wien_zero_cost
The theorem establishes that J-cost vanishes at the unit ratio, anchoring the zero-cost condition for Wien's displacement law in the Recognition Science derivation of black-body radiation. Researchers verifying thermal spectra from recognition cost would cite it to confirm matched configurations incur no penalty. The proof reduces directly to the unit lemma for Jcost via a one-line term application.
claim$J_ {cost}(1) = 0$, where $J_{cost}(x) = (x-1)^2/(2x)$ for positive ratios $x$.
background
Jcost is the cost function on positive ratios given by the squared expression $J(x) = (x-1)^2/(2x)$, which is nonnegative and zero only at $x=1$. This module derives the three canonical black-body laws as zero J-cost statements on thermal ratios, with the local setting being a structural theorem (zero sorry, zero axiom) that bundles Wien, Stefan-Boltzmann, and Planck results from the Recognition Composition Law. It depends on the upstream lemma Jcost_unit0, which states Jcost(1) = 0 by direct simplification of the squared-ratio form.
proof idea
The proof is a one-line term wrapper that applies the lemma Jcost_unit0 from the Cost module. That lemma itself reduces by simp on the definition of Jcost.
why it matters in Recognition Science
This zero-cost result supplies the wien_zero field of the master certificate blackBodyRadiationDeepCert, which bundles the three black-body statements. It fills the Wien component of the structural theorem for black-body radiation from J-cost, aligning with the Recognition Science forcing chain where matched configurations have vanishing cost at the self-similar fixed point. It closes the anchor for the displacement law without touching open questions.
scope and limits
- Does not derive the numerical value of the Wien constant.
- Does not prove the full displacement law relating wavelength and temperature.
- Does not establish positivity of cost away from the matched ratio.
- Does not incorporate physical units or measured constants.
formal statement (Lean)
36theorem wien_zero_cost : Jcost 1 = 0 := Jcost_unit0
proof body
Term-mode proof.
37
38/-- Stefan-Boltzmann: matched configuration has zero cost. -/