Gravity Analysis Regge4 Dtensor Algebraic Closer Regge4 Ddistinct Hinge Pinned V

The open factor of four

In the framework's machine-checked library of formal theorems, the declaration Regge4DDistinctHingePinnedVsEHFactor4_status_open is a status flag, not a mathematical result. It records, as a proved fact about the current state of the library, that the property regge4DTorusContinuumLimitStatus.ehTendstoInhabited is false. In plain language: the library has not yet constructed a geometric proof that a certain factor of four, appearing when a discrete four-dimensional gravity model is compared with the continuum Einstein-Hilbert action, is forced by the geometry rather than put in by hand.

The arithmetic behind this flag is settled. A theorem in the same file proves that the continuum-facing coefficient is exactly four times the distinct-hinge value: einsteinHilbertTTCoefficient4D = 4 * (-1/16). The same theorem also records that the distinct-hinge value is not equal to the Einstein-Hilbert value, and that a dictionary factor is 1. So the number 4 is real, it is measured in the discrete model, and it is not an artifact of a scaling convention. What remains open is a geometric explanation, one that would show the factor arises from local incidence structure, in the style of a 3D identity already in the library.

The declaration is one of several flags in a status structure. The structure also records that ray evaluations are banked, homogeneity is closed, the tensor closed form is open, and an axis-mode isotropy blocker is negated. The flag does not flip a separate property called gap_action_recovery, which stays false. So the open factor is isolated: it does not undo other completed work in the file.

What the declaration does not claim is as important as what it does. It does not claim the factor of four is wrong, or that the discrete model fails to approach the continuum. It does not install a multiplier of 4 into the continuum sequence. It only says that a certain geometric proof obligation, named Regge4DDistinctHingePinnedVsEHFactor4, is still a target. The flag is an honest ledger entry: it marks a specific unfinished step, so that a future proof, when it exists, can be checked against it.

/-- Status flag: factor-4 geometric closure still open. -/
theorem Regge4DDistinctHingePinnedVsEHFactor4_status_open :
    regge4DTorusContinuumLimitStatus.ehTendstoInhabited = false :=
  rfl

/-- Arithmetic residual (THEOREM side): pinned continuum face vs EH. -/
theorem residual_factor_four_arithmetic :
    einsteinHilbertTTCoefficient4D = (4 : ℝ) * (-1 / 16 : ℝ) ∧
      DistinctHingePinnedMomentVsEH ∧
        survivingDictionaryFactor4D = 1 :=
  ⟨by rw [einsteinHilbertTTCoefficient4D_eq]; norm_num,
    distinctHinge_pinned_ne_eh, rfl⟩

theorem does_not_flip_gap_action_recovery :
    regge4DTensorAlgebraicCloserStatus.gapActionRecovery = false :=
  rfl

What this page does not claim

The factor of four is geometrically explained; the flag only records that the explanation is missing. The discrete model fails to approach the continuum; the arithmetic residual is settled, only the geometric account is open. A multiplier of 4 has been installed into the continuum sequence; the declaration explicitly says no magic-4 multiplier is installed.

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: