Gravity Analysis Regge Exact Flat Hessian Bloch M2 Rayleigh4 D

The exact midpoint identity

In numerical analysis, a Rayleigh quotient measures the ratio of a quadratic form to a squared norm. For a matrix H and a vector k, the quotient Q(H,k) = (kᵀHk)/(kᵀk) is a scalar that often estimates an eigenvalue. This framework studies a particular four-dimensional version of that idea, where H is a 4x4 matrix and k is a four-component wave vector.

The framework's ledger, a discrete record of recognition events, models the gravitational field as a collection of such matrices. The framework examines two special cases. First, when H has unit Frobenius norm, meaning the sum of the squares of all its entries equals 1, and k satisfies a transversality condition with H, the Rayleigh quotient equals exactly -1/8. Second, when H is a pure gauge configuration, a transformation that leaves the physics unchanged, the quotient equals exactly 0.

In Recognition Science, the framework proves these identities in a machine-checked library of formal theorems. The library verifies that the computation of the midpoint Bloch m² term, a specific quadratic form appearing in the discrete gravity action, reduces to these two algebraic faces. The proof uses a rational table of coefficients and a closed-form transport argument, all checked by the kernel.

The practical consequence is a sharp classification. For any admissible field configuration in this discrete model, the normalized Hessian term can only take one of two exact rational values. This is not an approximation or a numerical coincidence; it is a theorem. The result pins down the behavior of the gravitational residual at the midpoint, a fact that constrains the dynamics of the discrete theory.

This exactness matters because it shows the discrete model does not drift. The Rayleigh quotient, a quantity that in generic settings varies continuously, collapses to a tiny set of values under the framework's constraints. This is a structural rigidity result: the model's dynamics are locked into a discrete spectrum of possibilities, a hallmark of the framework's approach to gravity.

/-- 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

This framework does not derive the value of the gravitational constant G. This framework does not prove that the discrete model converges to general relativity. This framework does not establish the existence of a unique vacuum state.

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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