Gravity Analysis Regge4 Dtransported Algebraic Closer

The transported closer

In Recognition Science, gravity analysis studies how a discrete ledger of recognition events, when transported across a 4-torus, produces continuum limits. The module Regge4DTransportedAlgebraicCloser is a status board: it banks every algebraic identity that is provable from the framework's axioms, and it names, without pretending to solve, the targets that remain open. The plain-language achievement is a precise boundary between what is closed and what is not.

The classical object here is a Regge calculus, a discretization of general relativity where spacetime is built from flat simplices and curvature lives on hinges. The module works with a 4D torus family and a concrete continuum symbol sequence, finiteTransportedSymbol, which is definitionally equal to a Bloch-fold over distinct hinges with weight 1/r_τ. Theorems establish that this sequence has a unique limit, is quadratically homogeneous under scaling, and decomposes into an orbit sum over hinge types. These are closed, machine-checked facts.

The central open target is Regge4DContinuumEHTarget: the claim that the normalized transported fold tends to -(1/4) on the Frobenius transverse-traceless subspace, the coefficient that would match the Einstein-Hilbert action. The module explicitly does not inhabit this target. It also proves a decoy strengthening: the one-orbit ray normalized coefficient for the axis TT direction is -3/2, which is not the frozen Einstein-Hilbert coefficient. And the unrestricted gauge-zero target is false as a transported distinct-hinge claim, with a counterexample at m=(1,1,0,0), v=e₂. The module banks the longitudinal, axis-parallel family where the transported Hessian does annihilate, as a theorem.

What this establishes, in plain terms, is a honest accounting. The framework's machine-checked library of formal theorems proves the algebraic scaffolding: uniqueness, homogeneity, orbit decomposition, and the area-convention match. It also proves what is not true, and it leaves the convergence-to-Einstein-Hilbert claim as a named open target, not as a fake inhabitant. The status structure records nine flags, with five closed and four open, and the axiom footprint is exactly the standard three, with no sorry or admit. The reader can now see precisely which algebraic identities about 4D gravity are forced by the framework, and which remain targets for future work.

theorem finiteTransportedSymbol_eq_blochFoldAllDistinctHinge
    (j : ℕ) (m : IntMode4) (E : Mat4) :
    finiteTransportedSymbol j m E =
      blochFoldAllDistinctHinge E (realMode (torusSide j) m) :=
  finiteTransportedSymbol_eq j m E

theorem finiteTransportedSymbol_smul (c : ℝ) (j : ℕ) (m : IntMode4)
    (E : Mat4) :
    finiteTransportedSymbol j m (c • E) =
      c ^ 2 * finiteTransportedSymbol j m E := by
  simp_rw [finiteTransportedSymbol_eq_blochFoldAllDistinctHinge,
    blochFoldAllDistinctHinge_smul]

theorem oneOrbitRayNormalizedCoeff_axisTTPlus :
    oneOrbitRayNormalizedCoeff axisTTPlus = (-3 : ℝ) / 2 := by
  unfold oneOrbitRayNormalizedCoeff
  rw [m2Symbol_axisTTPlus, symbolDir_normSq]

/-- Honesty: banked (1,1) identities do not inhabit continuum EH Tendsto,
and the ledger flag stays false. -/
theorem banked_does_not_inhabit_eh_or_flip_gap :
    regge4DTransportedAlgebraicCloserStatus.transportedTTIsotropyClosed =
        false ∧
      regge4DTransportedAlgebraicCloserStatus.gapActionRecovery = false ∧
        oneOrbitRayNormalizedCoeff axisTTPlus ≠
          einsteinHilbertTTCoefficient4D :=
  ⟨rfl, rfl, oneOrbit_ray_normalized_ne_eh_coefficient⟩

theorem Regge4DContinuumGaugeZeroTargetLongitudinal_closed :
    Regge4DContinuumGaugeZeroTargetLongitudinal :=
  ⟨m2TransportedAllOrbitMomentDistinctHinge_decoyGauge_symbolDir,
    FoldAlongM2Tendsto_of_decoyGauge⟩

What this page does not claim

This module does not prove convergence to the Einstein-Hilbert coefficient -(1/4); that remains an open target. This module does not establish that the unrestricted gauge-zero target holds; it is false as a transported distinct-hinge claim. This module does not claim any new axioms beyond the standard three; its footprint is exactly [propext, Classical.choice, Quot.sound].

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/Regge4DTransportedAlgebraicCloser.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: