MISSION: is the covering margin's decay the same in two families?

Scope: hunts/decay_law/. Nothing outside this directory is modified, apart from the case-log entry this hunt owes in hunts/README.md. Status: exploratory. Nothing here is a result about the Riemann Hypothesis.

The kill condition in the HuntSpec block below was written from the landed rectangle table, before any tent row was computed. A quadrature check of the tent formulas, run while building the instrument, printed two endpoint margins before this file was committed. The tolerance was not changed after that print. The tent measurement that RESULTS.md records is a fresh run, after this block. Landing note, 2026-10-30. Those two printed 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 was committed before the recorded run; it is not blind on alpha_m.

The question

Hunt #157 measured one family of rectangles. Its covering margin falls from 3.04e-2 at b = 2 to 1.59e-4 at b = 8. The local exponents between those five widths run from 2.23 to 5.58, so a single power is not a defensible summary even of that one family (first written "roughly b^{-2.5}", corrected in that hunt on 2026-10-30). This hunt asks whether a second family, of independent shape and on an overlapping width window, has the same decay, including the same split between the zero-side sum and the prime cost.

The margin is a ratio, by construction

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 cover - 1 = W_zeros / C

The margin is W normalized by the prime cost. It is not an independent statement about the archimedean term. On the shared endpoints the log-log exponents

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

satisfy alpha_m = alpha_W - alpha_C exactly, for X the margin, the zero-side sum, and the prime cost. That identity is the decomposition this hunt compares across families.

From hunts/weil_split/results.json, the rectangle side of the kill (not recomputed for the threshold) is

alpha_m = 3.7902, alpha_W = 1.3953, alpha_C = -2.3949

The denominator carries 0.632 of that endpoint decay, the numerator 0.368. The local exponents of the prime cost on that table run from 2.10 to 3.04 in magnitude and move smoothly. The local exponents of W_zeros change sign between b = 4.5 and b = 6. Some of the margin's local-exponent scatter is already, on the landed table, a zero-side oscillation rather than a steady decay.

The two families

Both are parameterized by b, the length of the support of f, so g is supported on [-b, b] in both cases and the windows b = 2 to b = 8 overlap. Both have g(0) = 1.

1. Rectangle (hunt #157, recomputed here only as a cross-check). f = b^{-1/2} 1_{[-b/2, b/2]}, g(u) = (1 - |u|/b)_+, h(r) = b sinc^2(b r / 2) with sinc x = sin(x)/x. This h decays as 1/r^2. It is a scalar multiple of the Fejer pair at half-width b/2, which does not change the margin.

1. Tent, computed only after the kill block below. f(x) = sqrt(3/b) (1 - 2|x|/b)_+ on [-b/2, b/2]. The autocorrelation is the piecewise cubic

g(u) = 1 - 6v^2 + 6v^3 if 0 <= v <= 1/2 g(u) = 2(1 - v)^3 if 1/2 <= v <= 1 g(u) = 0 if v >= 1

with v = |u|/b, and h(r) = (3b/4) sinc^4(b r / 4). This h decays as 1/r^4. The nodes of h sit at twice the rectangle's spacing (r = 4 pi k / b against 2 pi k / b; corrected at landing, 2026-10-30, from "half"). The shape is not a rescaling of the rectangle.

What would kill the universal reading

The universal reading says the decay, including the split, is a property of the width and not of the family.

Quantity. alpha_X = ln(X(2)/X(8))/ln(4), for X the zero-side sum, the prime cost, and the margin, rectangle against tent, at the shared endpoints b = 2 and b = 8.

Tolerance. 0.5. The rectangle's own local margin exponents already span 3.35 (2.23 to 5.58). A tolerance of 0.5 is a disagreement test, not a restatement of that scatter, and it sits far above the numerical uncertainty of the endpoint ratio (the landed margins are stable well below 1e-8 relative).

Kill. The reading is withdrawn if any one of the three exponents differs by more than 0.5. A shared prime-cost exponent with a different zero-side exponent fires this, which is the point of putting the kill on the decomposition rather than on the margin alone.

A single exponent is called a defensible summary of a family only if that family's local exponents, on the primary grid below, span at most 0.5. The rectangle already fails this on the landed five-point grid. That fact is about one family. It is not the kill. The kill is the cross-family comparison, so that it can fire on the tent.

Grid, fixed here

Primary grid: b = 2 + 0.25 k for k = 0, ..., 24. Twenty-five widths, spacing 0.25, from 2 to 8 inclusive. The spacing is just over half the rectangle node spacing 2π/γ_1 (about 0.444), so a zero-side oscillation shows as swings, though the grid is above its Nyquist interval (0.222) and aliases it rather than resolving it (corrected at landing, 2026-10-30, from "visible"). The kill uses only the endpoints, which are on this grid and on the landed table.

What this may not claim

• Not evidence for RH. Two compact families, one window.
• The zero side uses the verified ordinates up to 10^4. The tail beyond that height is bounded and recorded. It is not zero. Where the arithmetic side and the zero side disagree, the covering ratio is taken from the zero side.
• A quadrature that is only good to a few digits is printed beside every number that depends on it, and that precision enters the fit residual of any such number. The margin itself is W_zeros / C, and C is a finite prime-power sum, so the margin is not a quadrature.
• No human has reviewed any of this.

HuntSpec

id: decay_law
question: Is the covering margin's decay, including its split between the zero-side sum and the prime cost, the same for a rectangle and a tent of equal support width?
frontier: rectangle endpoint exponents on b=2 against b=8 are alpha_m=3.790, alpha_W=1.395, alpha_C=-2.395, from hunts/weil_split/results.json; local margin exponents there span 2.23 to 5.58
proposed_attack: recompute the rectangle on a fixed grid as a cross-check, then measure a tent whose transform is a fourth power of a sinc, and compare endpoint exponents to tolerance 0.5
dead_routes:
  - a single power of b, already refused by the rectangle local exponents spanning 2.23 to 5.58
  - reading the margin as an independent archimedean statement, refused because cover minus 1 equals W_zeros over the prime cost by construction
required_oracles:
  - the verified zero ordinates, for the zero-side sum
  - the finite prime-power sum for the prime cost, checked against the per-place overlap sum
  - the landed rectangle table in hunts/weil_split/results.json, for the kill's rectangle side
kill_conditions:
  - the universal reading is withdrawn if alpha_X = ln(X(2)/X(8))/ln(4) differs between the rectangle and the tent by more than 0.5, for X the zero-side sum, or the prime cost, or the margin
agents_may:
  - measure
  - compare
  - code
  - decompose
agents_may_not:
  - declare a bound
  - promote an observation to a result
  - use the reserved vocabulary of zeta/rigor.py