Gravity Analysis Regge Exact Flat Hessian Symbol4 D Exact Hessian S Rs Converges

The flat-space check

The declaration ExactHessianS_RS_converges_EH_4d_closed proves a precise statement about a discrete model of gravity. In this model, spacetime is built from flat four-dimensional blocks, and the curvature is concentrated along their shared faces. The declaration establishes that, at the flat background where all curvature vanishes, the second variation of the discrete action matches the corresponding piece of Einstein's theory. Specifically, it proves two algebraic identities: one says the transverse-traceless part of the discrete Hessian is isotropic, and the other says the gauge part is exactly zero. These are the two conditions needed for the discrete model to reproduce the linearized Einstein-Hilbert action in four dimensions.

The proof is a machine-checked theorem in the framework's library of formal theorems. It builds on earlier certificates: the transverse-traceless coefficient is exactly -1/4, matching the continuum Einstein-Hilbert value, and the gauge coefficient is exactly 0. The declaration also records that a numerical simulation at lattice size N=6 agrees with these exact values to within 2 parts in 100,000. This is the flat-space check: it confirms that the discrete model has the right linear behavior in the simplest regime, where the spacetime is just Minkowski space.

In Recognition Science, this result is a stepping stone toward a larger target. The framework models gravity as a discrete ledger of recognition events, and this declaration verifies one piece of that model against classical physics. It does not, however, inhabit the ledger S_RS_converges_EH_4d, which would be the full statement that the discrete action converges to Einstein-Hilbert in the continuum limit. That remains an open target, as does the question of whether the discrete Hessian tends to the continuum symbol for all modes. The declaration also does not provide a general algebraic table of all coupling coefficients; it proves only the specific face coefficients and gauge identities.

What the result changes is the confidence one can place in the discrete model. Before this proof, the match between the discrete and continuum theories at the flat background was a numerical observation. Now it is a theorem, at least for the transverse-traceless and gauge sectors. The reader can see that the model is not merely fitted to look like Einstein's theory at one point; it reproduces the linearized structure exactly in the cases checked. The remaining gap, the full convergence statement, is now precisely delimited rather than vague.

theorem ExactHessianS_RS_converges_EH_4d_closed :
    ExactHessianS_RS_converges_EH_4d :=
  ⟨ExactHessianTTIsotropyTarget_closed, ExactHessianGaugeZeroTarget_algebraic_face⟩

theorem exactHessianM2AxisTTPlus_eq_EH :
    exactHessianM2AxisTTPlusCoeff = einsteinHilbertTTCoefficient4D := by
  simp [exactHessianM2AxisTTPlusCoeff, einsteinHilbertTTCoefficient4D]

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

What this page does not claim

The declaration does not prove that the discrete action converges to Einstein-Hilbert in the continuum limit; that remains open. The declaration does not provide a general algebraic table of all coupling coefficients, only the specific face coefficients and gauge identities. The declaration does not establish anything about the full nonlinear dynamics of the discrete model, only the linearized flat-space Hessian.

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

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