BITKernel4DeepCert
plain-language theorem explainer
Certificate packing three analytic facts for the BIT deep-v4 kernel built from J-cost: the domain cost vanishes on the diagonal for nonzero reals, stays nonnegative for positive mass/energy arguments, and the canonical threshold is strictly positive. Downstream code cites the inhabited instance to discharge kernel hypotheses. The declaration is a pure structure, not a proved theorem.
Claim. A certificate is a triple of properties: (i) for every real $r \neq 0$, the domain cost satisfies $\mathrm{domainCost}(r,r)=0$; (ii) for all $m>0$ and $e>0$, $\mathrm{domainCost}(m,e)\ge 0$; (iii) the canonical threshold is strictly positive.
background
The module builds a deep BIT (bosonic vacuum-fluctuation) kernel from the Recognition Science J-cost. Module status is structural: zero sorry, zero axioms. BIT here means bosonic fields carry vacuum fluctuations $J(\phi)/45$ per recognition tick, with cumulative cosmic Z-aging $\sum J(\phi)/45 = J(\phi),N_{\mathrm{ticks}}/45$.
Domain cost is the local cost functional on mass/energy pairs used by the kernel; the diagonal identity $\mathrm{domainCost}(r,r)=0$ encodes zero cost for matched recognition, while nonnegativity mirrors the global fact that every recognition event has nonnegative cost (upstream cost_nonneg: "The cost of any recognition event is non-negative," via $J$-cost nonnegativity). The canonical threshold is the positive cutoff that gates the deep kernel.
The structure packages exactly those three obligations so later kernel lemmas can assume a single certificate rather than three separate hypotheses.
proof idea
No proof body: this is a structure declaration. The three fields are Prop-valued obligations (diagonal vanishing of domain cost, nonnegativity on the positive quadrant, positivity of the canonical threshold). Inhabitation is supplied separately by the noncomputable cert definition, which wires in the sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos.
why it matters
Gives the typed interface for the BIT Kernel Family Deep v4 derived from J-cost (Plan v7 deep session). Downstream, cert fills the three fields and cert_inhabited proves Nonempty BITKernel4DeepCert, so any lemma that needs the deep kernel can assume a certificate rather than re-proving diagonal vanishing, nonnegativity, and threshold positivity.
In the broader Recognition chain this sits under Foundation cost calculus: J-uniqueness (T5) forces the cost shape, and nonnegativity of recognition cost is already established upstream. The BIT story (vacuum $J(\phi)/45$ per tick, huge but suppressed cumulative aging by recombination) uses this certificate as the analytic gate before cosmic-scale sums. It does not itself compute masses, $\alpha$, or the eight-tick octave; it only certifies the cost/threshold side conditions those later steps rely on.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.