IndisputableMonolith.Information.AlgorithmicProb3FromJCost
Defines domain cost from the RS J-functional, a positive canonical threshold, and an inhabited AlgorithmicProb3 certificate package. Information theorists connecting Solomonoff-style priors to the unique RS cost cite this module. The file is mostly definitions plus nonnegativity and positivity lemmas over Constants and Cost.
claimFrom the RS cost $J$, the module introduces a domain cost $C$, proves $C \ge 0$, fixes a canonical threshold $\theta > 0$, and packages an inhabited certificate that algorithmic probability (level 3) is recovered from $J$-cost.
background
Recognition Science forces a unique nonnegative cost $J(x) = (x + x^{-1})/2 - 1$ (T5). The Cost import supplies that functional; Constants supplies the RS-native tick. Algorithmic probability (Solomonoff-style) assigns mass by shortest programs; here the program length proxy is rebuilt from $J$ rather than from an arbitrary universal machine.
The module works on abstract domains: a domain cost is the $J$-derived penalty attached to a domain element, with an equality lemma at distinguished points and a nonnegativity proof. A single positive canonical threshold then marks the cutoff used by the certificate.
Local setting is the Information domain of the monolith: convert the forced cost into a probability-facing certificate without reopening uniqueness of $J$.
proof idea
Definition-led module, not a deep theorem chain. domainCost is introduced from $J$; domainCost_at_eq and domainCost_nonneg discharge the basic algebraic and sign obligations. canonicalThreshold is a closed-form positive constant; canonicalThreshold_pos is the positivity lemma. AlgorithmicProb3Cert bundles the data; cert and cert_inhabited supply a concrete inhabitant so downstream code can assume the certificate without constructing it.
why it matters in Recognition Science
Places algorithmic probability on the same $J$-rail as the rest of RS (T5 J-uniqueness, RCL). No downstream edges are recorded yet; the inhabited certificate is the export surface for later Information or complexity results that need a $J$-native prior or threshold. Closes a scaffolding gap between pure Cost and probability-facing claims without touching mass formulas, alpha, or the forcing chain beyond the cost import.
scope and limits
- Does not prove uniqueness of algorithmic probability independent of $J$.
- Does not construct an explicit universal Turing machine or Kolmogorov complexity.
- Does not derive particle masses, alpha, or forcing steps T6–T8.
- Does not assert numerical bounds beyond threshold positivity and cost nonnegativity.
- Does not feed any recorded downstream theorem yet.