IndisputableMonolith.Foundation.BITKernel4_DeepFromJCost
Module deriving a four-bit kernel certificate from the J-cost functional rather than from an ad-hoc discrete model. It introduces a domain cost, a canonical positive threshold, and an inhabited certificate bundle (BITKernel4DeepCert). Foundation authors cite it when grounding discrete bit structure in continuous cost geometry. The argument is definitional plus elementary nonnegativity and positivity lemmas.
claimFrom the J-cost $J(x)=(x+x^{-1})/2-1$, define a domain cost on positive reals, a canonical threshold $T>0$, and a certificate package asserting the deep four-bit kernel properties (nonnegativity of the domain cost, positivity of $T$, and inhabitance of the certificate type).
background
Recognition Science forces the unique symmetric cost $J(x)=\cosh(\log x)-1$ (T5). Discrete bit and tick structure is meant to sit on top of that continuous cost, not beside it. This module lives in the Foundation layer and imports only Constants (RS-native units, including the tick quantum) and Cost (the J-cost API).
The local objects are a domain cost pulled back from $J$, equality and nonnegativity facts for that cost, and a canonical threshold with a positivity proof. These feed a single certificate record BITKernel4DeepCert whose inhabitance packages the deep-from-J-cost claim for a four-bit kernel.
proof idea
Definition module with short supporting lemmas, not a long derivation. Domain cost is defined from $J$; domainCost_at_eq and domainCost_nonneg are direct consequences of J's algebraic properties. The canonical threshold is a named positive constant; positivity is a one-line arithmetic fact. The certificate type bundles those facts, and cert_inhabited supplies a canonical inhabitant. No heavy tactic scripts or external analytic machinery.
why it matters in Recognition Science
Closes a Foundation gap: the four-bit kernel is not postulated as a discrete gadget but certified as a consequence of J-cost geometry. That aligns with the forcing chain (T5 J-uniqueness, T7 eight-tick octave built from bit structure). Downstream pages that need a deep BIT-kernel hypothesis can import the inhabited certificate instead of re-proving cost nonnegativity and threshold positivity. No further used-by edges are recorded yet; the module is a leaf certificate supplier for later kernel and ledger constructions.
scope and limits
- Does not prove uniqueness of the four-bit kernel among all discrete models.
- Does not derive the eight-tick octave or D=3; those remain upstream forcing steps.
- Does not compute physical constants (alpha, masses) from the certificate.
- Does not supply an independent discrete dynamics; only cost-side certificate data.
- Does not claim experimental falsification criteria beyond the mathematical package.