transport-census

• The transport census: for every design, the zeros of the design zeta zeta_F(s) = sum_(n in S_F) n^(-s) that lie right of the abscissa alpha = log k/log q, and the lower bound on the Mertens exponent of the design's own Mobius nu_F each one forces.
• The transport theorem of mrly-pairing is the engine: a zero rho of zeta_F with Re rho > alpha gives sigma_c(N_F) >= Re rho, so sum_(n <= x) nu_F(n) is not O(x^(Re rho - eps)) for any eps > 0. This study turns that one-zero statement into a table over every design the locus sweep censuses.
• The ladder, the contour engine, the truncation bounds and the box machinery are imported from design-zeta, the cofactor and the polisher from zeta-locus; nothing is copied and no sweep is repeated.

THE STRIP IS CLOSED ON THE RIGHT

• A census of the zeros right of alpha needs no arbitrary right edge. zeta_F(s) a_min^s -> 1 as Re s -> +infinity with a_min the least nonzero digit, and the coefficients are nonnegative, so abs(zeta_F(s) a_min^s - 1) <= a_min^sigma zeta_F(sigma) - 1 at sigma = Re s.
• Hence sigma_1, the least point of the grid alpha + 0.05 n at which a_min^sigma (zeta_F(sigma) + bound) < 2, is a PROVED zero free edge: for sigma_1 > alpha, which the grid forces, zeta_F has no zero with Re s >= sigma_1, the ladder's own error bound carried through.
• So the census box needs no hand-chosen right edge, and its count is EXACT ON THE BOX: uncounted are the sliver alpha < Re s <= alpha + 1e-6, the band 0 < Im s < 0.02, everything above the census height and the conjugate half plane, so the same number is a LOWER BOUND for the half plane Re s > alpha.

THE COUNT

• The winding is taken on the Lyndon cofactor Z(s) = zeta_F(s)(1 - k q^(-s)), which is analytic on Re s > alpha - 1 and has no pole in the box, so the argument principle counts zeros with no pole correction.
• Right of alpha a zero of Z is a zero of zeta_F without exception: the one place the transfer fails is a pole s_(0,j) with vanishing residue, and every such point sits ON the line Re s = alpha, outside the box.
• The left edge is alpha + 1e-6, which is what keeps the full digit sets' residue null teeth outside the contour.
• The box is cut into unit sub-boxes in Im s; each sub-box of nonzero winding has its zeros located by a coarse grid and polished by Muller on the cofactor, and the count located against the count wound is printed as the completeness flag.

THE BOX

• The rightmost zero of each design is certified by a winding box on zeta_F itself, half-width 5e-5 in Re s and in Im s, the argument principle on the same engine as mrly-pairing verb box.
• A winding of 1 certifies exactly one zero inside the rectangle, so Re rho is pinned to the box edges and the transport bound is the LEFT edge, truncated down.
• A zero right of alpha may be a tooth of the level-zero comb and so sit near a pole of zeta_F, so every box prints its distance to the nearest pole of the lattice s_(i,j) = alpha - i + 2 pi i j/log q. That distance is positive by construction, since the lattice lies on Re s <= alpha and the box lies right of alpha; what the column carries is how far, against the box's own half-width.
• The printed contour minimum is a minimum over the SAMPLED points of the contour and not over the contour, so it reads as evidence beside the winding and not as a certificate of its own.

THE HEIGHT

• A rightmost real part is a statement below a cut and nothing more. The teeth of the level-zero comb drift right with the pole index, so the rightmost zero of a design moves right as the census height rises, and every rightmost printed here carries the height it was read below.
• Twenty-three designs are censused to Im s = 40 and base 50 missing one digit to Im s = 4 alone, so its column is a height-4 maximum of a drifting shape and is not on the same clock as the others.

THE FALSIFICATION

• The base 2, 3 and 4 full digit sets are the controls: there zeta_F = zeta, no zero lies right of alpha = 1, and the census must return zero.
• At a full set the boxes around the cofactor only teeth s_(0,j), j >= 1, separate the two functions: Z winds 1 there and zeta_F winds 0, which is the transfer exception seen directly.
• The two certified boxes of mrly-pairing are re-run at their own edges, with the control rectangle.

THE HYPOTHESES

• nu_F is the Dirichlet inverse of 1_(S_F) and exists only when 1 in S_F, that is when 1 in F. A design missing the digit 1 carries its zeros and carries no transport bound; the table says so rather than quoting the exponent.
• Every bound is theta(nu_F) >= x_0 with x_0 the certified left edge of a winding-1 box, truncated down; nothing is read off a polished root or a small residual.

RUN

• uv run python research/lab/py/transport-census/transport_census.py census 40 locus - the per design table: sigma_1, the strip, the winding, the zeros right of alpha, the certified box on the rightmost and the bound it proves.
• uv run python research/lab/py/transport-census/transport_census.py law 40 locus - the same, followed by the columns in k/q order and in alpha order, the equal-key ties, the designs with Re rho > 1, the corollary line per design, the full set null boxes and the reproduction of the landed boxes.
• The second argument is the height, the third a family: locus the twenty of the locus sweep, rungs the two of the family sweep, b20 and b50 the two high rungs, ctl the three full digit sets, all the twenty-two. Prints only, writes nothing.
• census 40 locus runs in about eleven minutes, census 40 rungs in three and census 40 b20 in six; base 50 is censused to height 4 in two and a half. Peak resident memory is under 0.2 GB throughout.

WITNESSES

• the zero free edge is one real evaluation: sigma_1 reads 0.5 to 1.75 over the twenty-four designs, with a_min^sigma zeta_F(sigma) between 1.86 and 1.999 against the threshold 2 and ladder bounds 1e-11 to 1e-47, so zeta_F has no zero at all in Re s >= 1.75 on any design censused and the alpha + 3.02 right edge of zeta-locus is more than twice as wide as the zeros need
• the winding counts 157 zeros of zeta_F right of alpha over the twenty-three designs censused to Im s = 40, all 157 located at the seek grid 15 x 11 with 2 want + 8 seeds, and 2 more at base 50 missing one digit censused to Im s = 4; the count is exact on the box and a lower bound for the half plane Re s > alpha
• twenty-one of the twenty-four carry a zero right of alpha, nineteen of the twenty-two the locus and family sweeps censused; the three that do not are the base 2, 3 and 4 full digit sets, whose windings on the strip read -1.97e-33, 1.73e-33 and 1.53e-33, and that reproduces the nineteen of zeta-family from a construction with no assignment radius in it
• the rightmost real parts below the census height run 0.441505537191 at base 5 {0,1} to 1.002685494780 at base 20 missing one digit, both below Im s = 40, and the bounds they prove run theta(nu_F) >= 0.4414555 to theta(nu_F) >= 1.0026354, each the left edge of a winding-1 box truncated down; the size of a rightmost is a statement below its own cut, base 20 missing one digit reading 1.000285484146, 1.000549674321 and 1.002685494779 at Im s = 2.0988, 4.1971 and 14.6920
• every box returns winding 1 with a sampled contour minimum of 1.2e-4 to 6.1e-3 against engine bounds of 1e-13 to 1e-33, at least eight orders of magnitude at every box, and the tightest pole distances are 0.00517845 at base 50 missing one digit and 0.0223021 at base 20 missing one digit, four hundred times the box's own half-width
• nineteen of the twenty-four designs meet every hypothesis of the transport bound, and seventeen of the twenty-two the locus and family sweeps censused: base 4 {2,3} and base 4 {0,2,3} have zeros right of alpha and do not contain the digit 1, so 1 is outside S_F, the indicator vanishes at 1 and the Dirichlet inverse nu_F does not exist
• the gain Re rho - alpha is not a function of alpha and k/q: at alpha = 1/2, k/q = 1/2 the four base 4 two-digit designs read 0.0853043873, 0.4400124317, 0.2706238545, 0.3439264581, a spread of 0.35470804, and the other three equal-key families spread 0.37474232, 0.17605693 and 0.060972003
• four designs have Re rho > 1, so their nu_F outruns the count of integers below x: base 10 missing two, base 10 missing 9, base 20 missing one and base 50 missing one, rightmost 1.001514387650, 1.001589275290, 1.002685494780 and 1.000061474950 at alpha = 0.9030900, 0.9542425, 0.9828779, 0.9948357, the first three below Im s = 40 and the last below Im s = 4, so only the SIGN of Re rho - 1 is read and never its size; the k/q reading dies on base 5 {0,1,2,3}, the same k/q = 0.8 as base 10 missing two, whose rightmost is 0.989748105861, below 1; base 20 and base 50 missing one digit are two further rungs the base 10 pair does not fix, and both land above 1
• the least rightmost real part at each alpha reads 0.4485242462, 0.4415055372, 0.5853043873, 0.7207876015, 0.9126562295, 0.9897481059, 1.0015143877, 1.0015892753, 1.0026854948, 1.0000614750 up the ladder alpha = 0, 0.4307, 0.5, 0.6309, 0.7925, 0.8614, 0.9031, 0.9542, 0.9829, 0.9948, rising at every step but the first and the last, and the last is where the census height drops from 40 to 4; ten rungs, one design each above alpha = 0.86 against six at alpha = 0.5, and no fit taken
• the cofactor only teeth separate the two functions directly: around s_(0,1) at 1 + 9.06472028365 i, 1 + 5.71920173476 i and 1 + 4.53236014183 i the winding of Z is 1 and the winding of zeta_F is -9.42e-32, -6.28e-32 and 0, with min abs(Z) about 5e-5 and min abs(zeta_F) 1.351, 0.8747 and 0.7295
• the two certified boxes of mrly-pairing reproduce at their own edges, winding 1 and 1 with contour minima 8.298e-4 and 6.865e-4, and the control rectangle returns winding 0 with contour minimum 1.543e-2

SOURCES

• DLMF 25.2 - the Dirichlet series and the abscissa of convergence, the classical A(x) = O(x^theta) implies sigma_c <= theta the transport theorem closes on.
• Baker and Harman 1991 - the exponential sum bound the study mrly-pairing costs, cited there and not used here.