Materials Ion Channel Conductance From Jcost

What the module proves

Ion channels are protein pores in cell membranes that let specific ions cross. Their conductance, how easily ions pass, is measured in picosiemens. Voltage-gated sodium channels sit near 20 pS, potassium channels near 10 pS, so the sodium-to-potassium ratio is about 2, close to the golden ratio raised to the 1.4 power. Recognition Science models this ratio as a difference in recognition cost, the price a system pays to tell two states apart.

The domainCost function, a cost assigned to a ratio of two quantities, is defined as Jcost (m / e), where m and e are positive real numbers. The machine-checked library proves three facts about this function. First, the cost is zero when the two quantities are equal. Second, the cost is never negative for positive inputs. Third, the number phi minus 3/2, called the canonical threshold, is positive.

These three facts are general properties of the cost function. They hold for any positive m and e, not just for ion channels. The function does not define what m and e mean for a specific channel. That step is left open. What would make this a theorem about ion channels is a definition of m and e in the channel's own terms, such as a conductance value or an ion concentration.

In Recognition Science, the framework models conductance differences between channels as scaling in phi-rungs, steps on a ladder built from the golden ratio. The ratio of about 2 between sodium and potassium conductance sits near phi to the 1.4, one rung on that ladder. The function's role is to certify the cost function's basic behavior so that later work can attach physical meaning to m and e without rechecking the foundation.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The function does not prove that sodium conductance is 20 pS or that potassium is 10 pS. The function does not define what m and e mean for an ion channel. The phi-rung scaling for conductance differences is a research note, not a proved theorem.

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

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