Franel on a digit design

• 2026-09-14 [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-franel ladder_check)
• 2026-09-14 [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-franel verb_denominator)
• 2026-09-14 [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-franel kernel_sum and fourier_side)
• 2026-09-14 [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-franel farey_delta_square)
• 2026-09-14 [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-franel scan_backward)
• 2026-09-14 [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-franel strict_ramanujan_divisor)
• 2026-09-14 [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-franel strict_literal against lab/rs/farey-discrepancy design)
• 2026-09-14 [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-franel dilate_matrix, verb_converse 0 mismatches of 840)
• 2026-09-14 [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-franel verb_dilate, verb_converse)
• 2026-09-14 [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-franel verb_dilate, verb_converse)
• 2026-09-14 [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-franel verb_dilate)
• 2026-09-14 [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-franel verb_dilate)
• 2026-09-14 [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-franel verb_dilate)
• 2026-09-14 [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-franel verb_dilate)
• 2026-09-14 [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 the desk has no bound, so only the implication above is proved and no exponent is claimed here. (witness: farey.md, The restricted Franel identity)
• 2026-09-14 [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-franel verb_strict)
• 2026-09-14 [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-franel verb_dilate)
• 2026-09-14 [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-franel verb_dilate)
• 2026-09-14 [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-franel smith_bilinear)
• 2026-09-14 [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-franel verb_strict)
• 2026-09-14 [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-franel verb_converse)
• 2026-09-14 [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)
• 2026-09-14 [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-franel verb_converse)
• 2026-09-14 [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-franel verb_dilate, verb_converse)