IndisputableMonolith.Foundation.GaugeSymmetry3_FromJCost
Packages the derivation of three-dimensional gauge structure from the Recognition J-cost: a domain cost, its nonnegativity and evaluation identities, a positive canonical threshold, and an inhabited certificate that the three-gauge symmetry holds. Foundation workers linking cost uniqueness to spatial dimension and gauge form would cite it. The module is definition-and-certificate style over Constants and Cost, not a single deep proof.
claimFrom the Recognition cost $J$, the module defines a domain cost, proves it is nonnegative and agrees with pointwise evaluation where required, fixes a positive canonical threshold, and supplies an inhabited certificate that three-dimensional gauge symmetry is realized from that cost structure.
background
Recognition Science forces a unique nonnegative cost $J$ on positive reals (T5): $J(x)=(x+x^{-1})/2-1$, equivalently $\cosh(\log x)-1$, obeying the Recognition Composition Law. The Cost import supplies that functional and its elementary calculus; Constants supplies the RS-native tick $\tau_0=1$ used as the discrete time unit.
Gauge structure in RS is not postulated. It is read off how $J$ grades mismatches on configuration domains once the eight-tick octave and the forcing of $D=3$ spatial dimensions (T7–T8) are in place. This module sits in Foundation and turns that reading into named Lean objects: a domain-level cost, a threshold that marks when a mismatch is gauge-relevant, and a certificate type asserting the three-fold gauge symmetry.
Sibling names indicate the local API: domainCost with evaluation and nonnegativity lemmas, canonicalThreshold with positivity, and RSGaugeSymm3Cert / cert / cert_inhabited as the inhabited proof package.
proof idea
Definition-and-certificate module, not one monolithic theorem. It builds a domain cost from the imported $J$-cost, records elementary identities (pointwise evaluation, nonnegativity), fixes a positive canonical threshold, then packages an RSGaugeSymm3Cert record whose inhabitation (cert_inhabited) is the module’s main claim token. Supporting lemmas are short algebraic or positivity facts over Cost; the certificate is the assembly point rather than a long tactic script.
why it matters in Recognition Science
Closes the Foundation bridge from J-uniqueness (T5) and the forced spatial dimension $D=3$ (T8) to an explicit three-gauge symmetry certificate. Downstream pages that need “gauge symmetry is forced, not assumed” can depend on the inhabited cert rather than re-deriving cost inequalities. No used_by edges are recorded yet, so the module currently acts as a Foundation leaf: it freezes the gauge-from-cost story for later particle and field layers. It does not itself prove the full Standard Model gauge group; it certifies the three-dimensional gauge skeleton implied by the cost.
scope and limits
- Does not derive SU(3)×SU(2)×U(1) or coupling constants; only a three-gauge skeleton from J-cost.
- Does not reprove J-uniqueness (T5) or D=3 (T8); those are upstream forcing facts.
- Does not supply dynamical Yang–Mills equations or renormalization.
- Does not claim experimental gauge measurements; it is a Foundation certificate module.
- Does not list downstream consumers yet (used_by is empty).