IndisputableMonolith.Information.Channel_Capacity_Qubit_RS
Recognition Science packaging of qubit channel capacity via a domain cost and a positive canonical threshold. Defines a certificate bundle that records nonnegativity of the cost and positivity of the threshold in RS-native units. Information theorists working in the RS ledger would cite the certificate when wiring qubit links into larger capacity or coding arguments. The module is mostly definitions plus elementary positivity lemmas, not a deep coding theorem.
claimThe module introduces a domain cost $C$ on the qubit channel setting, proves $C \ge 0$ and evaluates it at equality cases, and fixes a canonical threshold $\theta > 0$. It packages these into a qubit channel certificate asserting the cost and threshold facts needed for RS capacity bookkeeping.
background
Recognition Science measures recognition effort with the $J$-cost from the Cost layer (the unique symmetric cost forced by the Recognition Composition Law). Channel capacity statements in this tree are therefore stated as cost and threshold facts rather than as Shannon mutual-information maximizations alone.
The Constants import supplies the RS time quantum $\tau_0 = 1$ tick, so rates and thresholds sit in RS-native units. The module name marks the qubit specialization: a two-level carrier whose admissible recognition cost and cutoff are recorded once and reused.
Sibling definitions include the domain cost, its value at distinguished points, nonnegativity, a canonical positive threshold, and an inhabited certificate type that bundles those facts for downstream capacity or coding lemmas.
proof idea
Definition-first module. Domain cost and canonical threshold are introduced as defs; nonnegativity and positivity are short lemmas (likely unfolding the Cost primitives and Constants). The certificate is a structure packing those propositions, with an inhabitation proof that assembles the elementary facts. No asymptotic coding or random-coding argument appears here.
why it matters in Recognition Science
Places qubit channel capacity inside the RS information stack so later ledger or network results can cite a single certificate rather than re-prove cost nonnegativity and threshold positivity. Imports only Constants and Cost, so it sits early: it does not yet close a full capacity theorem, but it freezes the qubit bookkeeping objects those theorems need. No downstream edges are recorded in the mirror graph yet; the natural parents are higher Information or network-capacity developments that consume QubitChannelCert.
scope and limits
- Does not prove the classical or quantum Shannon capacity formula.
- Does not derive multi-letter or asymptotic coding theorems.
- Does not treat qudit or continuous-variable channels.
- Does not connect the threshold to alpha, mass rungs, or T0–T8 forcing.
- Does not claim operational achievability beyond the packaged inequalities.