IndisputableMonolith.Foundation.Ecology
Foundation module for Recognition Science ecology: domain-level cost functionals built from the J-cost, a positive canonical threshold, and a four-way biotic interaction certificate. Supplies nonnegativity and positivity lemmas plus an inhabited certificate type. Cited by anyone wiring ecological admissibility into the forcing chain or mass/ladder constraints. Structure is mostly definitions with short algebraic positivity proofs.
claimThe module defines a domain cost $C_{\mathrm{dom}}$ (nonnegative, with an evaluation identity), a canonical threshold $\theta>0$, and a biotic four-interaction certificate type together with an inhabited witness asserting ecological admissibility under Recognition cost constraints.
background
Recognition Science measures mismatch by the J-cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the Recognition Composition Law. The Cost import exposes that functional; Constants supplies the RS time quantum $\tau_0=1$ tick.
Ecology here means domain-scale bookkeeping: aggregate cost of a biotic or multi-agent configuration, compared against a fixed positive threshold. The module sits in Foundation, so it only packages the cost interface and certificate shape; it does not yet derive species dynamics or population equations.
Sibling objects include domainCost with equality-at-point and nonnegativity lemmas, canonicalThreshold with positivity, and BioticInteract4Cert with an inhabited cert. These are the local vocabulary for later ecological forcing steps.
proof idea
Definition-heavy module. Domain cost is introduced as a functional of the underlying J-cost; nonnegativity and the evaluation identity are short algebraic consequences of J-properties from Cost. Canonical threshold positivity is a one-line constant fact. The biotic four-interaction certificate is a structure (Prop bundle) with an explicit inhabited instance, not a deep existence proof.
why it matters in Recognition Science
Gives Foundation a typed hook for ecological constraints without leaving the RS cost language. Downstream graph is currently empty, so the module is a leaf that future ecology or multi-agent forcing theorems are expected to import. Aligns with the broader program in which admissible configurations must sit above Berry-scale thresholds and respect eight-tick cadence; the four-way biotic certificate is the local stand-in for that admissibility gate. Does not yet close any T0–T8 step; it only prepares the cost and certificate surface those steps would use.
scope and limits
- Does not derive population dynamics or reaction-diffusion equations.
- Does not prove uniqueness of the canonical threshold from first principles.
- Does not connect domain cost to the mass ladder or alpha band.
- Does not discharge any T0–T8 forcing obligation.
- Does not model spatial ecology in D=3; certificate is combinatorial.