blackBodyRadiationDeepCert
This definition constructs the master certificate BlackBodyRadiationDeepCert that packages the zero J-cost conditions for Wien's law and the Stefan-Boltzmann law with the positivity of J-cost for all positive ratios away from unity. Researchers deriving thermodynamic radiation laws from the Recognition Science J-cost framework cite it as the central bundling structure. The construction is a direct record definition that populates the three fields from the prior zero-cost theorems and the off-match positivity result.
claimThe black-body radiation deep certificate is the structure whose Wien field asserts $J(1)=0$, whose Stefan-Boltzmann field asserts $J(1)=0$, and whose off-match field asserts $∀x>0, x≠1 → J(x)>0$, where $J$ denotes the J-cost function on positive reals.
background
In the Recognition Science setting, J-cost quantifies deviation from the self-similar fixed point on thermal ratios, with zero cost precisely at the matched configuration. The module establishes that Wien's displacement law, the Stefan-Boltzmann law, and the Planck distribution each correspond to a J-cost reading that vanishes or stays positive exactly when the ratio equals one. The structure BlackBodyRadiationDeepCert collects these three properties into a single certificate. Upstream results supply the individual facts: wien_zero_cost and stefan_boltzmann_zero_cost each reduce to Jcost_unit0, while off_match_positive applies Jcost_pos_of_ne_one to any positive ratio not equal to one.
proof idea
The definition is a direct record constructor. It assigns the wien_zero field to wien_zero_cost, the sb_zero field to stefan_boltzmann_zero_cost, and the off_match field to off_match_positive. No further tactics or reductions are required.
why it matters in Recognition Science
This certificate is the master bundling object for the three black-body laws derived from J-cost. It is consumed by the downstream theorem blackBodyRadiationDeepCert_inhabited to witness that the certificate type is nonempty. Within the framework it confirms that the canonical radiation laws arise as zero-cost configurations under the Recognition Composition Law, completing the structural theorem for thermal radiation.
scope and limits
- Does not derive explicit numerical values for the Wien or Stefan-Boltzmann constants.
- Does not recover the full Planck spectral form beyond the positivity condition.
- Does not link to the forcing chain T0-T8 or the emergence of three spatial dimensions.
- Does not evaluate J-cost at concrete temperatures or frequencies.
Lean usage
theorem example_inhabited : Nonempty BlackBodyRadiationDeepCert := ⟨blackBodyRadiationDeepCert⟩formal statement (Lean)
50noncomputable def blackBodyRadiationDeepCert : BlackBodyRadiationDeepCert where
51 wien_zero := wien_zero_cost
proof body
Definition body.
52 sb_zero := stefan_boltzmann_zero_cost
53 off_match := off_match_positive
54