Gravity Analysis Regge Exact Midpoint M2 Ttidentity4 Dkernel Glue Coupling Table

The coupling table's list identity

In the framework's machine-checked library of formal theorems, a coupling table is a structured array of numbers that encodes how certain interaction terms combine. The declaration couplingTable_toList_eq_ofFn proves, in the formal system, that the table's contents, when read out in order, exactly match a list built by indexing each position of the table. This is a statement about data representation: it says the two ways of viewing the table, as an array and as a linear list, are interchangeable.

The proof itself is a routine matter of list and array manipulation. It shows that for every position in the table, the element obtained by direct lookup equals the element produced by the list construction. The theorem is private, meaning it is an internal step used to support other results, not a standalone physical law. It is the kind of administrative bridge that lets later theorems freely switch between the array and list forms of the same coupling data.

In Recognition Science, this declaration does not introduce new physics. It does not say what the coupling values are, how they were derived, or that the table is complete. It only certifies that a particular representation change is valid. The significance is in what it enables: other theorems in the same file, such as sum_couplingTable_eq_toList_sum, rely on this identity to equate sums over the table with sums over its list form, which is a necessary step in connecting the table to the framework's broader calculations.

The practical consequence is that any later formal argument can treat the coupling table as a list without loss of information. This is a bookkeeping fact, not a discovery about gravity. It is the kind of result that makes a large formal library trustworthy: every step, even a trivial one, is checked by the machine.

private theorem couplingTable_toList_eq_ofFn :
    couplingTable.toList =
      List.ofFn (fun i : Fin couplingTable.size => couplingTable[i]) := by
  apply List.ext_getElem
  · rw [Array.length_toList, List.length_ofFn]
  · intro i h1 h2
    simp only [Array.getElem_toList, List.getElem_ofFn]
    rfl

What this page does not claim

The theorem does not assign physical meaning to the coupling values. It does not assert that the coupling table is complete or correct. It does not derive any gravitational law or constant.

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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