Thermodynamics Forced Response Sign Blind Mobility

The sign-blind premise

In electrochemistry, a Tafel plot graphs the logarithm of the current against the overpotential, the extra voltage beyond the equilibrium value. The slope of that graph, the transfer coefficient, describes how easily a reaction proceeds in one direction versus the other. The classical Butler-Volmer equation writes the current as the difference of two exponentials, one for the forward reaction and one for the reverse, each with its own coefficient.

The framework's recognition cost, the price reality pays to register an event, is symmetric under reversing the ratio of the two rates. That symmetry forces the response to be an odd function of the drive: reversing the overpotential must exactly negate the current. The question is what this requires of the medium. The answer, proved in the machine-checked library of formal theorems, is that the medium may respond to the strength of the drive but not to its sign. A mobility that is any even function of the drive preserves the oddness, because an even function times an odd function is odd.

This distinction has a measurable consequence. Marcus theory predicts the transfer coefficient drifts with overpotential as alpha = 1/2 + A/(2*Lambda), where A is the dimensionless drive and Lambda the reorganization energy. The mobility that reproduces this drift is M(A) = 4 * exp(A^2/(4*Lambda)), which is even in A. The theorem marcus_drift_preserves_oddness proves this drift is fully consistent with reciprocity. Observed curvature in Tafel plots is therefore not evidence against the recognition response law; an earlier audit that read it as a wall on the law is retracted.

What reciprocity does forbid is an odd component in the mobility. If the response is written with independent anodic and cathodic coefficients, exp(a*A) - exp(-(b*A)), then oddness holds exactly when a = b. The theorem eq_of_twoCoefficientShape_odd proves this, and the theorem unequal_coefficients_exclude_sign_blind_mobility shows that unequal measured coefficients exclude every sign-blind mobility at every bridge scale. The laboratory prediction is that the two branches of a Tafel plot have slopes of equal magnitude at every overpotential.

A later correction within the framework, ForcedResponseDetailedBalanceNormalForm, sharpens this. The sign-blind framing is empty: for any odd response there exists an even mobility that reproduces it, so the content is oddness alone. And local detailed balance already forces a + b = 1, so the prediction a = b with constant coefficients collapses to alpha = 1/2, which is Marcus 1965. The real prediction needs drive-dependent coefficients, relating one direction at two drives rather than two directions at one drive.

/-- **A sign-blind mobility keeps the flux odd.** This is the premise the reciprocity argument
actually requires: the medium may respond to the strength of the drive, but not to its sign. -/
theorem generalFlux_odd_of_even_mobility
    {M : ℝ → ℝ} (hM : ∀ A : ℝ, M (-A) = M A) (k A : ℝ) :
    generalFlux M k (-A) = -generalFlux M k A := by
  rw [generalFlux, generalFlux, hM A]
  rw [show k * -A = -(k * A) by ring, Real.sinh_neg]
  ring

/-- The Marcus mobility is even in the drive. -/
theorem marcusMobility_even (Lambda A : ℝ) :
    marcusMobility Lambda (-A) = marcusMobility Lambda A := by
  simp only [marcusMobility, neg_sq]

/-- **The Marcus drift preserves oddness.** A transfer coefficient drifting with overpotential
exactly as Marcus theory says is fully consistent with reciprocity. Observed Tafel curvature is
therefore not evidence against the recognition response law, and the earlier audit conclusion
that it bounded the law above `0.3*Lambda` is retracted by this theorem. -/
theorem marcus_drift_preserves_oddness (Lambda k A : ℝ) :
    generalFlux (marcusMobility Lambda) k (-A) = -generalFlux (marcusMobility Lambda) k A :=
  generalFlux_odd_of_even_mobility (marcusMobility_even Lambda) k A

/-- **An odd response forces equal anodic and cathodic coefficients.** The observable content of
reciprocity: the two branches must have Tafel slopes of equal magnitude, at every drive. -/
theorem eq_of_twoCoefficientShape_odd
    {a b : ℝ} (ha : 0 < a) (hb : 0 < b)
    (hodd : ∀ A : ℝ, twoCoefficientShape a b (-A) = -twoCoefficientShape a b A) :
    a = b := by
  -- Oddness at drive one turns into equality of two hyperbolic cosines.
  have h1 := hodd 1
  simp only [twoCoefficientShape, mul_one, mul_neg, neg_neg] at h1
  have hcosh : cosh a = cosh b := by
    rw [Real.cosh_eq, Real.cosh_eq]
    linarith [h1]
  have habs : |a| = |b| := abs_eq_of_cosh_eq hcosh
  rwa [abs_of_pos ha, abs_of_pos hb] at habs

/-- Unequal measured coefficients exclude every sign-blind mobility at every bridge scale. A
system with genuinely asymmetric anodic and cathodic Tafel slopes is not one channel driven both
ways. -/
theorem unequal_coefficients_exclude_sign_blind_mobility
    {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (hne : a ≠ b) :
    ¬ ∃ (M : ℝ → ℝ) (k : ℝ), (∀ A : ℝ, M (-A) = M A) ∧
        ∀ A : ℝ, twoCoefficientShape a b A = generalFlux M k A := by
  rintro ⟨M, k, hM, h⟩
  exact hne (even_mobility_match_forces_equal_coefficients hM ha hb h)

What this page does not claim

The sign-blind framing is not claimed to be the strongest possible premise; a later correction shows it is equivalent to oddness alone. The prediction a = b with constant coefficients is not claimed to be physically distinct from the Butler-Volmer convention; local detailed balance collapses it to alpha = 1/2. No claim is made that any real electrode obeys sign-blind mobility; the theorems are about the logical consequences of the premise.

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

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