mertens-numerology

• Computes the kernel-sum constants attached to a digit set in base base that omits m of the base digits, and the exponent bookkeeping they force.
• With F the kept digits, the digit symbol is g_F(t) = sum_{d in F} e(d t) and the Dirichlet kernel is D_base(t) = sum_{d=0}^{base-1} e(d t); the one-step constant of the shifted-grid l^1 recursion is B_base(F) = sup_t sum_{r mod base} |g_F((t+r)/base)|.
• Splitting |g_F| <= |D_base| + |g_E| over the excluded set E and using Parseval on the base shifted points gives the elementary upper bound B_base(F) <= base PB_base(m) with PB_base(m) = sqrt(m) + Phi_base/base.
• The kernel constant is Phi_base = (4/pi) base + (2 base/pi) H(ceil((base-2)/2)) + (1 - 2/pi)(base - 2) + 0.727 with the harmonic upper bound H(n) = ln n + gamma + 1/(2n); it is ~ (2/pi) base ln base, so PB_base(m)/base -> 0 but only logarithmically.
• Prints per base: the digit-mass exponent alpha_base = log(base-m)/log base, the exponent bookkeeping constant c_base = log PB_base(m)/log base, the normalized slack delta_base = (alpha_base - 3/4 - c_base)/alpha_base, and the yes/no test 3/4 + c_base < alpha_base.
• The test has an equivalent constant-space form gap_base(m) = (base-m) base^(-3/4) - PB_base(m) > 0; both are computed and their agreement is asserted at every printed row and across 3 <= base < 20000.
• Bases printed: 1000, 2000, 3000, 3689, 3690, 5000, 10^4, 10^5, 10^6, 10^9 at m = 1, then the largest m at each of 10^4, 10^5, 10^6 with gap_base(m) > 0.
• The sign change of gap_base(1) between base 3689 and base 3690, the stepwise increase of gap_base(1) across every step of 3690..10^5, and the three maxima m = 6, 78, 451 are asserted in the binary and pinned in tests.
• At the wall the margin is printed at ten significant digits from the cancellation-reduced form delta_base = ln(1 + gap_base(m)/PB_base(m)) / (alpha_base ln base), which is algebraically identical to (alpha_base - 3/4 - c_base)/alpha_base but never subtracts two numbers of size 1 to reach one of size 10^-6.
• A ladder block replaces the fixed exponent 3/4 by b(a), the exponent Baker-Harman and Zhang buy from a common zero-free half plane sigma > a for Dirichlet L-functions, and prints the wall base_0(a), the least base >= 3 with (base-1) base^(-b(a)) > PB_base(1).
• b(a) is carried as an exact rational and compared by cross multiplication, never in floating point: a + 1/4, 4/5 and (a+1)/2 on the three ranges of the first table, (8a - 7a^2)/(4 - 2a) on [1/2, 4/7] for the second, the row printing whichever is smaller and both where they meet.
• The ladder also prints Q(b), the monotone floor: PB_{base+1}(1) - PB_base(1) < 1.291/(base-2) for base >= 40 while (base-1) base^(-b) gains at least (1-b)(base+1)^(-b) per step, so gap_base(a, 1) steps up at every base with (1-b)(base-2)(base+1)^(-b) >= 1.291, and that quantity increases in base.
• Below the floor the ladder closes the range by hand-free means: an exhaustive scan clears 3 <= base < 3690 at every rung, and on [3690, Q(b)] the smooth majorant base^(1-b) - PB_base^-(1) dominates the gap.
• The majorant's derivative (1-b) base^(-b) - (2/pi)/(base-2) - 2(1-2/pi)/base^2 crosses zero once, from negative to positive, so the majorant has one interior minimum and its maximum on the range sits at an endpoint; both endpoint values are negative.
• The floor bound is not per-rung: at every exponent b in [3/4, 1) the same constants give gap_{Q(b)}(b, 1) < -1.56 and a negative majorant at both ends of [3690, Q(b)], and those constants are pinned in tests alongside a b-grid that reproduces the bound.
• Q(b) < base_0(a) at every printed rung, so each wall is the least base, and gap_base(a, 1) steps up from it on without any scan.
• A ladder m-corollary block prints the largest m with PB_base(m) < (base-m) base^(-b(a)) at base 10^7, for the rungs whose wall lies below 10^7.
• A cost-out table sets delta_base beside the defect exponent m/(2(base-m) ln base) carried by the level-x^(alpha/2) distribution bound for digit strings, at the m = 1 bases from base 3689 on and at the three corollary maxima, with the least base of the scan 3690..10^5 at which the saving exceeds the defect.
• A sharpened cost-out block re-costs the GRH rung at every one-step constant the desk proves, so no wall and no crossing is ever inherited: 1 + Phi_base/base from step 3 at base >= 3, the phase sharpening (4/pi) base + Psi_base + base/2 - sec(pi (e_0 - (base-1)/2)/base)/2 at base >= 17, and the same with the chord kernel constant Psi'_base in place of Phi_base at base >= 36, each at the worst excluded digit and at e_0 in {0, base-1}. Both Psi floors read H as the harmonic upper bound above: the chord floor is base >= 36 under that reading and base >= 37 under the harmonic number itself, and the lowest wall is 1032, so no printed wall moves with the reading.
• Psi_base = (2 base/pi) H(ceil((base-2)/2)) + (1 - 2/pi) base and Psi'_base = (base/pi)(2 H(P-1) - 1 + 1/P) + (1 - 2/pi) base/2 at even base, P = floor(base/2), with (base/pi)(2 H(P-1) - 1 + 2/P) + (1 - 2/pi)(base/2 + 1/(2 base)) at odd base; at even base the second is exactly Psi_base - base/2 + 2/pi.
• Each row prints the wall, the crossing, delta_base and the defect at the crossing, and whether both are up-sets over their scans; the step 3 row reproduces the ladder's 3690 and the cost-out's 3692 and is the block's own control.

ROUNDING

• Arithmetic is f64; every quantity is a smooth composition of ln, sqrt and powf on inputs exact in f64, so the relative error of an undifferenced quantity stays near 10^-15, and only gap_base and delta_base lose digits, to cancellation, by the bounded amount below.
• The five-digit columns carry a directional guard of 10^-12, three decades above that error and seven below the fifth printed digit.
• Upper-bound columns round up: c_base prints ceil((c_base + 10^-12) * 10^5) / 10^5.
• Lower-bound columns round down: delta_base prints floor((delta_base - 10^-12) * 10^5) / 10^5, alpha_base the same at six digits, so alpha_base never prints as 1.000000.
• Printed strings are built from the scaled integers, never by formatting the float again, so no second rounding can move a digit.
• delta_base is formed from the unrounded c_base, then rounded once; the sign test uses the unrounded values.
• Scientific rows carry a relative guard of 10^-10 instead: sci(x, d, up) scales x by 1 +/- 10^-10, then rounds the mantissa away from or toward zero so the printed string is a true upper or lower bound on x.
• The guard sits five decades below the last digit of a six-digit cost-out row and one decade below the last digit of a ten-digit margin row, and above the worst cancellation loss in the file, which is the direct form of delta_base at base 3689: terms of size 1 differencing to -2.4 * 10^-6, a relative loss of about 5 * 10^-11; gap_base(1) there loses about 5 * 10^-12.
• A ladder wall prints as an exact integer only when it sits below 2^53 and both neighbouring gaps exceed 1024 ulps of the terms differenced; otherwise the row prints <= and a scientific upper bound, which is what the certificate supports.
• At base 3690 the margin is delta_base >= 5.863425182 * 10^-6, below the fifth digit: the rounded columns cannot display the sign there, and the certificate is the margin block and the gap assertion, not the five-digit columns.

RUN

• CARGO_BUILD_JOBS=4 cargo run --release -p mertens-numerology
• Milliseconds; prints only, writes nothing; the tables are emitted as markdown rows by the generator itself.
• cargo test -p mertens-numerology pins every table row as a rendered string, the four margin strings and their bound direction, the ten cost-out rows, the crossover base and its step count, the agreement of the two delta_base forms to 10^-9 relative, the sign change at 3689 -> 3690, the exhaustive step sweep on 3690..10^5 with its smallest step, the three m maxima, the directionality of every rounding, the floor c_base >= 0, the harmonic bound against the exact harmonic numbers to n = 2000, and Phi_base against the exact shifted-grid kernel sum on a 4001-point grid at base 50, 101, 200.
• The sharpened cost-out adds: the five rendered rows, the step 3 row against the ladder wall and the crossover, PB_base(1) against the step 3 constant to 10^-12 relative, the identity Psi'_base = Psi_base - base/2 + 2/pi at even base and Psi'_base < Psi_base and Psi'_base >= (1 + pi) base/2 on 36..4000, the failure of that floor at base 35, the same floor under the harmonic number itself failing at 36 and holding on 37..4000, the five walls strictly decreasing and every crossing an up-set within five steps of its own wall.
• The ladder adds: the ten ladder rows and the six m-corollary rows as rendered strings, the GRH rung reading b = 3/4 and base_0 = 3690, Zhang strictly below Baker-Harman at every rational of denominator <= 200 inside (1/2, 4/7) and equal at both ends, Q(b) least and below every wall, PB_base^- below PB_base on 3 <= base < 20000, no rung closing below 3690, each wall below 4 * 10^6 reproduced by an exhaustive scan from base 3, the log10 base_0(a) trend line, the constants of the general-b floor bound with the b-grid behind them, and the last gap down-step 662 -> 663 below the floor at b = 3/4.

WITNESSES

• mobius.md a power saving under GRH at large base, the l^1 floor: c_base >= 0 at every base, since Parseval forces sum_r |g_F((t+r)/base)|^2 = base(base-m) for every t, so the recursion never contracts and a negative c_base is an arithmetic error.
• mobius.md a power saving under GRH at large base, step 3: the kernel bound is loose but not absurd, the exact sup_t sum_r |D_base((t+r)/base)| sitting within 20% of Phi_base at base 50, 101, 200 on the sampled grid, with the sup attained near t = 1/2 and not at the singular point.
• mobius.md a power saving under GRH at large base, step 5 and the wall: gap_base(1) is negative at base 3689 and positive at base 3690, and steps up at all 96310 steps of 3690..10^5, the smallest step >= 0.00003172 at the top of the range, where the base^(-3/4) growth of the mass term is closest to the 4/(pi base) jump of the harmonic term.
• mobius.md a power saving under GRH at large base, the m-corollary: the excluded-digit budget grows like sqrt(base) up to the log loss, the largest admissible m reading 6, 78, 451 at 10^4, 10^5, 10^6 against sqrt(base) = 100, 316, 1000.
• mobius.md a power saving under GRH at large base, the shape at large base: c_base tracks (ln ln base + ln(2/pi))/ln base to within 0.01 at base 10^12, and delta_base climbs toward 1/4 from below across the printed bases.
• mobius.md a power saving under GRH at large base, the margin at the wall: delta_base <= -2.395807653 * 10^-6 at base 3689 and delta_base >= 5.863425182 * 10^-6 at base 3690, the two delta_base forms agreeing to 10^-9 relative at every printed base, which is the independent check on the digits printed.
• mobius.md a power saving under GRH at large base, the rungs: the ladder is monotone in a, base_0 reading 3690, 8578, 33547, 92317, 92317, 3107080, 6939524168 and then three scientific bounds, so a wider zero-free half plane costs a higher base and nothing else.
• mobius.md a power saving under GRH at large base, the input b(a): the GRH rung a = 1/2 reproduces the wall 3690 exactly and Zhang meets Baker-Harman there, so the sharper second table moves no GRH number.
• mobius.md a power saving under GRH at large base, what a weaker half plane spends first: the m-budget at base 10^7, each rung printed with its a, its b(a) and its source, reads 1971 at 1/2, 3/4, both; 1002 at 13/25, 1417/1850, Zhang; 365 at 11/20, 913/1160, Zhang; 176 at 4/7, 4/5, both; 176 at 3/5, 4/5, BH; and 8 at 2/3, 5/6, BH.
• mobius.md a power saving under GRH at large base, the cost-out, sharpened: the wall and the crossing read 3690 and 3692 at step 3, 2446 and 2450 at the phase sharpening, 1812 and 1815 at its extreme-digit form, 1499 and 1502 at the chord and 1032 and 1036 at the chord's extreme-digit form, so every lower wall pays for itself within five steps and the b = 3/4 comparison is the only one run.
• mobius.md a power saving under GRH at large base, the cost-out: the saving is below the defect at both base 3689 and base 3690 and above it from base 3692 on, the crossover two steps past the wall, the difference rising at every one of the 96310 steps of 3690..10^5 without a proof of monotonicity beyond the scan.