Gravity Analysis Freudenthal Stencil Preflight Scaled Canonical Energy Eq Scaled

The stencil identity

In numerical analysis, a stencil is a fixed pattern of neighboring points used to approximate a derivative or a differential operator. The declaration scaledCanonicalEnergy_eq_scaled_stencil proves, for every lattice side length N greater than 2, that a certain quadratic energy defined from a Regge calculus Hessian equals a stencil sum over seven displacement classes. The weights are the square roots of the squared displacement lengths: 1 for the three axis directions, √2 for the three face diagonals, and √3 for the one body diagonal.

This is a theorem, not a postulate. The chain of reasoning starts with the canonical Regge Hessian, a matrix of second derivatives of a discrete gravitational action on a periodic Freudenthal triangulation. The proof shows that the quadratic form of this Hessian is exactly the graph Dirichlet energy, then reindexes the edges into the seven displacement classes. The final identity is

scaledCanonicalEnergy N u = stencilNormalization N * freudenthalStencilEnergy N u

where the normalization is 1/N, declared a priori from dimensional analysis. This identity holds for every N greater than 2, not just for the N = 5 certificates checked earlier. The proof is axiom-clean in the machine-checked library of formal theorems, with no unproved assumptions.

The same module computes the moment tensor of the stencil exactly. It is not isotropic: the off-diagonal entries are √2 + √3, strictly positive, so the continuum quadratic form is anisotropic. This corrects an earlier expectation of a rational symmetric matrix. The entries live in ℚ[√2, √3], and the tensor is positive semidefinite and nonzero.

What this declaration does not claim is as important as what it proves. It does not establish the continuum limit of the energy; that is a separate stage. It does not claim the stencil is isotropic, and it does not claim the normalization is fitted to data. The identity is a preflight check, a necessary step toward a larger goal that remains open.

theorem scaledCanonicalEnergy_eq_scaled_stencil (hN : 2 < N)
    (u : Vertex N N N → ℝ) :
    scaledCanonicalEnergy N u =
      stencilNormalization N * freudenthalStencilEnergy N u := by
  unfold scaledCanonicalEnergy
  rw [hessianQuadratic_canonical_eq_freudenthalStencil N hN u]

/-- Explicit values of the seven stencil weights:
`√1, √1, √1, √2, √2, √2, √3` for the three axis classes, the three
face-diagonal classes, and the body-diagonal class. -/
theorem stencilWeight_values :
    stencilWeight 0 = 1 ∧ stencilWeight 1 = 1 ∧ stencilWeight 2 = 1 ∧
      stencilWeight 3 = Real.sqrt 2 ∧ stencilWeight 4 = Real.sqrt 2 ∧
      stencilWeight 5 = Real.sqrt 2 ∧ stencilWeight 6 = Real.sqrt 3 := by
  refine ⟨?_, ?_, ?_, ?_, ?_, ?_, ?_⟩ <;>
    norm_num [stencilWeight, periodicDispSqEdge, Real.sqrt_one]

/-- FINDING (kernel-checked): the moment tensor is ANISOTROPIC. The
off-diagonal entry is `√2 + √3 > 0`, so `A₀ ≠ c·I` for every scalar `c`.
Isotropy is NOT claimed; the exact anisotropic value is
`stencilMomentTensor_eq`. -/
theorem stencilMomentTensor_not_isotropic (c : ℝ) :
    stencilMomentTensor ≠ fun i j => if i = j then c else 0 := by
  intro h
  have hne : (0 : Fin 3) ≠ 1 := by decide
  have h01 := congrFun (congrFun h 0) 1
  rw [if_neg hne] at h01
  exact absurd h01 (ne_of_gt (stencilMomentTensor_offDiag_pos 0 1 hne))

What this page does not claim

The declaration does not establish the continuum limit of the energy; that remains a separate open stage. The declaration does not claim the stencil is isotropic; the opposite is proved. The declaration does not claim the normalization is fitted to data; it is declared a priori.

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/FreudenthalStencilPreflight.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: