IndisputableMonolith.Foundation.ExistenceUniquenessFromCost
The module establishes that the cost-zero set equals exactly the singleton containing 1. Foundation researchers in Recognition Science cite it to anchor uniqueness results for the cost function before building existence and uniqueness certificates. The argument assembles imported properties from the Cost and Constants modules into targeted lemmas such as cost_zero_set_singleton.
claimThe set $\{x \mid \text{cost}(x)=0\}$ equals $\{1\}$.
background
The module imports the RS time quantum $\tau_0=1$ tick from Constants and the cost function from the Cost module. It focuses on the zero-cost locus and shows this locus collapses to a single point. Sibling declarations inside the module include cost_zero_set_singleton, jcost_log_symmetric, and ExistenceUniquenessCert.
proof idea
The module collects supporting lemmas that together isolate the zero-cost element; no single top-level proof body is present.
why it matters in Recognition Science
This module supplies the uniqueness fact that ExistenceUniquenessCert and existenceUniquenessCert rely on. It sits at the base of the forcing chain by fixing the zero-cost point before higher-level existence statements are derived.
scope and limits
- Does not address cost values away from zero.
- Does not derive the explicit form of the cost function.
- Does not connect to spatial dimensions or the phi ladder.