Gravity Analysis Regge4 Dflat Second Variation Schlaefli Candidate Vanishes On D

The vanishing test

In the framework's study of gravity, researchers work with a discrete model of spacetime called a Regge lattice, where space is built from flat pieces like a geodesic dome. The recognition ledger, the framework's discrete record of events, tracks how these pieces fit together. A key question is whether the model's second variation, a measure of how the action responds to small changes, matches the known result from Einstein's theory in a flat background.

The theorem schlaefliCandidate_vanishes_on_decoyGauge proves that a specific candidate expression, called the Schläfli candidate, evaluates to zero on a particular test configuration named decoyGauge. This is a precise, machine-checked statement: the candidate's zero-moment quadratic form is zero on that gauge. The result is a sanity check, confirming the candidate behaves as expected on this one test input, but it is not a general proof.

The larger goal, elevating the nonlinear Regge action to a form that matches Einstein-Hilbert in four dimensions, remains open. The framework's library contains a status record showing that the pathwise Schläfli identity is not yet present, and the elevation to the candidate is marked as open. The theorem here does not flip the gap action recovery flag, nor does it establish convergence to Einstein-Hilbert in four dimensions. It is one verified fact within a larger unfinished program.

theorem schlaefliCandidate_vanishes_on_decoyGauge :
    schlaefliCandidateZeroMom decoyGauge = 0 :=
  trueWeightZeroMomQuadratic_decoyGauge

What this page does not claim

The theorem does not prove the full Schläfli elevation to the candidate, which remains open. The theorem does not establish convergence of the Regge action to Einstein-Hilbert in four dimensions. The theorem does not apply to all configurations, only to the specific decoyGauge test case.

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/Regge4DFlatSecondVariation.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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