870. Witness: lab/py/restricted-franel verb accepting.
Proved For every base >= 3, F = {0,1}, t >= 2, R_t = (base^t - 1)/(base - 1) and x_t = floor(base^(2t)/R_t) = (base - 1)(base^t + 1): R_t^(-1) S_F intersect [1, x_t] = {(base - 1) m + 1 : m in B_t} union {base^t + 1} with B_t the 2^t padded t-strings, so N_F(base^(2t); R_t) = 2^t + 1 at every base and M_F(y; R_t) for y <= x_t is the Mobius sum over the affine image (base - 1) B_t + 1, odd when base is odd and mixed when base is even, plus [y >= base^t + 1] mu(base^t + 1); the shift by base^t and the complement m -> R_t - m cancel. Witness: lab/py/restricted-franel verb repunit, literal enumeration of S_F below base^(2t) at t <= 10
Proved At base 3 {0,1}, 2 B_t + 1 is the set of n <= 3^t whose lowest nonzero digit is 1 and whose other digits lie in {0,2}, equal to {3^k (6w + 1) : 0 <= k < t, w in B_(t-1-k)} union {3^t}, so M_F(x_t; R_t) = T_(t-1) - T_(t-2) + mu(3^t + 1) with T_s = sum over w in B_s of mu(6w + 1), T_0 = 1, and mu(3^t + 1) = 0 at odd t; at a general base T_(t-1) + mu(base) T_(t-2) + mu(base^t + 1) with T_s = sum over B_s of mu(base (base - 1) w + 1). Witness: lab/py/restricted-franel verb repunit, checks marked and shifted at every sieved t
Proved The transform of the affine image is e(theta) hat F_t((base - 1) theta), and with g = gcd(d, base - 1) the l^1 sum over the d-th roots of unity equals g times the block's l^1 sum over the (d/g)-th roots, since (base - 1)/g is coprime to d/g: the block's own at g = 1, and d 2^t at d dividing base - 1, where the image sits in the class 1 mod d, so at base 3 the image is odd and the dilate below x_t has exactly one even element, 3^t + 1; the multiplicative-energy bound of mobius.md holds for any 2^t integers below 3^t with the same miss (1 - alpha)/2. Witness: farey.md, the restricted Franel identity, the repunit paragraphs
Proved At base 3, S_(0,2) = 2 S_(0,1) gives d^(-1) S_(0,2) = (d/2)^(-1) S_(0,1) at even d and 2 (d^(-1) S_(0,1)) at odd d; the {0,2} repunit 3^t - 1 = 2 R_t carries the {0,1} repunit dilate and its readings exactly. Witness: lab/py/restricted-franel verb repunit, the scaled design block, t = 2 to 6
Verified At base 3 {0,1} the endpoint M_F(x_t; R_t) reads 0, -1, -1, -1, 3, -6, 0, -22, 6, -4, -27, -67, -49, -10, -78, 88, 82, 209, 543, 858, 335, -407, 3030 at t = 2 to 24 and the running maximum max over y <= x_t of abs(M_F(y; R_t)) reads 1, 2, 2, 5, 5, 6, 6, 23, 25, 28, 36, 83, 107, 142, 142, 228, 361, 366, 1023, 1435, 1435, 1435, 3484. Witness: lab/py/restricted-franel verb repunit, the PARI walk over the 2^t elements as generator, the literal enumeration of S_F below 3^(2t) divisible by R_t at t <= 10 as the independent witness
Verified peak/N^(1/2) with N = 2^t + 1 at base 3 {0,1} reads 0.4472, 0.6667, 0.4851, 0.8704, 0.6202, 0.5283, 0.3743, 1.0155, 0.7809, 0.6186, 0.5624, 0.9170, 0.8359, 0.7844, 0.5547, 0.6298, 0.7051, 0.5055, 0.9990, 0.9909, 0.7007, 0.4955, 0.8506 at t = 2 to 24, above 1 once, largest 1.0155 at t = 9; peak/N^(0.6) is largest at 0.6136 at t = 5 and reads 0.1612 at t = 24; the falsification threshold 2 by t = 14 is not reached at either base. Witness: lab/py/restricted-franel verb repunit
Verified At base 4 {0,1} with R_t = (4^t - 1)/3 the endpoint reads -1, 0, -2, -2, 2, -3, -2, 27, 30, -12, 15, -26, -30, 22, -305, -492, -273 at t = 2 to 18, the running maximum 1, 1, 2, 2, 3, 3, 8, 27, 37, 37, 37, 37, 77, 103, 319, 519, 577, and peak/N^(1/2) is largest at 1.4335 at t = 17. Witness: lab/py/restricted-franel verb repunit, the PARI walk over the 2^t elements as generator, the literal enumeration of S_F below 4^(2t) divisible by R_t at t <= 10 as the independent witness
Verified The shifted sums T_s = sum over w in B_s of mu(6w + 1) at base 3 read 0, -1, -3, -4, 0, -6, -5, -27, -21, -25, -51, -118, -168, -178, -257, -169 at s = 1 to 16, never positive, T_13 = -168 against 2^(13/2) = 90.5. Witness: lab/py/restricted-franel verb repunit
Conjecture (SR) at the repunits, abs(M_F(x_t; R_t)) <= C_eps (2^t + 1)^(1/2 + eps) for every t, is square-root cancellation of mu on the marked design 2 B_t + 1, open at the same wall as d = 1 and independent of it; it fails if max over y <= x_t of abs(M_F(y; R_t))/N^(1/2) climbs without bound along t, the threshold set here being 2 by t = 14, and the readings stay under 1.02 to t = 24. Witness: lab/py/restricted-franel verb repunit
Refuted Cross-reference to the earlier [Conjecture] row on (U'): as stated there, abs M_F(x; d) = O_eps(d_base^((alpha-1)/2) x^(alpha/2 + eps)) uniform in d >= 1 and x >= 1, (U') is false on every design carrying both 0 and 1 with alpha < 1, so the earlier [Proved] converse from (U') held only vacuously. At x = 1 and d = base^k + 1, which lies in S_F and is coprime to the base, M_F(1; d) = mu(1) = 1, while d_base^((alpha-1)/2) = d^((alpha-1)/2) tends to 0 as k grows; at base 3 {0,1} the repunits R_t are witnesses too, and at base 10 without 9 the d = 10^j + 1. Witness: franel.md, The converse under one hypothesis.
Conjecture Cross-reference narrowing the earlier [Refuted] row that the converse is one implication and not an equivalence: its witness, abs M_F(Q/d;d) <= (sigma(d)/d)(pi^2 G_F(Q)/6)^(1/2), is an upper bound that fails to decay and shows only that the natural route does not reverse the implication; it refutes nothing. With (U') read with its constant term, not an equivalence is the statement that the threshold holds and (U') fails, which contains the open denominator lane conjecture and so stands at Conjecture at most: the threshold is what (SR) gives, and the repunit peaks, peak/N^(1/2) in [0.3743, 1.0155] at t = 2 to 24 at base 3 {0,1}, sit above the (U') yardstick by a factor growing like (3/2)^(t/2) if they keep that size. Witness: franel.md, The converse under one hypothesis and The repunits; lab/py/restricted-franel verb repunit.
Proved The converse from (U') with a constant term: if for every eps > 0 there is C_eps with abs M_F(x; d) <= C_eps (1 + d_co^((alpha-1)/2) x^(alpha/2 + eps)) for every d >= 1 and x >= 1, d_co the part of d coprime to the base, then G_F(Q) = O_eps(Q^(alpha + eps)) and sum_j delta_j^2 = O_eps(Q^(-1+eps)) on the denominator-restricted set of every digit design; the hypothesis is not conjectured, and the converse rests on (SR). Witness: franel.md, The converse under one hypothesis; lab/py/restricted-franel verb repunit.
Proved On a design carrying 0, (U') forces abs M_F(x) <= C_eps (1 + x^(alpha - 1/2 + eps)) at every x, below the square-root ceiling x^(alpha/2) and bounded when alpha < 1/2, since at d = base^k + 1 and x < base^k the dilate is S_F itself; (U') is square-root cancellation against the accepting-set law's main term, not against each dilate's own mass. Witness: franel.md, The converse under one hypothesis.
Conjecture (U') fails at base 3 {0,1}, so that with the denominator lane's conjecture the threshold holds and (U') fails, the statement of the row narrowing the natural route: the design's meter reads log max abs M_F(y) / log x = 0.311823 at x = 3^12 against the 0.130930 (U') allows. Witness: franel.md, What stays open; lab/py/restricted-franel verb dilate.
Refuted The shifted sums T_s = sum_{w in B_s} mu(6w + 1) at base 3 {0,1} are one-signed: T_17 = -88 and T_18 = 121. Witness: lab/py/shifted-sums verb walk, and independently the endpoint readings of lab/py/restricted-franel verb repunit through M_F(x_t; R_t) = T_(t-1) - T_(t-2) + mu(3^t + 1).
VerifiedT_s reads -88, 121, 665, 1523, 1857, 1450, 4479, 4939, 1417, 12641 at s = 17 to 26; it is negative at s = 2 to 4 and 6 to 17, zero at s = 1, 5, positive at s = 18 to 26, and abs T_s / 2^(s/2) <= 1.8562 at every s <= 26, the maximum at s = 13. Witness: lab/py/shifted-sums verb walk.
Verified The prefix walk of mu(6w + 1) over B_26 in ascending w changes the sign of its last nonzero value 132 times, the last at n = 238418 where 6w + 1 = 1128943015, and is never negative from there to n = 2^26, touching 0 at n = 238419. Witness: lab/py/shifted-sums verb walk.
Verified The whole class M(x; 6, 1) = sum_{n <= x, n = 1 mod 6} mu(n) at x = 3^(s+1) - 2 is negative at s = 2 to 13 and reads 8 and 265 at s = 14 and 15, with 3996 sign changes below 3^16. Witness: lab/py/shifted-sums verb control.
Conjecturemu in the class 1 mod 6 has no bias at the square-root scale: its Dirichlet series (1/2)(1/(zeta(s)(1 - 2^-s)(1 - 3^-s)) + 1/(L(s, chi_-3)(1 + 2^-s))) has no pole at s = 1/2, and its one non-oscillating term, from s = 0, is 3/4 - Psi_(2,3)(x) with Psi_(2,3) the count of 3-smooth numbers, of size (log x)^2 / (2 log 2 log 3), reading -173.25 against the class sum -169 at x = 3^14 - 2; this corrects the row whose second term read 1/L(s, chi_-3), which counts even n, and whose constant read 3/2; T_s has no Euler product and no such term. Witness: lab/py/shifted-sums verb control.
Proved Correction to the row that (U') forces the cap x^(alpha - 1/2 + eps): on every design carrying 0 with alpha < 1, (U') forces abs M_F(x) <= C_eps at every x, so M_F bounded, since at d = base^k + 1 and x < base^k the dilate is S_F itself and letting k grow at fixed x drives d^((alpha-1)/2) to 0. Witness: franel.md, The converse under one hypothesis.
Conjecture Correction to the row that (U') fails at base 3 {0,1}: (U') fails there as soon as M_F is unbounded, and M_F is conjectured unbounded, the design's meter reading log max abs M_F(y) / log x = 0.311823 at x = 3^12 against the exponent 0 that (U') allows. Witness: franel.md, What stays open; lab/py/restricted-franel verb dilate.
Verified Cross-reference to the row on the printed ratio peak/N^(1/2) at t = 5: the paper's Fact 6.2 now reads at most 0.8839 over the law-meter cells at every level L = 8 to 16, odd levels included, the largest at L = 11, d = 364, N_F = 32, peak 5; 0.8704 is the maximum over the even levels. Witness: lab/py/restricted-franel verb accepting.