Constants Alpha Genesis Spectral Forcing Spectral Forcing Cert

The certificate

The eight-tick cycle is a discrete loop of eight positions, and the gap weight is a way of assigning importance to each position. The oscillation factor, the recognition term for the sine-squared pattern that modulates those weights, always looked like a modeling choice. The certificate SpectralForcingCert in the framework's machine-checked library of formal theorems establishes that it is not a choice at all: it is the spectrum of the one-step difference operator on that cycle.

The one-step difference operator measures how much a signal changes when you move one tick forward. Its spectrum, the set of possible change magnitudes, is computed on the standard eight-point Fourier basis. The key identity is that for mode k, the squared change magnitude equals 4 sin²(kπ/8). The certificate bundles three theorems: this trigonometric closure, the spectrum identity for every mode, and the factorization that the mode weight equals one quarter of that spectrum times the forced lattice measure. Both factors are now theorem-backed; neither is an input.

The certificate does not claim to derive the fine-structure constant itself. The exact value of alpha remains an open target. It does not claim to derive the normalization that scales the whole weight, the Parseval / 64-cell factor, which is inherited without re-derivation. It also does not claim that the eight-tick cycle is physically real; the cycle is a definitional model within the framework.

What the certificate changes is the status of the interior structure. Pattern, envelope, oscillation factor, and dressing form are all forced by theorems. The last unforced joint in the gap weight is the normalization. That is the one place where an input still enters, and the certificate names it plainly.

/-- **The spectrum identity.** The difference energy of DFT mode k equals
`4 sin²(kπ/8)`: the oscillation factor of the gap weight is exactly one
quarter of the difference-operator spectrum. -/
theorem diffEnergy8_mode_eq_four_sin_sq (k : Fin 8) :
    diffEnergy8 (dft8_mode k) =
      4 * (Real.sin ((k.val : ℝ) * Real.pi / 8)) ^ 2 := by
  rw [diffEnergy8_mode k]
  exact normSq_omega8_pow_sub_one k.val

/-- **SPECTRAL FORCING CERTIFICATE.** Bundles the M6 closure:
1. the trig closure `|ω₈ᵏ − 1|² = 4 sin²(kπ/8)`;
2. the spectrum identity for every DFT mode;
3. the full factorization of the mode weight into spectrum × measure. -/
structure SpectralForcingCert where
  deriving Inhabited

What this page does not claim

No claim that the fine-structure constant is derived; its exact value remains open. No claim that the eight-tick cycle is physically real; it is a definitional model. No claim that the normalization factor is derived; it is inherited without re-derivation.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Constants/AlphaGenesis/SpectralForcing.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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