Gravity Analysis Regge Ttbucket Aggregation Raw Jacobian Coefficient Eval

The rational stencil table

In the Recognition Science framework's study of gravity, a stencil is a fixed set of weights used to approximate a derivative from neighboring values. The declaration rawJacobianCoefficient_eval proves that every one of the 36 raw stencil coefficients, expressions that carry square roots, equals an exact rational number: a fraction of whole numbers. The values are drawn from the set {0, 1/12, +-1/8, +-1/4}. This is not a numerical approximation; the theorem states an exact equality, checked by the framework's machine-checked library of formal theorems.

The coefficients arise from the flat angle Jacobian, a matrix of partial derivatives describing how the angles of a tetrahedron change as its edge lengths vary. The theorem shows that when this Jacobian is evaluated at the flat tuple, a specific configuration where the tetrahedron degenerates, the radical expressions simplify completely. The proof route uses a proved bridge: the square root of an edge length times the derivative of an angle with respect to another edge equals a rational cofactor polynomial divided by a constant. At the flat tuple, that constant is 4, so every square root cancels, leaving only rational numbers.

The theorem is the headline of a larger aggregation result. It establishes that the raw coefficient on every bucket, meaning every slot pair and integer phase key, equals the real cast of an independent literal table called rationalStencilWeight. That table was defined separately, not derived from the Jacobian, so the equality is a genuine theorem: 36 kernel-checked radical cancellations. The table is also symmetric under swapping the two slot indices and negating the phase, a property proved in a companion theorem.

What the declaration does not claim is broader physical content. It does not assert that gravity itself is rational, or that the stencil approximates any particular physical quantity to a given accuracy. It does not derive the fine-structure constant or any other coupling. It does not claim that the flat tuple is physically realized in nature; it is a mathematical configuration used for evaluation. The theorem is purely algebraic: it simplifies a set of expressions. Its significance is that it closes a gate in the framework's campaign, showing that a previously open aggregation step is now proved exactly.

/-- All 36 raw coefficients evaluated to exact rationals.  Each entry is
the radical-free normal form of `rawJacobianCoefficient_eq_norm_div`
evaluated by kernel rational arithmetic on the flat integer tuple. -/
theorem rawJacobianCoefficient_eval (f g : Fin 6) :
    rawJacobianCoefficient f g =
      ((rationalStencilWeight ⟨f, g, fun _ => 0⟩ : ℚ) : ℝ) := by
  rw [rawJacobianCoefficient_eq_norm_div]
  fin_cases f <;> fin_cases g <;>
    norm_num [rationalStencilWeight,
      Geometry.SchlaefliTetrahedronProof.schlaefliPolySummandNorm,
      Geometry.CofactorPolynomial.cmCofactor3Poly,
      Geometry.CofactorPolynomial.cmCofactorPartial,
      freudenthalTetSqEdges]

/-- All 36 raw coefficients evaluated to exact rationals.  Each entry is
the radical-free normal form of `rawJacobianCoefficient_eq_norm_div`
evaluated by kernel rational arithmetic on the flat integer tuple. -/
theorem rawJacobianCoefficient_eval (f g : Fin 6) :
    rawJacobianCoefficient f g =
      ((rationalStencilWeight ⟨f, g, fun _ => 0⟩ : ℚ) : ℝ) := by
  rw [rawJacobianCoefficient_eq_norm_div]
  fin_cases f <;> fin_cases g <;>
    norm_num [rationalStencilWeight,
      Geometry.SchlaefliTetrahedronProof.schlaefliPolySummandNorm,
      Geometry.CofactorPolynomial.cmCofactor3Poly,
      Geometry.CofactorPolynomial.cmCofactorPartial,
      freudenthalTetSqEdges]

/-- **GATE C-A2f HEADLINE (THEOREM): on EVERY bucket (every slot pair,
every integer phase key), the actual radical-bearing raw stencil
coefficient `J_{fg} / (2 * sqrt a*_f)` equals the real cast of the
independent literal rational table.**  The left side is
`rawJacobianCoefficient` of the interface audit (built from the
kernel-proved flat angle Jacobian and the flat tuple); the right side is
the bare literal table of §1.  The two sides are independently defined;
their equality is 36 kernel-checked radical cancellations. -/
theorem aggregate_raw_weight_eq_rational (b : Bucket) :
    rawJacobianCoefficient b.left b.right =
      ((rationalStencilWeight b : ℚ) : ℝ) := by
  have h := rawJacobianCoefficient_eval b.left b.right
  have htbl : rationalStencilWeight ⟨b.left, b.right, fun _ => 0⟩ =
      rationalStencilWeight b := rfl
  rw [htbl] at h
  exact h

What this page does not claim

The theorem does not claim that gravity itself is rational or that the stencil approximates any physical quantity to a given accuracy. It does not derive the fine-structure constant or any other coupling constant. It does not claim that the flat tuple is physically realized in nature.

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/ReggeTTBucketAggregation.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.

Derived articles

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