domainCost
plain-language theorem explainer
Domain cost assigns to a message size m and encoding resource e the recognition cost of their ratio: J(m/e). Compression and coding arguments in the RS information layer cite it as the excess cost of mismatching payload to budget. The definition is a one-line specialization of the unique J-cost functional to that ratio.
Claim. For real $m$ and $e$, the domain cost is $J(m/e)$, where $J(x)=\frac{x+x^{-1}}{2}-1$ is the recognition cost of a positive ratio.
background
The ambient module develops a structural theorem on data-compression rate from J-cost: an RS-optimal code pays the Shannon entropy H plus a fixed J-overhead per symbol (illustrated at H=1 by average length 1+J(φ)≈1.118 bits/symbol).
The cost functional is the unique J forced by the Recognition Composition Law, J(x)=(x+x^{-1})/2-1 (equivalently cosh(log x)-1). Upstream copies of Jcost in Cost, Cosmology, Gravity, and Spiral all use this same formula; a genuine distinction (ratio ≠ 1) has strictly positive cost, and J is nonnegative on positives.
Domain cost simply evaluates that functional on the dimensionless ratio of message length to encoding budget, so mismatch of m to e is scored exactly as a recognition defect.
proof idea
Pure definitional abbreviation: domainCost m e is defined to be Jcost (m/e). No lemmas or tactics; the body is the single application of the imported J-cost functional to the ratio.
why it matters
Gives the information module a named handle for payload-versus-budget mismatch measured in the same units as the rest of Recognition Science. It sits under the module's compression-rate story (H plus J-overhead) and inherits uniqueness of J from the forcing chain (T5) and the Recognition Composition Law. Sibling facts (evaluation identity, nonnegativity) and the canonical threshold / DataCompr3DeepCert certificate build on this def; no external used_by edges are recorded yet. Landmark tie-in is the unique cost J, not the eight-tick or D=3 steps.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.