IndisputableMonolith.Information.Holevo_Bound_RS
Module packaging a Recognition-Science form of the Holevo bound: classical extractable information is controlled by a nonnegative domain cost built from the RS cost functional, with a positive canonical threshold and a certificate type. Information theorists working in the RS ledger would cite HolevoBoundRS and the domainCost lemmas. The file is mostly definitions plus elementary nonnegativity and positivity facts, not a deep inequality proof.
claimDefine a domain cost $C$ on the RS cost side, prove $C\ge 0$ and an evaluation identity, introduce a positive canonical threshold $\theta>0$, and package a Holevo-type bound object $\mathrm{HolevoBound}_{\mathrm{RS}}$ together with an inhabited certificate type asserting the bound holds in RS-native units.
background
Recognition Science measures mismatch with the cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced uniquely in the T5 step of the unified forcing chain and obeying the Recognition Composition Law. The Cost import supplies that functional; Constants supplies the RS time quantum $\tau_0=1$ tick against which rates are normalized.
Classical Holevo theory bounds accessible classical information by a quantum entropy functional. This module transplants that idea into the RS ledger: a domain cost aggregates $J$-type mismatch over a communication or measurement domain, and a canonical threshold marks the scale at which extractable information is forced below that cost. Sibling names indicate the cost is nonnegative and has a stable evaluation identity at equality cases.
The setting is the Information domain of the monolith: discrete tick structure, $\phi$-native constants, and cost-first accounting rather than Hilbert-space entropy as the primitive.
proof idea
Definition-heavy module. domainCost and canonicalThreshold are introduced as defs; domainCost_nonneg and canonicalThreshold_pos are short positivity/nonnegativity arguments from the Cost layer. domainCost_at_eq records an evaluation identity. HolevoBoundRS packages the bound statement; cert and cert_inhabited supply a Prop-level certificate and a trivial inhabitant so downstream code can assume the bound without re-proving local algebra. No multi-step analytic inequality is developed here beyond those elementary facts.
why it matters in Recognition Science
Places a Holevo-style capacity ceiling inside RS bookkeeping so later information and measurement results can quote a single named bound rather than rebuild cost estimates ad hoc. Feeds the Information domain's certificate pattern (cert / cert_inhabited) used when classical readout is limited by recognition cost. Ties to framework landmarks only indirectly: $J$-cost uniqueness (T5) and tick normalization from Constants; it does not itself force $\phi$, the eight-tick octave, or $D=3$. Downstream use edges are empty in the graph snapshot, so this file is presently a leaf supplier for future capacity or measurement theorems rather than a parent of existing proved results.
scope and limits
- Does not prove the classical Holevo theorem in full quantum Shannon form.
- Does not derive numerical alpha, mass, or G values from the bound.
- Does not establish channel capacity achievability or coding theorems.
- Does not force dimension, phi, or eight-tick structure inside this file.
- Does not supply a sorry-free analytic inequality beyond nonnegativity and threshold positivity.