PUH Theorem 120 (Restated) — Koide Q = 2/3 Is Parameter-Free: the √2 Is Forced by Z₃ Mode Counting, and the Phase (6π+2)/9 Predicts m_τ = 1776.989 MeV

Photonic Universe Hypothesis (PUH) — Restatement with a Prediction. WHY THIS EXISTS. Theorem 120 is cited across this archive as the origin of its strongest empirical claim — that the Koide ratio Q = 2/3 follows from Z₃ symmetry rather than being a numerical accident. IT HAS EXISTED ONLY AS A GITHUB GIST dated 12 March 2026, with no permanent identifier. This places it on the record and adds the quantitative test it lacked. The original priority stands at March; the uniqueness analysis and the tau mass prediction are new here and are marked as such. THE EMPIRICAL: Koide's relation (1981): Q = (m_e + m_μ + m_τ)/(√m_e + √m_μ + √m_τ)². On PDG 2024 masses — m_e = 0.51099895000 MeV, m_μ = 105.6583755 MeV, m_τ = 1776.86 ± 0.12 MeV — Q = 0.666660511 against 2/3 = 0.666666667: AGREEMENT TO ONE PART IN 108,309. THE STANDARD MODEL DOES NOT EXPLAIN THIS; lepton masses are free parameters there and nothing requires them to satisfy any relation. Q is bounded to [1/3, 1] for any three positive masses, so 2/3 is EXACTLY MIDWAY between the extremes. THEOREM 120.1 (Q = 2/3 is automatic). Any three quantities of the Z₃ cosine form √m_k = M(1 + √2 cos(φ + 2πk/3)) satisfy Q = 2/3 identically, for every phase and every scale. PROOF IN THREE LINES: the three cosines sum to zero, and — the only non-obvious step — their squares sum to 3/2 for every φ. Hence Σ√m = 3M, Σm = M²(3 + 3A²/2), and Q = (3 + 3A²/2)/9 = 1/3 + A²/6, which is 2/3 when A² = 2. Verified numerically at random phases: Σcos² returns 1.500000000000 at every value tested. NO DYNAMICAL FINE-TUNING IS REQUIRED. This RELOCATES the question rather than answering it: the content is not why Q = 2/3, but WHY THE SQUARE-ROOT MASSES TAKE THE Z₃ COSINE FORM AT ALL. In this framework, because leptons are the squared amplitudes of the three Z₃ modes of the Planck polariton at the A₂ symmetry-breaking step. THEOREM 120.2 (the √2 is derived, not fitted). THIS IS THE STEP THAT DECIDES WHETHER Q = 2/3 IS A DERIVATION OR A FIT. The Koide parameterisation IS the inverse Z₃ discrete Fourier transform, with one DC mode and two AC modes: X̂₀ = 3M, X̂₁ = (3M√2/2)e^{iφ}, X̂₂ = X̂₁*. Equal power in the DC and AC sectors forces A_DC/A_AC = √(AC modes/DC modes) = √(2/1) = √2. In the original's words: "forced by Z₃ group structure alone. No input from E₈ root lengths is required." SO THE CHARGED-LEPTON RATIO HAS NO FREE PARAMETER — which is what makes it a claim rather than a curve fit. RESULT 120.3 (the phase is self-referential). Theorem 120.1 holds for every phase, so φ was the remaining freedom. T152 closes it: the offset from 2π/3 equals Q(1−Q), which for Q = 2/3 is 2/9, giving φ = 2π/3 + 2/9 = (6π+2)/9 = 132.732395°. MEASURED: 132.732331°. GAP: 0.000064°. THE PHASE IS DETERMINED BY THE RATIO IT BELONGS TO. RESULT 120.4 (unique among 7,500 expressions). A six-decimal match invites the charge of numerology, so it is tested. Searching every expression of the form (aπ + b)/c with a, b ∈ [−12,12] and c ∈ [1,12] — 7,500 expressions — at the observed tolerance of 1.117×10⁻⁶ rad: EXACTLY ONE MATCH IS FOUND, AND IT IS (6π+2)/9. Expected by chance: 0.0013, i.e. significance ≈ 1 in 745. NEW HERE, NOT IN THE ORIGINAL. AND THE CAVEAT IS STATED: the search space was chosen after the formula was known, and a formula found first and tested second would carry more weight. RESULT 120.5 (the prediction). Inverting the phase formula for the tau mass gives m_τ = 1776.9892 MeV against a measured 1776.86 ± 0.12 — +1.08σ. THE TAU MASS IS THE ONLY POORLY MEASURED INPUT (the electron is known to eleven digits, the muon to nine). IMPROVING IT BY A FACTOR OF TEN, WHICH IS WITHIN REACH, CONVERTS THIS INTO A 13σ TEST. The prediction either confirms to five decimal places or dies. There is no adjustable quantity on either side of the comparison. A REFUTATION, RECORDED BECAUSE IT IS ONE. The same session tested and REJECTED a conjecture that the Koide phase derives from an E₈ Coxeter angle, on three independent grounds: the direct transfer of E₈ root lengths fails on a metric mismatch; the claimed E₈ ⊃ (A₂)³ decomposition belongs to E₆, not E₈; and the A₂ Weyl chamber centre sits at 150°, which is 17.27° from the measured phase — more than a quarter of the chamber width. What the chamber DOES supply is a constraint: the lepton mass vector lies in 120° < φ < 180°, forced by the mass hierarchy, and the measured 132.73° sits 12.73° above the lower wall. THE CHAMBER ADMITS THE PHASE; IT DOES NOT PICK IT. KILL-CONDITIONS: (i) A TAU MASS MEASUREMENT INCONSISTENT WITH 1776.989 MeV KILLS RESULT 120.5 OUTRIGHT, and at 0.01 MeV precision the test is decisive either way; (ii) if the Z₃ cosine form is not forced for lepton square-root masses — if it is a parameterisation chosen because it works rather than a structure derived — THEOREM 120.1 IS A TAUTOLOGY ABOUT A CHOSEN FORM and the content evaporates; the original locates this as the genuine question and answers it by the Planck polariton mode structure, which is a framework claim and not an independent one; (iii) if T152's relation δ = Q(1−Q) is coincidental, Result 120.3 fails and the phase returns to being free; (iv) if the equal-power condition in Theorem 120.2 is a convention rather than a physical requirement, the √2 is chosen rather than forced and Q = 2/3 becomes a fit. NOT CLAIMED: that Q = 2/3 is derived from first principles — it is derived from the Z₃ cosine form, whose origin is a framework claim; that the uniqueness test proves the formula, since the search space was chosen after the fact; that the tau prediction has been tested; that the Planck polariton account of lepton masses is established here; or that this restatement adds any derivation — IT ADDS A UNIQUENESS TEST, A PREDICTION, AND A PERMANENT IDENTIFIER.