Gravity Analysis Regge Exact Flat Hessian Bloch Symbol Zero4 D Q Coeff Eq Kernel

A bridge of coefficients

In the Recognition Science framework's machine-checked library of formal theorems, a calculation about gravity at zero momentum ends with a strong claim: a certain quantity, the exact midpoint Bloch symbol, is zero for every possible input. The declaration qCoeff_eq_kernelGlue is a small but load-bearing step inside that proof. It establishes that a coefficient qCoeff a b c d, defined as a rational number in the framework's own code, is equal to a corresponding coefficient KernelGlue.qCoeff a b c d defined in a separate, trusted kernel certificate. The equality is proved by unfolding both definitions and showing they are the same by reflexivity, a trivial step in formal logic.

The point of this bridge is to connect two different ways of computing the same thing. The framework's qCoeff is a readable, human-authored definition that sums over a table of coupling data. The kernel certificate's qCoeff is a precomputed list of rational numbers, cleared of denominators and scaled, that a separate arithmetic checker has already verified to be zero. By proving the two definitions agree, the framework can import the kernel's certificate of vanishing and apply it to its own more abstract coefficient. This is how the proof avoids relying on a fast but unverified computation method; it uses only the kernel's own certificate, lifted through the glue theorem.

What the declaration does not claim is just as precise. It does not claim that the coefficients are zero; that is the job of the later theorem qCoeff_eq_zero, which uses this glue as a lemma. It does not claim anything about the physical meaning of the coefficients, about gravity itself, or about the framework's broader forcing chain. It is purely a statement about the equality of two formal definitions. The declaration is also marked private, meaning it is an internal proof detail, not a public interface that other parts of the library are meant to rely on directly.

The practical consequence is that a reader can trust the final theorem without trusting a black-box computation. The bridge makes the proof auditable: every step, including the import of the kernel's arithmetic certificate, is checked by the machine. This is the framework's answer to the question of how a large calculation earns confidence, not by appeal to a faster solver, but by making the connection between the human-readable definition and the verified certificate explicit and formally proved.

private theorem qCoeff_eq_kernelGlue (a b c d : Fin 4) :
    qCoeff a b c d = KernelGlue.qCoeff a b c d := by
  unfold qCoeff KernelGlue.qCoeff couplingS
  rfl

private theorem qCoeff_eq_kernelGlue (a b c d : Fin 4) :
    qCoeff a b c d = KernelGlue.qCoeff a b c d := by
  unfold qCoeff KernelGlue.qCoeff couplingS
  rfl

/-- **THEOREM:** every rational quartic coefficient vanishes. -/
theorem qCoeff_eq_zero : ∀ (a b c d : Fin 4), qCoeff a b c d = 0 := by
  intro a b c d
  rw [qCoeff_eq_kernelGlue]
  exact KernelGlue.qCoeff_eq_zero a b c d

What this page does not claim

This answer does not claim that the coefficients are zero, only that the two definitions are equal. This answer does not claim anything about the physical interpretation of the coefficients or about gravity itself. This answer does not claim that the declaration is part of the framework's public interface; it is marked private.

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/ReggeExactFlatHessianBlochSymbolZero4D.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.

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