IndisputableMonolith.Foundation.CostFirstExistence
Cost-first existence in Recognition Science: a positive real is recognized as existing exactly when its J-cost vanishes. The module packages the predicate, the iff with the unit point, positivity of cost off the unit, and a small certificate bundle. Foundation authors cite it when tying ontology to the unique cost functional. The argument is definitional plus elementary calculus on J.
claimIn Recognition Science, a positive real $x$ is said to exist precisely when its cost vanishes: $J(x)=0$. Equivalently, existence holds if and only if $x=1$. Non-existence forces strictly positive cost, and the cost diverges as one approaches the singular direction $x\to 0^+$.
background
Recognition Science treats the unique cost functional $J$ (forced at T5 of the unified forcing chain) as ontologically primary. On positives one has $J(x)=(x+x^{-1})/2-1$, equivalently $\cosh(\log x)-1$, with global minimum $J(1)=0$ and $J(x)>0$ for $x\neq 1$. The Recognition Composition Law and the self-similar fixed point $\varphi$ sit upstream; this module does not re-derive them.
The local setting is cost-first ontology: existence is not postulated separately but read off the vanishing locus of $J$. The module imports the cost API and RS constants (including the tick $\tau_0$) and introduces a predicate for RS-existence together with elementary consequences of the shape of $J$.
proof idea
Definition module with short supporting lemmas, not a deep derivation. Existence is defined as $J(x)=0$. The equivalence with $x=1$ is the standard uniqueness of the global minimizer of $J$ on $\mathbb{R}_{>0}$. Strict positivity off the unit is the same minimizer fact. Divergence as $x\to 0^+$ is the elementary blow-up of $(x+x^{-1})/2$. A thin certificate record packages these facts for downstream consumers.
why it matters in Recognition Science
Places existence inside the cost layer rather than as an extra axiom: only the $J=0$ locus is admitted. That is the foundation slogan "recognition existence: $x$ exists iff $J(x)=0$." No downstream edges are recorded yet; the natural consumers are later foundation and phenomenology modules that need a cost-gated existence predicate, certificates, or the positivity/divergence lemmas when ruling out singular or off-ladder configurations. Ties directly to T5 J-uniqueness and the RCL-normalized cost, without touching the eight-tick, $D=3$, or mass-ladder steps.
scope and limits
- Does not derive uniqueness of J; assumes the cost API from T5.
- Does not construct physical objects, only the cost-vanishing existence predicate on positives.
- Does not address the phi-ladder, masses, alpha, or dimensional forcing.
- Does not prove dynamical stability or measurement postulates beyond J=0.
- Certificate is a packaging record, not an independent existence proof.