Gravity Analysis Regge4 Dalgebraic Closer Full Ttisotropy Target Mentions Eh Coe

The isotropy target

The declaration fullTTIsotropyTarget_mentions_eh_coefficient is a small but precise piece of bookkeeping inside a larger research program. It establishes two facts. First, the coefficient that appears in the Einstein-Hilbert action, a central object in classical gravity, is fixed at -1/4 in the framework's four-dimensional setting. Second, the framework's full isotropy target, the statement that a certain gravitational moment matches this coefficient for every direction and polarization, is definitionally identical to the continuum target it aims to reproduce. In plain terms, the theorem says: the target is the right target, and it is aimed at the right number.

The theorem does not prove that the target is true. The full isotropy claim, that the all-orbit moment divided by |k|² equals -1/4 for every nonzero direction and every Frobenius-normalized transverse-traceless polarization, remains an open proposition with a status flag set to false. The declaration is a definitional bridge, not a result. It connects the framework's discrete, combinatorial gravity construction to the continuum Einstein-Hilbert coefficient, but it does not close the gap between them.

What the surrounding module does prove is more modest but still substantial. It shows that certain one-orbit symbols do not match the Einstein-Hilbert coefficient: a decoy one-orbit symbol equals -3, not -1/4. It proves that the plus and cross polarization witnesses are genuine transverse-traceless polarizations. It shows that a gauge mode's symbol vanishes, and that the full zero-momentum moment vanishes on the axis TT plus mode and on the decoy gauge. These are banked witnesses, algebraic identities that hold in the discrete construction, but they do not establish the full isotropy target.

The distinction matters because the framework's honesty protocol requires that open targets be named as open. The status structure records fullTTIsotropyClosed as false, pureGaugeVanishesClosed as false, and plusCrossAgreeClosed as false. The theorem banked_does_not_flip_gap_or_isotropy explicitly confirms that the banked results do not flip the isotropy flag or the gap-action recovery flag. The module does not prove S_RS_converges_EH_4d, the statement that the discrete sum converges to the Einstein-Hilbert action in four dimensions. That remains a target, not an achievement.

The practical consequence is that a reader can trust the framework's bookkeeping: the target is well-defined, the coefficient is correct, and the open status is honestly recorded. What the framework has not yet done is prove that its discrete gravity construction reproduces the continuum Einstein-Hilbert coefficient in full generality. That is the open problem this declaration helps to frame, not the one it solves.

/-- Frozen EH coefficient value (definitional). -/
theorem eh_tt_coefficient_eq :
    einsteinHilbertTTCoefficient4D = -(1 / 4 : ℝ) :=
  einsteinHilbertTTCoefficient4D_eq

/-- Convenience: full isotropy implies the frozen coefficient value. -/
theorem fullTTIsotropyTarget_mentions_eh_coefficient :
    einsteinHilbertTTCoefficient4D = -(1 / 4 : ℝ) ∧
      Regge4DFullTTIsotropyTarget = Regge4DContinuumEHTarget :=
  ⟨einsteinHilbertTTCoefficient4D_eq, rfl⟩

theorem regge4DAlgebraicCloserStatus_flags :
    regge4DAlgebraicCloserStatus.decoyOneOrbitClosed = true ∧
      regge4DAlgebraicCloserStatus.plusCrossWitnessesClosed = true ∧
        regge4DAlgebraicCloserStatus.gaugeM2SymbolClosed = true ∧
          regge4DAlgebraicCloserStatus.fullMomentZeroMomClosed = true ∧
            regge4DAlgebraicCloserStatus.fullTTIsotropyClosed = false ∧
              regge4DAlgebraicCloserStatus.pureGaugeVanishesClosed = false ∧
                regge4DAlgebraicCloserStatus.plusCrossAgreeClosed = false ∧
                  regge4DAlgebraicCloserStatus.srsConvergesEH4d = false ∧
                    regge4DAlgebraicCloserStatus.gapActionRecovery = false := by
  decide

/-- Decoy: the single-orbit `(1,1)` m² coefficient is not the continuum EH
target. -/
theorem decoy_one_orbit_m2_ne_eh_coefficient :
    ReggeBlochM2Symbol4D.m2Symbol axisTTPlus = -3 ∧
      einsteinHilbertTTCoefficient4D = -(1 / 4 : ℝ) ∧
        (-3 : ℝ) ≠ -(1 / 4 : ℝ) :=
  decoy_one_orbit_m2_is_not_continuum_target

What this page does not claim

The full isotropy target is not proved; it remains an open proposition. The theorem does not establish that the discrete sum converges to the Einstein-Hilbert action in four dimensions. The declaration does not imply that the framework's gravity model is physically correct.

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)

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: