Charge Is the Leader Winding Minus a Third of the Depth

Abstract

Abstract On the closed 39-seat lattice of Wavon Mechanics — letters {a, b, c}, winding numbers q = 0·+1·−1,engine Q = (2/3)·Re[Σⱼ q(tⱼ)·ωʲ] (ω = e^(2πi/3)) — we show that the identity Q = q₀ − Σ/3 holds onevery seat (q₀ = winding of the leader letter, Σ = #b − #c). The proof is one line: Re ω = Re ω² = −1/2.Corollary: Q ∈ ℤ ⟺ Σ ≡ 0 (mod 3). This single identity yields five things. (1) Why the lattice'scompositional classification — bright 1+6+12, dark 2+6 — coincides exactly with integer/fractionalcharge: it is the mod-3 shadow of Σ, not color, which is why the classification survived the retractionof the color reading (V3) — debt D-2 closed. (2) The free-state criterion Σ ≡ 0 (mod 3) selects threequarks (baryon) and quark–antiquark (meson) as the minimal bound combinations, the samecriterion as QCD triality (the Z₃ center of SU(3)) [borrowed correspondence]; "mesons have baryonnumber zero" is a theorem of conjugation flipping Σ. (3) No exchange that changes a single lettercan preserve both charges (exhaustive: 810 nontrivial exchanges between quark-seat pairs, 0preserving) — the quantitative reason for the color retraction, and a constraint that a strong-binding carrier must not touch letters. (4) In the two-letter room the free carriers are the threeinteger-charged seats (aa·bc·cb) → selection rule ΔQ ∈ {0, ±1}; the six fractional seats are internalonly. (5) The six charged dark seats belong to the same class as quarks (Σ ≢ 0); applyingconfinement [borrowed] to that class, free dark composites have integer charge and neutral onesexist — an exit from the objection "the dark sector is charged, hence cannot be dark matter". As aby-product, isospin is the leader slot (proton and neutron share the letters {a:6, b:3}). One line: thislattice contains not the full color SU(3) but only its center Z₃. Confinement, binding strength,and masses are not derived. All figures reproduce from the enclosed code.Added in version 1.1: wavon entanglement — one or two levels below quantum entanglement, anentanglement that carries energy (wavelength and amplitude) — is recorded as the author's hunch,and the geometry of the identity is added as [Theorem]. Taking the three sites as three mutuallyperpendicular imaginary axes, the identity Q = q₀ − Σ/3 is the projection of a seat's three-dimensional arrow onto the shadow direction (2,−1,−1)/3 of the site-0 axis; from 3 × (number ofb·c letters) = 2|L|² + Σ² follows Q ∈ ℤ ⟺ |L|² is a multiple of 3; and the three free carriers of thetwo-letter room (aa·bc·cb) are the three terms of the maximal entanglement formed by a pair ofseparate wavons, bound with a leader.