Gravity Analysis Regge4 Dalgebraic Closer Full Moment Orbit Contribution Eq Bili

A bilinear identity

In the Recognition Science framework's campaign to recover general relativity from a discrete, ledger-like model of spacetime, the declaration fullMomentOrbitContribution_eq_bilinear proves a structural fact about how one piece of the calculation behaves. The piece is the full-moment orbit contribution, a number attached to each type of hinge in a four-dimensional Regge calculus, a discrete approximation to spacetime made of flat simplices. The theorem states that this contribution, which is defined as a quadratic form, is in fact a bilinear form: it depends on the metric-like matrix H in two separate linear slots, written as orbitZeroMomBilinear ty H H, rather than through any cross terms that would break linearity in each argument.

This is a technical but meaningful result. Bilinearity is the algebraic property that makes the contribution tractable: it means the function respects addition and scaling in each of its two arguments separately, so the whole object can be studied one linear factor at a time. The theorem is proved in the framework's machine-checked library of formal theorems, with no unproved assumptions beyond the standard axioms of the ambient type theory. It is one of several "banked witnesses," identities that are established and stored even though the larger goal they serve is not yet reached.

What the theorem does not claim is just as important as what it proves. It does not establish that the full-moment contribution matches the Einstein-Hilbert coefficient of -1/4, nor that the sum over all hinge orbits recovers the continuum Einstein tensor. Those targets, named Regge4DFullTTIsotropyTarget and S_RS_converges_EH_4d, remain open, with status flags set to false. The bilinear identity is a stepping stone, not the destination: it shows the algebraic shape of one term, while the question of whether the whole discrete sum converges to general relativity stays unresolved.

theorem fullMomentOrbitContribution_eq_bilinear (ty : HingeOrbitType)
    (H : Mat4) :
    fullMomentOrbitContribution ty H = orbitZeroMomBilinear ty H H :=
  orbitZeroMomQuadratic_eq_bilinear ty H

/-- **OPEN**: algebraic packaging of the preflight continuum EH target.
For every nonzero integer mode and Frobenius-normalized TT polarization,
the `|k|²`-normalized **concrete transported** continuum symbol equals
`einsteinHilbertTTCoefficient4D = -1/4`. -/
def Regge4DFullTTIsotropyTarget : Prop :=
  Regge4DContinuumEHTarget

What this page does not claim

This theorem does not prove that the discrete model converges to Einstein's equations. It does not establish that the plus and cross polarization symbols agree. It does not show that the pure-gauge moment vanishes for every direction.

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)

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Derived articles

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