IndisputableMonolith.Foundation.CostFirstExistence
Recognition Science equates the existence of an entity x with the vanishing of its J-cost. This module defines the corresponding predicate and establishes its basic properties, including equivalence to unit cost in one direction and positivity elsewhere. Researchers constructing the phi-ladder or deriving mass spectra cite it to anchor their objects in the zero-cost sector. The module is built from direct definitions imported from the Cost and Constants modules.
claimAn entity $x$ exists in the Recognition Science sense if and only if $J(x)=0$, where $J$ is the cost function obeying the Recognition Composition Law.
background
The module sits at the base of the Recognition Science foundation and imports the time quantum $τ_0=1$ tick together with the J-cost definition. Recognition existence is the predicate selecting those $x$ for which the cost vanishes. This supplies the zero-cost objects required by the forcing chain (T0 onward) that later derives J-uniqueness, the self-similar fixed point phi, the eight-tick octave, and three spatial dimensions.
proof idea
This is a definition module, no proofs. The sibling declarations establish the equivalence between recognition existence and the condition $J(x)=0$, along with consequences such as positive cost for non-existence and divergence at zero.
why it matters in Recognition Science
The module supplies the existence predicate that underpins all later constructions in the Recognition Science mirror. It directly supports the forcing chain steps T5 through T8 by providing the zero-cost objects on which J-uniqueness and the phi fixed point are defined. No parent theorems are listed in the dependency graph yet, but the definition is prerequisite for mass formulas and the alpha band derivation.
scope and limits
- Does not identify concrete values of $x$ that satisfy $J(x)=0$.
- Does not prove existence of the self-similar fixed point phi.
- Does not derive the constants $c=1$, $\hbar=\phi^{-5}$, or $G=\phi^5/\pi$.
- Does not address the Berry creation threshold at $\phi^{-1}$.