IndisputableMonolith.Physics.DiracEquationFrom_JCost
Module linking the Recognition Science J-cost to a Dirac-spinor certificate: a domain cost on spinor data, a positive canonical threshold, and an inhabited certificate that the cost stays below threshold. Physicists deriving fermionic kinematics from the unique cost functional would cite it. The argument is mostly definitional packaging plus nonnegativity and positivity lemmas over the Cost and Constants imports.
claimOn spinor-valued domain data one defines a nonnegative domain cost $C$ built from the RS cost $J$, a positive canonical threshold $\theta_*>0$, and a Dirac-spinor certificate asserting $C\le\theta_*$. The module packages these objects and proves $C\ge 0$, $\theta_*>0$, and that the certificate type is inhabited.
background
Recognition Science forces a unique nonnegative cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) via the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that functional; Constants supplies the RS time quantum $\tau_0=1$ tick used to normalize discrete update scales.
This physics module sits downstream of that cost layer. It introduces a domain cost on data meant to stand for Dirac spinor configurations, together with a canonical numerical threshold against which that cost is compared. The intended reading is that staying under the threshold is the certificate that the configuration is admissible as a free Dirac mode in the RS ledger.
Sibling names indicate the local API: domainCost and its evaluation and nonnegativity facts, canonicalThreshold with positivity, and a DiracSpinorCert bundle with an inhabited cert.
proof idea
Definition-heavy module rather than a long derivation. Domain cost is defined from the imported $J$-cost on the spinor domain; nonnegativity is inherited from nonnegativity of $J$. The canonical threshold is a positive constant (positivity lemma). The Dirac-spinor certificate is a Prop-valued structure packaging the inequality domain-cost $\le$ threshold, and inhabitation is shown by exhibiting a witness (or by a trivial default configuration). No deep PDE analysis appears at this layer; the module is the cost-side certificate interface for later Dirac kinematics.
why it matters in Recognition Science
Gives the cost-theoretic gate for treating Dirac spinors inside Recognition Science: fermionic degrees of freedom are admitted when their domain cost lies under the canonical threshold built from $J$. That matches the broader program in which continuum wave equations arise as continuum limits or certificates of discrete recognition dynamics forced by T5–T8 (unique $J$, $\phi$, eight-tick octave, $D=3$).
No downstream consumers are wired in the graph yet (used_by empty), so this module is presently a leaf interface in Physics. It is the natural attachment point for later theorems that recover the Dirac operator, dispersion, or spinor bundle structure from the same $J$-cost ledger. Until those land, the certificate is the formal placeholder that “Dirac from $J$” has a checkable cost bound rather than an ansatz.
scope and limits
- Does not derive the continuum Dirac PDE or gamma-matrix algebra from first principles.
- Does not prove uniqueness of the spinor representation or fix mass terms.
- Does not connect the certificate to measured fermion spectra or coupling constants.
- Does not discharge eight-tick or $D=3$ forcing inside this module.
- Does not supply dynamics; only a static cost-threshold certificate API.