Gravity Analysis Regge Exact Midpoint M2 Ttidentity4 Dkernel Glue Sum Coupling T

The summation bridge

In mathematics, when you have a finite collection of numbers, there are often several ways to write down their sum. You might use a summation sign with an index, or you might literally write out the list and add the entries one by one. A natural question is whether these two descriptions always agree. The declaration sum_couplingTable_eq_toList_sum answers this question for a specific list of objects called couplings: it proves that summing over the index positions in the table gives exactly the same result as adding up the entries of the list obtained from that table.

The proof is a formal one, checked by a machine. It works by showing that the table's list form is the same as a list built from its own entries, and then using a standard property of how sums behave when you map a function over a list. The statement is general in one important way: it holds for any function that assigns a rational number to each coupling. You could replace the function with anything else, and the equality would still hold. This makes the result a structural fact about the table itself, not about any particular values in it.

This theorem is a small piece of a larger effort to verify a physical identity in gravity, but the theorem itself does not depend on any physics. It is purely about lists, indices, and sums. It does not claim that the table contains any particular couplings, nor that the sum has any particular value. It does not say anything about the meaning of the couplings or about the physical theory. It simply guarantees that two ways of computing a total will not disagree, which is a necessary consistency check for any larger proof that relies on summing over the table.

In Recognition Science, this kind of formal bridge is part of how the framework builds confidence in its calculations. The ledger, a discrete record of events, often requires summing contributions from many entries. A proof that different summation methods agree is a basic hygiene check: it ensures that the order or style of adding does not change the outcome. This particular theorem is a kernel-level detail, not a headline result, but it is the kind of fact that makes larger proofs trustworthy.

theorem sum_couplingTable_eq_toList_sum (f : Coupling → ℚ) :
    (∑ idx : CouplingIdx, f couplingTable[idx]) =
      (couplingTable.toList.map f).sum := by
  have h2 :
      List.ofFn (fun i : Fin couplingTable.size => f couplingTable[i]) =
        couplingTable.toList.map f := by
    rw [couplingTable_toList_eq_ofFn]
    exact (List.map_ofFn (fun i : Fin couplingTable.size => couplingTable[i]) f).symm
  refine
    ((List.sum_ofFn
        (f := fun i : Fin couplingTable.size => f couplingTable[i])).symm).trans ?_
  rw [h2]

theorem sum_couplingTable_eq_toList_sum (f : Coupling → ℚ) :
    (∑ idx : CouplingIdx, f couplingTable[idx]) =
      (couplingTable.toList.map f).sum := by
  have h2 :
      List.ofFn (fun i : Fin couplingTable.size => f couplingTable[i]) =
        couplingTable.toList.map f := by
    rw [couplingTable_toList_eq_ofFn]
    exact (List.map_ofFn (fun i : Fin couplingTable.size => couplingTable[i]) f).symm
  refine
    ((List.sum_ofFn
        (f := fun i : Fin couplingTable.size => f couplingTable[i])).symm).trans ?_
  rw [h2]

What this page does not claim

This theorem does not claim that the coupling table contains any particular values or that the sum has a specific numerical result. It does not assert any physical meaning for the couplings or the sum. It does not establish any property of the larger gravity identity beyond the consistency of summation methods.

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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