Gravity Analysis Regge Exact Midpoint M2 Ttidentity4 Dkernel Glue To Cz Mem Coup

The coupling conversion

The declaration toCZ_mem_couplingZList_of_mem_toList is a formal statement inside the Recognition Science framework's machine-checked library of formal theorems. It proves a bookkeeping fact about two ways of storing the same list of coupling coefficients, which are the rational numbers that weight how different parts of a spacetime geometry interact in a gravity calculation. The theorem says: if a coupling belongs to the table used for the main calculation, then its converted form belongs to the integer list used for exact verification. In plain words, the conversion function toCZ does not drop or misplace any entry when moving from one representation to the other.

The proof is direct. The theorem rewrites the integer list as the image of the table under the conversion map, then applies the standard fact that mapping a list preserves membership. The library then uses this membership result to show that every converted coupling has a denominator dividing 16, a property needed for the exact rational arithmetic that follows. This is the glue that lets the framework trust that the integer-based verification covers exactly the same couplings as the original table, with nothing left out and nothing invented.

What the declaration does not claim is just as important. It does not prove that the gravity calculation itself is correct, nor that the couplings are physically meaningful, nor that the table is complete. It only certifies a structural correspondence between two lists. The theorem is a small, precise piece of plumbing: it guarantees that a conversion step preserves membership, nothing more. The actual physics, the derivation of the coupling values, and the verification that the calculation matches experiment all live in other declarations and remain separate questions.

private theorem toCZ_mem_couplingZList_of_mem_toList
    {coup : Coupling} (hc : coup ∈ couplingTable.toList) :
    toCZ coup ∈ couplingZList := by
  rw [← couplingZList_bridge]
  exact List.mem_map_of_mem hc

What this page does not claim

This theorem does not prove the correctness of the gravity calculation itself. It does not establish that the coupling values are physically meaningful or complete. It only certifies that the conversion between two list representations preserves membership.

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/ReggeExactMidpointM2TTIdentity4DKernelGlue.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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