IndisputableMonolith.Foundation.Determinism
Foundation.Determinism proves that strict convexity of the J-cost forces a unique minimizer for any constrained recognition problem, which yields deterministic resolution of projections. It is imported by variational dynamics, probability structure, and information conservation modules. The argument chains the second-derivative positivity lemma through the unique-minimizer principle to the determinism-resolution theorem.
claimFor the cost $J(x) = ½(x + x^{-1}) - 1$ on $x > 0$, the constrained minimization problem has a unique solution whenever $J''(x) = x^{-3} > 0$; this implies that every lossy projection resolves deterministically.
background
The module imports Cost and Cost.Convexity, whose doc states: 'This module proves that: 1. Jlog(t) = cosh t - 1 is strictly convex on ℝ 2. Jcost(x) = ½(x + x⁻¹) - 1 is strictly convex on ℝ₊. These are foundational for the uniqueness theorem T5.' It also imports LawOfExistence, whose core statement is 'x exists ⟺ defect(x) = 0'. Local definitions include ConstrainedProblem, Observer, project, projection_lossy, and determinism_resolution.
proof idea
The module first records the second-derivative fact Jcost_second_deriv_positive. Strict positivity is fed to unique_minimizer_principle for any constrained problem. The same convexity is then applied to projection_lossy to obtain determinism_resolution.
why it matters in Recognition Science
The module supplies determinism_resolution, which is imported by VariationalDynamics to define the ledger update rule, by ProbabilityMeaningStructure to equate probability with J-cost projection weight, and by InformationConservation to resolve the black-hole information paradox. It closes the convexity-to-determinism step required by the forcing chain.
scope and limits
- Does not derive the explicit form of J from the forcing chain.
- Does not treat multi-observer or relativistic consistency.
- Does not compute numerical values for constants such as alpha or G.
- Does not address quantum measurement or collapse.