Encyclopedia Gravity Gravity Analysis Regge Ttgate Bbridge Regge Ttmoment Eq Raw Triple Sum

ARTICLE 4 claims 4 theorems

Gravity Analysis Regge Ttgate Bbridge Regge Ttmoment Eq Raw Triple Sum

A machine-checked proof shows that a coarse-grained sum over grouped data equals a fine-grained sum over all 216 individual entries, with no loss or double counting.

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The bucket fold identity

The declaration reggeTTMoment_eq_rawTripleSum is a theorem in the framework's machine-checked library of formal theorems. It establishes an identity between two ways of computing a certain moment in a lattice gravity analysis. The first way is a bucket fold, a procedure that groups the 216 raw data triples into buckets by a key, sums the weights inside each bucket to get an amplitude, then combines these amplitudes. The second way is the plain sum over all 216 individual triples, without any grouping. The theorem proves these two results are exactly equal.

The identity is not a numerical approximation or a heuristic. It is a formal statement, checked by the kernel of the proof assistant, that the grouping operation loses no information and counts nothing twice. The proof uses the standard property Finset.sum_image', which states that summing over the image of a function is the same as summing over the original set, provided the function is well-defined on the buckets. In plain terms, if you sort a pile of receipts into folders and then add up each folder's total, the grand total is the same as adding every receipt directly.

The theorem is a bridge between two conventions in the framework's gravity analysis. The bucket fold uses the raw stencil data, the literal geometric objects from the periodic lattice. The plain triple sum is the starting point for a separate, more abstract representation called the spike. The identity shows that the two conventions agree at this stage, before any further physical assumptions are applied. This is a technical but necessary step: it guarantees that the framework's two computational routes are consistent with each other.

What the theorem does not claim is equally important. It does not assert that the spike representation is physically correct, nor does it invoke the seven TT hypotheses that appear as antecedents in the locked target proposition. The identity holds unconditionally, for any choice of the external field E and direction x. It is a statement about the algebra of the moment, not about the physics that the moment is meant to describe. The framework's own disclosure notes that the stronger unconditional statement is exported separately, so no hypothesis strength is hidden.

THEOREM reggeTTMoment_eq_rawTripleSum · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridge.lean
reggeTTMoment_eq_rawTripleSum · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridge.lean:182 
/-- The bucket fold with fiber-aggregated amplitudes equals the plain
sum over all 216 raw triples: no collision of bucket keys is dropped or
double counted (`Finset.sum_image'` on the key map). -/
theorem reggeTTMoment_eq_rawTripleSum (E : Fin 3 → Fin 3 → ℝ)
    (x : Fin 3 → ℝ) :
    reggeTTMoment rawMomentSupport (rawPhaseQuadratic x)
        (rawBucketAmplitude E) =
      ∑ p : Fin 6 × Fin 6 × Fin 6,
        -(rawPhaseQuadratic x (bucketKeyOf p)) / 2 * rawTripleWeight E p := by
  unfold reggeTTMoment reggeTTBlochFold rawMomentSupport
  rw [Finset.sum_image' (fun p : Fin 6 × Fin 6 × Fin 6 =>
    -(rawPhaseQuadratic x (bucketKeyOf p)) / 2 * rawTripleWeight E p)]
  intro p _
  unfold rawBucketAmplitude
  rw [Finset.mul_sum]
  refine Finset.sum_congr rfl fun q hq => ?_
  have hkey : bucketKeyOf q = bucketKeyOf p := (Finset.mem_filter.mp hq).2
  rw [hkey]
THEOREM reggeTTMoment_eq_rawTripleSum · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridge.lean
reggeTTMoment_eq_rawTripleSum · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridge.lean:182 
/-- The bucket fold with fiber-aggregated amplitudes equals the plain
sum over all 216 raw triples: no collision of bucket keys is dropped or
double counted (`Finset.sum_image'` on the key map). -/
theorem reggeTTMoment_eq_rawTripleSum (E : Fin 3 → Fin 3 → ℝ)
    (x : Fin 3 → ℝ) :
    reggeTTMoment rawMomentSupport (rawPhaseQuadratic x)
        (rawBucketAmplitude E) =
      ∑ p : Fin 6 × Fin 6 × Fin 6,
        -(rawPhaseQuadratic x (bucketKeyOf p)) / 2 * rawTripleWeight E p := by
  unfold reggeTTMoment reggeTTBlochFold rawMomentSupport
  rw [Finset.sum_image' (fun p : Fin 6 × Fin 6 × Fin 6 =>
    -(rawPhaseQuadratic x (bucketKeyOf p)) / 2 * rawTripleWeight E p)]
  intro p _
  unfold rawBucketAmplitude
  rw [Finset.mul_sum]
  refine Finset.sum_congr rfl fun q hq => ?_
  have hkey : bucketKeyOf q = bucketKeyOf p := (Finset.mem_filter.mp hq).2
  rw [hkey]
THEOREM reggeTTMoment_eq_rawTripleSum · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridge.lean
reggeTTMoment_eq_rawTripleSum · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridge.lean:182 
/-- The bucket fold with fiber-aggregated amplitudes equals the plain
sum over all 216 raw triples: no collision of bucket keys is dropped or
double counted (`Finset.sum_image'` on the key map). -/
theorem reggeTTMoment_eq_rawTripleSum (E : Fin 3 → Fin 3 → ℝ)
    (x : Fin 3 → ℝ) :
    reggeTTMoment rawMomentSupport (rawPhaseQuadratic x)
        (rawBucketAmplitude E) =
      ∑ p : Fin 6 × Fin 6 × Fin 6,
        -(rawPhaseQuadratic x (bucketKeyOf p)) / 2 * rawTripleWeight E p := by
  unfold reggeTTMoment reggeTTBlochFold rawMomentSupport
  rw [Finset.sum_image' (fun p : Fin 6 × Fin 6 × Fin 6 =>
    -(rawPhaseQuadratic x (bucketKeyOf p)) / 2 * rawTripleWeight E p)]
  intro p _
  unfold rawBucketAmplitude
  rw [Finset.mul_sum]
  refine Finset.sum_congr rfl fun q hq => ?_
  have hkey : bucketKeyOf q = bucketKeyOf p := (Finset.mem_filter.mp hq).2
  rw [hkey]
THEOREM rawMoment_eq_committedSpikeLHS · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridge.lean
rawMoment_eq_committedSpikeLHS · IndisputableMonolith/Gravity/Analysis/ReggeTTGateBBridge.lean:215 
/-- **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 assert that the spike representation is physically correct. The theorem does not invoke the seven TT hypotheses as assumptions. The theorem does not establish any numerical value for the moment.

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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