Gravity Analysis Regge Exact Midpoint M2 Ttidentity4 Dkernel Cert Sym Full Z Exp

A certificate of identity

In the mathematics of gravity, a single physical quantity can often be written in several forms. One form may be compact and elegant, another sprawling and explicit. The declaration symFullZ_explicit_eq_closed is a machine-checked certificate that two such forms, for one specific expression, are the same. It states that for any choice of the six indices a, b, c, d, i, j, each ranging over four values, the explicit version of the expression symFullZ equals the closed version. The proof is not a hand-waving argument; it is a direct computation, checked by the kernel of the machine-checked library of formal theorems.

The two forms are not just different notation. The explicit form is built from a list of coupling terms, each with its own integer coefficients, summed together. The closed form is a single, compact expression. The declaration proves they agree for all 4^6 = 4096 possible index combinations. This is a finite, exhaustive check, not an inductive proof. The computation uses integer arithmetic and a technique called decide, which means the kernel itself evaluates both sides and confirms they match.

This is a certificate, not a discovery. It does not say what the expression means physically, why it has this form, or whether it is the correct description of gravity. It only says that two mathematical objects, defined in the framework's library, are identical. The value of such a certificate is trust: a researcher can rely on the equivalence without redoing the arithmetic, because the check is mechanically verified. It is the kind of guarantee that makes large formal developments possible, where a single human error in a long computation could otherwise go unnoticed.

In Recognition Science, this declaration is a small but concrete example of the framework's method. The framework models physical structure through a discrete record of events, and its library of formal theorems is built to be audited to the kernel's axioms. This particular theorem does not advance the forcing chain or derive a constant. It is a verification step, ensuring that two expressions used in the gravity analysis are interchangeable. It is a piece of plumbing, not a new law.

theorem symFullZ_explicit_eq_closed :
    ∀ a b c d i j : Fin 4,
      symFullZ explicitZ a b c d i j = symFullZ closedZ a b c d i j := by
  decide

theorem symFullZ_explicit_eq_closed :
    ∀ a b c d i j : Fin 4,
      symFullZ explicitZ a b c d i j = symFullZ closedZ a b c d i j := by
  decide

What this page does not claim

This theorem does not state what the expression symFullZ means physically. It does not claim the expression is the correct description of gravity. It does not derive any new physical constant or advance the forcing chain.

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/ReggeExactMidpointM2TTIdentity4DKernelCert.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

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