Constants Gap Weight Projection Phi Dftenergy Total

The φ-pattern energy denominator

The declaration phiDFTEnergyTotal is a definition in the framework's machine-checked library of formal statements. It sets a number: the total energy of a specific discrete pattern, the φ-pattern, summed over the eight ticks of the framework's fundamental octave clock. In plain terms, it is the denominator used when converting a dimensionless fraction into a per-cell weight. The definition is explicit: it is the sum, over all eight ticks, of the squared amplitude of the pattern in the frequency domain, a standard Parseval-style total energy.

The framework's library establishes that this total energy is never negative, a basic sanity property. More importantly, the definition is part of a deliberate act of claim hygiene. The module it lives in, GapWeight.Projection (a module closing the historical "weights plus normalization" ambiguity), makes explicit two previously implicit choices. First, why the geometric weights carry a sin²(πk/8) factor: it is the spectral weight induced by the discrete derivative on the 8-tick cyclic shift, forced by shift-diagonalization. Second, why the normalization uses a factor of 64: that is 8 ticks times 8 spatial vertices in a Q₃ cell, the cardinality of the fundamental interface cell. The declaration phiDFTEnergyTotal is the denominator in that normalization, and the library states it as a definition, not as a theorem.

What the declaration does not claim is just as important. It does not assert that this total energy equals any particular numerical value, nor does it establish that the projected weight built from it equals the pipeline constant w8_from_eight_tick, which evaluates to about 2.49056927545. That equality is a tractable but nontrivial algebraic and trigonometric reduction problem, tracked as a follow-up theorem, not yet established in the library. The definition also does not claim that the 8-tick structure or the 64-cell normalization is physically forced; those are modeling choices, explicit and auditable, not derived results.

/-- Total DFT energy of the φ-pattern (Parseval denominator). -/
noncomputable def phiDFTEnergyTotal : ℝ :=
  Finset.univ.sum fun k : Fin 8 => phiDFTAmplitude k

/-- Total DFT energy of the φ-pattern (Parseval denominator). -/
noncomputable def phiDFTEnergyTotal : ℝ :=
  Finset.univ.sum fun k : Fin 8 => phiDFTAmplitude k

lemma phiDFTEnergyTotal_nonneg : 0 ≤ phiDFTEnergyTotal := by
  unfold phiDFTEnergyTotal
  apply Finset.sum_nonneg
  intro k _
  exact phiDFTAmplitude_nonneg k

What this page does not claim

The declaration does not assign a numerical value to the total energy. The declaration does not establish that the 8-tick structure is physically forced. The declaration does not establish equality with the pipeline constant w8_from_eight_tick.

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/GapWeight/Projection.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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