Foundation Three Substrate Validation Cert Shared Symmetry

Shared symmetry

In mathematics, a function is symmetric under reciprocals when it gives the same output for a number and its reciprocal: f(r) = f(1/r). The Recognition Science framework's cost function, which measures the price of a recognition event, obeys exactly this rule. The declaration shared_symmetry proves that for any positive real number r, the cost at r equals the cost at its reciprocal. This is a theorem in the framework's machine-checked library of formal theorems, meaning it follows from the axioms with no gaps.

The symmetry is not an isolated curiosity. It is one of three properties that the framework's certificate bundles together: a fixed point at r = 1 where the cost is zero, a descent property ensuring costs are positive away from that fixed point, and this reciprocal symmetry. The certificate, named ThreeSubstrateCert, assembles all three into a single structure. The framework's library proves each property holds for the cost function, and the certificate records that proof.

The certificate's name refers to three experimental substrates: language models, photonic qubits, and magnetized plasma. The framework's documentation reports that all three showed behavior consistent with these properties in experiments during 2026. The language model experiments showed the cost outperforming cross-entropy in 96.4% of layers, the photonic experiments achieved a code rate of 7/8 with 0.02% leakage, and the plasma experiment converged to a value near 1. These are empirical results, not formal theorems.

What shared_symmetry does not claim is equally important. It does not claim that the experiments themselves are formally verified. The certificate is explicitly graded as a hypothesis, meaning the experimental results are reported but not machine-checked. The theorem only establishes the mathematical property of the cost function, not that any particular physical system implements it. The symmetry holds for the abstract function; whether real substrates obey it is a separate, empirical question.

/-- All three validate J-cost symmetry: J(r) = J(1/r). -/
theorem shared_symmetry {r : ℝ} (hr : 0 < r) :
    Jcost r = Jcost r⁻¹ := Jcost_symm hr

def threeSubstrateCert : ThreeSubstrateCert where
  three_substrates := validationSubstrateCount
  fixed_point := shared_fixed_point
  descent := shared_descent
  symmetry := shared_symmetry
  lm_alignment := lm_fraction_eq
  photonic_rate := photonic_code_rate_rfl
  f2_cube_connection := seven_eighths_from_F2_cube

structure ThreeSubstrateCert where
  three_substrates : Fintype.card ValidationSubstrate = 3
  fixed_point : Jcost 1 = 0
  descent : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
  symmetry : ∀ {r : ℝ}, 0 < r → Jcost r = Jcost r⁻¹
  lm_alignment : languageModelAlignmentFraction = 7/8
  photonic_rate : photonicCodeRate = 7 / 8
  f2_cube_connection : languageModelAlignmentFraction = (2^3 - 1 : ℚ) / 2^3

What this page does not claim

The experimental results are formally verified theorems. The symmetry theorem implies any specific physical system implements the cost function. The certificate proves the cost function is unique among all possible cost functions.

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/Foundation/ThreeSubstrateValidationCert.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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