cert_inhabited
plain-language theorem explainer
The deep BIT-kernel certificate type is inhabited: there exists a packed witness that domain cost vanishes on the diagonal, is nonnegative off it, and the canonical threshold is positive. Foundation users cite this to discharge Nonempty obligations when wiring the J-cost BIT kernel into larger certificates. The proof is a one-line term that injects the concrete `cert` witness.
Claim. There exists a deep BIT-kernel certificate: a triple of facts that the domain cost $C(r,r)=0$ for all $r\neq 0$, that $C(m,e)\ge 0$ whenever $m>0$ and $e>0$, and that the canonical threshold is strictly positive.
background
The module builds a structural certificate for the BIT kernel family (deep v4) from the Recognition J-cost. BIT here means bosonic vacuum fluctuations of size $J(\varphi)/45$ per recognition tick, with cumulative Z-aging linear in tick count; the module status is a structural theorem with no sorry and no axioms.
Domain cost is the local cost functional on mass/energy-like pairs used by the kernel. The certificate structure packages three elementary properties: diagonal vanishing ($C(r,r)=0$ for $r\neq 0$), nonnegativity for positive arguments, and positivity of a canonical threshold constant. Those three fields are exactly the data of BITKernel4DeepCert.
Upstream, the structure itself is the only dependency; the concrete witness cert (a sibling) already assembles the three proofs. This theorem only records that the structure type is nonempty.
proof idea
One-line term proof. The sibling cert is already an element of BITKernel4DeepCert, so ⟨cert⟩ is a term of Nonempty BITKernel4DeepCert. No tactics, no further lemmas.
why it matters
In the Recognition foundation layer this closes the existence side of the deep BIT-kernel certificate derived from J-cost. The module frames BIT as vacuum fluctuations $J(\varphi)/45$ per tick with cosmic cumulative aging; the certificate guarantees the cost side of that kernel is well-behaved (zero on match, nonnegative, positive threshold).
No downstream consumers are wired yet in the graph, so the immediate role is local: any later theorem that assumes Nonempty BITKernel4DeepCert can invoke this. It sits under the J-uniqueness landmark (T5: $J(x)=(x+x^{-1})/2-1$) because the domain cost is built from that cost. It does not itself force $\varphi$, the eight-tick octave, or $D=3$.
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