Gravity Analysis Regge Exact Flat Hessian Symbol4 D Exact Hessian M2 Unit F Time

The flat Hessian identity

Regge calculus is a way of doing general relativity without a smooth spacetime: you replace the continuum with a flat, piecewise-linear mesh of simplices, and let the geometry live in the edge lengths. The action is a sum over hinges, and its second variation, the Hessian, controls how small perturbations behave. On a flat background all the deficit angles vanish, and the second variation reduces to a pure cross term between area variations and angle variations. The declaration exactHessianM2_unitF_times_two_eq_axisTTPlus records a specific algebraic fact about that flat Hessian: for a unit-Frobenius-norm transverse-traceless (TT) polarization, twice the per-mode coefficient equals the axis-TT-plus coefficient, and both equal -1/4.

That number -1/4 is not arbitrary. It is exactly the coefficient that appears in the continuum Einstein-Hilbert action for the same TT mode in four dimensions. The framework's library proves this by pure algebra: recognition, the discrete record of events, is not involved in this step. The identity is a discrete-bookkeeping fact, banked as a theorem with no ledger inhabitation. The measured finite-N value from a Python oracle, -0.24434, sits within 0.01 of -1/4, and the relative error against that oracle is below one part in ten thousand. The exact algebraic identity, however, does not depend on any measurement.

What the declaration does not claim is as important as what it proves. It does not establish that the discrete Regge action converges to the continuum Einstein-Hilbert action in any limit. That convergence, the statement that the recognition ledger S_RS converges to the Einstein-Hilbert action in four dimensions, remains an open target. The algebraic table for a general coupling is also absent: the declaration explicitly records that no general algebraic table is present, only certificates for named modes. The gauge face, where the perturbation is pure gauge, is exactly zero, which is a consistency check, not a derivation of the TT value.

The practical consequence is a sharpened target. The discrete geometry reproduces the continuum coefficient exactly for the specific modes that were checked, and the residual gap to a full convergence theorem is now precisely located: it is the limit statement, not the algebraic identity. A reader who wants to see the discrete theory become continuum gravity knows exactly which step remains.

theorem exactHessianM2UnitFrobeniusTTCoeff_eq :
    exactHessianM2UnitFrobeniusTTCoeff = -(1 / 8 : ℝ) := rfl

theorem exactHessianM2AxisTTPlusCoeff_eq :
    exactHessianM2AxisTTPlusCoeff = -(1 / 4 : ℝ) := rfl

/-- Banked: discrete bookkeeping times unit-F m² recovers frozen EH. -/
def ExactHessianTTIsotropyTarget : Prop :=
  (2 : ℝ) * exactHessianM2UnitFrobeniusTTCoeff = einsteinHilbertTTCoefficient4D

theorem einsteinHilbertTTCoefficient4D_eq :
    einsteinHilbertTTCoefficient4D = -(1 / 4 : ℝ) := rfl

theorem measuredTTNormCoeffN6_near_quarter :
    |measuredTTNormCoeffN6 - einsteinHilbertTTCoefficient4D| < (1 / 100 : ℝ) := by
  norm_num [measuredTTNormCoeffN6, einsteinHilbertTTCoefficient4D]

theorem measuredTTRelErrVsOracleN6_lt_1e4 :
    measuredTTRelErrVsOracleN6 < (1 / 10000 : ℝ) := by
  norm_num [measuredTTRelErrVsOracleN6]

What this page does not claim

The discrete Regge action converges to the continuum Einstein-Hilbert action in any limit. A general algebraic coupling table for all modes is present in the framework's library. The gauge face being zero is a derivation of the TT value rather than a consistency check.

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/ReggeExactFlatHessianSymbol4D.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: