Physics Black Body Peak From Phi Ladder

The phi ladder and the peak

Wien's displacement law is a classical result in thermal physics. It says the wavelength at which a black body radiates most intensely is inversely proportional to its temperature. A hotter object peaks at a shorter wavelength, which is why a heating coil glows red before it turns white. The constant of proportionality is measured: wavelength times temperature equals 2.898 millimeters times kelvin. For the Sun, with a surface temperature near 5778 kelvin, the peak lands in visible light. For the cosmic microwave background at 2.725 kelvin, the peak lands in microwaves. The ratio of those two peak wavelengths is the inverse ratio of the temperatures, about 4.72 times ten to the minus four.

Recognition Science proposes a different route to that ratio. The framework's central object is the ledger, a discrete record of recognition events whose cost is forced by a proved theorem. From that cost function, the framework derives the golden ratio phi, about 1.618, as the unique self-similar scaling. The phi ladder is the sequence of powers of phi: phi squared, phi cubed, and so on. The research note in the module suggests that phi to the twentieth power, divided by ten thousand, lands near the Sun-to-CMB peak ratio. Phi to the twentieth is about 6765, so the expression gives roughly 0.68, which the note calls close to 4.72 times ten to the minus four times a scale factor. That scale factor is not defined in the module.

What the module itself establishes is narrower and fully explicit. It defines a cost function on the ratio of a measured wavelength to an expected one. The machine-checked library of formal theorems establishes three facts about that function. First, the cost is zero when the measured and expected values are equal. Second, the cost is never negative for positive inputs. Third, a threshold built from phi minus three halves is positive. These three facts are assembled into a certificate structure, and the library establishes that such a certificate exists. The module does not establish that the phi ladder produces the black body peak. The docstring says plainly that the paragraph about Wien's law is a research note recording where the idea was meant to go, not a result.

The plain-language takeaway is this. The framework has a working cost function with sensible properties: it rewards agreement and penalizes disagreement. The phi ladder is a candidate structure for connecting that cost to real spectra. The formal proof stops at the cost function's general properties. Connecting those properties to a specific physical prediction, like the Sun's peak wavelength, remains a target. The module is an honest scaffold: it certifies the tools, not the application.

structure WienPeakCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

theorem domainCost_at_equilibrium (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 phi ladder predicts the measured black body peak wavelength. The module derives Wien's displacement law from Recognition Science principles. The number 0.68 is a measured or derived value for the Sun-to-CMB peak ratio.

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

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