Materials Electrical Resistance3 From Jcost

Resistivity and the cost function

Electrical resistivity measures how strongly a material opposes the flow of electric current. Metals like copper have very low resistivity, around 1e-8 ohm·m, while insulators like quartz can reach 1e16 ohm·m, a span of about 24 orders of magnitude. The framework's research note observes that this range is roughly phi^46, where phi is the golden ratio, and that phi^8 per decade would compress the range to about six decades. That observation is a structural sketch, not a proved statement.

The module cost, a measure of recognition effort, defines a function on two real numbers m and e as J(m/e), where J(x) = (x + 1/x)/2 - 1. The machine-checked library of formal theorems proves three general facts about this function: it vanishes when m equals e, it is nonnegative when both inputs are positive, and the quantity phi - 3/2 is positive. These facts are proved for any positive real inputs, with no reference to resistivity or any material property.

In Recognition Science, the framework models this as a ledger, a discrete record of recognition events, where the cost of recognition is forced by the function J. The module defines a certificate structure that bundles the three proved facts, and proves that such a certificate exists. The plain-language meaning is that the cost function behaves sensibly: it is zero at equality, never negative, and has a positive threshold related to the golden ratio.

The module proves nothing specific to electrical resistance. The definition of domainCost as J(m/e) does not define what m and e mean for a material; the docstring states this explicitly. The resistivity ladder idea, that material resistivities fall on a phi-power ladder, remains a research note recording where the idea was meant to go, not a result. What would make this a theorem about resistivity is a definition of m and e in the subject's own terms, which the module does not provide.

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The phi-power ladder for resistivity is not a proved theorem. The module does not identify which material property m or e represents. The cost function's behavior is not specific to electrical resistance.

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/Electrical_Resistance3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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