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.