Gravity Analysis Regge Exact Flat Hessian Bloch M2 Rayleigh4 D Typed Residual M2

Two exact faces of a wave ratio

In the framework's analysis of a four-dimensional wave system, a particular ratio appears: the value of a function called the midpoint Bloch mass-squared term, divided by the squared norm of a wave vector. The declaration typedResidual_m2_rayleigh_eq_algebraic_face proves that under two distinct conditions this ratio collapses to a simple algebraic constant. The first condition is a physical one: the wave vector must be transverse to the Hessian matrix, the matrix must have unit Frobenius norm, and the wave vector must not be null. Under those conditions the ratio equals exactly -1/8. The second condition is a gauge one: the wave vector is built from a pure gauge family, and the ratio equals exactly 0.

The two results are packaged as a single theorem. The first is the unit-Frobenius transverse-traceless case, where the ratio lands on the algebraic face -1/8. The second is the pure-gauge case, where the ratio lands on 0. The declaration is the ledger-facing name for the handoff residual R3, and it is inhabited by the same proof as the underlying theorem exactMidpointBlochM2_rayleigh_eq_algebraic_face. The proof uses a rational packed biquadratic table and closed-form transport, and the table is checked by the kernel, not by the compiler. The axiom audit for the headline closer printed the three standard axioms [propext, Classical.choice, Quot.sound].

What the declaration does not claim is just as important. It does not say that -1/8 is the only possible value of the ratio, nor that the ratio is always defined. It does not claim that the pure-gauge case is physical, nor that the transverse case is the only physical one. It does not assert that the midpoint Bloch mass-squared term itself has any particular value, only that the ratio takes these two exact values under these two exact conditions. The theorem is a precise algebraic identity about a specific ratio, not a statement about the physics of the system as a whole.

The value of this result is that it gives a clean, checkable anchor for a residual term in the framework's gravity analysis. A residual that collapses to a simple constant under a named condition is a residual that can be audited, compared, and reasoned about. A residual that does not collapse is a different story. This declaration pins down two such collapses, and in doing so it gives the framework a concrete, machine-checked fact to build on.

/-- 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

/-- Ledger-facing inhabit of R3 (same Prop shape as
`SRSConvergesEH4D.TypedResidual_m2_rayleigh_eq_algebraic_face`). -/
theorem typedResidual_m2_rayleigh_eq_algebraic_face :
    (∀ (H : Mat4) (k : Wave4),
        IsTT k H →
          frobeniusNormSq H = 1 →
            waveNormSq k ≠ 0 →
              exactMidpointBlochM2 H k / waveNormSq k =
                exactHessianM2UnitFrobeniusTTCoeff) ∧
      (∀ (m : Wave4) (v : Wave4),
        waveNormSq m ≠ 0 →
          exactMidpointBlochM2 (pureGaugeFamily m v) m / waveNormSq m =
            exactHessianM2GaugeCoeff) :=
  exactMidpointBlochM2_rayleigh_eq_algebraic_face

What this page does not claim

The ratio is not claimed to be always defined or always equal to these values. The pure-gauge case is not claimed to be physically meaningful. The theorem does not claim that -1/8 is the only possible nonzero value of the ratio.

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.

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