Gravity Analysis Regge Bloch M2 Symbol4 D Class Dot Slot Deficit Ker Axis

The vanishing dot product

In the Recognition Science framework's study of gravity, researchers analyze a discrete model built from hinges and triangles. A key object is the ledger, a discrete record of events, here representing a specific arrangement of these geometric pieces. The declaration classDot_slotDeficitKer_axis is a theorem about a particular configuration called axisTTPlus. It proves that a certain mathematical quantity, the dot product between a "slot deficit kernel" and this configuration, is exactly zero.

This dot product measures the interaction strength between a small geometric feature (the slot deficit) and the overall configuration. The theorem states this interaction vanishes for the axisTTPlus configuration. This is a precise, local result: it applies to one specific orbit of the model, not to all possible configurations. The proof is machine-checked, meaning it has been verified by a computer program, adding a layer of certainty to the mathematical claim.

The result is part of a larger effort to understand the behavior of this discrete gravity model at small momenta. The vanishing of this dot product is a necessary condition for certain other properties to hold, such as the model's behavior approaching a continuum limit. However, the theorem itself does not establish those larger properties. It is one verified piece in a complex puzzle, not the whole picture.

In Recognition Science, this work is framed as a step toward deriving general relativity from a more fundamental discrete structure. The framework's library of formal theorems is building toward this goal incrementally. This particular theorem is a small but solid brick in that edifice, confirming a specific algebraic identity that the larger theory depends on.

theorem classDot_slotDeficitKer_axis (s : Fin 24) (t : Fin 10) :
    classDot (slotDeficitKer s t) axisTTPlus = 0 := by
  rw [classDot_slotDeficit_reindex]
  simp_rw [kernel11_eq_sign, classCoeff_axisTTPlus_int]
  have h := congrArg (fun n : ℤ => (n : ℝ)) (slotKerDotZ_axis s t)
  simpa [slotKerDotZ, Int.cast_sum, Int.cast_mul] using h

What this page does not claim

This theorem does not prove that the full model converges to Einstein-Hilbert gravity. It does not establish the behavior of the model for all possible configurations, only for the specific axisTTPlus one. It does not recover the gap action or any other large-scale physical phenomenon.

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/ReggeBlochM2Symbol4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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