cert
The cert definition constructs a concrete RecogLattice3Cert instance for the phi-lattice by supplying the zero-cost identity, non-negative cost, and positive threshold properties. Researchers deriving mass spectra from J-cost would invoke this certificate to confirm the lattice satisfies the required metric axioms. It is realized as a direct structure constructor delegating each field to a prior lemma establishing the corresponding property.
claimLet $domainCost(m,e)$ denote the J-cost of the ratio $m/e$ on the recognition lattice generated by powers of $phi$. The structure cert asserts that $domainCost(r,r)=0$ for all $r neq 0$, that $domainCost(m,e) geq 0$ whenever $m>0$ and $e>0$, and that the canonical threshold is strictly positive.
background
The RecognitionLattice3 module constructs a lattice metric on the integers indexed by powers of the golden ratio $phi$, where the distance between rungs $k$ and $j$ is given by the absolute J-cost $J(phi^{k-j})$. The ground state sits at zero cost when the ratio is unity. Domain cost is defined as the J-cost of the mass-to-energy ratio, and the canonical threshold marks the minimal positive cost separating the ground state from the first excited state. Upstream results establish that J-cost is non-negative for any recognition event, that domainCost vanishes on the diagonal, and that the threshold exceeds zero by direct comparison with $phi > 1.5$.
proof idea
The definition is a record constructor that directly assigns the three fields of RecogLattice3Cert to the theorems domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos. No additional reasoning is required beyond these prior results.
why it matters in Recognition Science
This definition supplies the concrete certificate required by the structural theorem for the three-dimensional recognition lattice in the J-cost framework. It closes the verification that the lattice metric satisfies non-negativity and identity axioms, feeding into downstream derivations of particle masses via the phi-ladder. The construction relies on the Recognition Composition Law and the forcing chain that fixes $phi$ as the self-similar fixed point.
scope and limits
- Does not establish the full metric properties of the lattice beyond cost non-negativity and diagonal vanishing.
- Does not derive explicit mass values for specific particle rungs.
- Does not address higher-dimensional extensions of the lattice.
- Does not prove uniqueness of the canonical threshold.
formal statement (Lean)
27noncomputable def cert : RecogLattice3Cert where
28 cost_at_eq := domainCost_at_eq
proof body
Definition body.
29 cost_nonneg := domainCost_nonneg
30 threshold_pos := canonicalThreshold_pos