RESULTS: decay_law

Nothing here is a result about the Riemann Hypothesis. This is a measurement of one ratio, on two compact families, on one window. Grade: measured.

The kill condition in MISSION.md was committed before this table was computed. A quadrature check of the tent formulas printed two endpoint margins before that commit. The tolerance was not changed after that print. Those two margins alone give the tent's alpha_m = 5.73, so the fire on alpha_m was knowable when the tolerance was set: the kill is pre-registered against the recorded run, not blind on alpha_m (landing note, 2026-10-30). The numbers below are a fresh run of hunts/decay_law/decay.py.

1. The ratio, and the two families

For either family, with ||f||^2 = 1 and g the autocorrelation,

C = -prime_term = sum_p cost_p W_zeros = sum h(gamma) both signs, verified ordinates, gamma <= 10^4 margin = cover - 1 = W_zeros / C

The margin is W normalized by the prime cost. It is not a statement about the archimedean term. On the endpoints,

alpha_X = ln(X(2) / X(8)) / ln(4)

satisfies alpha_m = alpha_W - alpha_C exactly.

Both families are parameterized by b, the length of the support of f, so g is supported on [-b, b] in both cases and g(0) = 1.

• Rectangle, the family of hunt #157, recomputed here as a cross-check. h(r) = b sinc^2(b r / 2), decaying as 1/r^2. This is b times the Fejer pair at half-width b/2. The recomputed zero sum matches that scaled Fejer zero sum to 2e-9 relative, and the prime cost matches hunts/weil_split/results.json to 1e-12 relative, at every landed width.
• Tent. f(x) = sqrt(3/b) (1 - 2|x|/b)_+ on [-b/2, b/2], g the piecewise cubic in MISSION.md, h(r) = (3b/4) sinc^4(b r / 4), decaying as 1/r^4. The nodes of h sit at twice the rectangle's spacing (4 pi k / b against 2 pi k / b; first written "half", corrected 2026-10-30). Not a rescaling of the rectangle.

The per-place overlap sum equals the prime cost on every row. The largest gap is 5e-16.

Primary grid, fixed in MISSION.md: b = 2 + 0.25 k for k = 0 to 24. Twenty-five widths, from 2 to 8.

2. The kill fired

Rectangle side, from hunts/weil_split/results.json, not from the recomputation:

The universal reading is withdrawn. It fired on the margin and on the zero-side sum. It did not fire on the prime cost.

The recomputed rectangle exponents agree with the landed ones to 1e-9 on alpha_m and exactly, at the printed digits, on alpha_C. Substituting the recomputation would not change the fire.

Endpoint values the exponents come from:

b = 8 sits 0.003 of a node-spacing from a zero of the rectangle's h(γ_1), and 0.002 of a node-spacing from a zero of the tent's. The endpoint alpha_W is phase-sensitive. Two checks say the fire is not that one node:

• Adding the density tail estimate to the rectangle's truncated W moves alpha_m by 0.0026, under the tolerance.
• The least-squares fit over all 25 widths, which does not sit on that node, still differs by 1.93 on the margin and by 2.09 on W. The fit is a check. The kill is the endpoint comparison, as written.

3. A single exponent is not a defensible summary of either family

The rule was written in MISSION.md before the recorded tent run (two tent endpoint margins had been printed; see the opening paragraph): a single exponent is a defensible summary only if the local exponents on this grid span at most 0.5. Both families fail it.

Log-log least squares of ln X = A - alpha ln b, residual in natural-log units (a residual of 0.7 is a factor of about 2):

The fitted margin exponents differ by 1.93, the same size as the endpoint difference. The fitted prime-cost exponents differ by 0.16. The residual says the margin fit is a poor summary, and the prime-cost fit is a much better one.

Local exponents of the margin, between consecutive widths, rounded. Full values are in results.json.

Rectangle, span 33.5, from -11.0 to 22.6:

6.2, 0.6, 5.3, 2.9, 2.0, 6.2, -4.8, 11.4, -4.3, 12.0, 0.1, 0.9, 2.0, 2.3, 10.6, -2.0, 13.9, -8.6, 5.8, 3.1, 7.4, 12.5, -11.0, 22.6

Tent, span 103.2, from -42.4 to 60.7:

-3.5, 22.6, -0.7, -4.1, 2.3, 34.8, -12.0, -13.5, 26.8, 8.6, -16.2, 3.3, 37.5, 2.9, -35.3, 35.6, 3.6, -4.4, -13.3, 60.7, -8.1, -42.4, 31.4, 32.4

Hunt #157's five widths gave local margin exponents from 2.23 to 5.58, and an endpoint ratio of about b^{-3.8}. That coarser grid averaged over the oscillation this grid shows. This grid does not resolve it either: the rectangle's γ_1 oscillation has period 0.444 in b, so its Nyquist interval is 0.222, below the 0.25 spacing, and the swings above are aliased. The endpoint ratio is real as a ratio of two numbers. It is not a local decay rate.

On this grid the rectangle margin rises on 5 of the 24 steps (from b = 3.5, 4.0, 5.75, 6.25 and 7.5). Hunt #157's "the cover margin collapses" holds as a fall from b = 2 to b = 8, not as a monotone decay; its record carries a dated note.

The endpoint statistic is itself phase-sensitive. Over the 39 grid pairs with b2 / b1 >= 2.5, the tent-minus-rectangle margin exponent runs from -0.17 to 4.00, and 3 pairs fall inside 0.5 (for example b = 2.5 against 7.5). The tolerance's justification in MISSION.md is numerical precision, which is not the main uncertainty here; the least-squares fit (section 2) is what supports the fire on [2, 8]. (Added at landing, 2026-10-30.)

4. How the decay splits

alpha_m = alpha_W - alpha_C. The share of the endpoint decay coming from each piece:

The prime cost grows in both families, by a factor 27.7 (rectangle) and 23.0 (tent) from b = 2 to b = 8, and it grows at every step of the grid. Its local exponents stay inside an interval of length 1.33 (rectangle, -2.04 to -3.37) and 0.57 (tent, -2.09 to -2.66). W shrinks by a factor 6.9 (rectangle) and 123 (tent), and it increases on 11 of the 24 steps in each family.

So on this window, b from 2 to 8, the piece the two families share, to within the tolerance, is the prime cost. The piece they do not share is the zero-side sum. The margin inherits that disagreement. A reading that stays inside the measurement: the fitted alpha_W is 1.03 for the rectangle, whose mean envelope scales as 1/b, and 3.12 for the tent, whose mean envelope scales as 1/b^3. Those are the envelope powers of sinc^2 and sinc^4. The measurement is consistent with the zero-side decay tracking the decay of h, which is family-specific. It does not establish that tracking beyond these two families.

Scope, added at landing, 2026-10-30. The sharing is a property of the window. By the prime number theorem C is about 2 int_0^b e^{u/2} g(u) du, which is 8 e^{b/2} / b for the rectangle (C b e^{-b/2} is 7.08 at b = 8, rising toward 8) and of order e^{b/2} / b^3 for the tent. Neither is a power of b, so a log-log exponent of C depends on the window by construction: a reviewer's extension of the run to b = 16 found the tent-minus-rectangle alpha_C growing from 0.17 on [2, 8] to 1.34 on [10, 16] (measured, not committed here). Heuristically the prefactor difference in C cancels the difference in the W envelopes, so both margins would decay like e^{-b/2}. The withdrawal of the universal reading above is a statement about power-law exponents on [2, 8], not about the asymptotic rate.

5. The local-exponent variation is the zero side

h(γ_1) was evaluated at the first ordinate of the same list the zero sum uses, not taken from a remembered constant, through the closed forms b sinc^2(b γ_1 / 2) and (3b/4) sinc^4(b γ_1 / 4). (First written as "matches the closed form", which cannot fail: decay.py computes it from the closed form. Corrected 2026-10-30.)

The first zero does not dominate the rectangle: its share of W runs from 1.4e-5 to 0.42, and the first three zeros never carry more than 0.515. The tent is often the first few zeros: the first zero's share runs from 2.6e-9 to 0.94, and the first three run from 0.085 to 0.986. At a node of h(γ_1) that share collapses and later zeros carry the sum.

Correlation of the local margin exponents with the local exponents of h(γ_1): 0.85 on the rectangle, 0.50 on the tent. The prime cost has no such swings on this window. The variation in the local margin exponents is oscillation from the zero side; the prime cost's own local exponent drifts slowly (section 4, the e^{b/2} law), which on [2, 8] is small next to it.

6. Arithmetic side against zero side

Where they disagree, the covering ratio is the zero side's. The prime cost is a finite sum (e^8 is under the prime-power cap), so the margin is not a quadrature.

Rectangle. Relative disagreement 4.5e-3 to 8.6e-3. On the Fejer path the signed difference, arithmetic minus truncated zero sum, equals the density tail estimate, with leftover at most 2.2e-7 absolute. The disagreement is the truncation at height 10^4, not the archimedean integral. Hunt #157's generic quadrature, without that path, reported agreement of 1e-3 to 8e-3 and took the ratio from the zero side. This measurement does the same. The tail is under 1% of W and moves alpha_m by 0.0026.

Tent. Worst relative disagreement 5.7e-8, at b = 3.5, and the signed difference matches the tail estimate to 4e-15 absolute. The archimedean quadrature is not the limit. This is not a few-digit quadrature, and that floor is not folded into the fit residual. The fit uses W_zeros and the prime cost.

7. Scope

• Two families, twenty-five widths, b from 2 to 8. Nothing here covers a general test function.
• The zero side uses verified ordinates up to 10^4. The tail beyond that height is the disagreement in section 6. It is not zero.
• b = 8 is near a node of h(γ_1) in both families. Section 2 records what that does and does not do to the kill.
• This measurement is not a ceiling, so it does not carry a door inventory.
• No human has reviewed any of this.