Gravity Analysis Regge4 Dtransported Algebraic Closer One Orbit Ray Normalized N

The one-orbit check

The declaration oneOrbit_ray_normalized_ne_eh_coefficient is a formal theorem in the framework's machine-checked library of formal theorems. It states a precise negative result: for a particular one-orbit ray in a four-dimensional gravity analysis, the normalized coefficient equals -3/2, which is provably not equal to the Einstein-Hilbert trace-trace coefficient. The theorem does not merely assert this difference; it derives it from the definitions and checks the arithmetic by computation. The coefficient comes from a ray-normalized expression, where the raw orbit sum is divided by the squared length of a direction vector, and the theorem confirms the division yields -3/2 for the chosen axis.

This result is part of a larger, deliberately honest accounting. The framework's library banks what it has proved and marks what it has not. The theorem explicitly does not establish that the full continuum limit of the transported fold converges to -1/4, the value associated with the Einstein-Hilbert action in this context. That convergence target remains open, with a named declaration Regge4DContinuumEHTarget whose status is false, meaning no inhabitant has been constructed. The one-orbit result is a strengthening of a decoy check: it shows that a single orbit, even after normalization, does not accidentally reproduce the frozen Einstein-Hilbert coefficient. This is a guard against a false positive, not a proof of the main convergence claim.

The theorem also clarifies the scope of what is claimed. It does not show that the transported Hessian annihilates every gauge vector; in fact, a counterexample exists for a specific mode and vector. It does not flip the status of gap_action_recovery or transportedGaugeZeroClosed, both of which remain false. The library's status record lists which doors are closed and which are open: the continuum symbol bound, quadratic homogeneity, the (1,1)-orbit limit, and the area-convention match are all closed as true; the transport isotropy and gauge-zero targets are open as false. This is the framework's way of preserving a precise ledger of what is known and what is not.

What the theorem changes is the shape of the search. A reader can now see that the path to the Einstein-Hilbert value does not run through this particular one-orbit ray, normalized or not. The negative result eliminates a candidate shortcut. The remaining work is the open target: showing that the full transported fold, not a single orbit, converges to -1/4 on the Frobenius transverse-traceless subspace. That is the question the library leaves for the next step, with the one-orbit check standing as a cleared obstacle rather than a finished journey.

theorem oneOrbit_ray_normalized_ne_eh_coefficient :
    oneOrbitRayNormalizedCoeff axisTTPlus ≠
      einsteinHilbertTTCoefficient4D := by
  rw [oneOrbitRayNormalizedCoeff_axisTTPlus, einsteinHilbertTTCoefficient4D_eq]
  norm_num

/-- 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⟩

/-- **OPEN / false as transported.** Alias of the unrestricted
`Regge4DContinuumGaugeZeroTarget`. Remains uninhabited here;
`transportedGaugeZeroClosed = false`. See
`Regge4DContinuumGaugeZeroTargetLongitudinal` for the banked restriction. -/
def Regge4DTransportedGaugeZeroOpen : Prop :=
  Regge4DContinuumGaugeZeroTarget

What this page does not claim

The theorem does not prove the continuum limit converges to -1/4. The theorem does not show the transported Hessian annihilates every gauge vector. The theorem does not claim the one-orbit ray is the only obstruction to the Einstein-Hilbert value.

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)

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Derived articles

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