Gravity Analysis Regge Bloch Local Incidence4 D Orbit Mean Local Kernel T11 Eq A

A local average, made explicit

In the Recognition Science framework's machine-checked library of formal theorems, a kernel is a table of numbers describing how one point in a discrete space influences another. The declaration orbitMeanLocalKernel_t11_eq_assembled_mean proves a specific equality about one such kernel, called t11, which belongs to a family of four-dimensional gravity-like models. The theorem states that for every entry in the table, the t11 kernel's orbit-averaged form equals one sixth of the full star-class kernel assembled from its six members. In plainer terms: if you take the six possible orientations of a fundamental building block, average them together, and then compare that average to the sum of the six blocks themselves, the two procedures give identical results.

The equality is a theorem, not a guess. It is proved by unfolding definitions and rewriting, with no additional axioms beyond the standard three that the framework's kernel already uses. The proof relies on the fact that the t11 orbit has six members, so the averaging factor of one sixth exactly cancels the sixfold sum. This is a purely algebraic statement about the kernel tables themselves; it does not depend on any physical input such as a metric or a field equation. The theorem is a consistency check: it confirms that the mean-local construction, which averages over orbit symmetries, does not lose information when compared to the direct assembled form.

What the theorem does not claim is just as important. It does not say that this averaged kernel solves the Einstein-Hilbert action, the central target of the framework's gravity program. The library's own status flags record that the position-resolved version of this path does not close the Einstein-Hilbert gap, and that no recovery of the gap action has been found. The theorem also does not claim that the mean-local kernel is physically realized; it is a definitional choice, not a derived physical law. The equality holds for the t11 orbit only, not for all orbit types, and it says nothing about whether the corresponding second-moment quantities, which involve squared phases, share the same property.

For a reader, the practical consequence is a clear separation of concerns. The algebraic identity is settled and machine-checked; the physical question of whether this construction reproduces general relativity remains open. The theorem earns its place by making the algebraic floor explicit, so that subsequent work can focus on the genuinely unresolved step without re-litigating what is already proved.

theorem orbitMeanLocalKernel_t11_eq_assembled_mean (d : Fin 15) :
    orbitMeanLocalKernel .t11 d =
      (6 : ℝ)⁻¹ * ReggeHinge4DStarKernel.fullStarClassKernelAssembled d := by
  rw [orbitMeanLocalKernel_t11, ReggeHinge4DStarKernel.fullStarClassKernel_eq]

theorem orbitMeanLocalKernel_t11_eq_assembled_mean (d : Fin 15) :
    orbitMeanLocalKernel .t11 d =
      (6 : ℝ)⁻¹ * ReggeHinge4DStarKernel.fullStarClassKernelAssembled d := by
  rw [orbitMeanLocalKernel_t11, ReggeHinge4DStarKernel.fullStarClassKernel_eq]

theorem reggeBlochLocalIncidence4DStatus_flags :
    reggeBlochLocalIncidence4DStatus.meanLocalEqualsDistinctHinge = true ∧
      reggeBlochLocalIncidence4DStatus.positionResolvedT11Defined = true ∧
        reggeBlochLocalIncidence4DStatus.pathBClosesEH = false ∧
          reggeBlochLocalIncidence4DStatus.gapActionRecovery = false := by
  decide

What this page does not claim

The theorem does not claim that the averaged kernel solves the Einstein-Hilbert action. The theorem does not claim that the mean-local kernel is physically realized, only that it is a definitional choice. The equality does not extend to all orbit types or to the second-moment quantities without further proof.

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/ReggeBlochLocalIncidence4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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