Gravity Analysis Regge4 Dalgebraic Closer Full Moment Orbit Contribution Of Defi

A zero condition for a moment

In the framework's machine-checked library of formal theorems, the declaration fullMomentOrbitContribution_of_deficit_zero proves a narrow algebraic fact about a four-dimensional gravity analysis. The object in question is a quadratic moment, a number built from a 4x4 matrix that represents a gravitational perturbation. The theorem states: if a certain dot product between the orbit's deficit kernel and the perturbation matrix equals zero, then that orbit's contribution to the full moment is exactly zero. The proof is a direct rewrite: it unfolds the definition of the orbit contribution, applies the zero hypothesis, and multiplies by zero.

This is a banked witness, a stored algebraic identity that the framework's campaign can use later. The theorem is tagged THEOREM, meaning it is proved in Lean with no axioms beyond the standard three. It is used to show that two specific perturbations, the axis TT plus polarization and the decoy gauge, each have zero contribution from every orbit. The full moment for those cases is then also zero, by summing the orbit contributions.

In Recognition Science, the framework models a discrete ledger of recognition events, and this analysis is part of a campaign to recover general relativity's Einstein-Hilbert action from a discrete structure. The theorem does not establish that the full moment equals the Einstein-Hilbert coefficient of -1/4. That target, named fullTTIsotropyTarget, remains OPEN with status flag false. The theorem also does not prove that the discrete action converges to the Einstein-Hilbert action in four dimensions; that claim, S_RS_converges_EH_4d, is explicitly not proved. The library's status record confirms: the decoy one-orbit symbol equals -3, not -1/4, and the isotropy and pure-gauge vanishing targets are still open.

What the theorem changes is the boundary of what is known. It gives a clean, checkable condition for a zero moment contribution, and it banks that condition for future use. It does not close the gap to the continuum theory. The reader can now see exactly which algebraic identities are settled and which large targets remain, without mistaking a local zero for a global victory.

/-- Per-orbit deficit annihilation implies per-orbit contribution zero. -/
theorem fullMomentOrbitContribution_of_deficit_zero
    (ty : HingeOrbitType) (H : Mat4)
    (h : classDot (orbitDeficitKernel ty) H = 0) :
    fullMomentOrbitContribution ty H = 0 := by
  unfold fullMomentOrbitContribution orbitZeroMomQuadratic
  rw [h, mul_zero]

/-- Per-orbit deficit annihilation implies per-orbit contribution zero. -/
theorem fullMomentOrbitContribution_of_deficit_zero
    (ty : HingeOrbitType) (H : Mat4)
    (h : classDot (orbitDeficitKernel ty) H = 0) :
    fullMomentOrbitContribution ty H = 0 := by
  unfold fullMomentOrbitContribution orbitZeroMomQuadratic
  rw [h, mul_zero]

/-- Honesty: banked one-orbit identities do not inhabit the OPEN isotropy
target, and the ledger flag stays false. -/
theorem banked_does_not_flip_gap_or_isotropy :
    regge4DAlgebraicCloserStatus.fullTTIsotropyClosed = false ∧
      regge4DAlgebraicCloserStatus.gapActionRecovery = false ∧
        ReggeBlochM2Symbol4D.m2Symbol axisTTPlus ≠
          einsteinHilbertTTCoefficient4D := by
  refine ⟨rfl, rfl, ?_⟩
  rw [m2Symbol_axisTTPlus, einsteinHilbertTTCoefficient4D_eq]
  norm_num

def regge4DAlgebraicCloserStatus : Regge4DAlgebraicCloserStatus where
  decoyOneOrbitClosed := true
  plusCrossWitnessesClosed := true
  gaugeM2SymbolClosed := true
  fullMomentZeroMomClosed := true
  fullTTIsotropyClosed := false
  pureGaugeVanishesClosed := false
  plusCrossAgreeClosed := false
  srsConvergesEH4d := false
  gapActionRecovery := false

What this page does not claim

The theorem does not prove that the full moment equals the Einstein-Hilbert coefficient for any perturbation. The theorem does not establish that the discrete action converges to the Einstein-Hilbert action in four dimensions. The theorem does not close the full TT isotropy, pure-gauge vanishing, or plus-cross agreement targets.

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/Regge4DAlgebraicCloser.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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