Gravity Analysis Regge Exact Midpoint M2 Ttidentity4 Dkernel Cert

The kernel certificate

In numerical analysis, a certificate is a record that lets you check a computation without trusting the tool that produced it. The module here is a ledger, a discrete record of events, of exactly that kind. It stores a large table of integer coefficients, the output of a script that expanded a complicated expression in four-dimensional gravity analysis. The table is too big for a human to read, but the module does not ask you to take it on faith.

The table is split into sixteen chunks, each a list of entries in a structure called CZ. For every chunk, a theorem states that the chunk produced by the script equals the chunk written in the module. Each proof is a single `decide` step, which means the machine checks the equality by direct computation on the integers. The module then assembles the chunks into one full table, `couplingZList`, and defines a function `m2Num` that folds over this list to sum contributions. A final theorem, `symFullZ_explicit_eq_closed`, states that the explicitly expanded table equals a closed-form expression for all indices. This is the payload: the identity holds, and the machine has checked it.

Why go to this trouble? The identity in question is a symmetry property of a four-index object in the framework's gravity analysis. It is the kind of algebraic fact that, if true, simplifies later work, but if false, quietly corrupts everything built on it. The certificate makes the truth of the identity a checked fact rather than an assumption. The script that generated the table could have made an error; the module exists to catch that error before the identity is used elsewhere.

The certificates use only the kernel's `decide` tactic, not a faster native code path. This is a deliberate choice: `decide` is slow but simple, so the check is easy to audit. The cost is a large file, but the benefit is that the identity is verified by the same core logic that checks every other proof in the library. The module does not prove a new physical law. It proves that a specific, large algebraic identity, already derived in the source material, is correctly encoded in the framework's language.

What this changes is the status of the identity. Before the certificate, the identity was a plausible claim from a script. After it, the identity is a theorem in the machine-checked library, available for other proofs to use without re-deriving it. The reader can now treat the symmetry as a safe building block, and the framework's gravity analysis can proceed on a checked foundation.

theorem czChunk0_bridge :
    couplingChunk0.toList.map toCZ = czChunk0 := 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

theorem czChunk0_bridge :
    couplingChunk0.toList.map toCZ = czChunk0 := by
  decide

What this page does not claim

No new physical law is proved by this module. The identity is not derived here; it is only checked. The module does not prove that the framework's gravity analysis is correct, only that this one identity is correctly encoded.

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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