Gravity Analysis Regge4 Dtensor Algebraic Closer Regge4 Dtensor Algebraic Closer

The status ledger

The declaration regge4DTensorAlgebraicCloserStatus_flags is a machine-checked status report, a ledger entry in the framework's library of formal theorems. It records the exact state of a specific calculation in four-dimensional gravity: which parts are closed, which parts remain open, and which parts are explicitly not being claimed. The report is a conjunction of six boolean flags, each one a yes-or-no answer to a precise mathematical question.

Two flags are set to true because the corresponding facts are proved. The calculation is homogeneous: scaling the input by a constant squares the output. And a set of specific ray evaluations, particular directions in the polarization space, have been computed and banked. Three more flags are true because the corresponding goals are open, meaning they are targets for future work, not achievements. There is no closed-form tensor expression matching the computed values on all inputs, no geometric explanation for a residual factor of four, and an isotropy condition is known to be blocked. The sixth flag is false: the calculation does not flip a separate recovery flag, meaning it leaves that other mechanism untouched.

The power of the declaration is that it is a single, checkable summary. A reader, human or machine, can look at this one theorem and know the complete status of the calculation without tracing through the underlying definitions. The flags are not vague labels; each one corresponds to a specific proposition in the library, and the theorem itself is proved by the kernel's decision procedure, so the summary is not just asserted but verified.

What the declaration does not claim is as important as what it does. It does not claim that the open problems are impossible; it merely records that they are open. It does not claim that the residual factor of four has a geometric origin, only that the arithmetic shows a ratio of four and that no geometric proof has been found. It does not claim that the calculation recovers the gap action; the flag for that is explicitly false. The declaration is a snapshot of honest, current knowledge, with the boundaries between proved and open drawn sharply.

theorem regge4DTensorAlgebraicCloserStatus_flags :
    regge4DTensorAlgebraicCloserStatus.rayEvaluationsBanked = true ∧
      regge4DTensorAlgebraicCloserStatus.homogeneityClosed = true ∧
        regge4DTensorAlgebraicCloserStatus.tensorClosedFormOpen = true ∧
          regge4DTensorAlgebraicCloserStatus.factor4GeometricOpen = true ∧
            regge4DTensorAlgebraicCloserStatus.axisIsotropyBlocked = true ∧
              regge4DTensorAlgebraicCloserStatus.gapActionRecovery =
                false := by
  decide

theorem distinctHingeMomentForm_smul (c : ℝ) (E : Mat4) (dir : Fin 4 → ℝ) :
    distinctHingeMomentForm (c • E) dir =
      c ^ 2 * distinctHingeMomentForm E dir :=
  m2TransportedAllOrbitMomentDistinctHinge_smul c E dir

/-- **OPEN**: a universal tensor closed form on TT × nonzero directions,
in the spirit of the 3D adjugate identity
`K = (1/2) xᵀ adj(E) x = -(1/4)|x|²‖E‖_F²`. -/
def Regge4DDistinctHingeTensorClosedFormOpen : Prop :=
  ∃ (Q : Mat4 → (Fin 4 → ℝ) → ℝ),
    (∀ (c : ℝ) (E : Mat4) (dir : Fin 4 → ℝ),
        Q (c • E) dir = c ^ 2 * Q E dir) ∧
      (∀ (E : Mat4) (dir : Fin 4 → ℝ),
        IsTTPolarization4D dir E →
          (∑ i : Fin 4, dir i * dir i) ≠ 0 →
            distinctHingeMomentForm E dir = Q E dir)

theorem does_not_flip_gap_action_recovery :
    regge4DTensorAlgebraicCloserStatus.gapActionRecovery = false :=
  rfl

What this page does not claim

The declaration does not prove that a closed-form tensor expression exists; it records that the search for one is open. The declaration does not install a factor of four into the continuum sequence; it only records the arithmetic residual. The declaration does not claim that the gap action recovery is achieved; the flag is 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/Regge4DTensorAlgebraicCloser.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: