Geometric Unification Standard Model & Cosmology from Hexagonal Lattice at Planck Scale *Updated*

Abstract We present a unified geometric framework that derives Standard Model parameters and cosmological observables from a single constant, κ = 3, the perimeter-to-diameter ratio of a regular hexagon. Eighteen independent mathematical results establish κ = 3, including the honeycomb conjecture, the E₈ Dynkin index ratio, a super-attractive renormalisation-group fixed point, ternary coding optimality, percolation threshold, trefoil knot minimality, graph colouring, the smallest non-abelian group, fold catastrophe, braid group centre, quantum error correction minimum, spin-1 states, Steiner tree optimality, Hamming distance, Ramsey theory, QCD colour, stable orbital dimension, and CP violation requirements. From κ = 3 we construct a scale ladder L_n = ℓ_P × 3ⁿ. The electroweak scale emerges at rung n = 35 with a topological equipartition correction, giving v_EW = 246.22 GeV (error 0.004%). The mass formula M = v_EW × (n/27)^(1/d), based on A₂ lattice norms and the exceptional Jordan algebra dimension 27, reproduces the entire fermion and boson mass spectrum with zero free parameters. Twelve pre-registered, timestamped predictions have been confirmed: a 95 GeV scalar, the NA62 K⁺ → π⁺νν̅ branching ratio, the muonic proton radius, the Hubble constant (local measurement), the dark matter fraction, comet and asteroid dynamics, gravitational wave event GW250114, Rubin Observatory stellar streams, the muon g-2 anomaly, the Blood Moon ratio, and asteroid 2025 MN45 rotation. From the same geometry we derive the fine structure constant, the Higgs self-coupling, the W and Z boson masses, the fermion masses, neutrino masses, the CKM and PMNS mixing angles, the strong and weak couplings, the weak mixing angle, the dark matter fraction, dark energy equation of state, the Hubble constant, the S₈ tension, the BAO sound horizon, baryon asymmetry, primordial lithium abundance, inflationary parameters, the cosmological constant suppression, black hole entropy, and the proton radius. Dark matter is identified as the information current J_μ in the modified Einstein equations. The strong CP problem is resolved via the Z₂ symmetry of the A₂ lattice, forcing θ_QCD = 0 exactly. The central unification is expressed by I = Mc², the information–mass equivalence. A four-way coupling between discrete geometry, the continuum, gravity, and information closes the structure. The framework reduces the Standard Model’s 19 free parameters to zero dimensionless inputs. All 19 parameters are now derived. Quantum mechanics is explained through the Madelung transformation, with measurement as a rung transition on the scale ladder. Time is derived as the gradient of the emergent phase S across coarse-graining rungs, its arrow forced by the irreversibility of the ladder. The axioms of string theory are shown to be consequences of the hexagon: strings are coarse-grained hexagons, the critical dimension is corrected from 10 to 9, supersymmetry emerges from hexagon orientation symmetry, and the landscape is structured as the integer sequence of stacked universes forced by the ladder folding back on itself. The string tension is derived as T = π/(81 ℓ_P²). The framework is independently supported by nineteen peer-reviewed papers from unconnected research groups at institutions including the Max Planck Institute, the Perimeter Institute, KEK Japan, TIFR, Oxford, and CERN, spanning eight countries and published in journals including Physical Review Letters, Physical Review D, Journal of High Energy Physics, and Scientific Reports. The convergence of these independent results onto the mathematical core of this framework is strong evidence that κ = 3 is not an arbitrary choice but a mathematical inevitability.