Question. The sharpest gate in this repository is the quasicrystal gate (zeta/quasicrystal.py, docs/18 §4): the zero measure's Fourier transform is atomic, and the atom pattern detects the Euler product. But that instrument has never been pointed at the one kind of aperiodic point set whose atomic diffraction is a theorem: a regular model set (cut-and- project). The golden-ratio chain is the canonical one. This hunt gives the quasicrystal lane what zeta/finitefield.py gives the RH lane — a universe where the answer is proved — and asks whether the lane's instrument reproduces the proved answer.
The golden thread, stated precisely and then pinned by computation. The Davenport–Heilbronn rival is built on the quartic character mod 5, whose square is the quadratic character χ₅ — the character of ℚ(√5), the golden field. Fibonacci arithmetic mod p is governed by that same χ₅: the Pisano period π(p) divides p − χ₅(p). So the lab's standing counterexample and the golden quasicrystal sit over the same conductor-5 arithmetic, one step apart. This hunt does not claim that connection does anything; it pins it exactly and uses the golden chain as ground truth for the instrument.
House rule applied to the author: derive, never remember. No diffraction formula is quoted from memory. The probe calibrates its tapered transform on ℤ itself — where the answer is Poisson summation, elementary and exact — and then derives the golden chain's predicted peak set and intensity law from the cut-and-project data (lattice matrix, dual, window Fourier transform) inside the probe, as code that can be read and checked.
Pre-registered predictions
- P1 (calibration on ℤ). The tapered transform of ℤ matches the Poisson- summation prediction at k ∈ 2πℤ to better than 0.5%, and the calibration constant transfers unchanged to the aperiodic runs.
- P2 (the theorem reproduced). For the golden model set (window [0, 1)), the twelve strongest predicted peaks appear within 10⁻³ of their predicted positions in the projected dual module, and the measured amplitude ratios match the derived window-transform law to a few percent (finite-size limited).
- P3 (silence off the module). At 200 random frequencies away from the predicted module, the median response is at least 30× below the weakest tested peak — the analog of the 26.8× prime-power/composite separation the gate measured on ζ.
- P4 (lesions). Gaussian jitter of size σ suppresses each peak by the factor e^{−σ²k²/2} (measured log-slope within 10%); a Poisson set of equal density shows no peaks above background.
- P5 (the golden thread, exact). In exact integer arithmetic: the square of the DH quartic character equals χ₅ pointwise mod 5, and for every prime p < 500, p ≠ 5, the Pisano period π(p) divides p − χ₅(p).
- P6 (precision response). Doubling the point-set extent and the taper width moves every measured peak intensity toward the derived law — the defect shrinks, as a real quantity must.
Scope
May touch: hunts/golden_control/ only. Reads zeta.epstein.chi5 and nothing else from the package's subject-matter modules; the transform is probe-local (the packaged one is specialized to one-sided ordinate lists). No ledger entry unless something survives hunts/README.md's checklist. Everything here is the accurate regime — numpy floats and exact integer checks; the strongest words used are measured and observed. Nothing in this hunt is about ζ's zeros, and nothing in it is evidence about RH; it is an instrument-validation study against a proved ground truth, plus one exactly pinned piece of conductor-5 arithmetic.