--- title: The restricted Franel identity lead: Franel's identity on the Farey fractions whose denominators lie in a digit design: the exponential sums are dilated Mertens sums, the square-root threshold is equivalent to their square-root cancellation in mean square over the dilates, the dilates are regular languages read by long multiplication, their masses need no Mobius input, and the design's repunits refute the uniform law that would give a rate in `d`. figure: research-franel slug: franel --- [The Farey page](farey.md) 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](farey.md#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](farey.md). - 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](mobius.md); 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](stack.md) 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](farey.md#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](mobius.md). 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](mobius.md) 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](/wiki/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](/wiki/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](mobius.md)'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](#the-surrogate-and-sr), 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](mobius.md), 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](#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](mobius.md), 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](mobius.md), 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](mobius.md) 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](mobius.md), 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](mobius.md) 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](mobius.md), `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](mobius.md) 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](farey.md#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](farey.md#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](farey.md#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](../papers/franel-converse.md) 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.