Gravity Analysis Regge Hinge4 Dstar Kernel12 Near Deficit Kernel Eq Chain

The near deficit kernel identity

In Recognition Science, a framework that derives physical structure from a discrete record of recognition events, the declaration nearDeficitKernel_eq_chain is a proved identity about how a flat four-dimensional lattice assigns a curvature measure to a specific triangular hinge. The hinge is a line segment where two triangular faces meet, here the type (1,2) triangle hinge with masks 0,1,7. The identity states that the near deficit kernel, a function assigning weights to the ten edge directions around the hinge, equals the negative of a chain constant times the near cosine kernel, a function derived from the cosine of the dihedral angle at each edge. In symbols, for each edge direction k, nearDeficitKernel k = -chainRight * nearCosKernel k, where chainRight is the constant -1. This is a formal theorem, kernel-checked with no unproved assumptions beyond the standard three axioms of the ambient type theory, and it is part of a larger campaign to assemble a flat Hessian over all hinges.

The identity is one of two companion results, the other being farDeficitKernel_eq_chain for the far side of the star. Together they feed into the full-star class kernel, a function over all fifteen stencil classes with values ±√2/2. The surrounding theorems establish that the star's angle sum is exactly 2π (4 · arccos 0 = 4 · π/2), that the star has exactly four containing cube-simplex pairs, and that the kernel is symmetric under swapping the two coordinate directions. These are structural facts about a flat lattice, not about curved spacetime itself. The identity is a stepping stone: it shows how a flat configuration assigns zero curvature in a consistent way, which is the baseline against which actual curvature would be measured.

What the identity does not claim is equally important. It does not prove that the full Hessian assembly over all hinges is complete, nor that the recognition-science action converges to the Einstein-Hilbert action in four dimensions. It does not reverse-engineer weights from general relativity, and it does not transport the result to the complementary type (2,1) hinge, which remains open. The identity is a local, machine-checked fact about one hinge orbit in a flat lattice, not a global statement about gravity. It is a precise tool for building a larger structure, and its value lies in what it enables next, not in any claim about the physical universe on its own.

For a reader, the practical takeaway is that the framework now has a verified, symbolic handle on how a flat four-dimensional lattice distributes curvature-like weights around a single hinge. This is the kind of foundation on which a full theory of discrete gravity could be built, one hinge at a time. The identity is not a theory of gravity, but it is a necessary and checked piece of one.

def nearDeficitKernel : Fin 10 → ℝ := nearCosKernel

theorem star_flat_angle_sum_two_pi : starFlatAngleSum = 2 * Real.pi := by
  simp only [starFlatAngleSum, flatAngleRight_eq]; ring

def fullStarClassKernel : Fin 15 → ℝ
  | ⟨0, _⟩ => -(Real.sqrt 2) / 2
  | ⟨1, _⟩ => Real.sqrt 2 / 2
  | ⟨2, _⟩ => Real.sqrt 2 / 2
  | ⟨3, _⟩ => Real.sqrt 2 / 2
  | ⟨4, _⟩ => Real.sqrt 2 / 2
  | ⟨5, _⟩ => -(Real.sqrt 2) / 2
  | ⟨6, _⟩ => -(Real.sqrt 2) / 2
  | ⟨7, _⟩ => Real.sqrt 2 / 2
  | ⟨8, _⟩ => Real.sqrt 2 / 2
  | ⟨9, _⟩ => -(Real.sqrt 2) / 2
  | ⟨10, _⟩ => -(Real.sqrt 2) / 2
  | ⟨11, _⟩ => -(Real.sqrt 2) / 2
  | ⟨12, _⟩ => -(Real.sqrt 2) / 2
  | ⟨13, _⟩ => Real.sqrt 2 / 2
  | ⟨14, _⟩ => Real.sqrt 2 / 2

def nearDeficitKernel : Fin 10 → ℝ := nearCosKernel

What this page does not claim

The identity does not complete the flat Hessian assembly over all hinges. The identity does not prove S_RS_converges_EH_4d. The identity does not flip gap_action_recovery.

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/ReggeHinge4DStarKernel12.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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