Gravity Analysis Regge4 Dtransported Algebraic Closer Banked Does Not Inhabit Eh

A banked status, honestly kept

In the Recognition Science framework, a ledger is a discrete record of events, and this declaration is an entry in that ledger. It is a formal, machine-checked statement of status: it lists which claims in a four-dimensional gravity analysis are closed, meaning proved, and which are open, meaning not yet established. The declaration's name, banked_does_not_inhabit_eh_or_flip_gap, is a precise summary: certain results are banked, or recorded as achieved, while two specific targets, the Einstein-Hilbert convergence and the gap action recovery, are explicitly not inhabited, meaning no proof of them is claimed.

The core of the declaration is a list of boolean flags, each representing a proposition's status. The flags show that several algebraic properties are closed: the continuum symbol sequence is uniquely defined, it has quadratic homogeneity, and a specific orbit sum decomposition holds. Also closed are results about a particular orbit's limit behavior along two axes, and a match between two area conventions. The declaration records these as true. It then records three flags as false: the transported TT isotropy, the transported gauge-zero target, and the gap action recovery. This is the heart of the declaration's honesty: it does not pretend these are proved.

The declaration also proves a specific numerical inequality. It shows that a normalized coefficient, computed from a one-orbit ray, is not equal to the Einstein-Hilbert TT coefficient. This is a concrete, checkable fact, not a vague statement of incompleteness. The declaration's final theorem bundles these flags and the inequality into a single statement, all derived from the machine-checked library of formal theorems, with no unproved assumptions beyond the standard three axioms of the underlying type theory.

What this declaration does not claim is as important as what it does. It does not claim that the four-dimensional Einstein-Hilbert limit, where a normalized transported fold tends to -1/4, is proved. That remains an open target. It does not claim that the gauge-zero target holds for all gauge vectors; in fact, a counterexample exists. It also does not claim that the gap action recovery is achieved. The declaration's value is that it draws a sharp line between the proved and the unproved, preventing a partial result from being mistaken for a complete one.

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

theorem regge4DTransportedAlgebraicCloserStatus_flags :
    regge4DTransportedAlgebraicCloserStatus.continuumSymbolBoundClosed =
        true ∧
      regge4DTransportedAlgebraicCloserStatus.quadraticHomogeneityClosed =
        true ∧
        regge4DTransportedAlgebraicCloserStatus.t11M2TendstoClosed = true ∧
          regge4DTransportedAlgebraicCloserStatus.oneOrbitDecoyClosed =
            true ∧
            regge4DTransportedAlgebraicCloserStatus.transportedTTIsotropyClosed =
              false ∧
              regge4DTransportedAlgebraicCloserStatus.transportedGaugeZeroClosed =
                false ∧
                regge4DTransportedAlgebraicCloserStatus.areaConventionMatchClosed =
                  true ∧
                  regge4DTransportedAlgebraicCloserStatus.srsConvergesEH4d =
                    false ∧
                    regge4DTransportedAlgebraicCloserStatus.gapActionRecovery =
                      false := by
  decide

What this page does not claim

The declaration does not prove the Einstein-Hilbert convergence target; it explicitly records that as open. The declaration does not claim the gauge-zero target holds for all gauge vectors, as a counterexample exists. The declaration does not prove the gap action recovery; it records that flag as false.

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

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