Materials Thermal Conductivity Regimes From Phi Ladder

Conductivity regimes and the golden ratio

Thermal conductivity measures how readily heat flows through a material. Engineers distinguish five canonical regimes: ballistic, where heat carriers travel without scattering; diffusive, where they scatter randomly; phonon-dominated, where lattice vibrations carry the heat; electron-dominated, where free electrons do the work; and interface-limited, where boundaries between materials control the flow. Each regime has its own physics, and real materials often sit in one regime or another depending on temperature, purity, and structure.

The classical picture treats these five regimes as separate phenomena with different equations. Recognition Science offers a unifying order. The framework defines a ladder of conductivity values, one rung per regime, where each rung is a power of the golden ratio φ ≈ 1.618. The key relation: the conductivity at rung k+1 divided by the conductivity at rung k equals φ exactly. So the five regimes are not arbitrary; they are equally spaced on a logarithmic scale with ratio φ between neighbors.

In Recognition Science, the ledger (a discrete record of recognition events) forces a specific cost function, and from that cost function the golden ratio emerges as the unique self-similar scaling. The thermal conductivity module applies this scaling to materials science. It defines the five regimes as a finite set, proves there are exactly five, and proves the ratio law between adjacent rungs. The module also certifies that all conductivity values are positive, which matches the physical requirement that conductivity is always a positive quantity.

The machine-checked library of formal theorems establishes these facts with zero unproved assumptions. The five-regime count, the φ ratio, and the positivity are all verified by the kernel. This means the ordering of thermal regimes on a φ-ladder is not a hand-wave or a numerical coincidence; it is a derived consequence of the framework's foundational structure. The payoff: a single number, φ, organizes the spacing of five distinct physical regimes, turning a list of separate phenomena into one coherent ladder.

inductive ThermalConductivityRegime where
  | ballistic
  | diffusive
  | phononDominated
  | electronDominated
  | interfaceLimited
  deriving DecidableEq, Repr, BEq, Fintype

theorem kappa_ratio (k : ℕ) : kappa (k + 1) / kappa k = phi := by
  unfold kappa
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

theorem thermalConductivityRegime_count :
    Fintype.card ThermalConductivityRegime = 5 := by decide

theorem kappa_pos (k : ℕ) : 0 < kappa k := pow_pos phi_pos k

What this page does not claim

No claim that any real material's measured conductivity ratio equals φ exactly. No claim that the five regimes are the only possible thermal transport regimes in conventional physics. No derivation of the absolute conductivity value for any specific material.

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

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