Recognition Science: A Closed–Form Solution to the Flavour Puzzle

The Standard Model reproduces every tested cross-section yet accommodates its 27 Yukawa magnitudes, two Dirac CP phases, and two Majorana phases as empirical accidents. Here we show that all quark and lepton masses and the complete CKM–PMNS structure follow from a single informational axiom of Recognition Physics. The axiom demands dual recognition of any stable link and minimises the total overhead carried by those links. Extremising the overhead functional determines a universal "golden-ratio" scale Xopt = φ/π and a fractional resonance index R= 7/12. Building on these primitives we prove:

1. The gauge skeleton SU(3)c ×SU(2)L ×U(1)Y with exactly three chiral generations is the unique minimal-overhead, anomaly-free solution.
2. A single arithmetic ladder (2,1,0) for left-handed recognition charges, dressed by a Z3 super-diagonal and a Z2 checkerboard mask, fixes the exponents (∆qe,∆qµ,∆qτ) = (18,10,6).
3. An electromagnetic exponent κ= 0.639±0.005 and a colour exponent λ=−1/ln Xopt convert those exponents into closed-form masses. All charged-lepton and quark pole values agree with the PDG-2024 central numbers to within 6 %.
4. A two-gap family-phase extremum sets (|Vus|,|Vcb|,|Vub|) = (λ,λ2,λ3) with λ= 0.2243 and predicts δCKM ≃+67°, θ13 ≃8.3°, δPMNS ≃−67°. Immediate phenomenology includes BR(µ→eγ) = 1.3×10−13, BR(τ→µγ) = 6.1×10−9, and a flavour-seesaw contact scale Λ ≃30 PeV reachable by FCC-hh or ultra-high-energy cosmic-ray dileptons. Ten light singlet scalars are shown to be inevitable for vacuum stability. The framework removes thirty-one empirical constan