Gravity Analysis Regge Ttbloch Interface Audit Worst Radical Flat Angle Jacobian

A single stencil value

The declaration worstRadical_flatAngleJacobian_value is a theorem in the framework's machine-checked library of formal theorems. It states that for one specific pair of slots in a six-slot stencil, the flat-angle Jacobian equals the negative of the square root of two divided by four. In symbols, that is flatAngleJacobian(1, 2) = -sqrt(2)/4. The value itself is a concrete number, not a symbolic identity: it is approximately -0.353553.

The context is a numerical audit of a Regge calculus calculation. Regge calculus is a discrete approach to general relativity where spacetime is built from flat tetrahedra, and the calculation here examines how a perturbation behaves on a periodic lattice. The stencil is a 6 by 6 by 6 table of terms, one for each ordered pair of the six slots in a tetrahedron. The theorem computes one cell of that table, the one for slot pair (1, 2), which the framework's own notes call the worst radical bucket because it is the entry that produces a square root rather than a rational number.

The proof is direct. The theorem unfolds the definition of the Jacobian, applies a previously established row-zero evaluation, and then simplifies the resulting expression with arithmetic. The key step is recognizing that the square root of two does not vanish, so the fraction can be cleared. The result is a closed form: the coefficient for this slot pair is exactly negative one eighth, after dividing by the edge-length factor.

In Recognition Science, this theorem is a small but necessary piece of a larger audit. The framework is testing whether a certain discrete gravity operator, built from these stencil terms, has the correct continuum limit. The audit is deliberately narrow: it checks that the raw stencil matches the reduced second variation, and this theorem verifies one specific entry in that stencil. The full aggregation over all buckets, the zero-mode cancellation, and the connection to the continuum limit are all explicitly left open in the framework's own status notes.

What the theorem does not claim is as important as what it proves. It does not establish that the entire stencil vanishes, nor that the assembled gravity operator is correct. The framework's notes state that the stencil-only constant block does not vanish, and that the continuum limit engine must use a different route after the constant block is connected. The theorem is a single verified number in a table, not a verdict on the whole calculation.

/-- Genuine radical-row audit: at the row-Jacobian layer, individual raw
coefficients ARE irrational here; the panel's rationality claim lives at
bucket-fiber-AGGREGATION level and remains OPEN.  For this selected entry the
Jacobian is `-sqrt 2 / 4`; the current `rawJacobianCoefficient` normalization
then exposes and cancels the same `sqrt 2` denominator. -/
theorem worstRadical_flatAngleJacobian_value :
    ReggeTTDerivativeGate.flatAngleJacobian (1 : Fin 6) (2 : Fin 6) =
      -(Real.sqrt 2) / 4 := by
  rw [ReggeTTDerivativeGate.flatAngleJacobian_cofactor_form]
  rw [Geometry.CofactorDerivatives.dihedralCos3SqClosedFormDeriv_eq_poly]
  norm_num [ReggeTTDerivativeGate.flatArccosFactor,
    Geometry.CofactorDerivatives.dihedralCos3SqPolyClosedFormDeriv,
    Geometry.CofactorDerivatives.dihedralDenom3PolyClosedDerivValue,
    Geometry.CofactorDerivatives.dihedralDenom3Poly,
    Geometry.CofactorPolynomial.cmCofactor3Poly,
    Geometry.CofactorPolynomial.cmCofactorPartial,
    Geometry.DihedralCayleyMenger.oppositeCMVertices,
    freudenthalTetSqEdges]
  rw [show Real.sqrt 32 = 4 * Real.sqrt 2 by
    rw [show (32 : ℝ) = 16 * 2 by norm_num]
    rw [Real.sqrt_mul (by norm_num : (0 : ℝ) ≤ 16),
      show Real.sqrt (16 : ℝ) = 4 by norm_num]]
  ring

/-- Genuine radical-row audit: at the row-Jacobian layer, individual raw
coefficients ARE irrational here; the panel's rationality claim lives at
bucket-fiber-AGGREGATION level and remains OPEN.  For this selected entry the
Jacobian is `-sqrt 2 / 4`; the current `rawJacobianCoefficient` normalization
then exposes and cancels the same `sqrt 2` denominator. -/
theorem worstRadical_flatAngleJacobian_value :
    ReggeTTDerivativeGate.flatAngleJacobian (1 : Fin 6) (2 : Fin 6) =
      -(Real.sqrt 2) / 4 := by
  rw [ReggeTTDerivativeGate.flatAngleJacobian_cofactor_form]
  rw [Geometry.CofactorDerivatives.dihedralCos3SqClosedFormDeriv_eq_poly]
  norm_num [ReggeTTDerivativeGate.flatArccosFactor,
    Geometry.CofactorDerivatives.dihedralCos3SqPolyClosedFormDeriv,
    Geometry.CofactorDerivatives.dihedralDenom3PolyClosedDerivValue,
    Geometry.CofactorDerivatives.dihedralDenom3Poly,
    Geometry.CofactorPolynomial.cmCofactor3Poly,
    Geometry.CofactorPolynomial.cmCofactorPartial,
    Geometry.DihedralCayleyMenger.oppositeCMVertices,
    freudenthalTetSqEdges]
  rw [show Real.sqrt 32 = 4 * Real.sqrt 2 by
    rw [show (32 : ℝ) = 16 * 2 by norm_num]
    rw [Real.sqrt_mul (by norm_num : (0 : ℝ) ≤ 16),
      show Real.sqrt (16 : ℝ) = 4 by norm_num]]
  ring

/-- Exact raw coefficient for the genuine radical-row entry.  The statement
keeps the radical-bearing numerator visible; Lean also proves the normalized
coefficient simplifies to `-1/8`. -/
theorem worstRadical_rawJacobianCoefficient_closedForm :
    rawJacobianCoefficient ⟨1, by decide⟩ ⟨2, by decide⟩ = -(1 / 8 : ℝ) := by
  unfold rawJacobianCoefficient
  change ReggeTTDerivativeGate.flatAngleJacobian (1 : Fin 6) (2 : Fin 6) /
      (2 * Real.sqrt (freudenthalTetSqEdges (1 : Fin 6))) = -(1 / 8 : ℝ)
  rw [worstRadical_flatAngleJacobian_value]
  norm_num [freudenthalTetSqEdges]
  have hsqrt2_ne : Real.sqrt 2 ≠ 0 := by positivity
  field_simp [hsqrt2_ne]
  norm_num

What this page does not claim

The theorem does not prove that the full stencil vanishes. The theorem does not establish the correctness of the assembled gravity operator. The theorem does not connect this single value to the continuum limit.

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)

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.

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