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theorem proved term proof high

bridge_chain_complete

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The theorem asserts that on any weighted ledger graph the linearized Regge deficit equals the J-cost Dirichlet energy scaled by the Einstein coupling, with exact identity and flat-vacuum vanishing. Recognition Science derivations of general relativity and discrete gravity models cite it as the unconditional bridge from the functional equation to the continuum Einstein equations. The proof is a term-mode composition of the cubic linearization discharge lemma with the field-curvature identity and the pre-proved coupling constants.

claimLet $n$ be a natural number, $a>0$ a real, and $G$ a weighted ledger graph on $n$ sites. The Regge deficit linearization hypothesis holds for the cubic deficit functional at scale $a$ with the cubic hinges of $G$. For every log-potential $ε$, the Laplacian action of $G$ on $ε$ equals $(1/κ_E)$ times the Regge sum of the cubic deficit evaluated on the conformal edge-length field generated by $a$ and $ε$. Both sides are zero when $ε$ is identically zero. The Einstein coupling satisfies $κ_E=8φ^5>0$.

background

The module completes Phase D of the program to render the paper's Theorem 5.1 (field-curvature identity) as an unconditional Lean theorem. It composes the cubic linearization discharge (Phase A) with the identification of the bridge normalization to the Einstein coupling $κ_E=8φ^5$ (Phase B) and the upstream quadratic-remainder bound $J_log_quadratic_approx$. The Laplacian action is the Dirichlet energy built from the J-cost functional; the Regge sum is the linearized deficit on the conformal edge-length field. The constants module supplies $κ_E=8φ^5$ via the RS-native derivation $G=λ_rec^2 c^3/(π ħ)$ with $ħ=φ^{-5}$.

proof idea

The proof is a term-mode one-line wrapper. It applies cubic_linearization_discharge to discharge the linearization hypothesis, then invokes field_curvature_identity_einstein for the exact scaling identity, and pairs laplacian_action_flat with flat_regge_sum_zero for the vacuum case. The two constant lemmas kappa_einstein_eq and kappa_einstein_pos are inserted directly.

why it matters in Recognition Science

This declaration supplies the single artifact (ContinuumFieldCurvatureCert) that downstream work cites when invoking the field-curvature identity with the derived Einstein coupling. It closes the chain from the Recognition Composition Law through J-uniqueness and the forced value of φ to the continuum limit of the Regge action. The result is unconditional and contains no new axioms or hidden hypotheses.

scope and limits

formal statement (Lean)

 116theorem bridge_chain_complete {n : ℕ} (a : ℝ) (ha : 0 < a)
 117    (G : WeightedLedgerGraph n) :
 118    -- Discharge: `ReggeDeficitLinearizationHypothesis` holds.
 119    ReggeDeficitLinearizationHypothesis
 120      (cubicDeficitFunctional n) a ha (cubicHinges G) G ∧
 121    -- Identity: J-cost Dirichlet energy = (1/κ_E) · Regge sum.
 122    (∀ ε : LogPotential n,
 123      laplacian_action G ε
 124      = (1 / Constants.kappa_einstein) *
 125          regge_sum (cubicDeficitFunctional n)
 126            (conformal_edge_length_field a ha ε) (cubicHinges G)) ∧
 127    -- Flat vacuum: both sides zero.
 128    (laplacian_action G (fun _ => (0 : ℝ)) = 0 ∧
 129      regge_sum (cubicDeficitFunctional n)
 130        (conformal_edge_length_field a ha (fun _ => (0 : ℝ))) (cubicHinges G) = 0) ∧
 131    -- Coupling value: κ_Einstein = 8 φ⁵.
 132    Constants.kappa_einstein = 8 * Constants.phi ^ (5 : ℝ) ∧
 133    -- Coupling positivity: κ_Einstein > 0.
 134    0 < Constants.kappa_einstein := by

proof body

Term-mode proof.

 135  refine ⟨cubic_linearization_discharge a ha G, ?_, ?_, Constants.kappa_einstein_eq,
 136          Constants.kappa_einstein_pos⟩
 137  · intro ε
 138    exact field_curvature_identity_einstein a ha G ε
 139  · refine ⟨laplacian_action_flat G, flat_regge_sum_zero a ha G⟩
 140
 141/-! ## §3. Master certificate -/
 142
 143/-- **CONTINUUM FIELD-CURVATURE CERTIFICATE.**
 144
 145    The single artifact to cite when invoking the field-curvature
 146    identity with the Einstein coupling. Combines:
 147
 148    - discharge of the linearization hypothesis,
 149    - exact identity `∑ψ/∑κ = 1/κ_Einstein · ∑regge`,
 150    - flat-vacuum consistency,
 151    - coupling value κ_Einstein = 8 φ⁵,
 152    - coupling positivity.
 153-/

depends on (33)

Lean names referenced from this declaration's body.

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