franel.md

The Farey page meters Franel and Landau on a digit design and finds that restricting the denominator alone leaves the shape reading the same. This page writes the identity under that reading: the objects first, then the identity, the dilates it produces, the equivalence it closes on, the one hypothesis that suffices, and the repunits where the uniform law fails.

The objects

• A digit design S_F is the set of whole numbers whose every digit in a fixed base lies in a digit set F with abs F >= 2, of dimension alpha = log abs F / log base, counted by A_F(Q) = #{n in S_F : n <= Q}; the two designs metered are base 3 {0,1} and base 10 without 9, and the full set, every digit allowed, is the control.
• The strict set F_Q(S_F) of the meter on a digit design takes both a and b in S_F, with phi_F(b) numerators at the denominator b and card nodes in all. The denominator set F_Q^d(S_F) = {a/b reduced : b in S_F, b <= Q, 1 <= a <= b} restricts b alone, and it needs the second symbol because the denominator convention has a Mobius face that the strict one does not.
• The denominator set has m_F(Q) = sum_{b in S_F, b <= Q} phi(b) nodes, rho_1 < ... < rho_m ascending, and the sawtooth delta_j = rho_j - j/m_F(Q), exactly as on the Farey page.
• Its exponential sums are S_F(k, Q) = sum_{r in F_Q^d(S_F)} e(kr) with e(x) = exp(2 pi i x), and the strict set's are S_F^s(k, Q).
• The dilate d^(-1) S_F = {c : dc in S_F} has the dilated Mertens sum M_F(x; d) = sum_{c <= x, dc in S_F} mu(c), its Mertens function, and the sums of the dilates are a family of Mobius sums the full set never separates. At d = 1 it is the design's own meter M_F(x) = sum_{n in S_F, n <= x} mu(n) from mobius; at d > 1 it is a new function and not a rescaling of the old one, since d^(-1) S_F is not S_F, is not a digit design and carries no digit test. On the full set every dilate is the whole of Z and all of them collapse to M.
• The mass of a dilate is N_F(Q; d) = #{m in S_F : m <= Q, d divides m} = A_d(Q/d) with A_d(x) = #{c <= x : dc in S_F}, since the sum for M_F(Q/d; d) runs over exactly those m.
• The identity's vector is x_d = M_F(Q/d; d) for d <= Q, and its form is G_F(Q) = sum_{d, e >= 1} (gcd(d,e)^2/(d e)) M_F(Q/d; d) M_F(Q/e; e), a finite sum of exact rationals, every term with d > Q or e > Q vanishing.
• d_co is the part of d coprime to the base, and Delta_F is the gcd of the differences of the digits in F.
• Three hypotheses on the dilates recur. (U) is abs M_F(x; d) = O_eps(d^((alpha-1)/2) x^(alpha/2 + eps)), uniform in d. (U') is: 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. (SR) is: for every eps > 0 there is C_eps with abs M_F(Q/d; d) <= C_eps N_F(Q; d)^(1/2+eps) for every 1 <= d <= Q and every Q >= 1.
• The surrogate of G_F(Q) is B(Q) = sum_{d, e} (gcd(d,e)^2/(d e)) sqrt(N_F(Q; d) N_F(Q; e)), the same kernel on the masses, with no mu in it.
• The repunit R_t = (base^t - 1)/(base - 1), t >= 2, is written with t ones; at base 3 it is R_t = (3^t - 1)/2.

The exponential sums are dilated Mertens sums

At frequency m the denominator set's exponential sum is S_F(m, Q) = sum_{d | m} d M_F(Q/d; d), and at m = 1 it is exactly M_F(Q). Proved, in three steps. Partition by denominator, so S_F(m, Q) = sum_{b in S_F, b <= Q} c_b(m) with c_b(m) = sum_{a mod b, gcd(a,b) = 1} e(ma/b) Ramanujan's sum, which depends on b alone and knows nothing of the design; substitute Kluyver's formula c_b(m) = sum_{d | gcd(m,b)} d mu(b/d); exchange the two sums and write b = dc, which turns the inner one into M_F(Q/d; d). Frequency 1 needs no Ramanujan input at all, sum_{a mod b, gcd(a,b) = 1} e(a/b) = sum_{d | b} mu(d) sum_{c mod b/d} e(c/(b/d)) = mu(b), the complete inner sum vanishing unless b/d = 1. Verified: every denominator b in S_F up to Q = 10^5 at base 3 {0,1}, and up to Q = 10^4 at base 10 without 9 and on the control, has its literal sum of phi(b) roots of unity equal to mu(b), worst deviation 1.09e-11 at b = 86293 on the first design and 1.36e-12 at b = 7247 on the second, with 0 denominators rounding to the wrong integer anywhere; the Q = 10^5 rung at base 10 without 9 costs 2.6e9 roots of unity, past the machine budget, so it is not walked and nothing is claimed at it. The frequency-m formula is exact against the literal sum at m = 1, 2, 3, 4, 5, 6, 12 on both designs and the control (lab/py/restricted-franel).

The identity

Squaring that against the Franel weight gives the identity in two forms, both exact. The Fourier form: sum_{k != 0} |S_F(k, Q)|^2/k^2 = (pi^2/3) G_F(Q). Proved: substitute the frequency-m sum, expand the square and exchange, so for fixed d, e the inner sum is sum_{k != 0, lcm(d,e) | k} k^(-2) = 2 zeta(2)/lcm(d,e)^2, and d e/lcm(d,e)^2 = gcd(d,e)^2/(d e). The kernel gcd(d,e)^2/(d e) is the Smith gcd matrix that already carries the moire correlation law of the stack and the same identity one field up, so digit restriction moves the entries and never the kernel.

The rank form: G_F(Q) = 12 m_F(Q) sum_j delta_j^2 + 1 when 1 in F and 12 m_F(Q) sum_j delta_j^2 - 2 otherwise, so either way G_F(Q) = 12 m_F(Q) sum_j delta_j^2 + O(1). Proved, by Parseval on the sawtooth and piecewise integration of D(v)^2, D(v) = A(v) - m_F(Q) v with A(v) = #{j : rho_j < v}, between consecutive nodes, the two boundary cubes vanishing at 0 and at 1. The one input left is the mean value: with c = sum_r rho_r - m_F(Q)/2 the integration gives int D^2 - c^2 = m_F(Q) sum_j delta_j^2 + 1/12 at c = 1/2 and - 1/6 at c = 0. Closure under r -> 1 - r away from the node 1 is one sufficient condition for the mean value, and the denominator set has that closure: a/b reduced with b in S_F gives (b-a)/b reduced with the same b, so the nodes with a < b pair under a -> b - a, a = b only at the node 1, and the node 1 is present exactly when 1 in F; so sum_r rho_r is (m_F(Q) + 1)/2 or m_F(Q)/2 and c is 1/2 or 0. Every proper strict set fails that closure. Verified: the constant reads 1 at every jump Q <= 40 on base 3 {0,1} and base 10 without 9 and -2 at every jump 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_F(Q) sum_j delta_j^2 = 7.043162; and both forms hold as identities of exact rationals at base 3 {0,1} Q = 81 and Q = 243, base 10 without 9 Q = 40 and the control Q = 40, true at all four, the Fourier side truncated at |k| <= 200000 landing inside its printed tail bound 2 m_F(Q)^2/K at each; the control at Q = 40 regenerates Edwards section 12.2, m = 490 and sum_j delta_j^2 = 0.0104270117 giving G_F(40) = 62.310829 (lab/py/restricted-franel).

The node count is Q^(1+alpha) up to ln ln Q: Q^(1+alpha)/ln ln Q << m_F(Q) << Q^(1+alpha) for any design with abs F >= 2. Proved, the upper bound from m_F(Q) <= Q A_F(Q) with A_F(Q) <= 2 abs F Q^alpha, the block count abs F^level at Q = base^level up to a constant depending on the design alone, and the lower from the abs F^(level-1) members of S_F with exactly level = floor(log_base Q) digits, each in [Q/base^2, Q] with phi(b) >> b/ln ln b.

Dropping every term but k = 1 and k = -1 from a sum of nonnegative terms carries the meter back out: 2 M_F(Q)^2 <= (pi^2/3) G_F(Q), which is 4 pi^2 m_F(Q) sum_j delta_j^2 + pi^2/3 when 1 in F. Proved, and the factor pi^2/3 is load-bearing rather than decorative: the weaker-looking 2 M_F(Q)^2 <= G_F(Q) is false, 2 M(5)^2 = 8 standing against G(5) = 64/15 at the control Q = 5 and 18 against G_F(37) = 14.230517 at base 3 {0,1} Q = 37. Verified at every integer Q rather than at a sample, both sides stepping only at Q in S_F so that scanning S_F covers every Q below the bound: 0 violations over Q <= 2187 at base 3 {0,1} and Q <= 400 at base 10 without 9 and on the control. The ratio 2 M_F(Q)^2/((pi^2/3) G_F(Q)) peaks at 0.607927 at the trivial Q = 1 on all three, and over Q >= 100 its maximum is 0.340071 at Q = 253 on base 3 {0,1}, 0.137645 at Q = 221 on base 10 without 9 and 0.086385 at Q = 114 on the control (lab/py/restricted-franel).

What the inequality buys is a ceiling, and it is bought with the conjectured exponent and never the measured one. The denominator lane's conjecture sum_j delta_j^2 = O(Q^(-1+eps)) of the meter on a digit design gives |M_F(Q)| = O(Q^(alpha/2+eps)), the square-root ceiling for the design's Mertens meter on mobius. Proved: the node count gives m_F(Q) << Q^(1+alpha), so the conjecture gives G_F(Q) = O(Q^(alpha+eps)) through the rank form, and the inequality gives the ceiling, the constant pi^2/3 absorbed and no unproved input entering. The measured exponent does none of this: e_2 reads -0.959 and -0.899 at the top rungs of that section's table with S2*Q still climbing there, and a proof of only S2 = O(Q^(-0.9)) would give |M_F(Q)| = O(Q^((alpha+0.1)/2)) and no ceiling at all. The converse needs the whole k sum controlled from Mertens bounds, hence the dilated sums M_F(x; d) at d > 1, and the sections below settle everything about those sums except cancellation. So digit restriction of the denominator is invisible to the shape and expensive to prove, and nothing here is evidence for the conjecture it runs from.

The dilates are regular languages

For a design carrying the digit 0 every dilate is a regular language, and its automaton is long multiplication. Proved. Read c in its base from the least significant digit; multiplying by d carries a value r that never reaches d, since floor((d(base-1) + d - 1)/base) = d - 1, so the d carries are the states of a deterministic automaton: from carry r the digit e writes the output digit (de + r) mod base, which must lie in F, and moves to the carry floor((de + r)/base). After level digits dc is the level output digits with the terminal carry r_level written above them, so the run accepts exactly when r_level lies in Acc_d = {0} union (S_F intersect [1, d)), a set of size A_F(d-1) + 1. So the dilated Mertens sums run over regular sets rather than over digit designs. The hypothesis is load-bearing rather than cosmetic: without 0 in F the run tests every one of the level padded output digits, and a leading output digit 0 is not a digit of dc, so the automaton recognises the padded set of mobius instead. At base 3 with F = {1,2}, d = 1 and level = 3 it reads 8 where the true count is 14, which is what A_F(base^level) = (|F|^(level+1) - |F|)/(|F|-1) gives there, and over bases 3, 4 and 5, every F, every d <= 6 and every level <= 5 there are 0 mismatches in the 840 cases carrying 0 and 399 in the 750 without it.

The automaton's transfer matrix is T_d(r, r') = #{e < base : (de + r) mod base in F, floor((de + r)/base) = r'}, and it counts the dilate, #{c < base^level : dc in S_F} = e_0 T_d^level 1_(Acc_d), for a design carrying 0. Every column of T_d sums to exactly |F|, in every base, at every digit set and every d, with no hypothesis at all. Proved: the pairs (e, r) in [0, base) x [0, d) are in bijection with v = de + r in [0, d base) 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, d base) for every r' < d. The all-ones vector is therefore a positive left eigenvector and the spectral radius of T_d is |F| for every d: a 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, and there #{c < base^level : dc in S_F} <= |F|^level with constant 1, again for a design carrying 0. Verified: over d <= 64 on base 3 {0,1} and base 10 without 9 no column is off |F|, rows are off |F| at 21 and 38 of the 64 and every one of those d shares a factor with the base, and at base 3 {0,1} with d = 2 the matrix [[1,1],[1,1]] with both carries accepting counts 2^level - 1 against literal enumeration of {c : 2c in S_F} at every level <= 12, both reading 4095 at x = 3^12 = 531441 (lab/py/restricted-franel).

One dilate is free: if 0 in F then M_F(x; base^j d) = M_F(x; d). Proved, since appending zero digits neither enters nor leaves S_F. So at base 3 {0,1} the d = 3 column is the d = 1 column, M_F(3^12; 3) = 56 with peak 61, and it is the lever that fixes the rate below.

What the matrix replaces is the digit symbol. The transform of a dilate is sum of e(ct) over c < base^level with dc in S_F, and decomposing over automaton paths gives e_0 M(t) M(base t) ... 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. Proved, for a design carrying 0, with the count identity above. That is the Mobius page's ladder prod_j g_F(base^j t) with the scalar symbol replaced by a matrix, and the replacement is what the route costs: an ordered product of non-commuting matrices does not factor, so the sup-over-shift l^1 exponent that carries a Type I estimate for a digit design has no scalar analogue on a dilate. The matrix form gives the exact count at t = 0, the exact mass constant, and exact evaluation at any t; it gives no cancellation in mu, and the Type II wall stands where it stands at d = 1.

The mass constant the matrix gives is the accepting set, and the exact hypothesis for that is a second coprimality, to Delta_F. For gcd(d, base) = 1 both sums make T_d/|F| doubly stochastic, so the stationary law is uniform on each closed class, and where the carry chain is irreducible 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. Conjecture under gcd(d, base Delta_F) = 1. Coprimality to the base alone is not enough: at base 3 with F = {0, 2} and Delta_F = 2 the dilate d = 2 has T_2 = [[2,0],[0,2]], carry 1 is unreachable from carry 0, the counts are 2, 4, 8, 16, 32, 64, 128, 256 at level = 1 to 8, exactly |F|^level, and the constant is 1 against #Acc_2/2 = 1/2. The split is clean where it is swept: over every base up to 7, every F carrying 0, every 2 <= d <= 24 coprime to the base, read at level = 400, there are 1747 agreements and 0 failures at gcd(d, Delta_F) = 1 and 0 agreements and 148 failures at gcd(d, Delta_F) > 1. Verified to three decimals at level = 24 at every coprime d metered, both 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 (lab/py/restricted-franel).

That saving is not uniform in d. Refuted, and the base-power ladder is what refutes it: 0 in F makes (base^j)^(-1) S_F equal to S_F, so the constant at d = base^j is 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: base 10 without 9 reads 1.1111, 1.1358, 1.1111, 1.1413, 1.0700, 1.0343 at d = 2, 4, 5, 8, 16, 32 against the ceilings 0.968781, 0.938537, 0.929003, 0.909237, 0.880851, 0.853352, and the accepting-set law fails there too, those d carrying #Acc_d/d = 1, 1, 1, 1, 0.9375, 0.90625. The last digit of an element of S_F is uniform on F and F is unbalanced modulo a prime dividing the base, so no equidistribution of S_F modulo d is available at base-smooth d. What survives is the constant on the base-smooth part alone: over the 29 base-smooth d <= 1000 at base 10 without 9 it lies in [0.9273, 1.1637] and over every d <= 200 the inflation of the constant over its value at the coprime part of d lies in [0.9375, 1.1413], bounded on the metered range and unmeasured past it.

The converse under one hypothesis

The converse then closes on one hypothesis, and getting its dependence on d right is the whole difficulty. (U), the uniform bound the accepting set suggests, is false for every design carrying both 0 and 1 with alpha < 1. Refuted, and that is both designs metered here. The free dilate above gives M_F(x; base^j) = M_F(x), so (U) at d = base^j demands |M_F(x)| <= C_eps base^(j(alpha-1)/2) x^(alpha/2+eps) for every j, and alpha < 1 drives the right side to 0 at fixed x, forcing M_F identically zero against M_F(1) = 1. At base 3 {0,1} and 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 to 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. The cause is the refutation above: the exponent (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 it is 1.

(U') is square-root cancellation against the accepting-set law's main term A_F(Q)/d_co, up to the base-smooth factor, and not against each dilate's own mass, which departs from that term; its d = 1 case is exactly the square-root ceiling the implication above already delivers. Its constant term is load-bearing: without it the bound fails for every design carrying 0 and 1 with alpha < 1. Refuted: d = base^k + 1 lies in S_F and is coprime to the base, so M_F(1; d) = mu(1) = 1, while d_co^((alpha-1)/2) = d^((alpha-1)/2) tends to 0 as k grows; the same witness kills (U) at x = 1.

On a design carrying 0 with alpha < 1, (U') forces M_F to be bounded, abs M_F(x) <= C_eps at every x. Proved: at d = base^k + 1 and c < base^k the product dc = c base^k + c has no carries, so the dilate below base^k is S_F itself and M_F(x; d) = M_F(x) for x < base^k, while d_co = d; (U') there reads abs M_F(x) <= C_eps (1 + d^((alpha-1)/2) x^(alpha/2+eps)), and letting k grow at fixed x drives the second term to 0. At base 3 {0,1} the design's own meter reads log max_(y <= x) abs M_F(y) / log x = 0.311823 at x = 3^12 (lab/py/restricted-franel, verb dilate), against the exponent 0 a bounded M_F allows.

(U') gives G_F(Q) = O_eps(Q^(alpha + eps)), hence sum_j delta_j^2 = O_eps(Q^(-1+eps)), the denominator lane's conjecture. Proved. The entries of the kernel are nonnegative and x_d = 0 unless N_F(Q; d) >= 1, so G_F(Q) <= C_eps^2 (h + z)^T K (h + z) with K the kernel, h_d = [N_F(Q; d) >= 1] and z_d = d_co^((alpha-1)/2) (Q/d)^(alpha/2+eps), both indexed by d <= Q like x. The kernel is positive semidefinite, by the Jordan form below, so Cauchy-Schwarz in it gives (h + z)^T K (h + z) <= (sqrt(h^T K h) + sqrt(z^T K z))^2. h^T K h <= B(Q), since h_d <= sqrt(N_F(Q; d)), and B(Q) = O_eps(Q^(alpha+eps)) by the surrogate theorem below. The terms of z^T K z are nonnegative, so it is at most the same sum over all d, e >= 1; for that sum write d = a d_co and e = b e_co with a and b supported on the primes dividing the base; the two parts have disjoint prime support, so gcd(d,e) = gcd(a,b) gcd(d_co,e_co) and the kernel sum factors. Each term is at most gcd(d,e)^2 (d e)^(-1-alpha/2-eps) (d_co e_co)^((alpha-1)/2) Q^(alpha+2eps); the coprime factor carries exponent -3/2-eps, and writing d_co = g u and e_co = g v with gcd(u, v) = 1 it is at most zeta(1 + 2eps) zeta(3/2 + eps)^2; the base factor is prod over p | base of sum over i, j >= 0 of p^(2 min(i,j) - (i+j)s) with s = 1 + alpha/2 + eps, which sums to prod over p | base of (1 + p^(-s))/((1 - p^(-s))(1 - p^(-alpha-2eps))), finite because alpha > 0. Then the node count's lower bound m_F(Q) >> Q^(1+alpha)/ln ln Q lets the rank form divide that down to Q^(-1+3eps). At (alpha-1)/2 + delta with delta > 0 the coprime g sum becomes sum of g^(-1+2delta), of size Q^(2delta), and the conclusion weakens to G_F(Q) = O(Q^(alpha + 2delta + eps)) with no threshold; at delta = 0 it is the harmonic sum and only the eps closes it, and the base factor never sees the exponent, so the criticality is unaffected by the refutation above.

That is one implication, and the natural route does not reverse it: the threshold gives |S_F(k, Q)| <= k (pi^2 G_F(Q)/6)^(1/2) termwise, and Mobius inversion of the frequency-m formula gives d M_F(Q/d; d) = sum over c | d of mu(d/c) S_F(c, Q), hence only |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') asks for decay in d_co down to its constant term. The hypothesis has a Mobius-free surrogate, since M_F(Q/d; d) sums over exactly the N_F(Q; d) members of S_F below Q divisible by d: square-root cancellation in that mass is (SR), and under (SR) the converse reduces to B(Q) = O(Q^(alpha+eps)), a divisor statement with no mu in it. That form stays consistent where (U) does not, reading |M_F(Q/base^j)| <= A_F(Q/base^j)^(1/2+eps) at d = base^j, which is the d = 1 ceiling again. The bound on B(Q) is Proved in the surrogate section, by the divisor bound, where it is also metered.

Twelve dilates at base 3 {0,1} and seven at base 10 without 9 are metered against (U') itself, by the ratio max |M_F(y; d)| over y <= x divided by d_co^((alpha-1)/2) x^(alpha/2), which (U') keeps bounded in d up to its x^eps wherever the yardstick is large. Verified: at base 3 {0,1} and x = 3^12 it 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, peak 1.0535 at d = 13; at base 10 without 9 and x = 10^7 it reads 1.1276, 0.9264, 0.5523, 0.6173, 0.8583, 1.2702, 0.4594 at d = 1, 2, 3, 4, 5, 7, 11, peak 1.2702 at d = 7. The exponent of (U) puts 1.1673 at d = 3 on base 3 against 1.0535 as the maximum over the coprime d, which is the ladder again. The raw readings log max |M_F(x;d)| over log x add nothing to this: 0.311823 at d = 1 and at most 0.292046 over d = 2, 4, 5, 7, 8, 11, 13, 16, 22, 31 at base 3, the d = 3 row being the d = 1 row by the free dilate rather than an independent reading, and 0.484570 at d = 1 against 0.489199 at d = 7 and 0.472377 at d = 2 at base 10, the crossing at x = 10^6 (0.495982 at d = 2 against 0.444731) reversing by x = 10^7; the local exponents between consecutive rungs swing over 0.24 to 0.845, so none of these readings is an exponent (lab/py/restricted-franel).

The threshold is a mean square

The equivalence that does hold is a mean-square one, and the Jordan totient is what writes it. With J_2(f) = f^2 prod_{p | f} (1 - p^(-2)) and y_f = sum_{m <= Q/f} x_{fm}/m, the kernel sum is a sum of squares, G_F(Q) = sum_{f <= Q} (J_2(f)/f^2) y_f^2, and (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. Proved: gcd(d,e)^2 = sum_{f | gcd(d,e)} J_2(f), and every weight lies in [6/pi^2, 1] because J_2(f)/f^2 = prod_{p | f} (1 - p^(-2)) is at least prod_p (1 - p^(-2)) = 1/zeta(2). The vector y is itself a family of dilated sums, y_f = sum_{n <= Q/f, fn in S_F} w(n) with w = mu * (1/id), w(n) = (1/n) sum_{c | n} mu(c) c = prod_{p | n} (1 - p)/n, so abs w(n) = phi(rad n)/n <= 1 and w(1) = 1: y_f is the Mertens sum of the dilate f^(-1) S_F with mu replaced by w. Proved, by writing n = mc in sum_m sum_c mu(c)/m.

The sandwich: (6/pi^2) (1 + ln Q)^(-2) sum_{d <= Q} M_F(Q/d; d)^2 <= G_F(Q) <= (1 + ln Q)^2 sum_{d <= Q} M_F(Q/d; d)^2 for every design and every Q >= 1. Proved. The map x -> y inverts by Mobius, x_f = sum_{m <= Q/f} mu(m) y_{fm}/m, since sum_m (mu(m)/m) sum_n x_{fmn}/n = sum_k (x_{fk}/k) sum_{m | k} mu(m) = x_f; 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 bounds the l^2 operator norm of a matrix by the geometric mean of its largest absolute row sum and its largest absolute column sum. Row f of either map sums to at most H_(Q/f) <= H_Q, the harmonic number; column d sums to sum_{m | d} 1/m = sigma(d)/d for the first map and to sum_{m | d} abs mu(m)/m = prod_{p | d} (1 + 1/p) <= sigma(d)/d for the inverse, and sigma(d)/d = sum_{k | d} 1/k <= H_d. So the squared norms of both maps are at most N_Q = H_Q max_{d <= Q} sigma(d)/d <= H_Q^2 <= (1 + ln Q)^2, giving sum_f y_f^2 <= N_Q sum_d x_d^2 and sum_d x_d^2 <= N_Q sum_f y_f^2, and with the weights above (6/pi^2) sum_d x_d^2/N_Q <= G_F(Q) <= N_Q sum_d x_d^2, which is the sandwich. Robin's bound sigma(d)/d < e^gamma ln ln d + 0.6483/ln ln d at d >= 3 sharpens N_Q to O(ln Q ln ln Q).

The ratio sum_f y_f^2 / sum_d x_d^2 has no constant bound over all vectors. Proved: for x_d = 1 at the divisors of the primorial N = prod_{p <= z} p <= Q and 0 elsewhere, y_f = sigma(N/f)/(N/f) at f | N and 0 elsewhere, so the ratio is prod_{p <= z} (1 + 1/p + 1/(2 p^2)) > sum_{p <= z} 1/p, which diverges with Q, of order ln ln Q. That witness rules out a constant in the sandwich and says nothing about whether N_Q is sharp; the gap between ln ln Q and ln Q ln ln Q is open. On the vectors a design produces the ratio is measured below.

The threshold is the mean square of the dilated sums: for every digit design with at least two digits, sum_j delta_j^2 = O_eps(Q^(-1+eps)) on the denominator set if and only if sum_{d <= Q} M_F(Q/d; d)^2 = O_eps(Q^(alpha+eps)), and if and only if sum_{f <= Q} y_f^2 = O_eps(Q^(alpha+eps)). Proved. By the rank form G_F(Q) = 12 m_F(Q) sum_j delta_j^2 + O(1), and the node count is Q^(1+alpha) up to ln ln Q; so the threshold is G_F(Q) = O(Q^(alpha+eps)), the weights move that to sum_f y_f^2 at no cost, and the sandwich moves it to sum_d x_d^2 and back at the cost of (1 + ln Q)^2, which the eps absorbs. On the full set every dilate is M and the statement collapses to M(x) = O(x^(1/2+eps)), since the d = 1 term is M(Q)^2 and M(x) << x^(1/2+eps) gives sum_{d <= Q} M(Q/d)^2 << zeta(1+2eps) Q^(1+2eps): that is Franel's theorem with M(x) = O(x^(1/2+eps)) standing for RH, and the two-sided form is what it reads one design over. On a design the d = 1 term alone is the ceiling above, and the rest of the sum is exactly what the ceiling misses: the threshold is square-root cancellation of the dilated Mertens sums in mean square over d <= Q, M_F(Q/d; d)^2 summed against the one yardstick Q^alpha and not against each dilate's own mass. (U') implies it, since it gives x_d^2 <= 2 C_eps^2 ([N_F(Q; d) >= 1] + d_co^(alpha-1) (Q/d)^(alpha+2eps)), the d with N_F(Q; d) >= 1 number at most sum_d N_F(Q; d) = sum_{m in S_F, m <= Q} tau(m) = O(Q^(alpha+eps)), and sum_{d <= Q} d_co^(alpha-1) (Q/d)^(alpha+2eps) << Q^(alpha+2eps), the coprime part summing d_co^(-1-2eps) and the base-smooth part a^(-alpha-2eps). The mean square is not shown to give (U') back, which is pointwise in d, and the natural route above fails exactly at that step; with the mean square in place of (U'), the biconditional holds.

Verified at base 3 {0,1}, Q = 3^4 to 3^8, and on the control at Q = 40, 81, 243. The Jordan form and the gcd double sum agree as exact rationals at every rung to Q = 729 and to 2.8e-14 and 5.7e-14 in floating point at Q = 2187 and 6561, the control at Q = 40 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] at all eight. G_F(Q)/sum_d x_d^2 reads 0.661794, 0.510529, 0.583192, 0.585221, 0.593465 on the design, sum_d x_d^2 running 19, 63, 113, 261, 673, and 1.093172, 1.337459, 1.582101 on the control, inside the proved corridor [(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 design's ratio shows no trend and the control's climbs, and no constant is claimed for either. The share M_F(Q)^2/sum_d x_d^2 of the d = 1 term reads 0.210526, 0.253968, 0.035398, 0.187739, 0.005944, so the dilates at d > 1 carry most of the mean square at every rung. The two maps' norms, read as Rayleigh quotients after 3000 power steps on the Q x Q matrices and so lower bounds, are 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 at Q = 81 to 6561, against N_Q^(1/2) = 3.733351, 4.338690, 4.906916, 5.412152, 5.995693 and 1 + ln Q = 5.3944 to 9.7889: both readings grow and both sit below the bound at every rung (lab/py/restricted-franel).

The surrogate and (SR)

The surrogate is a theorem, and it asks nothing of any single dilate. For every 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), with tau the divisor count and sigma the divisor sum; 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)). Proved. The lower bound is the diagonal d = e, where the kernel is 1 and every term of B(Q) is nonnegative, and sum_{d >= 1} N_F(Q; d) = sum_{m in S_F, m <= Q} tau(m) by exchanging the sums. For the upper bound sqrt(N_F(Q; d) N_F(Q; e)) <= (N_F(Q; d) + N_F(Q; e))/2 and the kernel is symmetric, so B(Q) <= sum_d N_F(Q; d) sum_{e <= Q} gcd(d,e)^2/(d e); grouping e by f = gcd(d, e) and writing e = f e', the inner sum is at most (1/d) sum_{f | d} f H_(Q/f) <= (sigma(d)/d)(1 + ln Q); exchanging again, sum_{d | m} sigma(d)/d <= tau(m) sigma(m)/m <= tau(m)(1 + ln m), since sigma(n)/n = sum_{k | n} 1/k grows along divisibility; and tau(m) <= C_eps m^eps with C_eps = (max_{k >= 0} (k+1) 2^(-k eps))^(2^(1/eps)), because tau(m) m^(-eps) = prod (k+1) p^(-k eps) over the prime powers p^k exactly dividing m, and every factor at p >= 2^(1/eps) is at most (k+1) 2^(-k) <= 1. Its readings against Q^alpha (ln Q)^2 fall for a reason. What the trivial bound lost is visible: N_F(Q; d) <= abs F (Q/d_co)^alpha, the column-sum lemma at the coprime part, summed over d gives Q, the g sum reading sum_g g^(-alpha), while the same sum read through m is sum_m tau(m) = Q^(alpha + o(1)), because at d near Q the count is [d in S_F] and vanishes off the design. 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 every Q <= 3^8; B(Q)/sum tau(m) climbs 2.9019 to 4.4174 and the bound over B(Q) climbs 2.728 to 5.402, both far under (1 + ln Q)^2; B(Q)/Q^alpha reads 12.5146, 17.8640, 24.7369, 31.5935, 39.0671 at Q = 3^4 to 3^8, local exponents 0.955, 0.927, 0.854, 0.824 falling toward alpha = 0.630930; and B(Q)/(Q^alpha (ln Q)^2) falls 0.6480, 0.5920, 0.5693, 0.5342, 0.5058 to Q = 3^8 and reaches 0.4007 at 3^12, consistent with Q^alpha times a power of a logarithm, no exponent claimed (lab/py/restricted-franel, verb accepting).

Under (SR), the pointwise square-root hypothesis, the denominator lane's conjecture holds, sum_j delta_j^2 = O_eps(Q^(-1+eps)), on every digit design with at least two digits. Proved. Square (SR) and sum over d: N_F(Q; d)^(1+2eps) <= A_F(Q)^(2eps) N_F(Q; d), so sum_{d <= Q} M_F(Q/d; d)^2 <= C_eps^2 A_F(Q)^(2eps) sum_{m in S_F, m <= Q} tau(m) = O(Q^(alpha+3eps)), and the mean-square equivalence above is the rest; through the kernel the same line reads G_F(Q) <= C_eps^2 A_F(Q)^(2eps) B(Q). So the converse rests on cancellation of mu alone, pointwise in each dilate against that dilate's own mass, and on no statement about how S_F sits in residue classes: not the accepting-set law, not (U'), not a Type I bound at any level. At d = 1 (SR) is the square-root conjecture for the meter on mobius, at d = base^j it is that conjecture again by the free dilate, and at every other d it is new. The threshold is not shown to return (SR), the natural route above stopping at log log d times Q^(alpha/2+eps), so the implication is one way as stated. The law's only office is to turn (SR) into (U'): N_F(Q; d) << A_F(Q)/d_co uniformly in d would make (SR) read abs(M_F(Q/d; d)) << d_co^(-1/2) Q^(alpha/2+eps), which is (U') up to the base-smooth factor a^(-alpha/2); the repunits refute that uniform bound on the coprime moduli, so (SR) and (U') ask for different things there and only the first is needed.

What the law says, read without the automaton's normalisation, is equidistribution. For a design carrying 0, A_F(d base^level) = #Acc_d abs F^level - 1 + [d in S_F] exactly, and the law's main term is A_F(Q)/d, never (#Acc_d/d) A_F(Q). Proved: the top digits of n <= d base^level being any element of Acc_d and the low level digits any padded string, which is where 0 in F enters, base 3 {1,2} reading A_F(9) = 6 against 4 at d = 1; so A_d(base^level)/abs F^level -> #Acc_d/d is the statement N_F(Q; d) ~ A_F(Q)/d read at Q = d base^level. At general Q only the second form survives: A_d(Q/d)/A_F(Q/d) is not #Acc_d/d, reading 0.5078 against 3/4 at d = 4, Q = 3^14, because the two counts carry different log-periodic ripples, while N_F(Q; d) d/A_F(Q) sits at 1, N_F(3^level; 4) being 2^(level-2) + 2^(level/2-1) - 1 at even level by the four fourth roots of unity, since 3^j = (-1)^j mod 4. Proved. Per class, for a design carrying 0: Delta_F = gcd(F), F = Delta_F F' with F' primitive, S_F = Delta_F S_(F') and N_F(Q; d) = N_(F')(Q/Delta_F; d/gcd(d, Delta_F)) exactly, so every class reduces to a primitive design, where for d = d_1 d_2 with d_1 | base^m and gcd(d_2, base) = 1 the law reads N_F(Q; d) ~ rho_F(d_1) A_F(Q)/d_2, rho_F(d_1) = N_F(m; d_1)/abs F^m the digit-string density of the split on mobius, independent of m once d_1 | base^m, equal to abs F^(-j) at d_1 = base^j by the free dilate; the ladder refutation above is this main term against base^(-j). Conjecture at every d_2 > 1 and fixed d, exact at d_2 = 1.

The repunits

The law is not uniform on the coprime moduli either, and the design's own repunits refute it. Proved. At base 3 {0,1} the repunit R_t is an element of S_F coprime to 3, and Delta_F = 1. Then N_F(3^(2t); R_t) = 2^t + 1 exactly: an element m <= 3^(2t) of S_F is m_0 + m_1 3^t with m_0, m_1 padded t-strings, or 3^(2t) itself, which is 1 mod R_t; R_t | m iff (3^t - 1) | 2(m_0 + m_1), and 0 <= m_0 + m_1 <= 3^t - 1 leaves m_0 + m_1 in {0, R_t, 3^t - 1}, which are the zero string, the 2^t complementary pairs, digit sums never carrying, and the pair of all-ones strings. So N_F(Q; R_t) d/A_F(Q) = (2^t + 1)(3^t - 1)/2^(2t+1) at Q = 3^(2t), unbounded like (3/2)^t, that is like d^(1 - alpha): at Q^(1/2) the count exceeds the law's main term by d^(1-alpha) and attains the column-sum lemma up to the constant 2^(-1-alpha).

The law fails as an upper bound uniform in d <= Q^theta at every theta = 1/(2r) down to 1/10, and at theta = 1/3. Proved. The same count at Q = 3^(kt) is the number of k-tuples of t-strings whose sum R_t divides, 2 3^t + 1 at k = 3, the triples with one or with two ones in every position and the triple of all-ones strings, and at k = 2r at least the pair count K_r(t) = #{(u_1..u_r, v_1..v_r) in D_t^(2r) : u_1 + ... + u_r = v_1 + ... + v_r} over the set D_t of the 2^t padded t-strings, by complementing the v strings, where K_1(t) = 2^t and K_2(t) = 6^t, the count of s in {0,1,2}^t weighted by 4^(ones of s). K_r(t) is the [0, 0] entry of the t-th power of a carry matrix on the signed carries reachable from 0, that class aperiodic since its diagonal is positive, so K_r(t) >= c_r rho_r^t with rho_r its Perron root and c_r > 0; 3 rho_r is the even-moment constant Lambda(2r) of mobius, whose grid moment is the same count with the terminal carry left free and differs from K_r (482 against 430 at r = 3, t = 2); and rho_r > 4^r/3 is certified exactly at r = 1 to 5 by a positive rational vector v with min_c (Mv)_c/v_c above 4^r/3, Lambda(2r)/4^r reading 1.5, 1.125, 1.026009, 1.005236, 1.001075, Lambda(10) = 1025.101288 from the cubic factor x^3 - 392 x^2 + 17469 x - 96228. So at d <= Q^(1/(2r)) the ratio is at least K_r(t) R_t/2^(2rt) >= c_r (3 rho_r/4^r)^t/2, unbounded, and at k = 3 the count 2 3^t + 1 against A_F(3^(3t)) = 8^t does the same at d < Q^(1/3). That Lambda(2r) > abs F^(2r) at every r, which would carry the failure to every fixed theta > 0, is the hair above fill^p that page measures and does not prove; the certified ratios stop at r = 5. Conjecture. The pointwise level of distribution of S_F at the residue 0 is therefore not a fixed power of Q, exactly as mobius proves for the supremum norm at the pinned moduli, the witness being a count and not a transform; averages over d are untouched, and (U') is not refuted there, but on this page's own numbers it is under strain. At d = R_t and x = x_t = floor(3^(2t)/R_t) = 2 3^t + 2 its yardstick Y_t = d^((alpha-1)/2) x^(alpha/2) is sqrt 2 (2/sqrt 3)^t up to a factor tending to 1, so (U'), for every eps > 0, forces the peak max_(y <= x_t) abs M_F(y; R_t) of the dilate's N = 2^t + 1 Mobius values to be at most C'_eps N^(c_0 + eps log_2 3) with c_0 = 1 - (log_2 3)/2 = 0.2075: peaks of size N^c with c > c_0 along a subsequence refute it. Square-root cancellation in the dilate's own mass is N^(1/2), above N^(c_0), because (U') asks there for cancellation against the law's main term and the mass exceeds that term by a factor of order (3/2)^t. The peaks read below sit at N^(1/2) times 0.3743 to 1.0155 to t = 24, and over Y_t they read 0.502 at t = 2, at most 1.719 to t = 8, then 4.456 at t = 9, 10.099 at t = 14, 40.736 at t = 20 and 78.037 at t = 24 (lab/py/restricted-franel, verb repunit, column peak/Y_t). No lower bound on the peaks is known, so this refutes nothing; the hypothesis the converse rests on is (SR), which asks at the repunits only for N^(1/2+eps). Verified: N_F(3^(kt); R_t) reads 5, 9, 17, 33, 65, 129, 257 at k = 2, 19, 55, 163, 487 at k = 3 and 71, 369, 2003 at k = 4, the last matching 6^t + 5^t + 3^t + 1, the carry count at k = 4; K_r(2) reads 4, 36, 430, 5796, 82404 at r = 1 to 5 against brute force; and the meter R(Q, d) = N_F(Q; d) d_co/A_F(Q) over every d <= Q^(1/2) at 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 d within 0.02 of the peak, and the peak over the coprime d is the same cell, where the Mobius values of the dilate stay at most 0.8839 of N_F(Q; d)^(1/2) in absolute partial sum, the largest at level = 11, d = 364, N_F = 32, peak 5 (lab/py/restricted-franel, verb accepting).

The Type I route to a pointwise bound never beats the column-sum lemma, and the pinned moduli make its failure exact. At Q = base^level and for a design carrying 0 the count is a sum over the d-th roots of unity, N_F(Q; d) = (1/d) sum_{a mod d} hat F_level(a/d) - 1 + [1 in F][d divides Q], hat F_level the digit transform of mobius, the a = 0 term the main term abs F^level/d; without 0 the count is a sum over the exact-length blocks and every bound below survives with the constant abs F/(abs F - 1). The l^1 norm that page certifies bounds the rest: sum_{a mod d} abs(hat F_level(a/d)) <= 2 e^(2 pi abs F) base^(a_1) abs F^level d^(a_1) with a_1 = log_base(B_base(F)/abs F) the bound the one-step constant B_base(F) certifies for the l^1 exponent, alpha_1 <= a_1, by the shifted-grid recursion of that page's step 2 run at the scale base^(k-1) < d <= base^k with the supremum over a window of width base^(-k) inside the sum, which costs the factor prod_{i <= k} (1 + 2 pi (base-1) abs F base^(1-i)/B_base(F)) <= e^(2 pi abs F) through the Lipschitz constant 2 pi (base-1) abs F of g_F, at most two of the points a/d falling in one window, and the trivial abs F^(level-k) on the remaining positions. Proved, and it reads N_F(Q; d) << Q^alpha d^(a_1 - 1), which the l^1 floor a_1 >= alpha_1 >= 1 - alpha of that page puts above the lemma's abs F (Q/d)^alpha at every d coprime to the base and every design, and by the free dilate at every d whose base-smooth part is a base power; at base 3 {0,1} the exact B_3(F) = 4 there gives a_1 = log 2/log 3 = alpha and the bound Q^alpha d^(alpha - 1), which is d^alpha above the law and feeds the kernel sum as sum_g g^(alpha - 1), of size Q^alpha, so B(Q) << Q^(2 alpha), above the trivial Q whenever alpha > 1/2. At d = base^t - 1 and level = 2t the loss is exact: base^t = 1 mod d folds the transform, hat F_(2t)(a/d) = hat F_t(a/d)^2, and Parseval mod d on t-strings, two of which agree mod d only when equal unless both 0 and base - 1 lie in F, gives sum_{a mod d} abs(hat F_(2t)(a/d)) = d (abs F^t + 2w), w = [0 in F][base - 1 in F], so the nonzero roots carry (base^t - 1) abs F^t - abs F^(2t) + 2wd, which is Q^((1-alpha)/2) times the main term: the l^1 sum over the d-th roots is of size A_F(Q) Q^((1-alpha)/2) at d = Q^(1/2) - 1, and the count there is 1 at base 3 {0,1}, the all-ones string, since hat F_t(a/d)^2 summed over a is d times the number of pairs of t-strings with u + v = 0 mod d. Proved: the failure is the triangle inequality's and not the law's. Verified: E_d, the nonzero roots' l^1 over abs F^level, reads 10.375 = 728/64 - 1 at d = 728, level = 12, and 4, 6.5625, 10.375, 16.0781, 24.625 at d = 3^t - 1, level = 2t, t = 4 to 8, while its maximum over d <= Q^(1/2) sits at the base power d = Q^(1/2) itself and reads 13.6785, 27.5490, 54.5261, 106.9952, 209.0447, doubling per two levels like d^alpha (lab/py/restricted-franel, verb accepting). What that page certifies as a Type I statement is an average and not a pointwise bound: under (E1), which Delta_F = 1 grants, the pair route's level of distribution on an initial segment holds to D <= Q^(1 - alpha_1) (log Q)^(-C) with a saving of any power of log Q summed over d <= D, which B_3(F) = 4 puts at Q^(0.369071) or above at base 3 {0,1}, and a sum over d is silent on any one d; nothing above asks for it.

(SR) at the repunits is the design's own Mobius sum through one affine map, and the map is exact. Proved. For every base >= 3, F = {0,1} and t >= 2, with x_t = floor(base^(2t)/R_t) = (base - 1)(base^t + 1), write B_t = {0} union (S_F intersect [1, base^t)) for the 2^t padded t-strings. Then R_t^(-1) S_F intersect [1, x_t] = {(base - 1) m + 1 : m in B_t} union {base^t + 1}, so N_F(base^(2t); R_t) = 2^t + 1 at every base and M_F(y; R_t) = sum_{m in B_t, (base - 1) m + 1 <= y} mu((base - 1) m + 1) + [y >= base^t + 1] mu(base^t + 1) for every y <= x_t. An m <= base^(2t) in S_F is m_0 + m_1 base^t with m_0, m_1 in B_t, or base^(2t) itself, which is 1 mod R_t; base^t = 1 mod R_t gives m = m_0 + m_1 mod R_t with 0 <= m_0 + m_1 <= 2 R_t, and 1 + 1 < base means the digit sums never carry, so m_0 + m_1 = R_t forces m_1 = R_t - m_0, the complement string, 2^t pairs, and m_0 + m_1 = 2 R_t forces both all ones. The complementary pair gives m = R_t base^t - m_0 (base^t - 1) = R_t (base^t - (base - 1) m_0) and base^t - (base - 1) m_0 = 1 + (base - 1)(R_t - m_0), the complement again: the shift by base^t and the reflection m -> R_t - m cancel, and the dilate is the affine image (base - 1) B_t + 1 of the block, odd at every odd base and mixed at an even one, base 4 and t = 2 reading {1, 4, 13, 16}; the all-ones pair gives R_t (base^t + 1) = R_(2t); and x_t = (base - 1)(base^t + 1) + floor((base - 1)/(base^t - 1)) with the floor 0 at t >= 2. At base 3 the image 2 B_t + 1 is exactly the set of n <= 3^t whose lowest nonzero digit is 1 and whose every other digit is 0 or 2, since 2m is a {0,2}-string and adding 1 turns its trailing twos into zeros and the first zero into a one; so 2 B_t + 1 = {3^k (6 w + 1) : 0 <= k < t, w in B_(t-1-k)} union {3^t}, mu(3^k (6w + 1)) = mu(3^k) mu(6w + 1) vanishes at k >= 2, and with the shifted sums T_s = sum_{w in B_s} mu(6 w + 1), T_0 = 1, the endpoint is M_F(x_t; R_t) = T_(t-1) - T_(t-2) + mu(3^t + 1), where mu(3^t + 1) = 0 at odd t because 4 | 3^t + 1 there; at a general base the same reading is T_(t-1) + mu(base) T_(t-2) + mu(base^t + 1) with T_s = sum_{w in B_s} mu(base (base - 1) w + 1). Proved. The base 3 {0,2} design is 2 S_(0,1), so d^(-1) S_(0,2) = (d/2)^(-1) S_(0,1) at even d and 2 (d^(-1) S_(0,1)) at odd d, and its own repunit 3^t - 1 = 2 R_t carries the base 3 {0,1} dilate and its readings exactly. Proved.

What the map costs the tree's machinery is nothing, and what it buys is nothing. The transform of the image is the block's transform read at (base - 1) theta with a phase, sum_{m in B_t} e(((base - 1) m + 1) theta) = e(theta) hat F_t((base - 1) theta), so every l^1, l^2 and supremum statement mobius certifies for hat F_t transfers with one fold: with g = gcd(d, base - 1), sum_{a mod d} abs(hat F_t((base - 1) a/d)) = g sum_{a' mod d/g} abs(hat F_t(a'/(d/g))), since (base - 1)/g is coprime to d/g and a -> ((base - 1)/g) a runs over the residues mod d/g exactly g times each; at g = 1 the l^1 sum over the d-th roots of unity is the block's own, and at d | base - 1 every term reads 2^t, the whole image sitting in the class 1 mod d, so at base 3 the image is odd throughout and the dilate below x_t has exactly one even element, 3^t + 1. The multiplicative-energy bound of mobius, E <= K^2 max_m r(m) with r(m) <= tau(m), holds for any K = 2^t integers below 3^t and misses the trivial bound by the same (1 - alpha)/2. Proved. So the Type II wall stands at the repunits exactly where it stands at d = 1, and 2m + 1 and m are coprime with neither factorisation determining the other, so nothing proved or metered about M_F at d = 1 transfers: (SR) at d = R_t and Q = 3^(2t) is square-root cancellation of mu on the marked design {v 1 0^k}, a two-state regular language of dimension alpha that the tree's Type I machinery reads exactly and its Type II machinery does not reach, at the same wall as the meter itself. The structure of the repunit converts the extreme case of (SR) into a fresh copy of its generic case, not into an identity that settles it, and the two conjectures stand side by side.

Verified: the generator is a PARI walk factoring the 2^t elements of the affine image one by one to t = 24 at base 3 and t = 18 at base 4; the independent witness is the literal enumeration of S_F below base^(2t) divisible by R_t, which reproduces the set and every reading at t <= 10; a Mobius sieve to 3^17 + 1 and 4^13 + 1 over the image is a second lane that agrees with the walk at every rung it reaches and asserts the set identity {base^t - (base - 1) m} = {(base - 1) m' + 1}, the marked decomposition and the T identity at each of them. At base 3 {0,1}, 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, the running maximum max_{y <= x_t} 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, and peak/N^(1/2) with N = 2^t + 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, above 1 once in twenty-three rungs, largest 1.0155 at t = 9, while peak/N^(0.6) is largest at 0.6136 at t = 5 and reads 0.1612 at t = 24; the shifted sums T_s read 0, -1, -3, -4, 0, -6, -5, -27, -21, -25, -51, -118, -168, -178, -257, -169, -88 at s = 1 to 17 and 121, 665, 1523, 1857, 1450, 4479, 4939, 1417, 12641 at s = 18 to 26, negative at every s from 6 to 17 and positive at every s from 18 to 26, so the early negative run is a finite run and not a sign law, and abs T_s / 2^(s/2) peaks at 1.8562 at s = 13 and stays under 2 at every s walked; the prefix walk over B_26 in ascending w changes sign 132 times, the last at 6w + 1 = 1128943015, and the whole class sum_{n <= x, n = 1 mod 6} mu(n) at x = 3^(s+1) - 2 is negative from s = 2 to 13 and positive at s = 14 and 15, an early negative run of its own (lab/py/shifted-sums, verbs walk and control); the endpoint at t = 14 is the difference -168 + 118 + 1 = -49. At base 4 {0,1} 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. peak/N^(1/2) stays under 2 at both bases at every t walked, so (SR) stands at the repunits on numerics with no exponent claimed (lab/py/restricted-franel, verb repunit).

The strict set

The strict set gets the same divisor identity and no Mertens face. S_F^s(1, Q) = sum_{b in S_F, b <= Q} sum_{d | b} mu(d) sum_{a <= b/d, da in S_F} e(a/(b/d)). Proved, by Mobius inversion of the coprimality condition followed by a -> da, and it is exact and inert: the inner sum is a digit-restricted exponential sum over an arithmetic progression, the Type II object mobius has no bound for. That sum is also not real, reading 1 + e(1/3) = 0.500000000 + 0.866025404 i already at base 3 {0,1} and Q = 3 and -1.809016994 + 0.587785252 i at base 10 without 9 and Q = 10, so no Mertens-type sum over S_F can equal it. Refuted, at once for M_F(Q), for the count-weighted sum_{b in S_F, b <= Q} mu(b) phi_F(b) and for the normalised sum_{b in S_F, b <= Q} mu(b) phi_F(b)/phi(b), all three real; the refutation rests on those two witnesses alone, and the observation beside them, that the strict set also fails the pairing a -> b - a which makes the denominator set's sum real, is not shown to force a non-real sum. Its modulus rides the node count card of the strict set instead, |S_F^s(1,Q)|/card reading 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, and reaching modulus 374203.231 at Q = 3^11 against M_F(3^11) = -10. Verified (lab/py/restricted-franel). That is the mass reading of the meter on a digit design seen at frequency 1: a set whose frequency-1 sum is proportional to its own count has no cancellation there and does not equidistribute.

What stays open

Three statements stay open, and none carries an exponent of its own. The denominator lane's conjecture of the meter on a digit design is readable on the kernel, G_F(Q) = O(Q^(alpha+eps)) saying exactly what sum_j delta_j^2 = O(Q^(-1+eps)) says once the rank form is in hand, which is the Franel threshold one design over, and equal by the sandwich to sum_{d <= Q} M_F(Q/d; d)^2 = O(Q^(alpha+eps)); it follows from (U') and from (SR) alone, the Mobius-free half of the converse being the theorem above, and what stays open is cancellation in mu in mean square over the dilates, the masses being fixed by the divisor identity sum_d N_F(Q; d) = sum_m tau(m) and none of the d > 1 content of (SR) following from its d = 1 content; the law that would sharpen (SR) to (U') is refuted as a uniform statement at every level down to Q^(1/10), and at the repunits, the one modulus where the dilate is explicit, (SR) is the meter of the marked design 2 B_t + 1, the d = 1 wall in a fresh copy, reading under 1.02 N^(1/2) to t = 24. Conjecture. And S_F^s(1,Q)/card converges to the first Fourier coefficient of a limit measure of the strict set, nonzero; that measure is not uniform at base 3 {0,1}, where the interval [1/2, 2/3] is empty at every Q by the meter on a digit design, and until it is named there is no Franel-type equivalence to state on the strict set at all. Conjecture.

(U') fails at base 3 {0,1}. It forces M_F to be bounded, so it fails as soon as M_F is unbounded there, and the design's meter reads log max_(y <= x) abs M_F(y) / log x = 0.311823 at x = 3^12 (lab/py/restricted-franel, verb dilate), while the repunit peaks stand over the (U') yardstick at 78.037 by t = 24. Conjecture. With the denominator lane's conjecture it says the implication from (U') does not reverse: the threshold holds and (U') fails.

The paper The Franel threshold is a mean square writes this page out for an outside reader: the three identities, the Jordan sandwich with its logarithms and the primorial witness, the mean-square equivalence with the full set as Franel and Landau, the surrogate and (SR), and the repunit refutation, every theorem with its proof and every number with its verb.