Gravity Analysis Regge Ttgate Bbridge Triple Term Ident

The triple term identity

The declaration tripleTerm_ident is a theorem in the framework's machine-checked library of formal theorems. It states that one signed term in a large sum, written in the raw stencil convention, is exactly equal to the corresponding term written in the core module's convention. The equality holds for every choice of the field E and direction x, with no case split on the triple needed. In plain terms: two descriptions of the same physical quantity, one built directly from the raw cell stencil and the other from the committed spike blocks, agree term by term.

The context is a gravitational moment calculation on a periodic lattice. The raw stencil is the fundamental object: a set of 216 triples, each a tetrahedron type and an ordered slot pair. The raw convention assigns each triple a weight from the Jacobian coefficient and edge coefficients, and a phase from the squared midpoint displacement. The core module uses a different, more compact table of weights and coefficients. The theorem proves that for every one of the 216 triples, the raw term and the core term are the same number. It is a pointwise identification of every table entry with its actual object, so nothing rests on a transcription of values.

The theorem is one step in a chain that closes a panel-locked target called Gate B. The chain shows that the bucket-folded moment, which aggregates raw terms by their phase bucket, equals the plain sum over all 216 triples, and that this sum equals the committed spike left-hand side. The final headline theorem, gateB_convention_bridge, states that the raw-stencil instantiation satisfies the target property for every E and x. The identity itself is unconditional: it does not require the seven TT hypotheses that the locked target Prop lists as antecedents. Those hypotheses are introduced and then closed a fortiori, because the equality already holds identically.

What the theorem does not claim is just as important. It does not invoke the spike's continuum certificate or its -1/4 conclusion; only the spike's block data are used. It does not assert that the seven TT hypotheses are true, only that the target holds without them. It does not derive any physical law about gravity in the continuum; it is a finite algebraic identity over the lattice tables. The framework's library also does not claim that this identity is a new physical discovery, only that the two conventions agree exactly, which is the precondition for trusting the committed spike calculation.

/-- Each signed raw moment term equals the core module's literal term
(pointwise identification of every table with its actual object; no
case split on the triple is needed). -/
theorem tripleTerm_ident (E : Fin 3 → Fin 3 → ℝ) (x : Fin 3 → ℝ)
    (p : Fin 6 × Fin 6 × Fin 6) :
    -(rawPhaseQuadratic x (bucketKeyOf p)) / 2 * rawTripleWeight E p =
      coreTripleTerm E x p := by
  unfold rawPhaseQuadratic bucketKeyOf rawTripleWeight
  unfold ReggeTTGateBBridgeCore.coreTripleTerm
  rw [coreWeight_eq_raw, corePolEdgeCoeff_eq, corePolEdgeCoeff_eq,
    slotDispCore_eq, slotDispCore_eq]

/-- Each signed raw moment term equals the core module's literal term
(pointwise identification of every table with its actual object; no
case split on the triple is needed). -/
theorem tripleTerm_ident (E : Fin 3 → Fin 3 → ℝ) (x : Fin 3 → ℝ)
    (p : Fin 6 × Fin 6 × Fin 6) :
    -(rawPhaseQuadratic x (bucketKeyOf p)) / 2 * rawTripleWeight E p =
      coreTripleTerm E x p := by
  unfold rawPhaseQuadratic bucketKeyOf rawTripleWeight
  unfold ReggeTTGateBBridgeCore.coreTripleTerm
  rw [coreWeight_eq_raw, corePolEdgeCoeff_eq, corePolEdgeCoeff_eq,
    slotDispCore_eq, slotDispCore_eq]

/-- Each signed raw moment term equals the core module's literal term
(pointwise identification of every table with its actual object; no
case split on the triple is needed). -/
theorem tripleTerm_ident (E : Fin 3 → Fin 3 → ℝ) (x : Fin 3 → ℝ)
    (p : Fin 6 × Fin 6 × Fin 6) :
    -(rawPhaseQuadratic x (bucketKeyOf p)) / 2 * rawTripleWeight E p =
      coreTripleTerm E x p := by
  unfold rawPhaseQuadratic bucketKeyOf rawTripleWeight
  unfold ReggeTTGateBBridgeCore.coreTripleTerm
  rw [coreWeight_eq_raw, corePolEdgeCoeff_eq, corePolEdgeCoeff_eq,
    slotDispCore_eq, slotDispCore_eq]

/-- **THE UNCONDITIONAL BRIDGE (THEOREM): the raw-stencil moment fold
equals the committed spike LHS IDENTICALLY in `(E, x)`.**  Chains the
fiber-aggregation fold, the pointwise identification, and the core
216-term identity at `s2 = sqrt 2`, `s3 = sqrt 3`, `p = pi` (the
sidecar's committed instantiation; the core identity holds for free
values).  The spike's `tt_continuum_certificate` and its `-1/4`
conclusion are never invoked: only the block data `tetBlock0..5`. -/
theorem rawMoment_eq_committedSpikeLHS (E : Fin 3 → Fin 3 → ℝ)
    (x : Fin 3 → ℝ) :
    reggeTTMoment rawMomentSupport (rawPhaseQuadratic x)
        (rawBucketAmplitude E) =
      committedSpikeLHS (spikeInput E x) := by
  rw [reggeTTMoment_eq_rawTripleSum]
  rw [Finset.sum_congr rfl fun p _ => tripleTerm_ident E x p]
  rw [coreTripleSum_eq_spikeSum E x (Real.sqrt 2) (Real.sqrt 3) Real.pi]
  rfl

What this page does not claim

The theorem does not invoke the spike's continuum certificate or its -1/4 conclusion. The theorem does not assert that the seven TT hypotheses are true. The theorem does not derive any continuum physical law about gravity.

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

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