Gravity Analysis Regge Exact Flat Hessian Bloch M2 Rayleigh4 D Wave4

The Wave4 declaration

In the Recognition Science framework's machine-checked library of formal theorems, Wave4 is a definitional choice: it is a name for a four-dimensional wave vector, a mathematical object that carries momentum and direction. The declaration itself is a definition, not a discovery. It sets up the vocabulary for a larger result about a quantity called the exact midpoint Bloch m² Rayleigh quotient, which is a technical ratio built from a curvature matrix and a wave vector.

The theorem that uses Wave4 is about a special case. When the curvature matrix has a unit Frobenius norm, meaning its total size is scaled to one, and the wave vector satisfies a transversality condition, the Rayleigh quotient equals exactly -1/8. This is a precise algebraic face, a fixed number that the ratio must land on under those conditions. The result is proved in the framework's library, and the proof is checked by the kernel, not by a compiler. The library's audit prints only the three standard axioms of the ambient type theory, with no framework-specific axioms added.

A second, related theorem covers the pure gauge case. A pure gauge configuration is a kind of redundancy in the description, a transformation that does not change the physical state. For such a configuration, the same Rayleigh quotient equals exactly 0. Both identities are packaged together in a single theorem named typedResidual_m2_rayleigh_eq_algebraic_face, which is the ledger-facing name for the handoff residual R3. The word ledger here means a discrete record of events, and the residual is a quantity that measures how far a candidate solution is from satisfying an equation.

What Wave4 does not claim is just as important as what it proves. It does not claim that the -1/8 value is a physical constant measured in an experiment; it is a proved identity within the framework's formal system. It does not claim that the pure gauge result applies to all possible configurations, only to those built from the pureGaugeFamily construction. And it does not claim that the framework's model of gravity is the same as conventional general relativity; it is a separate, self-contained account that happens to use similar vocabulary.

abbrev Wave4 := Regge4DContinuumPreflight.Wave4

/-- Unit-Frobenius TT Rayleigh equals the algebraic face `-1/8`. -/
theorem exactMidpointBlochM2_rayleigh_eq_unitFrobeniusTTCoeff
    (H : Mat4) (k : Wave4)
    (hTT : IsTT k H)
    (hF : frobeniusNormSq H = 1)
    (hk : waveNormSq k ≠ 0) :
    exactMidpointBlochM2 H k / waveNormSq k =
      exactHessianM2UnitFrobeniusTTCoeff := by
  have hF' : ReggeExactMidpointM2TTIdentity4D.frobeniusNormSq H = 1 := by
    simpa [frobeniusNormSq_preflight_eq_identity] using hF
  have hk' : ReggeExactMidpointM2TTIdentity4D.waveNormSq k ≠ 0 := by
    simpa [waveNormSq_preflight_eq_identity] using hk
  have h :=
    exactMidpointBlochM2_rayleigh_eq_neg_eighth_of_TT H k hTT hF' hk'
  simpa [exactHessianM2UnitFrobeniusTTCoeff, waveNormSq_preflight_eq_identity]
    using h

/-- Pure-gauge Rayleigh equals the algebraic face `0`. -/
theorem exactMidpointBlochM2_rayleigh_eq_gaugeCoeff
    (m v : Wave4) (hm : waveNormSq m ≠ 0) :
    exactMidpointBlochM2 (pureGaugeFamily m v) m / waveNormSq m =
      exactHessianM2GaugeCoeff := by
  have hm' : ReggeExactMidpointM2TTIdentity4D.waveNormSq m ≠ 0 := by
    simpa [waveNormSq_preflight_eq_identity] using hm
  have h := exactMidpointBlochM2_gauge_rayleigh_eq_zero m v hm'
  simpa [pureGaugeFamily, exactHessianM2GaugeCoeff,
    waveNormSq_preflight_eq_identity] using h

What this page does not claim

The -1/8 value is a measured physical constant, not a proved identity. The pure gauge result applies to configurations outside the pureGaugeFamily construction. The framework's gravity model is equivalent to conventional general relativity.

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/Gravity/Analysis/ReggeExactFlatHessianBlochM2Rayleigh4D.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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