IndisputableMonolith.Foundation.ExistenceUniquenessFromCost
The module proves the cost-zero set equals exactly the singleton {1}. Researchers deriving uniqueness of the identity in the J-cost function for the Recognition forcing chain would cite these lemmas. The argument assembles imported symmetry and isolation properties from the Cost module into a sequence of singleton lemmas.
claimThe set $\{ x \mid \mathrm{Jcost}(x) = 0 \}$ equals $\{1\}$.
background
Recognition Science derives all structure from the J-cost function obeying the Recognition Composition Law. The module imports IndisputableMonolith.Cost for the cost definition and IndisputableMonolith.Constants. The Constants module states: "The fundamental RS time quantum (RS-native). τ₀ = 1 tick."
No new definitions appear in the module itself. Its doc-comment states the target claim directly: "The cost-zero set is exactly {1}." The listed sibling lemmas (cost_zero_set_singleton, jcost_isolated_from_zero, ExistenceUniquenessCert) together establish this kernel property.
proof idea
The module is a collection of lemmas rather than a single theorem. It proceeds by importing symmetry and isolation results from the Cost module, then applies them in cost_zero_set_singleton and jcost_isolated_from_zero to isolate the zero element and conclude the set is exactly {1}.
why it matters in Recognition Science
This module supplies the uniqueness of the zero-cost element required by the ExistenceUniquenessCert sibling and the broader Foundation domain. It fills the J-uniqueness step (T5) in the forcing chain by showing the kernel is trivial, enabling the later derivation of the self-similar fixed point phi and the eight-tick octave.
scope and limits
- Does not derive the explicit algebraic form of J.
- Does not prove the Recognition Composition Law itself.
- Does not address elements outside the positive reals.
- Does not connect the zero set to spatial dimension D=3.