Gravity Analysis Regge Exact Midpoint M2 Ttidentity4 Dkernel Glue Term Q Eq Cont

The kernel bridge

The declaration termQ_eq_contrib_div256_of_mem is a formal theorem inside the Recognition Science framework's machine-checked library of formal theorems. It concerns a specific object called termQ, which is a rational number built from a coupling constant and six indices, each ranging over four values. The theorem states: for any coupling that appears in a particular list, the value of termQ equals the integer contribution of a related coupling, divided by 256. In plain words, it establishes an exact scaling relationship between two different ways of computing a term in a gravitational identity.

The proof is computational and exhaustive. The theorem relies on a companion result that every denominator in the coupling list divides 16, which is checked by direct computation. From that, the theorem derives the division by 256 through algebraic manipulation. The final statement is a universal quantifier over all six indices, meaning it holds for every combination of the four-valued indices. This is not a heuristic or an approximation; it is a proved equality in the framework's formal system.

The theorem does not claim anything about the physical meaning of the terms. It does not say that termQ represents a force or a mass. It does not assert that the gravitational identity itself is physically correct. The theorem only establishes a formal algebraic relationship between two defined quantities. It also does not claim that the coupling list is complete or that the identity holds for couplings outside the list. The theorem is a bridge: it connects a rational-term formulation to an integer-based formulation, enabling a larger proof to proceed.

In the context of the Recognition Science framework, this theorem is part of a chain that forces the structure of gravity from a cost function. The framework models reality as a discrete ledger of recognition events, and from that model it derives constants and laws. This particular theorem is a technical step in that derivation, not a standalone physical claim. Its value is that it closes a formal gap, allowing the larger identity to be certified by the machine-checked library.

The practical consequence is that a reader can trust the algebraic step without rechecking it by hand. The theorem is verified by the framework's kernel, which means it is as certain as the framework's own axioms. This is a small but necessary piece in a large formal edifice. It does not, by itself, prove anything about gravity; it only ensures that one part of the proof is sound.

theorem termQ_eq_contrib_div256_of_mem
    {coup : Coupling} (hc : coup ∈ couplingTable.toList)
    (a b cd d i j : Fin 4) :
    termQ coup a b cd d i j = (contrib (toCZ coup) a b cd d i j : ℚ) / 256 := by
  have hz : toCZ coup ∈ couplingZList := toCZ_mem_couplingZList_of_mem_toList hc
  have hd : (toCZ coup).den ∣ 16 := couplingZ_den_dvd_16 _ hz
  have hd' : coup.den ∣ 16 := by simpa [toCZ] using hd
  exact termQ_eq_contrib_div256 coup hd' a b cd d i j

theorem couplingZ_den_dvd_16 : ∀ z ∈ couplingZList, z.den ∣ 16 := by
  decide

What this page does not claim

This theorem does not claim that termQ has any physical meaning. This theorem does not claim that the coupling list is complete or unique. This theorem does not claim that the gravitational identity is physically correct.

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)

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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