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.
• 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 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 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, 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; 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, 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 - the periodic zeta and Hurwitz's formula, the kernel of the position identity this study splits.
• Baker and Harman 1991 - 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 - the l^1 norm of the digit transform, the quantity whose level profile this study measures.