zeta-family

• The SECOND FAMILY of the design zeta zeta_F(s) = sum_(n in S_F) n^(-s): the zeros that survive when every comb of the pole lattice is stripped at a stated radius, and the law they obey.
• On the full digit set the second family is exactly the critical-line zeros of zeta, so it is the half of the zero set that carries the Riemann hypothesis; on a design it has no law.
• The ladder, the contour engine and the truncation bounds come from ../design-zeta, the cofactor, the Laurent data and the residue comb from ../zeta-locus; neither is copied. This study adds the split into second family and comb, the level-one comb, and three derived tests.

THE SPLIT

• The pole lattice of zeta_F is s_(i,j) = alpha - i + 2 pi i j/log q, one vertical line per level i = 0, 1, 2, ..., and Z(s) = zeta_F(s)(1 - k q^(-s)) kills the level-zero line and is analytic on Re s > alpha - 1.
• Every zero of Z in the census strip is a zero of zeta_F, except at a lattice point where the residue vanishes: there Z vanishes and zeta_F does not.
• THE ASSIGNMENT. A zero within abs(u) < 0.45 of a pole with nonvanishing residue is a tooth of that pole's comb; a zero sitting at a pole with vanishing residue is a zero of the cofactor alone. Everything else is SECOND FAMILY.
• The next cofactor Z_m(s) = zeta_F(s) prod_(i <= m) (1 - k q^(-(s+i))) has exactly the zeros of Z inside the strip, since the extra factors vanish only on Re s = alpha - 1, ..., alpha - m: the survivors are the same set under every comb. What Z_m adds is the level-one comb, whose teeth reach into the strip.
• THE LEVEL-ONE RESIDUE, derived and not measured. Z(s) = E_1(s) + sum_(l >= 1) binom(-s,l) q^(-s-l) gamma_l zeta_F(s+l) is singular at s_(1,j) through its l = 1 term alone; q^(-s_(1,j)-1) = q^(-s_(0,j)) = 1/k and 1 - k q^(-s_(1,j)) = 1 - q, so zeta_F has residue r_(1,j) = s_(1,j) gamma_1 r_(0,j)/(k(q - 1)) at the level-one pole.
• Two readings for free: the level-one comb is empty wherever the level-zero comb is, and at the full digit set s_(1,0) = alpha - 1 = 0 makes r_(1,0) vanish, which is zeta having no pole at s = 0.
• A level-one tooth lies within abs(u) < 0.45 of Re s = alpha - 1, so it is visible in the strip only in the band alpha - 0.92 < Re s < alpha - 0.55, and there the assignment is applied against both lattices.
• THE TRANSPORT READING. A zero of zeta_F at Re rho > alpha forces the design's own Mobius nu_F, the Dirichlet inverse of 1_(S_F), to have Mertens exponent at least Re rho. So the rightmost second-family real part is a proved lower bound on that exponent, printed per design.

THE THREE TESTS

• The expectations are derived first, from the full digit set and from the classical formulas, and never from the design sweep.
• SYMMETRY. On the full set the functional equation puts a zero at (1 - sigma) + i t for every zero at sigma + i t, so the real parts are symmetric about Re s = 1/2 = alpha/2 and each zero is its own partner. The test reads c_F as the midpoint of the real parts of the two second-family zeros of least Im s, then asks of every remaining zero whether (2 c_F - sigma) + i t is again a second-family zero, and compares c_F against alpha/2, alpha and alpha - 1/2.
• The design has no functional equation: the reflection moves the kernel and not the design, so the expectation is failure at the third zero, and the falsifier is a design where the pairing holds.
• COUNT. On the full set N(T) = (T/2 pi) log(T/2 pi e) + O(log T), hence N(T) = c T log T + d T with c = 1/(2 pi) = 0.15915494 and d = -(1 + log 2 pi)/(2 pi) = -0.451662163. Reading c and d from two heights rather than the closed form costs the control about two percent, which is the accuracy of the test; the question is whether c_F is a function of alpha, of the fill, or of the design alone.
• The count needs no zero located: N_2(T) is the winding of Z on the box less the zeros of Z inside every pole disc, both read by the argument principle.
• THE FULL-SET LIMIT. As alpha -> 1 through designs missing one digit and then two, the second family's real parts contract to alpha/2 or they do not. A contraction is the rule that makes RH a special case of MrlyMath; a non-contraction says the critical line is a property of the Euler product alone. The statistic is the mean and the spread of Re s - alpha/2 per design, ordered by k/q.
• The controls are the base 2, 3 and 4 full digit sets, where the answers are known: c_F = 1/2, N(T) classical, spread zero.

THE BLIND BAND

• The ladder does not reach tolerance in alpha - 1 < Re s < alpha - 0.92, a band against the cofactor's own pole line, so every census here runs on alpha - 0.92 < Re s < alpha + 3.02 and every count is a lower bound for the strip to alpha - 1.
• The box height is nudged up to the midpoint between two poles whenever a pole sits within 0.5 of it, so no pole disc straddles the top edge; the height actually used is printed with every count.

WHAT THE SWEEP FOUND

• Twenty-two designs: the twenty of ../zeta-locus, plus base 5 {0,1,2,3} and base 10 missing two digits as the rungs that carry alpha toward 1. Each is censused to its own printed height, 40 except the four base 3 designs at 42.894, base 9 {0,1,2} at 41.464 and base 10 missing two at 25.923, every height nudged off the pole lattice.
• 377 zeros wound on the strips, 351 located, 171 teeth of which 9 are level-one teeth, 19 cofactor-only zeros at null-residue poles, and 161 second family, all at rho = 0.45. Where the located count falls short of the winding the printed N_2 is a lower bound, base 4 {2,3} at 12 of 18 being the worst.
• THE RADIUS IS AN UNDEFENDED CONSTANT AND THERE IS NO GAP AT IT. rho = 0.45 is fixed only by the pole discs not overlapping, while the tooth law is accurate only inside abs(u) < 0.2 and says nothing past 0.3. The nearest live pole to a second-family zero is 0.45510938 away at base 4 {2,3}, 0.45909168 at base 3 {0,1}, 0.48696667 at base 4 {0,1,2} and 0.50481072 at base 4 {1,3}, five of base 4 {2,3}'s eight sitting inside 0.45 < abs(u) < 0.6, so the cut lands in a pile-up and every count is conditional on rho.
• Both radii are printed. N_2 reads 8, 7, 13, 9, 14 at rho = 0.45 and 7, 6, 12, 8, 7 at rho = 0.6 on base 3 {0,1} and the four base 4 two-digit designs. The full digit set is where the gap is real: at base 2 the nearest live pole is 14.143566 away and no radius below 0.9 moves a count.
• Base 3 {0,1} reads N_2 = 7 below Im 40 and 8 to its nudged height 42.894; the seventh zero -0.273079611 + 39.262315320 i sits 0.0160 right of the strip edge, inside the censused box and outside the blind band.
• THE LEVEL-ONE COMB BITES. Nine of the twenty-two designs carry a zero the level-zero comb calls second family and the level-one comb claims, all in the band against the cofactor's pole line; without it the second family would be overcounted by nine.
• THE CONTROLS. Base 2, 3 and 4 full digit sets give N_2 = 6, 7, 6 with every real part 0.5 to 1.4e-22, 1.1e-21 and 1.8e-22, the comb empty and every tooth a cofactor-only zero.
• THE TRANSPORT READING. Nineteen of the twenty-two designs carry a zero right of alpha, twelve of them from the second family alone, and the seventeen of those whose digit set contains 1 get a proved lower bound on the Mertens exponent of their own Mobius nu_F; base 4 {2,3} and {0,2,3} omit the digit 1 and have no nu_F.
• The strongest is base 10 missing two digits, alpha = 0.903089987, with a zero at 1.00151438765 + 2.77402670058 i: the Mertens exponent of nu_F exceeds 1 at a second base-10 column. Base 4 {1,2} at alpha = 1/2 has a SECOND-FAMILY zero at 0.940012431696 + 13.0678968771 i, an exponent of 0.94 against a design mass exponent of 0.5.

WHAT DIED

• NO SYMMETRY, AND NOTHING PAIRS AT ALL. No second-family zero in any design has a reflection partner: the branch needs two second-family zeros within 0.05 in Im s, and the smallest ordinate gap inside a design is far above that on every design tested, so it cannot fire. Every recorded pair is a self-pair, a real part landing within 0.05 of c_F, and self-pairs run BELOW chance: 8 of 47 with 0 partners over the ten designs recensused, a rate of 0.170213 against the 0.229904 that a uniform draw from each design's own band predicts, and 22 of 117 over the full sweep. The three full-set controls pair 13 of 13 at c_F = 1/2 to 1e-22, where the functional equation makes every zero its own partner.
• c_F is not a quantity: c_F - alpha/2 runs -0.28413232 to +0.47788515 and c_F - 1/2 runs -0.78413232 to +0.28664994. The reflection moves the kernel and not the design, and the census says so zero by zero.
• NO COUNTING LAW IN alpha OR FILL, AT EITHER RADIUS. The four base 4 two-digit designs share alpha = 1/2 and fill/base = 1/2 exactly and read N_2(40) = 7, 13, 9, 14 at rho = 0.45 and 6, 12, 8, 7 at rho = 0.6, with N_2(80) = 20, 30, 22, 29 and 17, 26, 21, 20: a factor of two at both radii, so the refutation is radius-robust even though the integers are not.
• The spread is carried by COMB OCCUPANCY and not by the second family: base 4 {0,1} and {2,3} differ 29 percent in total winding, 14 against 18, and a factor of two in N_2 because 7 of 8 poles are occupied against 4 of 8, and by the pile-up above an unoccupied pole is a pole whose zero drifted past rho. Every winding is the nearest integer to an integrated phase whose largest surviving step runs 0.9205 to 0.9998 against a cap of 1, so these counts are Verified and not Proved.
• Read as N_2(T) = c_F T log T + d_F T from the two heights, c_F is 0.10820213, 0.072134752, 0.072134752, 0.018033688 at alpha = 1/2, a spread of 0.09016844, while the two full-set controls read 0.15486803 and 0.16230319 against the classical 1/(2 pi) = 0.15915494 and spread 0.0074351582. The equal-alpha spreads are 0.071807313, 0.09016844, 0.013413767, 0.09016844 against that control spread.
• NO CONTRACTION OF THE REAL PART. The refuted law is max abs(Re s - alpha/2) -> 0 as alpha -> 1. Undivided the statistic stays FLAT along the ladder, reading 0.2275679549, 0.5549589411, 0.5885444877, 0.4233198337, 0.5365616661, 0.3151426744 at alpha = 0.430676558, 0.5, 0.630929754, 0.792481250, 0.861353116, 0.903089987, and then collapses to 1.43e-22, 1.10e-21 and 1.76e-22 at the base 2, 3 and 4 full sets. At base 10 missing 9, alpha = 0.954242509, the real parts read 0.216084781875 to 0.70401657869 about alpha/2 = 0.477121255, a band of 0.488 against 1 - alpha = 0.0458.
• Divided by 1 - alpha the same statistic runs 0.39971647 to 5.7047812 with NO monotone in alpha, falling from 3.8699872 to 3.2519104 on the last two rungs that carry alpha toward 1, so the refutation rests on the undivided spread and the ratio is not quoted for it.
• So the critical line is not reached by contraction. It appears at alpha = 1 as a jump from a band of width 0.49 to 1e-22, and that is the Euler product speaking, not the digit set.

WHAT THE ORDINATES DO INSTEAD

• The heights converge even though the real parts do not. Against the derived null - a quarter of the mean gap between consecutive zeta ordinates in the range, exact for an equally spaced set of the same density and about 18 percent conservative for one with gap variance - the mean distance from a second-family ordinate to the nearest zeta ordinate divided by that null reads 2.0495374 at alpha = 0.430676558, 1.8953371 at 0.5, 0.75419266 at 0.630929754, 0.51648744 at 0.792481250, 0.32356636 at 0.861353116, 0.090501352 at 0.903089987 and 1.0429899e-23 at alpha = 1.
• Monotone in alpha over seven rungs with a control that reads zero exactly, and the base and fill confounds are dead: the fall is monotone at FIXED BASE, 2.0495374 to 0.32356636 inside base 5 and 1.8953371 to 0.51648744 inside base 4, and at FIXED fill 2 across bases, 2.0495374, 1.8953371, 0.75419266, 1.0429899e-23. The nulls move only 1.0425839 to 1.3595166 across the ladder while the raw mean distance falls 2.4032315 to 0.12303809, so the denominator is not driving it.
• alpha is a trend and not a function: the four base 4 two-digit designs at one alpha = 1/2 spread 0.79050661 to 2.8404536. Over the ladder mean abs(Re s - 1/2) reads 0.36482392, 0.39426128, 0.3901396, 0.25540269, 0.31452367, 0.20473972 and 2.4065966e-23 and does not fall monotonically at all.
• Reading: A FILLING DESIGN LEARNS WHERE zeta'S ZEROS ARE BEFORE IT LEARNS THEY LIE ON A LINE. At alpha = 0.903 the heights are pinned to 2.3% of the mean gap while the real parts are still off 1/2 by 0.20. The Riemann hypothesis is the last thing to appear and it appears only at the full digit set.
• One caveat printed with the rows: the matching is nearest-ordinate and not injective, 3 distinct zeta ordinates for 4 design zeros at base 10 missing two and 5 for 6 at base 5 missing one, so the design carries a few zeros with no zeta partner.

WHAT STANDS AND WHAT IS OWED

• The second family is comb-independent and censused at a stated radius; it is not symmetric, it has no counting law in alpha or fill at either radius, and its real parts do not contract to alpha/2.
• What does converge is the ordinate set, and that is the open object: the rate of the ordinate convergence in 1 - alpha, and whether the extra zeros with no zeta partner are the design's own.
• Owed: a defended assignment radius, which the tooth law does not supply past abs(u) = 0.3; the shadow row at base 10 missing 9, one command; a Rouche certificate for the disc counts; and the blind band alpha - 1 < Re s < alpha - 0.92, which Z_1 removes as soon as the ladder can stop one level higher.

RUN

• uv run python research/lab/py/zeta-family/zeta_family.py family 40 all - the second family split from the comb: per design the counts N_2 at both radii, the real parts, the rightmost real part with its Mertens reading, and per zero the residue of Im s against 2 pi/log q.
• uv run python research/lab/py/zeta-family/zeta_family.py symmetry 40 all - the family sweep plus the reflection test about c_F.
• uv run python research/lab/py/zeta-family/zeta_family.py count 40 all - the winding count at two heights and the c_F T log T + d_F T read.
• uv run python research/lab/py/zeta-family/zeta_family.py limit 40 ladder - the same on the rungs that carry alpha toward 1, followed by the ordinate shadow against the zeros of zeta and its derived null.
• The second argument is the height, the third a family: q3, q4, q5, half, wide, ctl, near, ladder, rungs, q34, all; tests runs symmetry, limit, shadow and transport together. Prints only, writes nothing.
• The full sweep to height 40 is about thirty-five minutes and the ladder about seven; base 10 costs about nine times base 3 per evaluation.