# 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.