Gravity Track1 Bcorrected Quadratic Canonical Periodic Mixed Axis Stencil Action

The axis stencil's nonnegativity

The declaration canonicalPeriodicMixedAxisStencilAction_nonneg is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It states that a particular mathematical expression, the axis stencil action, a discrete sum defined on a periodic three-dimensional grid, always evaluates to a number greater than or equal to zero. In plainer terms, for any configuration of values assigned to the grid's vertices, this specific way of adding up local contributions can never yield a negative total.

This nonnegativity is not an isolated curiosity. The axis stencil action is the corrected quadratic term in a local correspondence with the Regge action, a discrete model of general relativity. The proof is a formal theorem, meaning it has been checked by a computer from the framework's axioms. It is a necessary property for the action to serve as a stable, energy-like quantity in the framework's damped-schedule closure, and it is proved for all grid sizes and all vertex configurations, not just for a few examples.

What the theorem does not claim is equally important. It does not say that the axis stencil action is the only correct quadratic, nor that it is the one used by the legacy edge stencil. In fact, a separate theorem, not_both_correspondences_of_quadratics_differ, proves that the legacy and corrected stencils cannot both satisfy the local correspondence unless they are identical, and a mismatch witness shows they are not. The nonnegativity theorem also does not assert that the corrected gate at N=5 is closed; that is a separate statement, now proved, but it is not part of this declaration. The theorem's scope is precisely the sign of the action, nothing more.

/-- The axis stencil is nonnegative. -/
theorem canonicalPeriodicMixedAxisStencilAction_nonneg
    (Nx Ny Nz : ℕ) [NeZero Nx] [NeZero Ny] [NeZero Nz]
    (hx : 2 < Nx) (hy : 2 < Ny) (hz : 2 < Nz)
    (ξ : VertexPotential
      (canonicalEncodedPeriodicFreudenthalTorus Nx Ny Nz hx hy hz).K) :
    0 ≤ canonicalPeriodicMixedAxisStencilAction Nx Ny Nz hx hy hz ξ := by
  unfold canonicalPeriodicMixedAxisStencilAction
  refine Finset.sum_nonneg fun base _ => Finset.sum_nonneg fun d _ => ?_
  dsimp only
  positivity

/-- **EXCLUSIVITY.**  Given the audit witness, the legacy seven-class endpoint
and the corrected axis endpoint are mutually exclusive: at most one of them is
the true cubic-Taylor statement for the Regge action. -/
theorem not_both_correspondences_of_quadratics_differ
    (Nx Ny Nz : ℕ) [NeZero Nx] [NeZero Ny] [NeZero Nz]
    (hx : 2 < Nx) (hy : 2 < Ny) (hz : 2 < Nz)
    (hdiff : AxisEdgeStencilQuadraticsDiffer Nx Ny Nz hx hy hz) :
    ¬(CanonicalPeriodicEdgeStencilLocalCorrespondence Nx Ny Nz hx hy hz ∧
      CanonicalPeriodicAxisStencilLocalCorrespondence Nx Ny Nz hx hy hz) := by
  rintro ⟨hLegacy, hCorrected⟩
  obtain ⟨ξ, hξ⟩ := hdiff
  exact hξ (both_correspondences_force_equal_quadratics
    Nx Ny Nz hx hy hz hLegacy hCorrected ξ)

What this page does not claim

This theorem does not establish that the axis stencil action is the unique or preferred quadratic. This theorem does not prove that the legacy edge stencil is incorrect in all contexts. This theorem does not assert that the corrected N=5 gate is closed; that is a separate theorem.

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/Track1BCorrectedQuadratic.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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