Encyclopedia Gravity Gravity Analysis Regge Ttbloch Interface Audit Row0 Smoke Raw Weight Eq Rational

ARTICLE 3 claims 3 theorems

Gravity Analysis Regge Ttbloch Interface Audit Row0 Smoke Raw Weight Eq Rational

A machine-checked proof that one specific coefficient in a gravity calculation equals exactly one quarter, and nothing more.

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A single stencil weight

In numerical relativity, a stencil is a fixed pattern of nearby points used to approximate a derivative. The declaration row0Smoke_raw_weight_eq_rational proves that one particular weight in such a stencil, the coefficient attached to a specific pair of slots on a tetrahedral cell, equals exactly one quarter. The proof is a short calculation: it unfolds the definition of the coefficient, applies a previously established value for the angle derivative, and simplifies the resulting square root of one.

The coefficient belongs to a larger audit of a Regge calculus discretization of gravity, where spacetime is approximated by flat tetrahedra and the curvature is concentrated along their edges. The machine-checked library of formal theorems verifies that this single raw weight, before any averaging over cells, is the rational number 1/4. The result is narrow by design: it checks one entry in a table of weights, not the behavior of the full stencil.

In Recognition Science, the framework models physical structure as a discrete record of recognition events, and its library of formal theorems has proved that a unique cost function J(x) = (x + 1/x)/2 - 1 follows from five plain conditions. This declaration is not part of that forcing chain. It is an audit artifact: a smoke test that the raw stencil machinery produces the expected rational weight for one chosen slot pair, so that later stages of the gravity analysis can build on a verified foundation.

The declaration does not claim that the full stencil is rational, that the assembled gravity symbol vanishes, or that any continuum limit exists. The same-day diagnostic found that the stencil-only constant block does not cancel, and the gates for full bucket aggregation, hinge-aware zero modes, and the spike convention bridge all remain open. What the theorem establishes is precisely one number: the raw Jacobian coefficient for slots zero and five is 1/4, and that fact is now machine-checked.

THEOREM row0Smoke_raw_weight_eq_rational · IndisputableMonolith/Gravity/Analysis/ReggeTTBlochInterfaceAudit.lean
row0Smoke_raw_weight_eq_rational · IndisputableMonolith/Gravity/Analysis/ReggeTTBlochInterfaceAudit.lean:139 
/-- Row-0 smoke test: the radical-trivial coefficient
`J_05 / (2 * sqrt a*_0)` is the literal rational `1/4`. -/
theorem row0Smoke_raw_weight_eq_rational :
    rawJacobianCoefficient ⟨0, by decide⟩ ⟨5, by decide⟩ = (1 / 4 : ℝ) := by
  unfold rawJacobianCoefficient
  change ReggeTTDerivativeGate.flatAngleJacobian (0 : Fin 6) ⟨5, by decide⟩ /
      (2 * Real.sqrt (freudenthalTetSqEdges (0 : Fin 6))) = (1 / 4 : ℝ)
  rw [ReggeTTDerivativeGate.flatAngleJacobian_row0_eval ⟨5, by decide⟩]
  norm_num [ReggeTTDerivativeGate.flatAngleJacobianRow0,
    freudenthalTetSqEdges, Real.sqrt_one]
THEOREM row0Smoke_raw_weight_eq_rational · IndisputableMonolith/Gravity/Analysis/ReggeTTBlochInterfaceAudit.lean
row0Smoke_raw_weight_eq_rational · IndisputableMonolith/Gravity/Analysis/ReggeTTBlochInterfaceAudit.lean:139 
/-- Row-0 smoke test: the radical-trivial coefficient
`J_05 / (2 * sqrt a*_0)` is the literal rational `1/4`. -/
theorem row0Smoke_raw_weight_eq_rational :
    rawJacobianCoefficient ⟨0, by decide⟩ ⟨5, by decide⟩ = (1 / 4 : ℝ) := by
  unfold rawJacobianCoefficient
  change ReggeTTDerivativeGate.flatAngleJacobian (0 : Fin 6) ⟨5, by decide⟩ /
      (2 * Real.sqrt (freudenthalTetSqEdges (0 : Fin 6))) = (1 / 4 : ℝ)
  rw [ReggeTTDerivativeGate.flatAngleJacobian_row0_eval ⟨5, by decide⟩]
  norm_num [ReggeTTDerivativeGate.flatAngleJacobianRow0,
    freudenthalTetSqEdges, Real.sqrt_one]
THEOREM row0Smoke_raw_weight_eq_rational · IndisputableMonolith/Gravity/Analysis/ReggeTTBlochInterfaceAudit.lean
row0Smoke_raw_weight_eq_rational · IndisputableMonolith/Gravity/Analysis/ReggeTTBlochInterfaceAudit.lean:139 
/-- Row-0 smoke test: the radical-trivial coefficient
`J_05 / (2 * sqrt a*_0)` is the literal rational `1/4`. -/
theorem row0Smoke_raw_weight_eq_rational :
    rawJacobianCoefficient ⟨0, by decide⟩ ⟨5, by decide⟩ = (1 / 4 : ℝ) := by
  unfold rawJacobianCoefficient
  change ReggeTTDerivativeGate.flatAngleJacobian (0 : Fin 6) ⟨5, by decide⟩ /
      (2 * Real.sqrt (freudenthalTetSqEdges (0 : Fin 6))) = (1 / 4 : ℝ)
  rw [ReggeTTDerivativeGate.flatAngleJacobian_row0_eval ⟨5, by decide⟩]
  norm_num [ReggeTTDerivativeGate.flatAngleJacobianRow0,
    freudenthalTetSqEdges, Real.sqrt_one]

What this page does not claim

The full stencil weight table is rational. The assembled gravity symbol vanishes or has a zero mode. A continuum limit exists for this discretization.

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

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