Constants Alpha Genesis Spectral Forcing Geometric Weight Eq Spectrum Mul Measur

The spectral factorization

The declaration geometricWeight_eq_spectrum_mul_measure is a theorem in the framework's machine-checked library of formal theorems. It states that, for every nonzero mode k on the framework's eight-tick recognition cycle, the mode's geometric weight equals the difference-operator spectrum divided by four, multiplied by the lattice weight. In plainer terms: the oscillation factor sin²(kπ/8) that appears inside the gap-weight mode weights is not a free modeling choice. It is forced, because it is exactly one quarter of the spectrum of the one-step difference operator on the cycle, evaluated on the discrete Fourier transform (DFT-8) eigenbasis.

The theorem is the final link in a chain that shows the gap weight's pattern, envelope, oscillation factor, and dressing form are all forced by the framework's structure. The chain works as follows. First, the DFT-8 modes diagonalize the cyclic shift, a standard fact. Second, the difference energy of mode k is the squared modulus of its shift eigenvalue minus one. Third, a trigonometric closure shows that this squared modulus equals 4 sin²(kπ/8). Fourth, the factorization theorem combines these to express the mode weight as the product of two theorem-backed factors: the spectrum term and the lattice weight. Both factors are now theorems; neither is an input. The only remaining ingredient inherited without re-derivation is the Parseval / 64-cell normalization, named in the premise section of the framework's documentation.

The theorem does not claim that the fine-structure constant α is derived. It does not claim that the overall normalization of the gap weight is forced. It does not claim that the eight-tick cycle itself is derived here; that is a separate part of the framework's forcing chain. The theorem's scope is narrow: it shows that, given the cycle and the lattice weight, the oscillation factor is forced by the difference operator's spectrum. The empirical comparison of the resulting α value against measured data is a separate check, not part of this theorem.

/-- **SPECTRAL FORCING.** For every nonzero mode, the gap-weight mode
weight factors as (difference-operator spectrum / 4) times the T9 forced
measure:
`geometricWeight k = (diffEnergy8(mode k)/4) · latticeWeight k`.
Both factors are theorems; neither is an input. -/
theorem geometricWeight_eq_spectrum_mul_measure (k : Fin 8) (hk : ¬ k.val = 0) :
    GapWeight.geometricWeight k =
      (diffEnergy8 (dft8_mode k) / 4) *
        Foundation.MeasureForcing.latticeWeight k.val := by
  rw [geometricWeight_eq_sin_mul_forced_measure k hk,
    diffEnergy8_mode_eq_four_sin_sq k]
  ring

/-- **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

What this page does not claim

The theorem does not derive the fine-structure constant α. The theorem does not force the overall normalization of the gap weight. The theorem does not derive the eight-tick cycle itself.

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)

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