Gravity Analysis Regge4 Dtransported Algebraic Closer Finite Transported Symbol

The transported symbol

The declaration is a theorem inside the Recognition Science framework's machine-checked library of formal theorems. It states that a certain finite, discrete approximation to a gravity-related quantity, called the finite transported symbol (a weighted sum built from a sequence of torus samples), is exactly equal to a single fold operation, blochFoldAllDistinctHinge, applied to the same data. In plainer terms: instead of summing many separate pieces, you can compute the whole thing in one pass, and the two procedures give the identical number.

This equality is not a numerical approximation. The theorem proves the two expressions are definitionally the same, meaning the finite symbol is, by construction, that fold. The framework's library records this as a closed result, one of several algebraic identities it banks. It also proves related facts: the symbol scales quadratically when you scale the input matrix, and it can be decomposed as a sum over orbit types, each weighted by the size of its orbit star.

What the declaration does not claim is the harder analytical target. It does not prove that this sequence of finite symbols converges to the value -1/4 on the Frobenius transverse-traceless slice, which is the open goal named Regge4DContinuumEHTarget. The library explicitly marks that target as open, with status false, and does not fabricate an inhabitant. It also does not establish the unrestricted gauge-zero target; a counterexample exists for a specific mode and gauge vector. The one-orbit ray's normalized coefficient is shown to be -3/2, which is provably not the Einstein-Hilbert coefficient, a deliberate decoy strengthening.

In practice, this theorem is a cleanup result. It tells a reader that the algebraic structure of the finite approximation is fully understood and matches the fold convention, so any future work on convergence can start from this exact identity. The open convergence question remains a target, not a failure, and the library's status record makes that distinction explicit.

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 finiteTransportedSymbol_eq_orbit_sum (j : ℕ) (m : IntMode4)
    (E : Mat4) :
    finiteTransportedSymbol j m E =
      ∑ ty : HingeOrbitType,
        (orbitStarSize ty)⁻¹ *
          blochFoldOrbit ty E (realMode (torusSide j) m) := by
  rw [finiteTransportedSymbol_eq_blochFoldAllDistinctHinge]
  rfl

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

This theorem does not prove the convergence of the finite transported symbol to -1/4. This theorem does not establish the unrestricted gauge-zero target. This theorem does not claim the finite transported symbol equals the Einstein-Hilbert coefficient.

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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