# mrly-pairing

- The identity that replaces `zeta(s) M(s) = 1` on a digit design `S_F`, with `F` its kept digits, and what the position pairing `M_F(base^level) = int_0^1 G_level(t) S_level(t) dt` costs when it is split by `base`-power denominator.
- `split`: the pairing written exactly on the grid, `M_F(base^level) = base^(-level) sum_(a mod base^level) G_level(a/base^level) S_level(a/base^level)`, with the second moments `int abs(G_level)^2 = fill^level` and `int abs(S_level)^2`, the level decomposition of the `l^1` mass `C_level = sum_j fill^(level-j) c_j`, the top-level share against the proved floor `m/base`, the one-step growth `C_level/C_(level-1)` against the grid sup, and the exponent the split proves against `alpha`, `alpha/2` and the measured meter exponent of [mobius](../../../notes/mobius.md).
- `onestep`: the exact one-step constant `B_base(F) = sup_t sum_(r mod base) abs(g_F((t+r)/base))` of the `l^1` recursion at one excluded digit, reduced to `base` real square roots at every `t` by `abs(g_F((t+r)/base)) = abs(A_r - e(c (t+r)/base))` with `A_r = (-1)^r sin(pi t)/sin(pi (t+r)/base)` and `c = e_0 - (base-1)/2`; the seat in `t`, the phase identity the sharpening runs on, the split defect against the triangle bound `K_base + base` carried by `abs(g_F) <= abs(D_base) + abs(g_E)`, the chord kernel bound `1/sin x <= 1/x + (2/pi)(1 - 2/pi) x` that replaces the constant `Phi_base` of [mobius](../../../notes/mobius.md) by `Psi'_base` inside step 3, the two sharpened bounds proved here and the bases `base_0(a)` each moves, all against `Phi_base` and against the measured constant's own base, and the price of the weight route the chord leaves owed.
- `perden`: the per-denominator Mobius input of Baker-Harman's PROPOSITION put into step 5 of [mobius](../../../notes/mobius.md) at the exact frequencies `a'/base^j`, which is its corollary at `(r,Q)` the frequency itself, the level decomposition `C_level = sum_j fill^(level-j) c_j` it weights, the proved floor `C_j >= base C_(j-1)` that pins the top level's share of the `l^1` mass above `m/base`, the exact rationals `a + 1/2 - b(a)` that make the top level charge the uniform constant at every rung, the composite-`base` frequencies whose true reduced denominator sits below their level, the exponent and the single factor the weighting buys against the uniform input, the same tool at full strength through `nu(a) = min_Q (Q + ||aQ||_(base^level))`, and the transform checked against brute force over the digit strings.
- `glue`: the coefficients `c_F(n) = sum_(d e = n, d and e in S_F) mu(e)` of `zeta_F M_F`, their partial sums `P(x) = sum_(e in S_F) mu(e) A_F(x/e)`, the ratio `P(x)/x^alpha`, and the limit test `M_F(sigma)` as `sigma -> alpha+` that decides the abscissa of `D_F = zeta_F M_F - 1`.
- `box`: winding boxes on `zeta_F` by the argument principle, evaluated on the engine [design-zeta](../design-zeta/README.md), each printing the box edges, the winding, the largest phase step, the contour minimum of `abs(zeta_F)` and the engine's own error bound; a winding of `1` certifies one zero inside the rectangle and so pins `Re rho` to the box edges, a winding of `0` certifies none.
- `inverse`: the Dirichlet inverse `nu_F = 1_(S_F)^(-1)`, computed by a blocked strided sieve whose divisor set carries `base^level` itself, the one element of `S_F` a level enumeration below `base^level` misses and the one index it would leave `1` too large, with the full digit set as the control where `nu_F = mu` term for term; the design Mertens `sum_(n <= x) nu_F(n)` and its running maximum against the design's own mass `A_F(x)`; the rightmost censused zero of `zeta_F` refined on the engine [design-zeta](../design-zeta/README.md); and the partial sums of `N_F(sigma)` against `1/zeta_F(sigma)`.
- Domain: `level 14` at base 3 `{0,1}`, `level 11` at base 4 `{0,1}`, `level 9` at base 5 `{0,1}` and `level 6` at base 10 missing `9` for the exact grid pairing, the ladder in `level` from `6` to `14`; `x` up to `3^17.75`, `4^13.75`, `5^11.75` and `10^6.75` for the glue, sampled at the four phases `log_base x = l, l + 1/4, l + 1/2, l + 3/4` because the phase `log_base x = l` aliases every Fourier mode of the log-periodic ratio onto one number; the integer `n` up to `3^16`, `4^12`, `5^10` and `10^7` for the Dirichlet inverse.

## RUN

- `uv run python research/lab/py/mrly-pairing/pairing.py split`
- `uv run python research/lab/py/mrly-pairing/pairing.py glue`
- `uv run python research/lab/py/mrly-pairing/pairing.py inverse`
- `uv run python research/lab/py/mrly-pairing/pairing.py box`
- `uv run python research/lab/py/mrly-pairing/pairing.py onestep`
- `uv run python research/lab/py/mrly-pairing/pairing.py perden`
- `split` runs in three seconds, `glue` and `inverse` in about thirty each, `box` in about twenty, `onestep` in twenty nine and `perden` in ten; peak resident memory is `0.78` GB in `split`, `0.41` in `inverse`, `0.30` in `glue`, `0.18` in `onestep`, whose rung scans run in blocks, and negligible in `box`.

## WITNESSES

- the grid pairing reproduces `M_F(base^level)` exactly at every family, `11`, `6`, `9` and `276` at base 3 `{0,1}` `level 14`, base 4 `{0,1}` `level 11`, base 5 `{0,1}` `level 9` and base 10 missing `9` `level 6`, with `int abs(G_level)^2 = fill^level` exact and `int abs(S_level)^2` the squarefree count below `base^level`
- the principal fibre `a = 0` carries `-0.31857` of `11`, `0.11133` of `6`, `-0.05924` of `9` and `112.66549` of `276`, shares `-0.028961`, `0.018555`, `-0.006583` and `0.408208`, and exactly all of it on the two full-set controls; its exponent is `alpha - 1/2` under RH against the conjectured `alpha/2` for the design meter, so the asymptotic gap is not yet visible at base 10 missing `9`, where the fibre still carries `40.8` percent of the meter and a factor of `10` between the two exponents `0.454243` and `0.477121` needs `x = 10^44`
- the top-level share of the `l^1` mass reads `0.485846`, `0.602606`, `0.687994` and `0.510055` against the proved floor `m/base = 0.333333`, `0.500000`, `0.600000` and `0.100000`, and levels `j >= level/2` carry `0.995116`, `0.996061`, `0.997043` and `0.942350` of it
- the per-denominator split saves exactly `log_base(C_level/c_level)/level`: at base 3 `{0,1}` that is `0.657068` divided by `level`, identical at every `level 6..14`, so the saving is a constant factor at most `base/m` and never an exponent
- the split exponents are `0.988106`, `0.912502`, `0.905006` and `1.012881` with the uniform GRH input and `0.941173`, `0.879287`, `0.879188` and `0.964150` with the per-denominator one, against `alpha = 0.630930`, `0.500000`, `0.430677` and `0.954243`; the base 2 and base 3 full-set controls give `0.500000`, the classical RH exponent
- the one-step growth `C_level/C_(level-1)` reads `3.889888518`, `5.032783116`, `6.410132461` and `18.369402635` at `level 9, 9, 9, 6`, each below the grid sup `B_base(F) = sup_t sum_(r mod base) abs(g_F((t+r)/base))` of `4.000000000`, `5.226251860`, `6.472135955` and `19.888543820`; the bound `C_level/C_(level-1) <= B_base(F)` holds at every `level` with no computation and is attained, `C_1/C_0 = 4 = B_base(F)` exactly at base 3 `{0,1}`, so strictness begins at `level >= 2`
- the one-step growth agrees between the two consecutive `level` printed to `8.5`, `7.4`, `10` and `5.0` digits at base 3 `{0,1}`, base 4 `{0,1}`, base 5 `{0,1}` and base 10 missing `9`, reading `3.889888507` then `3.889888518`, `5.032782920` then `5.032783116`, `6.410132461` twice and `18.369226930` then `18.369402635`, so the constant is stable family by family and two values of `level` are all that is measured
- the one-step reduction at one excluded digit reproduces the direct sum over `F` to `12` digits at every `(base, e_0, t)` checked, and reproduces the grid sup of `split`, printing the floored `4.0000000000` at base 3 `{0,1}` and `19.8885438199` at base 10 missing `9` against `split`'s `4.000000000` and `19.888543820`
- the phase identity `sum_r (1 + sign(A_r) cos(2 pi c (t+r)/base)) = base + cos(2 pi c (t - 1/2)/base)/cos(pi c/base)` is exact at every `(base, e_0, t)` printed, and its right side is at least `base + 1` because `abs(2 pi c (t - 1/2)/base) <= abs(pi c/base) < pi/2`
- the `t -> 0` endpoint carries `2(base-1)` exactly, so `B_base(F) >= 2(base-1)` at every `base` and every `e_0`, above the `l^1` floor `base`, and that endpoint is the seat at base 3 `{0,1}`
- the seat of `B_base(F)` is `t = 1/2` at every family printed from `base 100` up and interior at `base 11` and `base 13` with `e_0 = 0`, where the sup on the cut reads `22.5094271855` and `27.9876970872` against `Sigma(1/2) = 22.4703926508` and `27.9570308138`, so `t = 1/2` is where the constant is read and not where it is proved to sit
- the split defect `(K_base + base) - B_base(F)` reads `1.441272 base` at `e_0 = 0` and `0.798914 base` at the middle digit at `base 3690`, flat to `1.3e-5` at `e_0 = 0` across `base 100, 1000, 2234, 3690` and agreeing there to six digits with `1 + (2/pi) ln 2 = 1.4412712`, while the middle digit reads `0.808644`, `0.799643`, `0.799091` and `0.798914` across the same four and so holds only to `1e-2`, so the triangle `abs(g_F) <= abs(D_base) + abs(g_E)` throws away a full `base` and more, and the proved `Phi_base` sits at most a further `0.600121 base` above the exact kernel sup `K_base`, the up-rounded `old gap/base` column
- the exact constant falls with the excluded digit: `B_base(F)/base` reads `5.750052` at `e_0 = 0` against `6.392410` at `e_0 = 1844` at `base 3690`, while the step 3 bound `1 + Phi_base/base = 7.791445` is one number for all of them
- the sharpened bound `B_base(F) <= (4/pi) base + Psi_base + base/2 - sec(pi (e_0 - (base-1)/2)/base)/2`, proved for `base >= 17` at `m = 1`, moves the base of the GRH rung from `3690` to `2446` at every excluded digit and to `1812` at `e_0 in {0, base-1}`, and the rungs at uniform exponent `b = 1417/1850` and `b = 913/1160` from `8578` and `33547` to `5700` and `22416`, and to `4242` and `16816`; each sharpened wall is an up-set over its whole scan, `3997555 = 4000000 - 2446 + 1` of the GRH rung's `base` meeting the condition and the five like counts printed beside it
- the measured constant itself would move the GRH base to `927` at every excluded digit, the last failure being `base 926` at `e_0 = 462` on a downward scan of `17..2000` over every digit, and to `304` at `e_0 in {0, base-1}`, last failure `303` on `17..8000`, the ceiling of this lever; the middle digit is not the maximiser at odd `base`, since `base 695` fails at `e_0 = 463` with `B_base(F)/base = 5.327344` against `(base-1) base^(-3/4) = 5.127089` while its middle digit passes, and both bases read `Sigma(1/2)` and are measurements and never a certificate
- no family reaches the sharpened bound: the worst ratio `B_base(F)` to bound is `0.807189` over every `e_0` at `base 17..60`, `0.876716` at the larger seats, and `0.872146` over `4000` seeded draws of `(base, e_0, t)` with `base` in `{17, 23, 60, 101, 333, 1000, 3690}`
- the chord `1/sin x <= 1/x + (2/pi)(1 - 2/pi) x` is the chord of the convex `csc x - 1/x` on `(0, pi/2]`, and pairing `r` with `base-1-r` holds every shifted-grid argument inside that interval at `t` in `(0, 1/2]`, while `K(t) = K(1-t)` carries the rest of the circle, so `K(t) <= (4/pi) base + sin(pi t) Psi'_base` with `Psi'_base = (base/pi)(2 H(P-1) - 1 + 1/P) + (1 - 2/pi) base/2` at even `base`, `P = floor(base/2)` and `H(n) = ln n + gamma + 1/(2n)` the harmonic upper bound of [mobius](../../../notes/mobius.md), and `(base/pi)(2 H(P-1) - 1 + 2/P) + (1 - 2/pi)(base/2 + 1/(2 base))` at odd `base`, whose extra `1/(2 base)` is there because the paired argument sum is `base^2/4` only at even `base` and `P(P+1) + t` at odd `base`
- `Psi'_base` is `Psi_base - base/2 + 2/pi` at even `base`, and it cuts the gap to the exact kernel sup by a factor `5.98`: `Psi_base - (K_base - (4/pi) base)` is at most `0.600121 base` at `base 3690` where the chord leaves at most `0.100293 base`, both columns rounded up and the lemma's own slack floored to `0.100292 base` at the same `base`; the chord column reads `0.100290, 0.100293, 0.100293` at `base 100, 1000, 2234`, so it is flat to `1e-5`, and the gap is attained at the seat `t = 1/2`
- the monotone step of the sharpening needs `Psi'_base >= (1 + pi) base/2`, first true at `base 36` under the harmonic upper bound `H` and at `base 37` under the harmonic number itself, and true at every `base` above either, so the chord bound `B_base(F) <= (4/pi) base + Psi'_base + base/2 - sec(pi (e_0 - (base-1)/2)/base)/2` is proved from `base 36` on the desk's reading of `H` and the scans start there; the hypothesis is sufficient and not necessary, since the max of `h(tau)` it exists to place at `tau = 0` sits there at every `e_0` from `base 8` up on the exhaustive scan `4..79`
- the chord bound moves the GRH base from `3690` at step 3 and `2446` at the phase sharpening to `1499` at every excluded digit and from `1812` to `1032` at `e_0 in {0, base-1}`, and the rungs `b = 1417/1850` and `b = 913/1160` from `5700` and `22416` to `3525` and `14078`, and from `4242` and `16816` to `2459` and `10013`, each an up-set over its own scan, `36..4000000` for the GRH rung and `36..8000000` and `36..40000000` for the two rungs above it
- no family reaches the chord bound either: the worst ratio `B_base(F)` to bound is `0.902124` over every `e_0` at `base 36..60`, `0.941239` at the larger seats and `0.936333` over the same `4000` seeded draws
- the weight route the chord leaves owed is priced and loses: the kept weight `w_r = |A_r|/(|A_r| + 1) >= s/2` is worth `base/2` at the seat, while dropping the singular terms by `(1 + sign(A_r) cos) <= 2` costs `2 sum_r 1/(|A_r| + 1)`, which is `0.726761 base` at `base 1000, 3690, 20000` against its limit `2(1 - 2/pi) = 0.726761`, a net `-0.226761 base`
- the level decomposition `C_j = sum_(i <= j) fill^(j-i) c_i` into primitive level sums is exact at every printed row, and the proved floor `C_j >= base C_(j-1)` holds at all of them, the first ratios reading `4.000000`, `18.000000`, `200.000000` and `2996.000000` at `base 3, 10, 101, 1499` with one excluded digit
- the top level carries the largest single share of the `l^1` mass and the mass decays geometrically downward: `c_j/C_j` reads `0.485846`, `0.510055`, `0.573574` and `0.676152` at `base 3` `j = 12`, `base 10` `j = 6`, `base 101` `j = 3` and `base 1499` `j = 2`, each above the proved floor `m/base = 0.333333`, `0.100000`, `0.009901` and `0.000667`, the last two short rows where `C_j/C_(j-1)` is still moving, `244.658399` then `234.507307` at `base 101`, and so not converged constants
- the levels of charge at most `x^(3/4)`, reduced denominator at most `x^(1/2)` with the tie at `j = level/2` included, carry `0.018474`, `0.117603`, `0.174294` and `0.323848` of the mass at the same four rows against the proved cap `(fill/base)^(j-J)` reading `0.087791`, `0.729000`, `0.980296` and `0.999333`, and the measured share falls geometrically in the level while only the cap is proved
- `a + 1/2 - b(a)` is `1/4`, `47/185`, `61/232`, `19/70`, `3/10` and `1/3` as exact rationals at the rungs `a = 1/2, 13/25, 11/20, 4/7, 3/5, 2/3`, so the top-level charge `x^(a + 1/2)` is strictly worse than the uniform `x^(b(a))` at every rung and the crossing `2(b(a) - a)` never exceeds `1/2`
- the per-denominator weighting saves at most `-log_base(c_level/C_level)/level` in the exponent, `0.657068/level` at `base 3` one digit off and `0.292383/level` at `base 10` missing `9`, against the proved cap `log_base(base/m)/level`, the largest term sitting at `j = level` at every printed row and not by proof; read on the whole bound the saving is `log_base(ratio)/level`, `-0.000434` and `-0.003444` at those two rows, and the weighted sum over `C_level x^(b(a))` reads `0.994295`, `0.953536`, `0.855931` and `0.842258` inside the proved bracket `[m/base, 1]`: one factor at most `base/m`, never an exponent
- the same tool at full strength buys no exponent either: with any reduced `r/Q` at any frequency the per-frequency constant is `min(x^(b(a)), x^a nu(a)^(1/2))`, `nu(a) = min_Q (Q + ||aQ||_(base^level))` checked against a full search over reduced `r/Q` at `base^level = 81` with `0` mismatches, and the honest ratio reads `0.817368`, `0.835986`, `0.851049`, `0.861910` at `base 3` `level 6, 8, 10, 12` and `0.684136`, `0.705013`, `0.737968` at `base 10` `level 4, 5, 6`, rising in `level` while `level` times the gain falls from `-0.183564` to `-0.135265` and from `-0.164858` to `-0.131963`, so the saving decays faster than `1/level`; that the honest ratio is bounded below in `level` is measured over these seven rows and not proved
- the level charge is an upper bound and not the pointwise truth: at `base 10` the top-level frequencies whose true reduced denominator is at most `x^(1/2)`, `5^6/10^6 = 1/64` among them, number `160` and carry `2.061134e-05` of `C_level` at `level 6`, and `80` carrying `1.305646e-04` at `level 5`
- the product form of the transform matches brute force over the digit strings: `C_j = 234.856179`, `913.566768` and `331.978584` with top shares `0.485833`, `0.485848` and `0.512017` at `base 3` `j = 4, 5` and `base 10` `j = 2`
- `zeta_F M_F` leaves `1` at `n = 4` for base 3 `{0,1}`, `n = 4` at base 4 `{0,1}`, `n = 6` at base 5 `{0,1}` and `n = 9` at base 10 missing `9`, and the full digit set has `P(x) = 1` at every `x`
- `P(x)/x^alpha` at the four phases settles at `0.493767, 0.699235, 0.758519, 0.587055` for base 3 `{0,1}`, `0.596436, 1.047015, 0.843833, 0.709576` for base 4 `{0,1}`, `0.611328, 1.048657, 0.860060, 0.723221` for base 5 `{0,1}` and `-0.011742, 0.033309, 0.058519, 0.073298` for base 10 missing `9`, bounded away from `0` and from infinity at every family; at base 10 missing `9` the lattice phase alone reads `-0.011742` and falls in magnitude, so a generator sampling only `x = base^level` there reads a false zero
- `M_F(alpha) = 0.519548`, `0.615960`, `0.601816` and `0.053132`, printed with truncation tails `1.95e-3`, `7.81e-3`, `1.56e-2` and `4.57e-4` that carry the exponent `base^(-level alpha/2)` of the conjecture and so bound nothing unconditionally; the unconditional tail from `A_F(base^l) = fill^l` is `(fill-1) base^(-level eps)/(1 - base^(-eps))`, which leaves no printed value at base 10 missing `9` distinguishable from `0`, and the reading `0.188542, 0.119886, 0.085776, 0.065917, 0.053132` down the `eps` column there falls monotonically
- the Dirichlet inverse equals `mu` term for term on the full digit set to `n = 131072` at `base 2` and `n = 177147` at `base 3`
- the rightmost censused zero of `zeta_F` refines to `0.7207876014768929 + 28.60567656491595 i` at base 3 `{0,1}` and to `1.001589275292455 + 2.739199500566845 i` at base 10 missing `9`; the printed residual sinks beneath the engine's noise floor and locates nothing on its own, so `Re rho` is pinned instead by winding boxes
- `box` returns winding `1` on `Re in [0.72074, 0.72084]`, `Im in [28.60563, 28.60573]` at base 3 `{0,1}`, contour minimum `abs(zeta_F) = 8.298e-4` against the engine bound `6.284e-30`, and winding `1` on `Re in [1.00150, 1.00168]`, `Im in [2.73915, 2.73925]` at base 10 missing `9`, contour minimum `6.865e-4` against `2.798e-23`, while the control rectangle `Re in [0.99900, 1.00050]`, `Im in [2.73810, 2.74030]` returns winding `0`; both certified boxes lie strictly right of `alpha = 0.6309297536` and `0.9542425094`, and the base 10 box lies strictly right of `Re s = 1`
- the running maximum of `sum_(n <= x) nu_F(n)` grows by `9.4474, 11.5000, 10.2220, 10.0354` per level at base 10 missing `9`, `level 4..7`, against `base^(Re rho) = 10.036661` and `fill 9`, with `max/A_F(base^level)` rising `0.1043, 0.1094, 0.1398, 0.1588, 0.1771`; at base 3 `{0,1}` the geometric mean of the four steps `level 12..16` is `2.059` against `base^(Re rho) = 2.207512` and `fill 2` while the arithmetic mean of the five printed level ratios is `1.9972`, below the trivial `2`, a census too short to separate them
- the transported statements are limsup statements and the census contradicts none of them pointwise: `max/A_F` reaches only `0.0738` at base 3 `{0,1}` `level 16` and `max/x` only `0.0847` at base 10 missing `9` `level 7`
- base 4 `{0,1}` and base 5 `{0,1}` produce the same running maxima `1, 1, 2, 3, 4, 7, 15, 23, 45, 86` at levels `1..10` while base 3 `{0,1}` leaves that sequence at level `9`, which is the carry-free polynomial structure of a two-digit design surviving until the base is small enough for a product to carry
- the partial sums of `N_F(sigma)` meet `1/zeta_F(sigma)` from the engine to `1.60e-3` at `sigma = Re rho + 0.08 = 0.8008` and `1.96e-4` at `sigma = Re rho + 0.20 = 0.9208` at base 3 `{0,1}`, and to `1.72e-2` at `sigma = 1.0816` and `2.39e-3` at `sigma = 1.2016` at base 10 missing `9`; nothing is evaluated below `Re rho`, so these pin `sigma_c(N_F)` from below only
- the support of `nu_F` lies inside the multiplicative semigroup generated by `S_F` and strictly inside it: `9`, `27` and `36` lie in the semigroup with `nu_F = 0`, and `16`, `48` and `52` lie in the semigroup and outside `S_F`

## SOURCES

- [DLMF 25.13](https://dlmf.nist.gov/25.13) - the periodic zeta and Hurwitz's formula, the kernel of the position identity this study splits.
- [Baker and Harman 1991](https://doi.org/10.1112/jlms/s2-43.2.193) - the uniform `x^(3/4 + eps)` bound for `sum mu(n) e(n theta)` under the generalised Riemann hypothesis, and the sharper `x^(a + eps) Q^(1/2) (1 + x abs(theta - r/Q))^(1/2)` at a rational frequency of denominator `Q`, which is the per-denominator input costed here.
- [Maynard 2019](https://link.springer.com/article/10.1007/s00222-019-00865-6) - the `l^1` norm of the digit transform, the quantity whose level profile this study measures.