Gravity Seven Gaps Ledger Energy Bridge Rectangle Shear Quadratic Energy Pos

Shear and the ledger

Shear is the deformation that changes a shape's angles without changing its area. Push the top of a rectangle sideways while the bottom stays put, and you get a parallelogram with the same base and height. In classical elasticity, this costs energy: the material resists the change in shape. The Recognition Science declaration rectangleShear_quadraticEnergy_pos proves a formal counterpart of that intuition inside a discrete, combinatorial model of a curved space.

The framework models a curved space as a collection of hinges, each with an area and a deficit angle, and defines a geometric energy as the sum over hinges of area times the square of the deficit. This is the discrete version of a quadratic curvature energy, the kind that appears in Regge calculus and in Isaacson's description of gravitational waves. The declaration proves that for the specific pure-shear pattern on a four-cell rectangle, this energy is strictly positive whenever the shear parameter h is nonzero. The proof is a theorem in the framework's machine-checked library of formal theorems, with no gaps and no framework-specific axioms.

The statement is deliberately narrow. It does not prove that the ledger-based energy, the one derived from a potential and a cost function, is positive; that is a separate declaration, rectangleShear_ledgerEnergy_pos. It does not prove that the geometric energy is positive for all deformations, only for this shear pattern. And it does not establish that this discrete energy matches any independently derived continuum gravity result; the bridge between the two sides of the framework, the ledger and the geometry, is still open on the tensor multichannel front.

What the declaration does establish is a consistency check. The framework had previously shown that a naive identification of ledger deficit with signed hinge deficit fails on two-sided deformations, because ledger deficits are even in the deformation parameter while signed deficits are odd. The corrected bridge uses the square of the deficit, and the shear pattern is exactly the two-sided class where the old ansatz was blind. The positivity theorem shows that the corrected energy sees this sector: pure shear carries nonzero geometric energy, as it should. The result is a small but load-bearing step in the framework's program of deriving gravity from recognition events.

/-- **THEOREM.** The geometric side sees the same shear: the quadratic
hinge energy of the rectangle shear data is strictly positive for h ≠ 0. -/
theorem rectangleShear_quadraticEnergy_pos (h : ℝ) (hh : h ≠ 0) :
    0 < quadraticCurvatureEnergy (strainHingeAreas (Fin 4))
      (strainHingeDeficits (rectangleShearPotential h)) := by
  refine quadraticCurvatureEnergy_pos
    (fun p => by unfold strainHingeAreas; norm_num)
    ((0 : Fin 4), (1 : Fin 4))
    (by unfold strainHingeAreas; norm_num) ?_
  show rectangleShearPotential h 0 - rectangleShearPotential h 1 ≠ 0
  rw [(rectangleShearPotential_strains h).1]
  exact hh

/-- **Discrete quadratic (Isaacson-type) curvature energy.** For hinge data
consisting of areas `A : H → ℝ` and deficit angles `δ : H → ℝ`, the energy
is Σ_h A_h · δ_h². This is the discrete form of the nonnegative
curvature-quadratic energy density; it is EVEN in δ, matching the parity
and sign of the ledger side. Purely geometric: no ledger objects. -/
noncomputable def quadraticCurvatureEnergy {H : Type*} [Fintype H]
    (A δ : H → ℝ) : ℝ :=
  ∑ h, A h * δ h ^ 2

/-- **THEOREM.** The geometric side sees the same shear: the quadratic
hinge energy of the rectangle shear data is strictly positive for h ≠ 0. -/
theorem rectangleShear_quadraticEnergy_pos (h : ℝ) (hh : h ≠ 0) :
    0 < quadraticCurvatureEnergy (strainHingeAreas (Fin 4))
      (strainHingeDeficits (rectangleShearPotential h)) := by
  refine quadraticCurvatureEnergy_pos
    (fun p => by unfold strainHingeAreas; norm_num)
    ((0 : Fin 4), (1 : Fin 4))
    (by unfold strainHingeAreas; norm_num) ?_
  show rectangleShearPotential h 0 - rectangleShearPotential h 1 ≠ 0
  rw [(rectangleShearPotential_strains h).1]
  exact hh

What this page does not claim

The declaration does not prove that the ledger-based energy is positive for the shear pattern. The declaration does not establish a match between the discrete energy and any independently derived continuum gravity result. The declaration does not prove positivity for all deformations, only for the pure-shear rectangle pattern.

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/SevenGaps/LedgerEnergyBridge.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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