IndisputableMonolith.Foundation.RS_No_Information_Loss
Foundation module packaging the Recognition Science no-information-loss principle: a nonnegative domain cost, a positive canonical threshold, and an inhabited certificate type. Theorists working the information-preservation side of the forcing chain cite it when they need a formal stand-in for lossless recognition dynamics. The module is largely definitional, with elementary nonnegativity and positivity lemmas supporting the certificate.
claimThe module introduces a domain cost $C \ge 0$, a canonical threshold $\theta > 0$, and a certificate type asserting that Recognition Science recognition dynamics incur no information loss relative to that cost and threshold.
background
Recognition Science derives physics from a single cost functional $J$, with $J(x) = (x + x^{-1})/2 - 1$ on positive reals (equivalently $\cosh(\log x) - 1$). The Cost import supplies that $J$-calculus; Constants supplies the RS-native tick $\tau_0 = 1$. Information loss would appear as a positive residual of $J$ (or a derived domain cost) under composition of recognition steps.
This module sits in the Foundation layer. It does not re-prove $J$-uniqueness (T5) or the Recognition Composition Law. Instead it defines a domain-level cost, records its nonnegativity and evaluation identities, fixes a positive canonical threshold, and wraps those facts in a certificate structure meant to stand for the claim that RS dynamics preserve information below that threshold.
proof idea
Definition-and-certificate module rather than a deep proof development. Domain cost is introduced as a def; companion lemmas record evaluation at equality and nonnegativity. The canonical threshold is a positive constant (positivity lemma separate). The main object is a certificate structure bundling those facts, with an inhabited instance so downstream code can assume the no-information-loss package without reconstructing the pieces.
why it matters in Recognition Science
No-information-loss is a structural demand of the Recognition framework: if recognition steps destroyed information, the forcing chain from the unique $J$ cost through the golden-ratio fixed point, eight-tick octave, and $D = 3$ would not close cleanly. This module gives that demand a named Lean home (cost, threshold, certificate) so later Foundation and physics layers can depend on a single interface.
Downstream use is not yet wired in the graph (no used_by edges). The certificate is the intended hook for theorems that need lossless composition of recognition updates, including any argument that residual $J$-cost cannot accumulate across ticks.
scope and limits
- Does not prove uniqueness of the $J$ cost or derive it from the Recognition Composition Law.
- Does not establish numerical bounds on physical constants or mass-ladder rungs.
- Does not claim a theorem that every concrete RS process is lossless; only packages cost, threshold, and certificate defs.
- Does not connect yet to downstream used_by theorems in the current graph.