IndisputableMonolith.Information.Bandwidth_Phi_RS
Module packaging φ-scaled bandwidth thresholds for Recognition Science information channels. It defines a domain cost from the J-cost, a canonical positive threshold fixed by φ, and a small certificate type that the φ-bandwidth relation holds. Workers in the RS information layer cite the certificate and the nonnegativity facts. Content is mostly definitions plus elementary positivity and evaluation lemmas.
claimIntroduce a domain cost $C$ built from the Recognition cost $J$, a canonical bandwidth threshold $\theta(\varphi)>0$, and a certificate asserting the $\varphi$-bandwidth relation for RS information channels (with $\tau_0=1$ tick in RS-native units).
background
Recognition Science measures mismatch with the unique cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced by the Recognition Composition Law and the T5 uniqueness step. The golden ratio $\varphi$ is the self-similar fixed point (T6). Time is counted in the fundamental tick $\tau_0=1$.
This module sits in the Information domain and imports the Constants and Cost layers only. It specializes those primitives to bandwidth: a domain-level cost assembled from $J$, and a single canonical threshold expressed in $\varphi$ that marks admissible channel width in RS units.
Sibling declarations cover evaluation of the domain cost, its nonnegativity, positivity of the threshold, and an inhabited certificate type packaging the φ-bandwidth claim.
proof idea
Definition-first module, not a deep theorem stack. Domain cost is defined from $J$; a one-line evaluation lemma records its value at a point; nonnegativity follows from the known sign of $J$. The canonical threshold is a closed φ-expression; positivity is a short arithmetic check. The certificate is a structure (or Prop bundle) with an inhabitation witness assembling those facts. No long tactic scripts or external analysis.
why it matters in Recognition Science
Gives the Information layer a single φ-native bandwidth gate rather than an ad-hoc cutoff. Downstream consumers (none linked yet in the mirror graph) can require the certificate instead of re-deriving threshold positivity. Ties channel capacity language to the same $J$ and $\varphi$ forced by T5–T6, so bandwidth statements stay inside the RS unit system ($c=1$, tick $\tau_0=1$). Does not itself close mass, α, or forcing-chain goals; it only standardizes the information-side threshold.
scope and limits
- Does not derive Shannon capacity or classical coding theorems.
- Does not prove uniqueness of the threshold beyond the chosen φ formula.
- Does not connect bandwidth to mass rungs, α, or the eight-tick octave.
- Does not supply numerical channel simulations or empirical fits.
- Does not discharge unresolved Cost or Constants obligations outside imports.