Information Fepbridge From Jcost Jcost Kl Same Second Order At Equilibrium

The local contact

The free energy principle (FEP) describes how a system persists by minimizing a quantity called variational free energy, often written as a Kullback-Leibler (KL) divergence, a measure of how one probability distribution differs from another. Recognition Science (RS) uses a different object, the reciprocal cost J(x) = (x + x⁻¹)/2 - 1, which measures the price of a recognition event. The theorem jcost_kl_same_second_order_at_equilibrium proves that, in a specific logarithmic coordinate, these two apparently unrelated quantities touch at equilibrium: they have the same value, the same first derivative, and the same second derivative at that point.

The bridge is built in log-ratio coordinates, where the recognition variable x is written as exp(u). In these coordinates, the reciprocal cost becomes exactly J(exp u) = cosh u - 1, a standard hyperbolic function. The KL divergence, when approximated locally, is represented by the quadratic klQuadratic(u) = u²/2. The theorem's content is that at u = 0, which corresponds to x = 1, the equilibrium point where the cost is zero, the two functions and their first two derivatives coincide. The value is 0, the first derivative is 0, and the second derivative, which measures curvature and is related to Fisher information, is 1 for both.

This is a statement about local geometry, not global equivalence. The theorem does not claim that the reciprocal cost and the KL divergence are the same function, nor that one can replace the other in all contexts. It establishes a precise, machine-checked contact of order two at a single point. This contact means that for small fluctuations around equilibrium, the two frameworks make the same predictions to second order, a fact that could be useful for translating results between FEP and RS in that regime.

In Recognition Science, this result is part of a larger effort to compare its framework with established theory. The library also defines a Markov blanket sparsity condition, a structural requirement that internal and external states do not couple directly, and proves that this condition is definitionally equivalent to a similar condition in FEP. However, the harder task of deriving this sparsity from the dynamics of the recognition cost, rather than assuming it, remains open. The local contact theorem is the theorem-grade part of the bridge; the derivation of the blanket is a target for future work.

/-- RS reciprocal cost and the KL quadratic have the same Fisher curvature
at equilibrium.  This is the exact local crossover with FEP-style free energy.
-/
theorem jcost_kl_same_second_order_at_equilibrium :
    Jlog 0 = klQuadratic 0 ∧
    deriv Jlog 0 = deriv klQuadratic 0 ∧
    deriv (deriv Jlog) 0 = deriv (deriv klQuadratic) 0 := by
  constructor
  · simp [Jlog_zero]
  constructor
  · simp
  · have hJ := hasDerivAt_deriv_Jlog_zero.deriv
    have hK := hasDerivAt_deriv_klQuadratic_zero.deriv
    rw [hJ, hK]

/-- In log coordinates, reciprocal J-cost is exactly `cosh u - 1`. -/
theorem jcost_log_exact (u : ℝ) :
    Jlog u = Real.cosh u - 1 :=
  Jlog_as_cosh u

/-- RS reciprocal cost and the KL quadratic have the same Fisher curvature
at equilibrium.  This is the exact local crossover with FEP-style free energy.
-/
theorem jcost_kl_same_second_order_at_equilibrium :
    Jlog 0 = klQuadratic 0 ∧
    deriv Jlog 0 = deriv klQuadratic 0 ∧
    deriv (deriv Jlog) 0 = deriv (deriv klQuadratic) 0 := by
  constructor
  · simp [Jlog_zero]
  constructor
  · simp
  · have hJ := hasDerivAt_deriv_Jlog_zero.deriv
    have hK := hasDerivAt_deriv_klQuadratic_zero.deriv
    rw [hJ, hK]

What this page does not claim

The theorem does not claim that the reciprocal cost and KL divergence are globally equivalent functions. It does not derive Markov blankets or Bayesian filtering from the recognition cost ledger. It does not claim that the free energy principle and Recognition Science are the same framework.

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

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