The K₇ Framework: Conditional Topological Relations for Standard-Model Parameters from G₂ Holonomy and E₈×E₈ Structure

The Standard Model requires 19 experimentally determined parameters lacking theoretical explanation. We present a geometric framework in which physical observables emerge as topological invariants of a seven-dimensional G₂ holonomy manifold K₇ coupled to E₈×E₈ gauge structure, with topological inputs treated as discrete and the metric determinant det(g) = 65/32 imposed as a normalization target (§2.3). The 33 Type I relations employ no continuously adjustable parameter after the declared structural ledger is frozen; Types II, III, and IV are conditional extractions using explicitly-enumerated inputs (§3.2). Building on a Newton–Kantorovich-certified neck/metric model ([A], [B]), a Joyce–Karigiannis route realizing (b₂, b₃) = (21, 77), and the companion analytic scheme [E] (discharged at a normalised datum $\mathcal D_0$), we derive 95 observables organized in four types: 33 direct algebraic (Type I, mean deviation 0.73%), 19 one-step physical extractions (Type II, 0.17%), 21 multi-step dynamical chains (Type III, 3.4%), and 22 structural diagnostics (Type IV). Of 66 experimentally comparable observables, 11 are exact matches (deviation < 0.01%) and 53 are within 1%. The existence question is discharged conditionally at the datum-level analytic layer by [E]; at the global-analytic layer it remains conditional on $\mathsf H_{\rm global}$ and hypothesis $(\mathrm J)$ of [E, §8.3] (two-layer boundary: §9).

Beyond the original 33 predictions, the framework adds: (1) a representation-branching chain E₈ ⊃ E₆ × SU(3) ⊃ ⋯ ⊃ SU(3) × SU(2) × U(1) compatible with the SM matter content (a chain of subgroup branchings, not a physical breaking mechanism, whose UV completion remains a load-bearing open question; §5.1, §9); (2) a combined lepton mass hierarchy mechanism achieving sub-percent precision from two independent geometric sources; and (3) a Sieve reading under which m_H/m_W (Route A) is the unique survivor at exact rank 1 budget-unique after Type I calibration (§7). Of the 95 observables, 55 are formally verified in Lean 4 (213 certificate conjuncts, 15 classified axioms across the A-F taxonomy, 0 sorry); Lean certifies the algebraic relations and internal consistency, not the physical interpretation or the existence of the complete compact construction (§8.3, §9).

DUNE will test the topological prediction δ_CP = 197°; a measurement outside [182, 212]° would create serious tension. NuFIT 6.1 (2025) reports δ_CP = $207^{+23}_{-20}$°, so the prediction now sits well within the 1σ band. We present this as an exploratory investigation emphasizing falsifiability, not a claim of correctness.