zeta-shadow

• The MECHANISM of the ordinate shadow: why the zeros of the design zeta zeta_F(s) = sum_(n in S_F) n^(-s), with F the digit set, sit near the zeros of zeta, and how far.
• The shadow is a first-order perturbation, so the offset from a zeta zero rho_0 is a computed number and not a trend: it is abs(zeta_F(rho_0))/abs(zeta_F'(rho_0)) with no constant, and abs(zeta_F(rho_0))/(c abs(zeta'(rho_0))) under a split zeta_F = c zeta + E_F.
• The evaluator, the truncation bounds and the pole data come from ../design-zeta; the second family and its census come from ../zeta-family and ../zeta-locus. This study adds the fibre weight, the per-zero prediction and two new rungs.

THE SPLIT

• The position identity of ../mrly-euler reads sum_(n in D_level, n >= 1) n^(-s) = int_0^1 G_level(t) Z(s,t) dt with G_level(t) = prod_(i<level) sum_(d in F) e(d base^i t) the position product, Z(s,t) the periodic zeta and Z(s,0) = zeta(s); D_level is the set of strings of level digits over F.
• Its discrete form is exact and finite: 1_(D_level)(n) = base^(-level) sum_(a mod base^level) G_level(a/base^level) e(-n a/base^level) for 0 <= n < base^level, so zeta_(F,level)(s) = base^(-level) sum_(a mod base^level) G_level(a/base^level) S_level(s, a/base^level) with S_level(s,x) = sum_(1 <= n < base^level) e(-nx) n^(-s). Verb mass reproduces 1_(D_level) from the transform to 8.326e-40 at level 2 on every design of the ladder.
• THE PRINCIPAL FIBRE. The term a = 0 is base^(-level) G_level(0) S_level(s,0) = (fill/base)^level times the partial sum of zeta to base^level, since G_level(0) = fill^level. That weight is exact and needs no arc, and the verb prints it against (fill/base)^level at every rung.
• THE LEVEL IS NOT FORCED. The identity splits the POLYNOMIAL at level level against a TRUNCATED zeta, while the object of this study is the continued zeta_F against the full zeta; (fill/base)^level falls to 0 with level, and to the right of the abscissa both series tend to 1, so the fibre gives fill/base only at level 1 and that reading is a choice.
• The continuous form is the mass of G_level on the principal arc abs(t) < 1/(2 base^level), an exact sinc sum 1/base^level + sum_(n in D_level, n > 0) sin(pi n/base^level)/(pi n); it equals kappa_level(F) (fill/base)^level with kappa_level running 0.6015221 to 0.96774464 over the ladder at level 1, 2, 3, a factor of 1.6, and the full set's own readings falling 0.81830989, 0.70926038, 0.65068647 toward Si(pi)/pi = 0.5894898722. It is the same weight up to that shape factor and adds no constant the fibre does not already give.

THE FIRST-ORDER SHADOW

• At a zero rho_0 = 1/2 + i gamma of zeta one has zeta(rho_0) = 0, hence for ANY constant c the split zeta_F = c zeta + E_F gives E_F(rho_0) = zeta_F(rho_0) exactly. That identity carries no information about c.
• A zero of zeta_F near rho_0 solves c zeta(s) = -E_F(s), so to first order s = rho_0 - zeta_F(rho_0)/(c zeta'(rho_0)).
• STEP, THE PRIMARY COLUMN. Reading c zeta'(rho_0) as zeta_F'(rho_0) removes the constant and gives s = rho_0 - zeta_F(rho_0)/zeta_F'(rho_0), Newton's own first step, which is Taylor's theorem at a simple zero of zeta_F and needs no split at all.
• PRED, THE SECONDARY COLUMN. The same law at the level 1 fibre reading c = fill/base. The lab prints both and lets the sweep choose.
• Both columns are computed from zeta_F, zeta and the digit density alone and see no design zero. The offset is one complex number, so at the zeros the law pairs the ordinate offset and the real-part offset are one quantity with no preferred phase.
• The full digit set is the control and it is exact: zeta_F = zeta, so E_F(rho_0) = 0 and both predicted offsets are 0.
• Proved: the discrete identity, the fibre weight, E_F(rho_0) = zeta_F(rho_0) for every c, and the first-order step at a simple zero. Verified: every ratio and every rate, resting on the ladder's propagated bound and on a resolved and uncertified Newton step.

THE FALSIFICATION PLAN

• Both predictions are computed BEFORE the comparison and from quantities that do not see the measured zero.
• The measured zero is reached by Newton from rho_0 and accepted only at abs(zeta_F) < 1e-16, within 1.5 of rho_0 and 0.02 clear of the pole lattice; the printed ratio is per zero, never a mean of the two sides. A miss is a zero the trust region does not reach, and the rung's medians are conditioned on that.
• A ratio far from 1 at the dense rungs kills the mechanism. The full-set control must read E_F = 0 and offset 0.
• The coupling zeta_F'(rho_0)/zeta'(rho_0) is a second reading of the same split and it decides between the candidate constants only if it resolves them; the lab prints its distance to fill/base and to 1 side by side.

WHAT THE SWEEP FOUND

• Nine designs on one ladder in alpha, the missing digit always the top one, so base 20 missing one digit is F = {0..18} and base 50 missing one digit is F = {0..48}. Twelve zeta zeros to Im s = 56.4462476971 on the seven rungs below alpha = 0.96 and six to Im s = 37.5861781588 on the two densest, so the rungs do not share one height. Largest ladder bound anywhere 9.001e-23.
• THE CONTROL IS EXACT. At the base 2 full set the evaluator reads abs(zeta_F(rho_0)) between 1.85e-34 and 1.329e-25 at all twelve zeros, so E_F = 0 and both predicted offsets are 0, with nothing fitted to make it so.
• THE CONSTANT-FREE STEP IS THE LAW. Its median ratio reads 1.3843088, 1.284225, 1.2481449, 1.2060106, 1.2042502, 0.89075541, 1.0195598, 1.005076, 0.99741809 at alpha = 0.430676558, 0.5, 0.630929754, 0.792481250, 0.861353116, 0.903089987, 0.954242509, 0.982877878, 0.994835739, and its band closes at the dense end: largest abs(ratio - 1) is 0.14041 at base 20 missing one digit and 0.01734 at base 50 missing one digit, band [0.94875, 1.14041] and [0.98266, 1.01144].
• IT SHARPENS WITH THE OFFSET, WHICH IS THE COMB LAW'S OWN SHAPE. Pooled over the ladder, the step's largest abs(ratio - 1) runs 0.01734, 0.0508884, 0.193158, 0.83912, 3.32327 over the buckets abs off < 0.05, < 0.1, < 0.2, < 0.4 and above, on 7, 4, 14, 18, 44 zeros.
• THE fill/base READING IS WEAKER AND IS NOT SELECTED. Its median ratio reads 1.4129353, 1.2842149, 1.0955991, 1.1806066, 1.1372453, 1.276577, 1.1347487, 1.0320127, 1.0507079 with largest abs(ratio - 1) 0.24964 and 0.0821168 at the two dense rungs, five times looser than the step at base 50. The coupling cannot decide it either: median abs(coupling - fill/base) is 0.24057225 and 0.08291158 there against median abs(coupling - 1) 0.27619434 and 0.079335871, so the reading flips between the two rungs while the candidates differ only by m/base = 0.05 and 0.02. c = fill/base stays the level 1 fibre reading and nothing more.
• THE RATE. Median offset over m/base = 1 - fill/base reads 1.6463532, 1.2495026, 1.8345578, 1.5731321, 2.2102406, 1.6634381, 2.2424916, 1.8779239, 1.051349, and over 1 - alpha it reads 1.7350628, 1.2495026, 1.6569184, 1.8951686, 3.1883019, 3.432954, 4.9008186, 5.4839111, 4.0716338. A least squares in the logs, a fit and not a theorem, gives (m/base)^1.04544 with R2 0.957842 against (1-alpha)^0.71691 with R2 0.944011; the first column spans 2.13297 and the second 4.38888, so m/base carries the exponent by a factor of 2.05764 inside the 4.28797 that (1-alpha)/(m/base) itself spans over this ladder.
• TWO NEW RUNGS BETWEEN 0.954 AND 1. Base 20 missing one digit at alpha = 0.9828778777 and base 50 missing one digit at alpha = 0.9948357391, all six zeros located at each: median abs(E_F(rho_0)) 0.11830158 and 0.028066806, median offset 0.093896196 and 0.021026979. The paired offset falls fast across the interval the family sweep left empty.
• THE SHADOW STATISTIC ON THE PAIRED ZEROS. Read in the form of the family row, the mean distance from a located design ordinate to the nearest zeta ordinate over a quarter of the mean gap between consecutive zeta ordinates in the range, the ladder reads 0.81218635, 0.57141859, 0.50488757, 0.37447728, 0.20954319, 0.29197634, 0.14794812, 0.052888241, 0.011098646 and 0 at the full set. The pairing is zeta-zero-first and the family row's is design-zero-first, so this is a parallel ladder and not that row recomputed.

WHAT DIED

• THE SHADOW DOES NOT SEPARATE ORDINATE FROM REAL PART AT THE ZEROS IT PAIRS. The offset is one complex number with no preferred phase, and per zero abs(Im off)/abs(Re off) spans 0.137681 to 6.11895 at base 20 missing one digit and 0.14167 to 18.7749 at base 50 missing one digit.
• Rung by rung the two medians are 0.55734029/0.43095421, 0.49670656/0.28603903, 0.48081533/0.22535435, 0.30633741/0.16748977, 0.12823995/0.36138728, 0.25093621/0.12181102, 0.11574693/0.19332005, 0.058239278/0.049215607, 0.011954894/0.010995712: the ordinate offset is the larger on six rungs and the smaller on three, and both fall along the ladder.
• The law binds only zeros Newton reaches from a zeta zero inside 1.5, so it constrains no other design zero and says nothing about a census taken design-zero-first.

WHAT IS OWED

• The census at base 20 and base 50 missing one digit, so the located zeros are named second family or teeth and the new rungs carry the family row's own statistic.
• The count of design zeros with no zeta partner per rung, which this lab cannot produce: the pairing is zeta-zero-first and it enumerates no design zero.
• The nine zeros at base 5 {0,1}, base 4 {0,1} and base 3 {0,1} where Newton reaches nothing inside the trust region: predicted offsets there run 0.95618855 to 3.0967393, outside the first order's own domain.
• A rigorous bound on zeta_F near rho_0, which turns the Newton step into a Rouche count and the Verified ratios into a proved enclosure.
• The two new rungs at twelve zeros, which would remove the height incomparability with the other seven.

RUN

• uv run python research/lab/py/zeta-shadow/zeta_shadow.py mass ladder - the discrete identity check, the principal fibre weight (fill/base)^level and the exact principal-arc mass at level 1, 2, 3, with the full-set control.
• uv run python research/lab/py/zeta-shadow/zeta_shadow.py predict ladder - per design and per zeta zero the remainder abs(E_F(rho_0)), the coupling, the constant-free step, the fill/base reading, the located zero and both ratios.
• uv run python research/lab/py/zeta-shadow/zeta_shadow.py rungs ladder - the same plus the rate in m/base and 1 - alpha with its log-log fit, the shadow statistic per rung, and the sharpening of both columns as the offset shrinks.
• The second argument is a family: ladder, old, new. Prints only, writes nothing. The full ladder is about thirteen minutes and mass about four minutes (217 s).