IndisputableMonolith.Foundation.ExistenceUniquenessFromCost
The module establishes that the cost-zero set under the J-cost is exactly the singleton {1}. Researchers deriving the Recognition Science structure from the functional equation cite this to fix the identity element before the forcing chain. The argument assembles log-symmetry, isolation from zero, and singleton lemmas into a single conclusion.
claimLet $J$ be the J-cost. Then $\{x \mid J(x)=0\}=\{1\}$.
background
The module imports the J-cost definition from IndisputableMonolith.Cost and the RS time quantum $\tau_0=1$ tick from IndisputableMonolith.Constants. It contains the sibling declarations cost_zero_set_singleton, jcost_log_symmetric, jcost_isolated_from_zero, and ExistenceUniquenessCert. These lemmas operate in the foundation layer that extracts uniqueness directly from the cost function before any forcing-chain steps.
proof idea
The module organizes its argument as a chain of lemmas: jcost_log_symmetric supplies the symmetry relation, jcost_isolated_from_zero shows no other points reach zero, and cost_zero_set_singleton concludes the set is exactly {1}. ExistenceUniquenessCert packages the result for downstream use.
why it matters in Recognition Science
This module supplies the uniqueness of the zero-cost element required by ExistenceUniquenessCert and anchors the identity before T5 J-uniqueness in UnifiedForcingChain. It closes the step that converts the Recognition Composition Law into a concrete singleton property.
scope and limits
- Does not derive the explicit algebraic form of $J$.
- Does not address nonzero cost values or the phi-ladder.
- Does not connect to spatial dimensions or the eight-tick octave.