Materials Memristor Switching From Jcost

Memristor switching and the cost threshold

A memristor is a two-terminal electrical component whose resistance is not fixed but remembers the history of voltage applied to it. The name blends "memory" and "resistor." In practice, a memristor can be switched between a high-resistance state and a low-resistance state by applying a voltage above a certain threshold. Below that threshold, the state holds; above it, the device changes. This is the basis for non-volatile memory and neuromorphic computing, where the device's state mimics a synapse's weight.

The classical model of this switching behavior involves a forming step: an initial, higher voltage V_forming that activates the device, and a lower operating threshold V_th at which it subsequently switches. The relationship proposed in the research note is V_th ≈ J(φ) × V_forming, where J is a specific cost function and φ is the golden ratio. This would mean the switching threshold sits at a fixed fraction of the forming voltage, about 11.8 percent. The number 0.118 comes from evaluating the cost function at the golden ratio and subtracting 1.5, which is the quantity the library calls the canonical threshold.

In Recognition Science, the cost function J(x) = (x + 1/x)/2 − 1 is not chosen freely. A proved theorem in the framework's machine-checked library of formal theorems shows that any cost function satisfying five plain conditions must equal this form. The conditions are reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity. The golden ratio φ arises as the unique self-similar scaling of the framework, and the number φ − 3/2 is positive, a fact the library also proves.

What the module materials memristor switching from jcost actually establishes is narrower than the research note suggests. The Lean file defines a domain cost as J(m/e), where m and e are real numbers, and proves three general facts about this cost: it vanishes when m equals e, it is nonnegative for positive inputs, and the canonical threshold φ − 3/2 is positive. These are true for any positive real numbers m and e. The file proves nothing specific to memristors, because the definition of domainCost does not reference voltage, resistance, or any physical quantity of the device.

The gap is the missing definition of m and e in the memristor's own terms. If m were the switching voltage and e the forming voltage, then the vanishing at m = e would say that no switching occurs when the two are equal, and nonnegativity would say the cost never goes negative. But the library does not supply those identifications. The research note records where the idea was meant to go, not a result about the device. The formal content is a general statement about a cost function, shared verbatim with 2383 sibling modules, and the physical bridge to memristor switching remains open.

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

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 module does not prove that any physical memristor switches at 11.8 percent of its forming voltage. The module does not derive the forming voltage or any other memristor parameter from first principles. The research note's voltage relationship is a hypothesis, not a theorem, and no falsifier is named in the pack.

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

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