# Franel on a digit design

- 2026-09-14 [Proved] At frequency 1 the exponential sum of the denominator-restricted Farey set IS the design's Mertens meter: with `S_F` the whole numbers whose every digit lies in a digit set, `sum of e(r) over r in {a/b reduced, b in S_F, b <= Q, 1 <= a <= b}` equals `M_F(Q) = sum of mu(b) over b in S_F, b <= Q`, since `sum over a mod b coprime to b of e(a/b) = mu(b)` by Mobius inversion against the complete sums; every denominator in `S_F` up to `Q = 10^5` has its literal sum of `phi(b)` roots of unity equal to `mu(b)`, worst deviation `1.09e-11` at `b = 86293` (base 3 digits `{0,1}`) and `1.36e-12` at `b = 7247` (base 10 without 9), 0 wrong roundings. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `ladder_check`)
- 2026-09-14 [Proved] At frequency `m` the same sum is `sum over d dividing m of d M_F(Q/d; d)`, where `M_F(x; d) = sum of mu(c) over c <= x with dc in S_F` is the Mertens function of the DILATED design `d^{-1} S_F`; the classical divisor-shifted Mertens sums are the case `S_F = Z`, where every dilate is `Z` and all of them collapse to `M`, while for a digit design `d^{-1} S_F` is not `S_F`, is not a digit design and carries no digit test, so each `d > 1` brings a new function; checked at `m = 1, 2, 3, 4, 5, 6, 12` on base 3 `{0,1}` at `Q = 2187`, base 10 without 9 at `Q = 1000` and the full-set control at `Q = 300`, exact integer against literal sum at every cell. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_denominator`)
- 2026-09-14 [Proved] The digit-restricted Franel identity, Fourier form: `sum over k nonzero of abs(S_F(k,Q))^2 / k^2 = (pi^2/3) G_F(Q)` with `G_F(Q) = sum over d, e of (gcd(d,e)^2/(d e)) M_F(Q/d; d) M_F(Q/e; e)`, a finite sum of exact rationals; the kernel is the Smith gcd matrix that already carries the moire correlation law and the Gaussian identity, so digit restriction moves the entries and never the kernel; checked against the literal Fourier side truncated at `abs(k) <= 200000` with the printed tail bound `2 m^2/K`, gap inside bound at base 3 `Q = 81` and `243`, base 10 `Q = 40` and the control `Q = 40`. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `kernel_sum` and `fourier_side`)
- 2026-09-14 [Proved] The digit-restricted Franel identity, rank form: if the node set has top node `1` and mean value `sum of rho_r = (m_F(Q)+1)/2`, then `G_F(Q) - 1 = 12 m_F(Q) sum_j delta_j^2` as exact rationals, with `m_F(Q) = sum of phi(b) over b in S_F, b <= Q` and `delta_j = rho_j - j/m_F(Q)`; closure under `r -> 1 - r` away from the node `1` is one sufficient condition for that mean value, holding for every denominator-restricted set and failing for every proper strict set; the proof is Parseval plus piecewise integration of `(A(v) - m v)^2`; `True` at base 3 `Q = 81, 243`, base 10 `Q = 40` and the control `Q = 40`, which regenerates Edwards section 12.2. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `farey_delta_square`)
- 2026-09-14 [Proved] The denominator lane's CONJECTURED shape implies the square-root ceiling for the design's Mertens meter: dropping every term but `k = 1` and `k = -1` from a sum of nonnegative terms gives `2 M_F(Q)^2 <= (pi^2/3) G_F(Q)`, which by the rank form is `4 pi^2 m_F(Q) sum_j delta_j^2 + pi^2/3`; with `A_F(Q) << Q^alpha` (the block count) and `m_F(Q) <= Q A_F(Q) << Q^(1+alpha)`, the conjecture `sum_j delta_j^2 = O(Q^(-1+eps))` forces `abs(M_F(Q)) = O(Q^(alpha/2+eps))`, the constant absorbed and no unproved input entering; the measured exponent is `-0.959` and `-0.899`, not `-1`, so only the conjecture yields the ceiling. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `scan_backward`)
- 2026-09-14 [Proved] The strict set's frequency-1 sum has an exact divisor form: it equals `sum over b in S_F, b <= Q of sum over d dividing b of mu(d) times sum of e(a/(b/d)) over a <= b/d with da in S_F`, by Mobius inversion of the coprimality condition followed by `a -> da`; the inner sum is a digit-restricted exponential sum over an arithmetic progression, the Type II object with no bound on the tree, so the identity is exact and inert; literal summation against the divisor route agrees to `6.28e-15` over the 64 denominators of base 3 `{0,1}` below 729 and to `1.95e-14` over the 162 of base 10 without 9 below 200. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `strict_ramanujan_divisor`)
- 2026-09-14 [Verified] Literal enumeration of the strict digit-restricted Farey set reaches the counts the sieve prints without enumerating a fraction: `278, 4286, 67561, 1080458` at `Q = 3^5, 3^7, 3^9, 3^11` on base 3 `{0,1}` and `1830, 147096, 11890654` at `Q = 10^2, 10^3, 10^4` on base 10 without 9, 7 rungs and no disagreement. (witness: `lab/py/restricted-franel` `strict_literal` against `lab/rs/farey-discrepancy` `design`)
- 2026-09-14 [Proved] For a design carrying the digit `0` the dilate `d^(-1) S_F = {c : dc in S_F}` is a regular language recognised least-significant-digit-first by a deterministic automaton whose `d` states are the carries of long multiplication by `d`: reading digit `e` from carry `r` writes the output digit `(de + r) mod base`, which must lie in `F`, and moves to carry `floor((de + r)/base)`, which stays below `d` by induction, and after `level` digits `dc` is the `level` output digits with the terminal carry `r_level` written above them, so acceptance is exactly that `r_level` has all its digits in `F`; the accepting set is `Acc_d = {0} union (S_F intersect [1, d))`, of size `A_F(d-1) + 1`. The hypothesis `0 in F` is load-bearing and not cosmetic: without it the run tests all `level` PADDED output digits, a leading output digit `0` is not a digit of `dc`, and the automaton recognises the padded set of the Mobius page instead, reading `8` at base 3 with `F = {1,2}`, `d = 1` and `level = 3` where the true count is `14`. Regularity of the dilate itself survives without the hypothesis; the count identity does not. So the dilated Mertens sums of the restricted Franel identity run over regular sets, not over digit designs. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `dilate_matrix`, `verb_converse` `0` mismatches of `840`)
- 2026-09-14 [Proved] The dilate's transfer matrix `T_d(r, r') = #{e < base : (de + r) mod base in F, floor((de + r)/base) = r'}` counts it, `#{c < base^level : dc in S_F} = e_0 T_d^level 1_(Acc_d)` for a design carrying `0`, and EVERY column of `T_d` sums to exactly `#F`, in every base, at every digit set and every `d`, with no hypothesis at all: the pairs `(e, r)` in `[0,base) x [0,d)` are in bijection with `v = de + r` in `[0, dq)` by the division algorithm, the column at `r'` counts the `v` with `v - base r'` in `F`, and the window `[base r', base r' + base)` lies inside `[0, dq)` for every `r' < d`, so exactly `#F` of them qualify. Hence the all-ones vector is a positive left eigenvector and the spectral radius of `T_d` is `#F` for every `d`: the dilate carries the design's own mass exponent as its Perron root. The rows sum to `g` times `#(F intersect (r + gZ))` with `g = gcd(d, base)`, so they equal `#F` whenever `gcd(d, base) = 1`, giving `#{c < base^level : dc in S_F} <= (#F)^level` at those `d` with constant `1`, again only for a design carrying `0`: base 3 with `F = {1,2}` and `d = 1` reads `14` at `level = 3` against `(#F)^level = 8`. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_dilate`, `verb_converse`)
- 2026-09-14 [Verified] The dilate automaton and its transfer matrix are checked against brute-force enumeration: over `d <= 64` at base 3 `{0,1}` and base 10 without 9 no column of `T_d` is off `#F`, while rows are off `#F` at 21 and 38 of the 64 respectively, every one of them at a `d` sharing a factor with the base; at base 3 `{0,1}` with `d = 2` the transfer matrix `[[1,1],[1,1]]` with both carries accepting counts `2^level - 1` at every `level <= 12`, agreeing with literal enumeration of `{c : 2c in S_F}` at every rung and reading `4095` at `x = 3^12 = 531441`; and the count identity's scope is exact on the sweep over `base = 3, 4, 5`, every `F`, every `d <= 6` and every `level <= 5`, with `0` mismatches in the `840` cases carrying the digit `0` and `399` in the `750` without it. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_dilate`, `verb_converse`)
- 2026-09-14 [Proved] The dilated meter is blind to the base's own powers: if `0 in F` then `M_F(x; base^j d) = M_F(x; d)` for every `j >= 0` and every `d`, since appending `j` zero digits neither leaves nor enters `S_F`, so the dilates repeat along every base-power ladder and only the base-prime part of `d` can move them. At base 3 `{0,1}` the `d = 3` column reproduces the `d = 1` column exactly, `M_F(3^12; 3) = M_F(3^12) = 56` with both peaks `61`. It is the lever that fixes the rate in the converse's hypothesis and that refutes the mass saving uniformly in `d`. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_dilate`)
- 2026-09-14 [Proved] The digit transform of a dilate is the transfer matrix in place of the digit symbol: for a design carrying `0`, `sum of e(ct) over c < base^level with dc in S_F` equals `e_0 M(t) M(qt) ... M(base^(level-1) t) 1_(Acc_d)` with `M(t)(r, r') = sum of e(et)` over the digits `e` carrying `r` to `r'`, and `M(0) = T_d`, by decomposing over automaton paths. That is the ladder of the Mobius page with the scalar symbol `g_F(base^j t)` replaced by a matrix, and the replacement is exactly what the route costs: the product no longer factors, so the sup-over-shift `l^1` exponent that carries a Type I estimate for a digit design has no scalar analogue here. The matrix form gives the exact count at `t = 0` and exact evaluation at any `t`, and gives no cancellation in `mu`; the Type II wall stands where it stands at `d = 1`. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_dilate`)
- 2026-09-14 [Proved] The converse of the restricted Franel identity, from (U') and with the dependence on `d` explicit. (U') gives `G_F(Q) = O_eps(Q^(alpha + eps))` and hence `sum_j delta_j^2 = O_eps(Q^(-1+eps))` on the denominator-restricted set, which is the denominator lane's conjecture. Write `d = a d_base` and `e = b e_base` with `a` and `b` supported on the primes dividing `base`; the two parts have disjoint prime support, so `gcd(d,e) = gcd(a,b) gcd(d_base,e_base)` and the kernel sum FACTORS. Each term is at most `gcd(d,e)^2 (de)^(-1-alpha/2-eps) (d_base e_base)^((alpha-1)/2) Q^(alpha+2eps)`; the coprime factor carries exponent `-3/2-eps` and, writing `d_base = g u` and `e_base = g v` with `gcd(u,v) = 1`, is at most `zeta(1 + 2eps) zeta(3/2 + eps)^2`; the base factor is the product over `p` dividing `base` of `sum over i, j >= 0 of p^(2 min(i,j) - (i+j)s)` with `s = 1 + alpha/2 + eps`, which sums in closed form to the product of `(1 + p^(-s))/((1 - p^(-s))(1 - p^(-alpha-2eps)))` and is FINITE because `alpha > 0`. Then `m_F(Q) >> Q^(1+alpha)/log log Q`, the `>> Q^alpha` members of `S_F` in the top block below `Q` each exceeding `Q/base` with `phi(b) >> b/log log b`, so the rank form divides the bound down to `Q^(-1+3eps)`. The exponent `(alpha-1)/2` on `d_base` is critical, not chosen: at `(alpha-1)/2 + delta` on `d_base` the same argument gives only `G_F(Q) = O(Q^(alpha + 2delta + eps))` and no threshold, the coprime `g` sum becoming `sum of g^(-1+2delta)` of size `Q^(2delta)`, and at `delta = 0` it is the harmonic sum, convergent only through the `eps`; the base factor never sees the exponent. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_dilate`)
- 2026-09-14 [Verified] The raw exponent readings on the dilates separate nothing and are not exponents: over base 3 `{0,1}` to `x = 3^12 = 531441` the reading `log max abs M_F(x;d)` over `log x` is `0.311823` at `d = 1` and at most `0.292046` over `d = 2, 4, 5, 7, 8, 11, 13, 16, 22, 31`, the `d = 3` row being the `d = 1` row by the free base powers rather than an independent reading; over base 10 without 9 to `x = 10^7` the reading is `0.484570` at `d = 1` against `0.489199` at `d = 7` and `0.472377` at `d = 2`, and the crossing seen at `x = 10^6`, `0.495982` at `d = 2` against `0.444731` at `d = 1`, reverses by `x = 10^7`, peaks `2026` against `2466`. The local exponents between consecutive rungs swing over `0.24` to `0.845`, so none of these readings is an exponent and none of them tests the converse's hypothesis, which is a statement about the ratio to `d_base^((alpha-1)/2) x^(alpha/2)` and is metered separately. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_dilate`)
- 2026-09-14 [Conjecture] The denominator-restricted set's Franel analogue is `sum_j delta_j^2 = O(Q^(-1+eps))`, equivalently `G_F(Q) = O(Q^(alpha+eps))`; the forward half of an equivalence with the square-root conjecture for `M_F` is open and needs the dilated sums `M_F(x; d)` for `d > 1`, for which the desk has no bound, so only the implication above is proved and no exponent is claimed here. (witness: farey.md, The restricted Franel identity)
- 2026-09-14 [Conjecture] The strict set's frequency-1 sum divided by its node count converges to the first Fourier coefficient of a limit measure of the strict set, nonzero; the readings are `0.335693, 0.343837, 0.345905, 0.346338` at `Q = 3^5, 3^7, 3^9, 3^11` and `0.015138, 0.012250, 0.011561` at `Q = 10^2, 10^3, 10^4`, four and three nested rungs and no exponent claimed; that limit measure has no definition on the tree, and until it is named there is no Franel-type equivalence to state on the strict set. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_strict`)
- 2026-09-14 [Conjecture] (U'), the surviving hypothesis with the dilate's true mass: `abs M_F(x; d) = O_eps(d_base^((alpha-1)/2) x^(alpha/2 + eps))` uniform in `d >= 1` and `x >= 1`, with `d_base` the part of `d` coprime to the base. It is square-root cancellation in each dilate's own mass read correctly, since `d_base^((alpha-1)/2)` is the square root of the accepting-set constant at `d_base` and the base-smooth inflation is bounded; it is consistent with the free base powers by construction because `(base^j d)_base = d_base`; and its `d = 1` case is exactly the square-root ceiling the forward implication already delivers. Metered as a ratio it does not fire: `max abs M_F(y;d)` over `y <= x` divided by `d_base^((alpha-1)/2) x^(alpha/2)` reads `0.9531, 0.7991, 0.9531, 0.6054, 0.9883, 0.3580, 0.5504, 0.8269, 1.0535, 0.4431, 0.5528, 0.3828` at `d = 1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 22, 31` on base 3 `{0,1}` at `x = 3^12`, and `1.1276, 0.9264, 0.5523, 0.6173, 0.8583, 1.2702, 0.4594` at `d = 1, 2, 3, 4, 5, 7, 11` on base 10 without 9 at `x = 10^7`, while the refuted exponent puts `1.1673` at `d = 3` against `1.0535` as the coprime maximum. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_dilate`)
- 2026-09-14 [Conjecture] The dilate's mass constant is read off the automaton's accepting set under a SECOND coprimality: writing `Delta_F` for the gcd of the differences of the digits in `F`, for `gcd(d, base Delta_F) = 1` the matrix `T_d` over `#F` is doubly stochastic, the carry chain is irreducible, its stationary law is uniform, and `A_d(base^level)/(#F)^level` converges to `#Acc_d/d = (A_F(d-1) + 1)/d`, which is `O(d^(alpha-1))` and is exactly the saving a level of distribution for `S_F` at the modulus `d` would give. Verified to three decimals at `level = 24` at every printed `d` coprime to the base, both metered designs having `Delta_F = 1`: base 3 `{0,1}` reads `1.0000, 1.0000, 0.7501, 0.8000, 0.5714, 0.5001, 0.5455, 0.5394, 0.5001, 0.3636, 0.3548` at `d = 1, 2, 4, 5, 7, 8, 11, 13, 16, 22, 31` against `1, 1, 0.75, 0.8, 0.571429, 0.5, 0.545455, 0.538462, 0.5, 0.363636, 0.354839`, and base 10 without 9 reads `1.0000, 1.0000, 1.0000, 0.9091, 0.9231, 0.8264, 0.8272, 0.8148` at `d = 1, 3, 7, 11, 13, 121, 243, 729` against `1, 1, 1, 0.909091, 0.923077, 0.826446, 0.827160, 0.814815`. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_dilate`)
- 2026-09-14 [Conjecture] The converse's hypothesis has a Mobius-free surrogate the lab can meter: square-root cancellation in each dilate's own mass is `abs M_F(Q/d; d) <= N_F(Q; d)^(1/2+eps)` with `N_F(Q; d) = #{m in S_F : m <= Q, d divides m}`, since the sum for `M_F(Q/d; d)` runs over exactly those `m`, so under that hypothesis the converse reduces to the divisor statement that `B(Q) = sum over d, e of gcd(d,e)^2/(de) times sqrt(N_F(Q;d) N_F(Q;e))` is `O(Q^(alpha+eps))`, which mentions no Mobius function at all. This form stays consistent where the `d`-uniform bound above does not, reading `abs M_F(Q/base^j) <= A_F(Q/base^j)^(1/2+eps)` at `d = base^j`, which is the `d = 1` ceiling again. Measured at base 3 `{0,1}`: `B(Q)/Q^alpha` reads `12.5146, 17.8640, 24.7369, 31.5935, 39.0671` at `Q = 3^4` to `3^8` with local exponents `0.955, 0.927, 0.854, 0.824` falling toward `alpha = 0.630930`, and `B(Q)` over `Q^alpha (ln Q)^2` falls `0.6480, 0.5920, 0.5693, 0.5342, 0.5058`, so the range is consistent with `Q^alpha` times a power of a logarithm and no exponent is claimed. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `smith_bilinear`)
- 2026-09-14 [Refuted] No Mertens-type sum over `S_F` equals the strict set's frequency-1 sum, because that sum is not real: at base 3 `{0,1}` and `Q = 3` it is `1 + e(1/3) = 0.5 + (sqrt 3/2) i` and at base 10 without 9 and `Q = 10` it is `-1.809016994 + 0.587785252 i`, both exact algebraic sums evaluated past `1e-9`, while `M_F(Q)`, the count-weighted `sum of mu(b) phi_F(b)` and the normalised `sum of mu(b) phi_F(b)/phi(b)` are all real; the two witnesses carry the refutation alone; beside them sits an observation and not a mechanism, that the strict set also fails the pairing `a -> b - a`, failure of which is not shown to force a non-real sum. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_strict`)
- 2026-09-14 [Refuted] (U), the `d`-uniform dilated bound `abs M_F(x; d) = O_eps(d^((alpha-1)/2) x^(alpha/2 + eps))`, holds for NO design carrying both `0` and `1`, so it cannot be the hypothesis of the converse. Base powers being free gives `M_F(x; base^j) = M_F(x)`, so (U) at `d = base^j` demands `abs M_F(x) <= C_eps base^(j(alpha-1)/2) x^(alpha/2+eps)` at every `j >= 0`, and `alpha < 1` drives the right side to `0` at fixed `x`, forcing `M_F` identically zero against `M_F(1) = mu(1) = 1`. Base 3 `{0,1}` at `x = 3^12`: the left side is `56` at every `j = 0` to `12` while `d^((alpha-1)/2) x^(alpha/2)` falls `64.0000, 52.2558, 42.6667, 34.8372, 28.4444, 23.2248, 18.9630, 15.4832, 12.6420, 10.3221, 8.4280, 6.8814, 5.6187` over `d = 3^0` to `3^12` and the ratio climbs `0.875, 1.072, 1.313, 1.607, 1.969, 2.411, 2.953, 3.617, 4.430, 5.425, 6.645, 8.138, 9.967`, unbounded in `j`. The cause is that `(alpha-1)/2` is the square root of the dilate's mass constant only where that constant is `d^(alpha-1)`, and on the base-power ladder the constant is `1`. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_converse`)
- 2026-09-14 [Refuted] The converse is one implication and not an equivalence: the threshold does not give (U') by the natural route. From `G_F(Q) = O(Q^(alpha+eps))` the Fourier form gives termwise `abs S_F(fill,Q) <= fill (pi^2 G_F(Q)/6)^(1/2)`, and Mobius inversion of `S_F(m,Q) = sum over d dividing m of d M_F(Q/d;d)` gives `d M_F(Q/d;d) = sum over c dividing d of mu(d/c) S_F(c,Q)`, hence only `abs M_F(Q/d;d) <= (sigma(d)/d)(pi^2 G_F(Q)/6)^(1/2)`, which is `<< log log d` times `Q^(alpha/2+eps)` and GROWS in `d` where (U') needs `d_base^(-1/2-eps)` decay. So no biconditional is available, and none is claimed. (witness: farey.md, The restricted Franel identity)
- 2026-09-14 [Refuted] Coprimality to the base alone does NOT give the accepting-set constant. At base 3 with `F = {0,2}`, where `Delta_F = 2`, the dilate `d = 2` is coprime to the base and carries `T_2 = [[2,0],[0,2]]`, so carry `1` is unreachable from carry `0`, the closed class is `{0}` and the uniform stationary law is read on the wrong class: exhaustive counts are `2, 4, 8, 16, 32, 64, 128, 256` at `level = 1` to `8`, exactly `(#F)^level`, so the constant is `1` against `#Acc_2/2 = 1/2`. The split is exact where it is swept, over every `base <= 7`, every `F` carrying `0` and every `2 <= d <= 24` coprime to `base`, read at `level = 400`: `1747` agreements and `0` failures at `gcd(d, Delta_F) = 1`, `0` agreements and `148` failures at `gcd(d, Delta_F) > 1`. That same constant `1` sits at a `d` coprime to the base, so it also kills `O(d^(alpha-1))` there, `1` against `2^(alpha-1) = 0.6444`, and the base-smooth mechanism is therefore one cause and not the only one. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_converse`)
- 2026-09-14 [Refuted] The mass saving `A_d(x) = O(d^(alpha-1) x^alpha)` does not hold uniformly in `d`, and the base-power ladder is what refutes it: with `0 in F` the dilate at `d = base^j` is `S_F` itself, so `K_d = 1` exactly at every `j` while the ceiling `base^(j(alpha-1))` tends to `0`, and `A_d(x)/(d^(alpha-1) x^alpha)` is at least `base^(j(1-alpha))`, UNBOUNDED. Off the ladder the base-smooth dilates are denser than the design in the same way: at base 10 without 9 `K_d = A_d(base^level)/(#F)^level` reads `1.1111, 1.1358, 1.1111, 1.1413, 1.0700, 1.0343` at `d = 2, 4, 5, 8, 16, 32` against the claimed ceilings `0.968781, 0.938537, 0.929003, 0.909237, 0.880851, 0.853352`, and the accepting-set law fails there too, those same `d` carrying `#Acc_d/d = 1, 1, 1, 1, 0.9375, 0.90625`. The dilate is denser because the last digit of an element of `S_F` is uniform on `F` and `F` is not balanced modulo a prime dividing the base, so no equidistribution of `S_F` modulo `d` is available at base-smooth `d`. What it costs is the converse's first hypothesis and not its conclusion, the repaired hypothesis asking the rate on the coprime part only: over the `29` base-smooth `d <= 1000` at base 10 without 9 the constant lies in `[0.9273` at `d = 512`, `1.1637` at `d = 625]` and over every `d <= 200` its inflation over the value at the coprime part of `d` lies in `[0.9375` at `d = 112`, `1.1413` at `d = 88]`, bounded on the metered range and unmeasured past it. (witness: farey.md, The restricted Franel identity; `lab/py/restricted-franel` `verb_dilate`, `verb_converse`)