Gravity Analysis Regge Ttgate Bbridge Raw Moment Eq Committed Spike Lhs

The spike bridge

In numerical gravity, a stencil is a fixed pattern of points used to approximate derivatives on a grid. The Recognition Science framework's library, a machine-checked collection of formal theorems, contains a proof about a specific stencil on a small periodic torus. The theorem rawMoment_eq_committedSpikeLHS states that a certain moment, a weighted sum over the stencil's 216 terms, computed directly from the raw stencil data, is identically equal to a separately defined expression called the committed spike left-hand side. This equality holds for every choice of the two free parameters, the field E and direction x, with no additional conditions.

The proof works by showing the bucket-folded moment, which groups the 216 terms into buckets by their phase, equals the plain sum of the raw terms. Each raw term is then shown to match a corresponding core term from a separate module, and the core sum is proven equal to the committed spike expression at the specific values s2 = sqrt 2, s3 = sqrt 3, and p = pi. The result is a bridge: it connects the raw stencil's own moment calculation to the committed spike form, which was the target of a convention audit. The theorem closes Gate B of that audit, confirming the two conventions agree.

A notable disclosure accompanies the proof: the equality holds identically, without using the seven TT hypotheses that the locked target proposition lists as antecedents. These hypotheses are introduced but not consumed, because the identity is unconditional. The stronger statement, without those hypotheses, is what the theorem exports. This means the bridge does not depend on the physical assumptions of the TT reduction, only on the algebraic structure of the stencil and the spike's block data. The spike's own continuum certificate and its -1/4 conclusion are never invoked; only the spike's six block terms are used as data.

What the theorem does not claim is equally precise. It does not assert that the TT hypotheses are true, nor that the spike's -1/4 conclusion follows from this bridge. It does not claim the moment calculation has any physical meaning beyond the algebraic identity. The theorem is a statement about two formal expressions being equal, not about the physics of gravity. The proof's axiom footprint is the standard trio, meaning it relies only on the ambient type theory's basic postulates, with no new axioms and no unproved assumptions.

/-- **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

/-- **GATE C-B HEADLINE (THEOREM): the panel-locked Gate B target is
closed on the raw-stencil instantiation.**  `GateBConventionTarget`
(exactly the sidecar's Prop) holds for the raw-stencil support, the
midpoint-displacement phase quadratic, and the bucket-fiber-aggregated
amplitude, for every `E` and `x`.  DISCLOSURE: the seven TT antecedents
of the locked Prop are introduced but not consumed, because the bridge
equality holds identically in `(E, x)`
(`rawMoment_eq_committedSpikeLHS`); the locked statement shape is closed
a fortiori and the stronger unconditional theorem is exported above. -/
theorem gateB_convention_bridge (E : Fin 3 → Fin 3 → ℝ) (x : Fin 3 → ℝ) :
    GateBConventionTarget rawMomentSupport (rawPhaseQuadratic x)
      (rawBucketAmplitude E) E x := by
  intro _ _ _ _ _ _ _
  exact rawMoment_eq_committedSpikeLHS E x

What this page does not claim

The theorem does not prove the seven TT hypotheses are true. The theorem does not derive the spike's -1/4 conclusion. The theorem does not assign physical meaning to the moment equality.

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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