# Digit strings across divisors

- 2026-09-01 [Proved] Orthogonality for digit strings against a divisor: `N_F(level; d, r) = (1/d) sum_{a mod d} e(-a r/d) prod_{j < level} g_F(a base^j/d)` with `g_F(t) = sum_{f in F} e(f t)`, by expanding the divisibility indicator in additive characters mod `d`, the digits being independent so the character sum factors over positions, the `a = 0` term giving `fill^level/d`; the attempt to break it rebuilds the whole residue vector by dynamic programming against brute-force string enumeration at four bases and checks its total against `fill^level` at every census cell, with no mismatch. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
- 2026-09-01 [Proved] The uniform geometric equidistribution bound: for `fill >= 2`, `d >= 2`, `(d, base) = 1` and `gcd(d, Delta_F) = 1` with `Delta_F` the digit-difference gcd, `|N_F(level; d, r) - fill^level/d| <= ((d-1)/d) fill^level (1 - 8/(fill^2 d^2))^level <= fill^level exp(-8 level/(fill^2 d^2))` for every `r` and `level >= 1`, since `|g_F(a/d)|^2 = fill^2 - 4 sum_{f < f'} sin^2(pi a (f' - f)/d)` and `d | a(f' - f)` at every pair would force `d/gcd(a, d) | Delta_F` hence `d | a`; the attempt to break it asserts the weaker form as an exact integer inequality at every census cell where the hypotheses hold, five bases and depths to `level = 96`, with no failure and largest observed-to-bound ratio `0.187` at `base = 100`, `F = {0,1}`, `level = 16`, and the hypothesis edge `d = 2`, `fill = 2` holds at bound factor `(1 - 1/2)^level`; the `d^(-2)` in the exponent is sharp in shape, since `d | base - 1` with `F` an arithmetic progression of difference `m'` and `a m' = 1 mod d` gives `|g_F(a/d)|/fill = sin(pi fill/d)/(fill sin(pi/d)) = 1 - Theta(fill^2/d^2)`. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
- 2026-09-01 [Proved] The dense-digit bound: for `F = {0..base-1}` minus `E` with `m = |E|`, `fill = base - m` and `(d, base) = 1`, `gamma_F(d) <= (d/2 + m)/fill` because `g_F` is the full Dirichlet kernel less `g_E`, `|D_base(a/d)| <= 1/(2||a/d||) <= d/2` and `|g_E| <= m`, so for `d/2 + m < fill` the error is at most `fill^level ((d/2 + m)/fill)^level` uniformly in `r`; the attempt to break it looks for the gain at fixed digit count, where the bound is vacuous and stays vacuous - at `d = 7` the per-digit rate falls `0.4869, 0.3312, 0.2484, 0.1104, 0.0167` as `fill` runs `2, 3, 4, 9, 99` but reads `0.4992` for `F = {0,1}` at `base = 100`, against the same ceiling `0.9010` that `F = {0,1}` carries at `base = 3`. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
- 2026-09-01 [Proved] A power saving over a whole level for dense digit sets: for `eps in (0,1)`, `base >= 4^(1/eps)`, `m <= base^(1-eps)/2` and `level >= 4/eps`, every `2 <= d <= base^(1-eps)` coprime to `base` has per-digit factor `(d/2 + m)/fill <= base^(-eps/2)`, so `sum over those d of |N_F(level; d) - fill^level/d| <= fill^level base^(1 - eps level/2) <= fill^level x^(-eps/4)` at `x = base^level`, a level of distribution `base^(1-eps)` with no conditional input; the attempt to break it pushes the level past a constant power of the base and fails, since summing the geometric bound alone caps the level at `d ~ sqrt(level)/fill`, and the census argmax at every family's deepest level is a divisor of `base^t - 1` with `t <= 8`, where no per-factor bound decays. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
- 2026-09-01 [Proved] The exact split across the base's own divisors: for `d = d1 d2` with `d1 | base^m`, `m <= level`, and `(d2, base) = 1`, the low `m` digits fix the value mod `d1` and reach the rest only through the invertible multiplier `base^m mod d2`, so `N_F(level; d) = sum over w in F^m with d1 | val(w) of N_F(level - m; d2, r_w)` with `r_w = -val(w) (base^m)^(-1) mod d2`, and the density splits as `rho_F(d1 d2) = (N_F(m; d1)/fill^m)(1/d2)`; the attempt to break it tests the natural guess `1/d1` for the base part and refutes it, the base part being a digit-string count, with the identity itself pinned against direct enumeration at `base = 6`, `d = 10`. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
- 2026-09-01 [Proved] The digit-gcd hypothesis is a wall: if `gcd(d, Delta_F) > 1` there is no equidistribution, witness `base = 3`, `F = {0,2}`, `d = 2`, where every value is even, `N_F(level; 2) = fill^level` and the normalized error `d |N_F(level; d) - fill^level/d| / fill^level` is exactly `1` at every `level`; the attempt to break the wall by sweeping the whole range instead of one divisor leaves it standing, the unrestricted worst error over `d <= 200` reading `1.0483` at `level = 32` pinned at `d = 164` against `0.019166` once `d` is required coprime to `Delta_F`, and such families reduce to a primitive one through the scaling bijection `S_(aF') = a S_(F')`. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
- 2026-09-01 [Proved] The second moment across residue classes: `sum_{r mod d} (N_F(level; d, r) - fill^level/d)^2 = (1/d) sum_{a not 0 mod d} prod_{j < level} |g_F(a base^j/d)|^2`, by Parseval mod `d` on the orthogonality identity, the mean being the `a = 0` term and no cross terms surviving; the attempt to break it looks for a hidden hypothesis and finds none, the identity holding for every `d >= 1` and every `F`, including the walls where the supremum bound is worthless, which is what makes it the one handle left at a pinned divisor. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
- 2026-09-01 [Verified] The divisor census of digit strings: exact dynamic-programming counts of `N_F(level; d)` over `2 <= d <= D` for ten families at `base = 3, 4, 5, 10, 100`, depths to `level = 96` and `D` to `500`, printing the worst normalized error, the multiplicative order of `base` at the argmax, the per-factor ceiling `gamma_F(d)` and the slack against the proved bound; the counts are pinned against brute-force string enumeration at four bases, the residue vector totals `fill^level`, and the argmax is a pinned divisor of `base^t - 1` with `t <= 8` at every family's deepest level, `d = 164` at `base = 3`, `d = 143` at `base = 10`, `d = 101, 303` at `base = 100`, with shallow depths straying (`d = 199`, `ord = 99`, at `base = 10`, `level = 6`). The slow column is the sparse one: `F = {0,1}` at `base = 100` reads worst normalized error `28.593, 14.590, 9.0340, 7.2034` at `level = 16, 32, 64, 96`, per-digit factor `0.9929`. Witness: lab/rs/rho-decoupling, mobius.md digit strings across divisors.
- 2026-09-01 [Conjecture] The orbit-mean law at a pinned divisor: for `F = {0..base-1}` minus one digit and `d = base^t - 1`, the worst orbit-mean damping is `fill^(-1/t) (1 + o(1))`, the orbit `a base^j mod d` carrying `t - 1` undamped points and one damped by `~ 1/fill`; at `base = 100`, `level = 12` the single-divisor probes read orbit mean `0.1059` at `d = base^2 - 1` against `fill^(-1/2)` and `0.2369` at `d = base^3 - 1` against `fill^(-1/3)`, with the proper divisor `d = 3367 | base^3 - 1` better at `0.0549` and `d = 101 | base + 1` pinned but harmless at `0.0261`, the kernel being flat across that whole orbit. Two values of `t` on one base with one dominant character are a check and not a law, and the `o(1)` is untested; `t = 4` needs the orbit product analysed rather than counted, the exact count at `d = base^4 - 1` being out of reach of the census. Witness: lab/rs/rho-decoupling, mobius.md digit strings across divisors.
- 2026-09-05 [Verified] The signed pinned sum against its absolute sum: over the squarefree moduli `e = (base^t - 1)/g`, `g | base - 1`, `e >= 2`, `t <= level <= 40`, with `T_level(e) = N_F(level; e) - fill^level/e`, the ratio `sum mu(e) T_level(e) / sum |T_level(e)|` reads `-0.211, -0.123, +0.069, -0.498` at `level = 10, 20, 30, 40` for `F = {0,1}`, `base = 3`, and `+0.812, -0.495, -0.127, -0.192` for one excluded digit at `base = 10`, swinging across `[-1, 1]` with no decay, `Abs_level/fill^level` at `2.1 * 10^-4` and `3.9 * 10^-12` at `level = 40`; counts exact by the carry DP pinned against brute force and the residue DP at every reachable `e <= 30000`, `mu` from a complete certified factorisation with zero unknown cofactors. Witness: mobius.md digit strings across divisors; lab/rs/rho-decoupling, the carry sweep and its five pinned tests.
- 2026-09-06 [Refuted] The adversarial pass on the divisor census: the geometric bound was attacked as an exact integer inequality at every census cell where its hypotheses hold, five bases and depths to `level = 96`, with zero failures and the closest cell at observed-to-bound ratio `0.187`; the hypothesis edges were attacked one at a time, `d = 2` with `k = 2` holding at bound factor `(1 - 1/2)^level`, the digit-gcd hypothesis breaking exactly where the proof says it must (`base = 3`, `F = {0,2}`, `d = 2`, normalized error `1` at every `level`, sweep worst `1.0483` at `level = 32`), and the base-coprimality hypothesis handled by the exact split rather than dropped; the search for decay at fixed digit count failed and is recorded as the slow column rather than smoothed away. The printed floats truncate at forty decimal digits, so every claim-bearing comparison runs in exact integers or fractions and no rate is quoted past what the exact columns carry. Witness: lab/rs/rho-decoupling.
- 2026-09-14 [Proved] The pinned-orbit law, two-sided at one excluded digit. For `t >= 1`, every `d` dividing `base^t - 1` with `d >= 2` and every `a` nonzero mod `d`, the full Dirichlet kernel's orbit product telescopes to `Prod_(j < t) abs(D_base(a base^j/d)) = 1` exactly, since `a base^t = a mod d` and no factor degenerates, so a closed shift orbit is invisible to the full digit set and every damping comes from the excluded digits. With `abs(g_F) <= abs(D_base) + m` and `abs(D_base(a base^j/d)) <= B = min(base, d/2)`, convexity of `log(e^y + m)` puts the maximum at a vertex of `{Sum_j log D_j = 0, log D_j <= log B}` and gives `Prod_(j < t) abs(g_F(a base^j/d)) <= (B + m)^(t-1) (m + B^(1-t))`. At one excluded digit that is sharp both ways: for `d = base^t - 1`, `t >= 2`, `base >= 10` and any single excluded digit, `fill^(-1/t) (1 - 9/base) <= max_(a not 0 mod d) (Prod_(j < t) abs(g_F(a base^j/d))/fill^t)^(1/t) <= fill^(-1/t) (1 + 3/(base-1))` uniformly in `t`, the lower bound witnessed by `a = 1`. So `t - 1` undamped positions and one damped by `~ 1/fill` is the truth and the `1 + o(1)` is a two-sided `O(1/base)` that does not grow with `t`; the constants `9/base` and `3/(base-1)` are stated at `base >= 10`. This supersedes the orbit-mean Conjecture row in OPEN, whose `base = 100`, `level = 12` probes `0.1059` and `0.2369` read the finite-depth error rate `(abs(N_F(level; d) - fill^level/d)/fill^level)^(1/level) d^(1/level)` and not the orbit maximum. Witness: mobius.md digit strings across divisors, lab/rs/rho-decoupling.
- 2026-09-14 [Proved] The `a`-average at a full pinned modulus is exact at every digit set and every number of excluded digits: for `d = base^t - 1`, `Sum_(a mod d) Prod_(j < t) abs(g_F(a base^j/d))^2 = d (fill^t + 2w)` with `w = 1` when both `0` and `base - 1` lie in `F` and `w = 0` otherwise, since `val` is injective on length-`t` strings with range `[0, d]`, so the congruent pairs are the diagonal plus the single wraparound pair of the all-`0` and all-`(base-1)` strings when both lie over `F`. Under the pinned-orbit law's hypotheses, one excluded digit and `base >= 10`, the worst orbit exceeds the average over all `a`, which is `fill^t + 2w`, by `fill^(t-2) e^(O(t/base))`; most of that average is its own `a = 0` term `fill^(2t)/d`, `970299/101` of `9803` at `base = 100`, `t = 2`, so the average a second moment sees, over `a` nonzero, is `(d (fill^t + 2w) - fill^(2t))/(d - 1)`, smaller again by `~ t/base` and `980298/4999` at the same cell, and the spread is wider than the exponent states rather than narrower. From `t = 3` on at most a `fill^(2-t)` fraction of residues sits near the worst orbit: the bad mass at a pinned divisor is spread, which is what the supremum over `r` gives up and an average over `a` buys. Witness: mobius.md digit strings across divisors.
- 2026-09-14 [Proved] The bisection bound and a level of distribution `x^(alpha/2)` for digit strings, up to one factor. For every `F` with `fill >= 1`, every `d >= 2` coprime to `base`, every `level >= 1` and uniformly in `r`, `abs(N_F(level; d, r) - fill^level/d) <= fill^(level/2) (1 + 2 base^((level+1)/2)/d)`: cut the string in the middle and bound each half's variance by `fill^b (1 + 2 base^b/d)`, an off-diagonal congruent pair of length-`b` strings needing `val(f) - val(f') = j d` with `0 < abs(j) <= (base^b - 1)/d` and each pair `(f', j)` fixing at most one `f`. Summed against `Sum_(d <= D) 1/d <= 1 + log D` this gives `Sum_(2 <= d <= D, (d,base) = 1) max_r abs(N_F(level; d, r) - fill^level/d) <= fill^level (D fill^(-level/2) + 2 sqrt(base) (1 + log D) (base/fill)^(level/2))` at every `base >= 3`, `level >= 1` and `D >= 2`, every `d` and not only the squarefree ones, supremum over the target residue and not only the residue `0`. At `D = x^theta` with `theta <= alpha/2` and `m = base - fill` excluded digits the whole sum is at most `3 sqrt(base) (1 + log x) fill^level x^(m/(2 fill log base))`: a level `x^(alpha/2 - o(1))` at every `theta` up to `alpha/2` at once, with a defect sub-power in `base` and a positive power in `x`, the exponent `1/(2(base-1) log base)` at one excluded digit sitting under `0.0011` at `base = 100`. At `d >= sqrt(base x)` the same bound reads `max_r abs(N_F(level; d, r) - fill^level/d) <= 3 fill^level x^(-alpha/2)`, asking nothing of `F` at all. The defect is the pair count's own overshoot `2 (base/fill)^b` over its mean `fill^(2b)/d` at the balanced depth `b = level/2`, and no cut point removes it. Witness: mobius.md digit strings across divisors.
- 2026-09-14 [Proved] The assembled level-of-distribution theorem for digit strings, and the one family it leaves. Fix `eps in (0,1)` and an integer `T_0 >= 2` and put `base_0(eps, T_0) = max(4^(1/eps), base_1)` with `base_1` any base satisfying `3 base_1^(-eps)/log base_1 <= eps/(16 T_0)`. For every `base >= base_0`, every `F = {0..base-1}` minus `E` with `1 <= m <= base^(1-eps)/2`, every `level >= max(6 T_0, 4/eps)` and `x = base^level`: at every level `D <= x` the sum of `max_r abs(N_F(level; d, r) - fill^level/d)` over `2 <= d <= D` coprime to `base` with `d <= base^(1-eps)` or `ord_d(base) <= T_0` is at most `(T_0 + 2)(1 + log x) fill^level x^(-eps/(8 T_0))`; at level `x^(alpha/2)` the full sum is at most `3 sqrt(base) (1 + log x) fill^level x^(m/(2 fill log base))`; every `d` coprime to `base` with `sqrt(base x) <= d <= x` has `max_r abs(N_F(level; d, r) - fill^level/d) <= 3 fill^level x^(-alpha/2)`; and at one excluded digit the full pinned moduli are damped together by `level fill^(2 - sqrt(2 level)) e^(4 level/base) fill^level`, superpolynomial in `level`, worst at `t ~ sqrt(2 level)`, and never a fixed power of `x`. No clause asks `d` squarefree and every clause is a supremum over the target residue. One family survives at a fixed level, the generic-order middle moduli `base^(1-eps) < d <= x^theta` coprime to `base` with `ord_d(base) > T_0`, where only the level-`alpha/2` clause applies and its single factor `x^(m/(2 fill log base))` is the whole distance to a fixed power; that factor is the wraparound overshoot shared by the pair-count certificate, the orbit moment and the additive large sieve over the Farey points, and no rearrangement of cuts, Cauchy-Schwarz or divisor bookkeeping tried here removes it. Witness: mobius.md digit strings across divisors.
- 2026-09-19 [Proved] The uniform geometric bound summed over a divisor range is microscopic: `D exp(-8 level/(fill^2 D^2)) < 1` fails past `D ~ sqrt(level)/fill`, so it certifies a level of that size and nothing like a power of `x`. Witness: exact arithmetic on the geometric bound of mobius.md DIGIT STRINGS ACROSS DIVISORS.
- 2026-09-19 [Proved] No moment past the second helps at a pinned divisor: at `d = base^t - 1` the `2r`-th orbit moment is `d` times an additive energy of length-`t` strings, and the pair-count certificate places it a factor `4 (base/fill)^(r t)` above its own mean `fill^(2 r t)/d`, a loss growing in `r`. Witness: the orthogonality and bisection bullets of mobius.md DIGIT STRINGS ACROSS DIVISORS, lab/rs/rho-decoupling.
- 2026-09-19 [Proved] The signed Type I weight is itself a Mertens-type sum: with `T_level(d) = N_F(level; d) - fill^level/d` and `P_level(e)` the primitive frequency sum, `sum_{d <= U} mu(d) T_level(d) = sum_{e >= 2} (mu(e)/e) M_e(U/e) P_level(e)` with `M_e(y) = sum_{f <= y, (f,e) = 1} mu(f)/f`, so `mu(e)` fixes only the sign and the signed route restates the wall one layer down. Witness: Mobius inversion over reduced denominators, carried out in the sentence that prints it.
- 2026-09-19 [Refuted] Any bound on `M_e(y)` uniform in `e` and tending to zero in `y`: at the primorial `e` of all primes up to `P` and `y = P` the only `f <= y` coprime to `e` is `f = 1`, so `M_e(P) = 1` exactly. Witness: that witness, exact.
- 2026-09-19 [Proved] No clause in the supremum norm is a fixed power of `x`, and clause (iv) is not slack: at one excluded digit and `base >= 10`, `d = base^t - 1` with `t = ceil(sqrt(level))` carries `max_r abs(N_F(level; d, r) - fill^level/d) >= fill^level x^(-O(1/sqrt(level)))` by the two-sided orbit law and the second moment across residues, so the sum over `2 <= d <= x^theta` is at least that at every fixed `theta > 0` and every `level >= max(9, 4/theta^2)`. Witness: the two-sided pinned orbit law of mobius.md DIGIT STRINGS ACROSS DIVISORS, lab/rs/rho-decoupling.
- 2026-09-19 [Proved] The wraparound defect is circular and not loose: with the pair-count certificate alone both halves of a cut need `base^b <= d`, so `b <= 2 log_base d` and the certificate reads `3 d^(1 - alpha) > 1`, while winning asks the depth `b ~ (2/alpha) log_base d` at which `fill^b ~ d^2` strings meet `d` classes and equidistribution there is the statement being proved. Witness: the bisection certificate of mobius.md DIGIT STRINGS ACROSS DIVISORS.