Cost Oscillatory Branch Audit

The oscillatory branch audit

The Recognition Composition Law is a functional equation that any recognition cost must satisfy. In log coordinates, the standard cost function becomes cosh t minus 1, where t is the log of a ratio. But the same equation admits another family: cos t minus 1. This oscillatory branch satisfies the law exactly, yet it fails the two physical side conditions that select the correct cost. The audit records this fact and its consequences.

In plain terms, the audit shows that the composition law alone does not pin down the cost function. The oscillatory branch passes the law, is normalized at exact balance, and is reciprocal-symmetric. But its second log-derivative at the origin is -1, not +1, so it fails the unit calibration condition. It also takes negative values for some positive ratios: at x = e^π, the cost is -2, which violates the requirement that recognition cost be nonnegative. These two failures are what reject the branch.

The audit corrects a claim in the literature. The earlier statement that cos is incompatible with the composition law is sharpened: cos is an RCL branch, but calibration and positivity reject it. This leaves the main cost uniqueness theorem unchanged, because that theorem includes the side conditions the oscillatory branch fails.

In Recognition Science, this matters because the cost function is the foundation of the framework. The audit closes a loophole: it proves that the uniqueness result does not depend on ignoring an alternative solution, but on the physical requirements that any real recognition cost must meet. The framework's library records this as a certificate, a machine-checked collection of formal theorems confirming both the branch's existence and its rejection.

/-- Therefore the oscillatory branch satisfies the RCL on positive ratios. -/
theorem oscillatory_satisfies_composition_law :
    SatisfiesCompositionLaw oscillatoryCost :=
  (composition_law_equiv_coshAdd oscillatoryCost).mpr
    oscillatory_cosh_add_identity

/-- Hence it fails the RS unit calibration. -/
theorem oscillatory_not_calibrated :
    ¬ IsCalibrated oscillatoryCost := by
  intro h
  have hneg := oscillatory_second_log_derivative
  rw [IsCalibrated] at h
  linarith

/-- The oscillatory branch is negative at one nonzero log-ratio. -/
theorem oscillatory_negative_at_exp_pi :
    oscillatoryCost (Real.exp Real.pi) = -2 := by
  simp [oscillatoryCost]
  norm_num

/-- Certificate inhabitant for the oscillatory-branch audit. -/
theorem oscillatory_branch_audit : OscillatoryBranchCert where
  satisfies_rcl := oscillatory_satisfies_composition_law
  normalized := oscillatory_normalized
  reciprocal := oscillatory_reciprocal
  second_log_derivative_eq_neg_one := oscillatory_second_log_derivative
  fails_calibration := oscillatory_not_calibrated
  fails_nonnegativity := oscillatory_not_nonnegative_on_positive

What this page does not claim

The composition law alone forces the unique cost function; the audit shows it does not. The oscillatory branch is a valid recognition cost; it is rejected by side conditions. The main cost uniqueness theorem is weakened by this audit; it remains unchanged.

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

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