Keys/Hadrons.lean — FrcCore

1import FrcCore.Theme.Interactions2import FrcCore.Theme.Fields3import FrcCore.Theme.Ladder4import FrcCore.Theme.Hadrons56/-!7# FrcCore.Keys.Hadrons — the keyed theorems of the hadrons theme (ledger migration, task LM19)89One declaration per key, `FRC.Ledger.p<key>`, generated by `make_keys.py` from the ledgers' Lean bindings: the theorem10that every row carrying the key asserts, proved from the themes alone. The paper's module keeps its declaration11`FRC.<Module>.p<key>` as an alias. Every declaration has no axioms (`check_core_axioms.py`).12-/1314namespace FRC.Ledger1516-- Keys of the hadrons theme (generated by make_keys.py from the ledgers' Lean bindings; edit the ledgers, not this file)17/-- p00070 — 00:G5. \textbf{Strong} (tension): the string tension in finite units, $\sigma=-\ln c_1(\beta)>0$, computed --- the finite-group area law gives $c_1$ as an exact character sum (no Haar, no $\Om\to\infty$); closed form on $\SU(2,\F_3){=}2T$ (matching continuum $\SU(2)$ through $O(\beta^3)$), colour $c_1^{\SU(3)}{=}\tfrac\beta{18}{+}\tfrac{\beta^2}{216}{+}\cdots$. -/18theorem p00070 : (FRC.Interactions.su2.length = (24 : Nat) ∧ (24 : Nat) = (3 : Nat) * ((3 : Nat) * (3 : Nat) - (1 : Nat)) ∧ (FRC.Interactions.su2.all fun x => FRC.Interactions.su2.all fun y => FRC.Interactions.su2.contains (FRC.Interactions.qmul x y)) = true ∧ (FRC.Interactions.su2.any fun x => FRC.Interactions.su2.any fun y => FRC.Interactions.qmul x y != FRC.Interactions.qmul y x) = true ∧ (FRC.Interactions.su2.all fun x => (x.fst + x.fst.conj).im == (0 : FRC.Shell (3 : Nat))) = true ∧ List.map (fun k => (List.filter (fun x => FRC.Interactions.qord x == k) FRC.Interactions.su2).length) [(1 : Nat), (2 : Nat), (3 : Nat), (4 : Nat), (6 : Nat)] = [(1 : Nat), (1 : Nat), (8 : Nat), (6 : Nat), (8 : Nat)] ∧ List.map (fun t => (List.filter (fun x => (x.fst + x.fst.conj).re == FRC.Shell.ofNat t) FRC.Interactions.su2).length) [(0 : Nat), (1 : Nat), (2 : Nat)] = [(6 : Nat), (9 : Nat), (9 : Nat)] ∧ FRC.Interactions.su2.contains ({ re := (0 : FRC.Shell (3 : Nat)), im := (1 : FRC.Shell (3 : Nat)) }, (0 : FRC.Extension.Ext (3 : Nat) (2 : FRC.Shell (3 : Nat)))) = true ∧ (List.filter (fun x => FRC.Interactions.qmul x ({ re := (0 : FRC.Shell (3 : Nat)), im := (1 : FRC.Shell (3 : Nat)) }, (0 : FRC.Extension.Ext (3 : Nat) (2 : FRC.Shell (3 : Nat)))) == FRC.Interactions.qmul ({ re := (0 : FRC.Shell (3 : Nat)), im := (1 : FRC.Shell (3 : Nat)) }, (0 : FRC.Extension.Ext (3 : Nat) (2 : FRC.Shell (3 : Nat)))) x) FRC.Interactions.su2).length = (4 : Nat)) ∧ FRC.Hadrons.moment6 (1 : Nat) (1 : Nat) = (6 : Int) * (1 : Int) ∧ FRC.Hadrons.moment6 (2 : Nat) (0 : Nat) = (0 : Int) ∧ FRC.Hadrons.moment6 (2 : Nat) (2 : Nat) - FRC.Hadrons.moment6 (1 : Nat) (1 : Nat) = (6 : Int) * (1 : Int) ∧ FRC.Hadrons.moment6 (3 : Nat) (1 : Nat) - FRC.Hadrons.moment6 (2 : Nat) (0 : Nat) = (0 : Int) ∧ (2 : Nat) * (3 : Nat) ^ (2 : Nat) = (18 : Nat) ∧ ((2 : Nat) * (3 : Nat)) ^ (2 : Nat) = (36 : Nat) :=19  And.intro @FRC.Interactions.tetrahedral_group (@FRC.Hadrons.adjoint_channel)20/-- p00073 — 00:G8. \textbf{Strong} (hadron spectrum): a baryon is the centre-neutral triality invariant $\varepsilon_{abc}$ of $\SU(3,\F_2)$; the $\SU(3)_F$ mass relations are exact scale-cancelling identities --- Gell-Mann--Okubo, decuplet equal spacing, Coleman--Glashow, the orderings $M_n{>}M_p$ and $M_\Sigma{>}M_\Lambda$, heavy-quark symmetry, the vector-nonet spacing; the decuplet--octet hyperfine is the colour-magnetic curvature invariant, $A_{\mathrm{light}}{=}\sqrt\sigma\,c_{\mathrm{mag}}(\beta)$ the spin-$2$ partner of the tension $c_1$ (G5). Formulas: 35-hadr. -/21theorem p00073 : (FRC.Hadrons.unitv.length = (36 : Nat) ∧ FRC.Hadrons.u3.length = (648 : Nat) ∧ FRC.Hadrons.su3.length = (216 : Nat) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.mm (FRC.Interactions.F4.dg g) g == FRC.Interactions.F4.one && FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ (List.filter (fun g => FRC.Interactions.F4.mm g FRC.Hadrons.D1 == FRC.Interactions.F4.mm FRC.Hadrons.D1 g && FRC.Interactions.F4.mm g FRC.Hadrons.U1 == FRC.Interactions.F4.mm FRC.Hadrons.U1 g) FRC.Hadrons.su3).length = (3 : Nat) ∧ ((List.filter (fun g => FRC.Interactions.F4.mm g FRC.Hadrons.D1 == FRC.Interactions.F4.mm FRC.Hadrons.D1 g && FRC.Interactions.F4.mm g FRC.Hadrons.U1 == FRC.Interactions.F4.mm FRC.Hadrons.U1 g) FRC.Hadrons.su3).all fun g => g == FRC.Interactions.F4.one || g == FRC.Interactions.F4.w.scal || g == FRC.Interactions.F4.w2.scal) = true ∧ ([FRC.Interactions.F4.o, FRC.Interactions.F4.w, FRC.Interactions.F4.w2].all fun x => FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.mm x.scal g == FRC.Interactions.F4.mm g x.scal) = true) ∧ (FRC.Hadrons.fixed1.length = (0 : Nat) ∧ FRC.Hadrons.fixed2.length = (0 : Nat) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ [(1 : Nat), FRC.Hadrons.fixed1.length, FRC.Hadrons.fixed2.length, (1 : Nat)] = [(1 : Nat), (0 : Nat), (0 : Nat), (1 : Nat)] ∧ (1 : Nat) + (3 : Nat) + (3 : Nat) + (1 : Nat) = (2 : Nat) ^ (3 : Nat)) ∧ (∀ {p : Nat} [FRC.Pos p] (m a c : FRC.Shell p), FRC.Hadrons.octN m a c + FRC.Hadrons.octN m a c + (FRC.Hadrons.octX m a c + FRC.Hadrons.octX m a c) + -(FRC.Hadrons.octL m a c + FRC.Hadrons.octL m a c + FRC.Hadrons.octL m a c) + -FRC.Hadrons.octS m a c = (0 : FRC.Shell p)) ∧ (∀ {p : Nat} [FRC.Pos p] (m b : FRC.Shell p), FRC.Hadrons.decD m b + -(FRC.Hadrons.decSs m b + FRC.Hadrons.decSs m b) + FRC.Hadrons.decXs m b = (0 : FRC.Shell p) ∧ FRC.Hadrons.decSs m b + -(FRC.Hadrons.decXs m b + FRC.Hadrons.decXs m b) + FRC.Hadrons.decO m b = (0 : FRC.Shell p)) ∧ (∀ {p : Nat} [FRC.Pos p] (δ : FRC.Shell p), δ + δ + -(δ + δ) = (0 : FRC.Shell p)) ∧ ((4 : Int) + (4 : Int) + (1 : Int) = (9 : Int) ∧ (4 : Int) + (1 : Int) + (1 : Int) = (6 : Int) ∧ (2 : Int) * (2 : Int) + (2 : Int) * (-1 : Int) + (2 : Int) * (-1 : Int) = (0 : Int) ∧ (2 : Int) * (-1 : Int) + (2 : Int) * (-1 : Int) + (-1 : Int) * (-1 : Int) = (-3 : Int) ∧ (9 : Int) - (6 : Int) = (3 : Int) ∧ (0 : Int) - (-3 : Int) = (3 : Int) ∧ (0 : Int) < (3 : Int)) ∧ ((2 : Int) * (0 : Int) * (1 : Int) - (3 : Int) = (-3 : Int) ∧ (2 : Int) * (1 : Int) * (2 : Int) - (3 : Int) = (1 : Int) ∧ (1 : Int) - (-3 : Int) = (4 : Int) ∧ (3 : Int) - (4 : Int) * ((0 : Int) * (1 : Int)) - (3 : Int) = (0 : Int) ∧ (3 : Int) - (4 : Int) * ((1 : Int) * (2 : Int)) - (3 : Int) = (-8 : Int)) ∧ (∀ {p : Nat} [FRC.Pos p] (a b : FRC.Shell p), FRC.Hadrons.vecKs a b + FRC.Hadrons.vecKs a b + -FRC.Hadrons.vecRho a b + -FRC.Hadrons.vecPhi a b = (0 : FRC.Shell p)) ∧ ((1 : Int) * ((1 : Int) + (2 : Int)) - (9 : Int) = (-6 : Int) ∧ (3 : Int) * ((3 : Int) + (2 : Int)) - (9 : Int) = (6 : Int) ∧ (6 : Int) - (-6 : Int) = (12 : Int) ∧ (-8 : Int) * (-6 : Int) = (2 : Int) * (24 : Int) ∧ (-8 : Int) * (6 : Int) = (-2 : Int) * (24 : Int)) ∧ ((3 : Int) * (3 : Int) - (1 : Int) = (8 : Int) ∧ (2 : Nat) * (3 : Nat) * (4 : Nat) = (3 : Nat) * (8 : Nat) ∧ (0 : Int) - (3 : Int) * (8 : Int) = (-24 : Int) ∧ (-24 / 12 : Int) = (-2 : Int) ∧ (4 : Int) * (-2 : Int) = (-8 : Int)) ∧ FRC.Hadrons.moment6 (1 : Nat) (1 : Nat) = (6 : Int) * (1 : Int) ∧ FRC.Hadrons.moment6 (2 : Nat) (0 : Nat) = (0 : Int) ∧ FRC.Hadrons.moment6 (2 : Nat) (2 : Nat) - FRC.Hadrons.moment6 (1 : Nat) (1 : Nat) = (6 : Int) * (1 : Int) ∧ FRC.Hadrons.moment6 (3 : Nat) (1 : Nat) - FRC.Hadrons.moment6 (2 : Nat) (0 : Nat) = (0 : Int) ∧ (2 : Nat) * (3 : Nat) ^ (2 : Nat) = (18 : Nat) ∧ ((2 : Nat) * (3 : Nat)) ^ (2 : Nat) = (36 : Nat) :=22  And.intro @FRC.Hadrons.colour_group (And.intro @FRC.Hadrons.colourless_content (And.intro @FRC.Hadrons.gmo (And.intro @FRC.Hadrons.spacing (And.intro @FRC.Hadrons.coleman_glashow (And.intro @FRC.Hadrons.charge_structure (And.intro @FRC.Hadrons.sigma_lambda (And.intro @FRC.Hadrons.vector_spacing (And.intro @FRC.Hadrons.spin_pattern (And.intro @FRC.Hadrons.colour_factor (@FRC.Hadrons.adjoint_channel))))))))))23/-- p00074 — 00:G9. \textbf{Strong} (the residue series): the observable baryon is the Carrier residue $1$, the determinant $\Lambda^3(\mathbf3){=}\varepsilon_{abc}$, forced by the gluon residue $0$ ($N{=}3$ from $\Lambda^N{=}\det$); the meson is residue $1$ (the $q\bar q$ trace); series closed: photon $2$, gluon $0$, baryon $1$, meson $1$ (G1). Selector, stability: the minimal-wrap state is the stable proton, decay frame-normalisation toward it (D9). -/24theorem p00074 : (FRC.Hadrons.fixed1.length = (0 : Nat) ∧ FRC.Hadrons.fixed2.length = (0 : Nat) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ [(1 : Nat), FRC.Hadrons.fixed1.length, FRC.Hadrons.fixed2.length, (1 : Nat)] = [(1 : Nat), (0 : Nat), (0 : Nat), (1 : Nat)] ∧ (1 : Nat) + (3 : Nat) + (3 : Nat) + (1 : Nat) = (2 : Nat) ^ (3 : Nat)) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.mm (FRC.Interactions.F4.dg g) g == FRC.Interactions.F4.one && FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ ((1 : Nat) + (2 : Nat)) % (3 : Nat) = (0 : Nat) ∧ (3 : Nat) % (3 : Nat) = (0 : Nat) ∧ (1 : Nat) % (3 : Nat) = (1 : Nat) ∧ (2 : Nat) % (3 : Nat) = (2 : Nat) :=25  And.intro @FRC.Hadrons.colourless_content (@FRC.Hadrons.meson_invariant)26/-- p00075 — 00:G10. \textbf{Strong} (the baryon scale): the absolute ground-state spectrum is the $\Om$-hard scale $\sqrt\sigma{=}\Lambda_{\mathrm{QCD}}$ (G6) times sub-horizon eigenvalues; the baryon mass-over-scale is the computed confinement eigenvalue $M_N/\sqrt\sigma{=}E_0{=}2.232$ (the finite operator $H{=}|p|{+}\sqrt\sigma\,r$, converged finite matrix; $4.6\%$); the constituent ratios $\{\lambda_l,\lambda_{\mathrm{hf}}\}$ are evaluated forward from $\{N{=}3,\beta\}$ with no measured baryon mass, so the light-hadron spectrum is parameter-free over $\sqrt\sigma$; no new $\Om$-hard residue. -/27theorem p00075 : (∀ {p : Nat} [FRC.Pos p] (l f : FRC.Shell p), FRC.Hadrons.splitN l f + FRC.Hadrons.splitD l f = l + l + l + l + l + l ∧ FRC.Hadrons.splitD l f + -FRC.Hadrons.splitN l f = f + f + f + f + f + f) ∧ (∀ {p : Nat} [FRC.Pos p] (l s u : FRC.Shell p), FRC.Hadrons.massL l s u + -FRC.Hadrons.massN l s u = s + -l) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (233 : Int) / (100 : Int)) = (0 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (234 : Int) / (100 : Int)) = (1 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (408 : Int) / (100 : Int)) = (1 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (409 : Int) / (100 : Int)) = (2 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (551 : Int) / (100 : Int)) = (2 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (552 : Int) / (100 : Int)) = (3 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (677 : Int) / (100 : Int)) = (3 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (679 : Int) / (100 : Int)) = (4 : Nat) :=28  And.intro @FRC.Hadrons.spin_split (And.intro @FRC.Hadrons.lambda_n_gap (@FRC.Hadrons.airy_sturm))29/-- p00091 — 00:P4. No fourth generation, no fifth force, an empty desert, no TeV supersymmetry, no new dark-matter particle, no beyond-SM $(g{-}2)_\mu$, and the proton effectively stable ($\tau_p\gtrsim10^{45}$ yr; G9). Entailed by the closed ladder of the rank-five frame ($\dim\Lambda^{\mathrm{even}}=16$, D6), given the role-to-matter realisation (27:B6, 27:C10) and completeness (A3). Falsifier: any excluded object observed. Confrontation: none observed. -/30theorem p00091 : (∀ {p : Nat} [FRC.Pos p] {κ : Nat} {g : FRC.Shell p}, FRC.Shell.Frame p κ g → ∀ (j : Nat), g ^ (g ^ j).val = g ^ ((g ^ j).val % (p - (1 : Nat)))) ∧ ((List.filter (fun S => FRC.Interactions.col S == (0 : Nat) && FRC.Interactions.wk S == (0 : Nat) && FRC.Interactions.y6 S == (0 : Int)) FRC.Interactions.gen16).length = (1 : Nat) ∧ (List.filter (fun S => FRC.Interactions.col S == (2 : Nat) && FRC.Interactions.wk S == (0 : Nat) && FRC.Interactions.y6 S == (-4 : Int)) FRC.Interactions.gen16).length = (3 : Nat) ∧ (List.filter (fun S => FRC.Interactions.col S == (1 : Nat) && FRC.Interactions.wk S == (1 : Nat) && FRC.Interactions.y6 S == (1 : Int)) FRC.Interactions.gen16).length = (6 : Nat) ∧ (List.filter (fun S => FRC.Interactions.col S == (0 : Nat) && FRC.Interactions.wk S == (2 : Nat) && FRC.Interactions.y6 S == (6 : Int)) FRC.Interactions.gen16).length = (1 : Nat) ∧ (List.filter (fun S => FRC.Interactions.col S == (3 : Nat) && FRC.Interactions.wk S == (1 : Nat) && FRC.Interactions.y6 S == (-3 : Int)) FRC.Interactions.gen16).length = (2 : Nat) ∧ (List.filter (fun S => FRC.Interactions.col S == (2 : Nat) && FRC.Interactions.wk S == (2 : Nat) && FRC.Interactions.y6 S == (2 : Int)) FRC.Interactions.gen16).length = (3 : Nat) ∧ List.map FRC.Interactions.q6 FRC.Interactions.gen16 = [(0 : Int), (-4 : Int), (-4 : Int), (-4 : Int), (4 : Int), (4 : Int), (4 : Int), (0 : Int), (-2 : Int), (-2 : Int), (-2 : Int), (-6 : Int), (6 : Int), (2 : Int), (2 : Int), (2 : Int)] ∧ FRC.Interactions.isum (List.map FRC.Interactions.y6 FRC.Interactions.gen16) = (0 : Int) ∧ FRC.Interactions.isum (List.map (fun S => FRC.Interactions.y6 S ^ (3 : Nat)) FRC.Interactions.gen16) = (0 : Int) ∧ FRC.Interactions.isum (List.map FRC.Interactions.y6 (List.filter (fun S => FRC.Interactions.col S == (1 : Nat) || FRC.Interactions.col S == (2 : Nat)) FRC.Interactions.gen16)) = (0 : Int) ∧ FRC.Interactions.isum (List.map FRC.Interactions.y6 (List.filter (fun S => FRC.Interactions.wk S == (1 : Nat)) FRC.Interactions.gen16)) = (0 : Int) ∧ (List.filter (fun S => FRC.Interactions.col S == (1 : Nat)) FRC.Interactions.gen16).length = (List.filter (fun S => FRC.Interactions.col S == (2 : Nat)) FRC.Interactions.gen16).length ∧ (List.filter (fun S => FRC.Interactions.wk S == (1 : Nat)) FRC.Interactions.gen16).length % (4 : Nat) = (0 : Nat) ∧ (FRC.Interactions.gen16.contains (0 : Nat) && FRC.Interactions.y6 (0 : Nat) == (0 : Int) && FRC.Interactions.q6 (0 : Nat) == (0 : Int) && FRC.Interactions.col (0 : Nat) == (0 : Nat) && FRC.Interactions.wk (0 : Nat) == (0 : Nat)) = true) ∧ ((24 : Nat) - ((8 : Nat) + (3 : Nat) + (1 : Nat)) = (12 : Nat) ∧ (12 : Nat) = (2 : Nat) * ((3 : Nat) * (2 : Nat)) ∧ (1 : Nat) + (5 : Nat) + (10 : Nat) = (16 : Nat) ∧ (10 : Nat) * (9 : Nat) = (2 : Nat) * (45 : Nat)) ∧ FRC.Hadrons.fixed1.length = (0 : Nat) ∧ FRC.Hadrons.fixed2.length = (0 : Nat) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ [(1 : Nat), FRC.Hadrons.fixed1.length, FRC.Hadrons.fixed2.length, (1 : Nat)] = [(1 : Nat), (0 : Nat), (0 : Nat), (1 : Nat)] ∧ (1 : Nat) + (3 : Nat) + (3 : Nat) + (1 : Nat) = (2 : Nat) ^ (3 : Nat) :=31  And.intro @FRC.Ladder.tower_reduces (And.intro @FRC.Interactions.one_generation (And.intro @FRC.Interactions.unification_counts (@FRC.Hadrons.colourless_content)))32/-- p35015 — 35:C1. Colour $\SU(3,\F_2)$ built exactly: order $216$, centre $\Z_3$ the triality centre; the colour-singlet baryon is the centre-neutral $\varepsilon_{abc}$ invariant $\Lambda^3\mathbf3=\mathbf1$. -/33theorem p35015 : (FRC.Hadrons.unitv.length = (36 : Nat) ∧ FRC.Hadrons.u3.length = (648 : Nat) ∧ FRC.Hadrons.su3.length = (216 : Nat) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.mm (FRC.Interactions.F4.dg g) g == FRC.Interactions.F4.one && FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ (List.filter (fun g => FRC.Interactions.F4.mm g FRC.Hadrons.D1 == FRC.Interactions.F4.mm FRC.Hadrons.D1 g && FRC.Interactions.F4.mm g FRC.Hadrons.U1 == FRC.Interactions.F4.mm FRC.Hadrons.U1 g) FRC.Hadrons.su3).length = (3 : Nat) ∧ ((List.filter (fun g => FRC.Interactions.F4.mm g FRC.Hadrons.D1 == FRC.Interactions.F4.mm FRC.Hadrons.D1 g && FRC.Interactions.F4.mm g FRC.Hadrons.U1 == FRC.Interactions.F4.mm FRC.Hadrons.U1 g) FRC.Hadrons.su3).all fun g => g == FRC.Interactions.F4.one || g == FRC.Interactions.F4.w.scal || g == FRC.Interactions.F4.w2.scal) = true ∧ ([FRC.Interactions.F4.o, FRC.Interactions.F4.w, FRC.Interactions.F4.w2].all fun x => FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.mm x.scal g == FRC.Interactions.F4.mm g x.scal) = true) ∧ ([FRC.Interactions.F4.o, FRC.Interactions.F4.w, FRC.Interactions.F4.w2].all fun x => x * x.conj == FRC.Interactions.F4.o && x * x * x == FRC.Interactions.F4.o) = true ∧ (FRC.Interactions.F4.z * FRC.Interactions.F4.z.conj == FRC.Interactions.F4.o) = false ∧ [FRC.Interactions.F4.o, FRC.Interactions.F4.w, FRC.Interactions.F4.w2].length = (3 : Nat) ∧ (2 : Nat) + (1 : Nat) = (3 : Nat) ∧ ([FRC.Interactions.F4.o, FRC.Interactions.F4.w, FRC.Interactions.F4.w2].all fun x => FRC.Interactions.F4.mm x.scal [FRC.Interactions.F4.z, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.z] == FRC.Interactions.F4.mm [FRC.Interactions.F4.z, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.z] x.scal && FRC.Interactions.F4.mm x.scal [FRC.Interactions.F4.z, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.w, FRC.Interactions.F4.z, FRC.Interactions.F4.w2, FRC.Interactions.F4.z, FRC.Interactions.F4.z] == FRC.Interactions.F4.mm [FRC.Interactions.F4.z, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.w, FRC.Interactions.F4.z, FRC.Interactions.F4.w2, FRC.Interactions.F4.z, FRC.Interactions.F4.z] x.scal) = true :=34  And.intro @FRC.Hadrons.colour_group (@FRC.Interactions.su3_centre)35/-- p35016 — 35:C2. Gell-Mann--Okubo $\tfrac12(N+\Xi)=\tfrac14(3\Lambda+\Sigma)$: $2N+2\Xi-3\Lambda-\Sigma=0$ identically, scale-cancelling. -/36theorem p35016 : ∀ {p : Nat} [FRC.Pos p] (m a c : FRC.Shell p), FRC.Hadrons.octN m a c + FRC.Hadrons.octN m a c + (FRC.Hadrons.octX m a c + FRC.Hadrons.octX m a c) + -(FRC.Hadrons.octL m a c + FRC.Hadrons.octL m a c + FRC.Hadrons.octL m a c) + -FRC.Hadrons.octS m a c = (0 : FRC.Shell p) :=37  @FRC.Hadrons.gmo38/-- p35017 — 35:C3. Decuplet equal spacing $\Sigma^{*}-\Delta=\Xi^{*}-\Sigma^{*}=\Omega-\Xi^{*}=\beta$: the second differences vanish identically. -/39theorem p35017 : ∀ {p : Nat} [FRC.Pos p] (m b : FRC.Shell p), FRC.Hadrons.decD m b + -(FRC.Hadrons.decSs m b + FRC.Hadrons.decSs m b) + FRC.Hadrons.decXs m b = (0 : FRC.Shell p) ∧ FRC.Hadrons.decSs m b + -(FRC.Hadrons.decXs m b + FRC.Hadrons.decXs m b) + FRC.Hadrons.decO m b = (0 : FRC.Shell p) :=40  @FRC.Hadrons.spacing41/-- p35018 — 35:C4. Colour factor of the singlet $\sum_{i<j}\bvec\lambda_i\!\cdot\!\bvec\lambda_j=-8$ ($-\tfrac83$ per pair) from $C_2(\mathbf3)=\tfrac43$, $C_2(\mathbf1)=0$. -/42theorem p35018 : (3 : Int) * (3 : Int) - (1 : Int) = (8 : Int) ∧ (2 : Nat) * (3 : Nat) * (4 : Nat) = (3 : Nat) * (8 : Nat) ∧ (0 : Int) - (3 : Int) * (8 : Int) = (-24 : Int) ∧ (-24 / 12 : Int) = (-2 : Int) ∧ (4 : Int) * (-2 : Int) = (-8 : Int) :=43  @FRC.Hadrons.colour_factor44/-- p35019 — 35:C5. Spin structure $g_{\mathrm{spin}}=\half S(S+1)-\tfrac98$, octet $-\tfrac34$, decuplet $+\tfrac34$; eigenvalues $+2,-2$; separation $M_{\mathbf{10}}-M_{\mathbf8}=\tfrac32A_{\mathrm{light}}$. -/45theorem p35019 : (1 : Int) * ((1 : Int) + (2 : Int)) - (9 : Int) = (-6 : Int) ∧ (3 : Int) * ((3 : Int) + (2 : Int)) - (9 : Int) = (6 : Int) ∧ (6 : Int) - (-6 : Int) = (12 : Int) ∧ (-8 : Int) * (-6 : Int) = (2 : Int) * (24 : Int) ∧ (-8 : Int) * (6 : Int) = (-2 : Int) * (24 : Int) :=46  @FRC.Hadrons.spin_pattern47/-- p35020 — 35:C6. $A_{\mathrm{light}}=\sig\,c_{\mathrm{mag}}(\beta)$, with $c_{\mathrm{mag}}$ a colour-curvature character sum of the $c_1$ family ($c_1=\beta/18+\cdots$ reproduced); the dimensionless pattern exact, the absolute scale $\Om$-hard. -/48theorem p35020 : FRC.Hadrons.moment6 (1 : Nat) (1 : Nat) = (6 : Int) * (1 : Int) ∧ FRC.Hadrons.moment6 (2 : Nat) (0 : Nat) = (0 : Int) ∧ FRC.Hadrons.moment6 (2 : Nat) (2 : Nat) - FRC.Hadrons.moment6 (1 : Nat) (1 : Nat) = (6 : Int) * (1 : Int) ∧ FRC.Hadrons.moment6 (3 : Nat) (1 : Nat) - FRC.Hadrons.moment6 (2 : Nat) (0 : Nat) = (0 : Int) ∧ (2 : Nat) * (3 : Nat) ^ (2 : Nat) = (18 : Nat) ∧ ((2 : Nat) * (3 : Nat)) ^ (2 : Nat) = (36 : Nat) :=49  @FRC.Hadrons.adjoint_channel50/-- p35021 — 35:C7. Coleman--Glashow $(n-p)+(\Xi^-{-}\Xi^0)=\Sigma^-{-}\Sigma^+$: an exact one-body identity, the two-body electromagnetic term cancelling. -/51theorem p35021 : ∀ {p : Nat} [FRC.Pos p] (δ : FRC.Shell p), δ + δ + -(δ + δ) = (0 : FRC.Shell p) :=52  @FRC.Hadrons.coleman_glashow53/-- p35022 — 35:C8. The ordering $M_n>M_p$: the $d-u$ seed exceeds the charge-squared electromagnetic term ($\sum Q^2=1$ for $p$, $\tfrac23$ for $n$), fixing $m_d>m_u$. -/54theorem p35022 : (4 : Int) + (4 : Int) + (1 : Int) = (9 : Int) ∧ (4 : Int) + (1 : Int) + (1 : Int) = (6 : Int) ∧ (2 : Int) * (2 : Int) + (2 : Int) * (-1 : Int) + (2 : Int) * (-1 : Int) = (0 : Int) ∧ (2 : Int) * (-1 : Int) + (2 : Int) * (-1 : Int) + (-1 : Int) * (-1 : Int) = (-3 : Int) ∧ (9 : Int) - (6 : Int) = (3 : Int) ∧ (0 : Int) - (-3 : Int) = (3 : Int) ∧ (0 : Int) < (3 : Int) :=55  @FRC.Hadrons.charge_structure56/-- p35023 — 35:C9. $M_\Sigma>M_\Lambda$: hyperfine fine structure, the light $(ud)$ pair spin-$1$ vs spin-$0$, $M_\Sigma-M_\Lambda=a_{ll}(1-m_l/m_s)>0$. -/57theorem p35023 : (2 : Int) * (0 : Int) * (1 : Int) - (3 : Int) = (-3 : Int) ∧ (2 : Int) * (1 : Int) * (2 : Int) - (3 : Int) = (1 : Int) ∧ (1 : Int) - (-3 : Int) = (4 : Int) ∧ (3 : Int) - (4 : Int) * ((0 : Int) * (1 : Int)) - (3 : Int) = (0 : Int) ∧ (3 : Int) - (4 : Int) * ((1 : Int) * (2 : Int)) - (3 : Int) = (-8 : Int) :=58  @FRC.Hadrons.sigma_lambda59/-- p35025 — 35:C11. Second-order decuplet relation $M_\Delta-3M_{\Sigma^*}+3M_{\Xi^*}-M_\Omega=0$ (vanishing third difference), exact in the linear-seed pairwise-hyperfine model. -/60theorem p35025 : ∀ {p : Nat} [FRC.Pos p] (m d a r : FRC.Shell p), FRC.Hadrons.hfD m d a r + -(FRC.Hadrons.hfSs m d a r + FRC.Hadrons.hfSs m d a r + FRC.Hadrons.hfSs m d a r) + (FRC.Hadrons.hfXs m d a r + FRC.Hadrons.hfXs m d a r + FRC.Hadrons.hfXs m d a r) + -FRC.Hadrons.hfO m d a r = (0 : FRC.Shell p) :=61  @FRC.Hadrons.third_difference62/-- p35026 — 35:C12. Octet--decuplet hyperfine links $M_\Sigma-M_\Lambda=\tfrac23[(M_\Delta-M_N)-(M_{\Sigma^*}-M_\Sigma)]$ and $M_{\Sigma^*}-M_\Sigma=M_{\Xi^*}-M_\Xi$, parameter-free. -/63theorem p35026 : ∀ {p : Nat} [FRC.Pos p] (m d a r : FRC.Shell p), FRC.Hadrons.hfS m d a r + -FRC.Hadrons.hfL m d a r + (FRC.Hadrons.hfS m d a r + -FRC.Hadrons.hfL m d a r) + (FRC.Hadrons.hfS m d a r + -FRC.Hadrons.hfL m d a r) = FRC.Hadrons.hfD m d a r + -FRC.Hadrons.hfN m d a r + -(FRC.Hadrons.hfSs m d a r + -FRC.Hadrons.hfS m d a r) + (FRC.Hadrons.hfD m d a r + -FRC.Hadrons.hfN m d a r + -(FRC.Hadrons.hfSs m d a r + -FRC.Hadrons.hfS m d a r)) ∧ FRC.Hadrons.hfSs m d a r + -FRC.Hadrons.hfS m d a r + -(FRC.Hadrons.hfXs m d a r + -FRC.Hadrons.hfX m d a r) = (0 : FRC.Shell p) :=64  @FRC.Hadrons.hyperfine_links65/-- p35027 — 35:C13. EM splitting structure: one-body $\sum Q_i^2$ and two-body $\sum_{i<j}Q_iQ_j$ exact ($p$: $1,0$; $n$: $\tfrac23,-\tfrac13$); $\Delta_{\mathrm{EM}}=\alpha\,\sig\times$structure, the absolute reducing to the imported $\alpha$ (A9) and $\sig$ (A3), no new residue. -/66theorem p35027 : (4 : Int) + (4 : Int) + (1 : Int) = (9 : Int) ∧ (4 : Int) + (1 : Int) + (1 : Int) = (6 : Int) ∧ (2 : Int) * (2 : Int) + (2 : Int) * (-1 : Int) + (2 : Int) * (-1 : Int) = (0 : Int) ∧ (2 : Int) * (-1 : Int) + (2 : Int) * (-1 : Int) + (-1 : Int) * (-1 : Int) = (-3 : Int) ∧ (9 : Int) - (6 : Int) = (3 : Int) ∧ (0 : Int) - (-3 : Int) = (3 : Int) ∧ (0 : Int) < (3 : Int) :=67  @FRC.Hadrons.charge_structure68/-- p35028 — 35:C14. $c_{\mathrm{mag}}$ the adjoint (colour-octet) plaquette character sum, $\mathbf8\subset\mathbf3\otimes\bar{\mathbf3}$ ($\mathbf8\not\subset\mathbf3\otimes\mathbf3$), leading $\beta^2/36$ at $O(\beta^2)$, one order finer than the tension $c_1=\beta/18$. -/69theorem p35028 : FRC.Hadrons.moment6 (1 : Nat) (1 : Nat) = (6 : Int) * (1 : Int) ∧ FRC.Hadrons.moment6 (2 : Nat) (0 : Nat) = (0 : Int) ∧ FRC.Hadrons.moment6 (2 : Nat) (2 : Nat) - FRC.Hadrons.moment6 (1 : Nat) (1 : Nat) = (6 : Int) * (1 : Int) ∧ FRC.Hadrons.moment6 (3 : Nat) (1 : Nat) - FRC.Hadrons.moment6 (2 : Nat) (0 : Nat) = (0 : Int) ∧ (2 : Nat) * (3 : Nat) ^ (2 : Nat) = (18 : Nat) ∧ ((2 : Nat) * (3 : Nat)) ^ (2 : Nat) = (36 : Nat) :=70  @FRC.Hadrons.adjoint_channel71/-- p35029 — 35:C15. Baryon $=$ Carrier residue $1$: the colourless content of $\Lambda^\bullet(\mathbf3)$ is $(1,0,0,1)$; the baryon is the determinant $\Lambda^3=\det$ at $k=N=3$ (residue $1$, baryon number), forced by the gluon residue $0$ (the fundamental has no invariant vector on $\SU(3,\F_2)$); parallel to photon $2$, gluon $0$. -/72theorem p35029 : FRC.Hadrons.fixed1.length = (0 : Nat) ∧ FRC.Hadrons.fixed2.length = (0 : Nat) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ [(1 : Nat), FRC.Hadrons.fixed1.length, FRC.Hadrons.fixed2.length, (1 : Nat)] = [(1 : Nat), (0 : Nat), (0 : Nat), (1 : Nat)] ∧ (1 : Nat) + (3 : Nat) + (3 : Nat) + (1 : Nat) = (2 : Nat) ^ (3 : Nat) :=73  @FRC.Hadrons.colourless_content74/-- p35030 — 35:C16. Single-scale reduction: $M_H=\sig\,\lambda_H$, the three anchored masses read as one $\Om$-hard scale $\sig$ and three sub-horizon numbers ($\lambda_l$, $m_s/m_l$, $\lambda_{\mathrm{hf}}$); $M_\Lambda-M_N=m_s-m_l$ exact (hyperfine-cancelling); the finite-stage eigenvalues $\varepsilon_k^{(G)}\in\mathrm{Spec}(H^{(N)})$ exact, the decimal Airy values the [approx] readout, $\Om$-stable. -/75theorem p35030 : (FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (233 : Int) / (100 : Int)) = (0 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (234 : Int) / (100 : Int)) = (1 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (408 : Int) / (100 : Int)) = (1 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (409 : Int) / (100 : Int)) = (2 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (551 : Int) / (100 : Int)) = (2 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (552 : Int) / (100 : Int)) = (3 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (677 : Int) / (100 : Int)) = (3 : Nat) ∧ FRC.Hadrons.sturm (200 : Nat) (16 : Nat) ((201 : Int) * (256 : Int) * (679 : Int) / (100 : Int)) = (4 : Nat)) ∧ (∀ {p : Nat} [FRC.Pos p] (l s u : FRC.Shell p), FRC.Hadrons.massL l s u + -FRC.Hadrons.massN l s u = s + -l) ∧ ∀ {p : Nat} [FRC.Pos p] (l f : FRC.Shell p), FRC.Hadrons.splitN l f + FRC.Hadrons.splitD l f = l + l + l + l + l + l ∧ FRC.Hadrons.splitD l f + -FRC.Hadrons.splitN l f = f + f + f + f + f + f :=76  And.intro @FRC.Hadrons.airy_sturm (And.intro @FRC.Hadrons.lambda_n_gap (@FRC.Hadrons.spin_split))77/-- p35031 — 35:C17. The baryon scale: the computed finite eigenvalue $E_0\simeq2.232$ (operator $H_G=T_G^{1/2}+R_G$, no baryon mass entering); $M_N=E_0\sig$ predicts the nucleon to $4.6\%$, the absolute prediction carrying the imported $\Om$-hard $\sig$ (A3). $\lambda_l,\lambda_{\mathrm{hf}}$ evaluated forward from $\{N,\beta\}$ (C19), $\Om$-stable. -/78theorem p35031 : ∀ {p : Nat} [FRC.Pos p] (l f : FRC.Shell p), FRC.Hadrons.splitN l f + FRC.Hadrons.splitD l f = l + l + l + l + l + l ∧ FRC.Hadrons.splitD l f + -FRC.Hadrons.splitN l f = f + f + f + f + f + f :=79  @FRC.Hadrons.spin_split80/-- p35032 — 35:C18. Meson Carrier residue: the $q\bar q$ colour singlet is the contraction of $\mathbf3\otimes\bar{\mathbf3}=\mathbf1\oplus\mathbf8$ (the Hermitian invariant), residue $1$, $B=0$ (triality $0$); the residue series closes (photon $2$, gluon $0$, baryon $1$, meson $1$); on the same frame the vector-nonet equal spacing $2M_{K^{*}}=M_\rho+M_\phi$ is an exact identity of linear strangeness. -/81theorem p35032 : ((FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.mm (FRC.Interactions.F4.dg g) g == FRC.Interactions.F4.one && FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ ((1 : Nat) + (2 : Nat)) % (3 : Nat) = (0 : Nat) ∧ (3 : Nat) % (3 : Nat) = (0 : Nat) ∧ (1 : Nat) % (3 : Nat) = (1 : Nat) ∧ (2 : Nat) % (3 : Nat) = (2 : Nat)) ∧ ∀ {p : Nat} [FRC.Pos p] (a b : FRC.Shell p), FRC.Hadrons.vecKs a b + FRC.Hadrons.vecKs a b + -FRC.Hadrons.vecRho a b + -FRC.Hadrons.vecPhi a b = (0 : FRC.Shell p) :=82  And.intro @FRC.Hadrons.meson_invariant (@FRC.Hadrons.vector_spacing)83/-- p35033 — 35:C19. Forward constituent eigenvalues: $\lambda_l,\lambda_{\mathrm{hf}}$ computed from $\{N{=}3,\beta\}$ with no measured baryon mass --- $\lambda_{\mathrm{hf}}$ the colour-magnetic contact ($c_{\mathrm{adj}}{=}\beta^2/36$, the exact overlap $|u'(0)|^2{=}1$, $\alpha_s(\sig)$ transmutation-pinned), $\lambda_l$ from $E_0$ and the exact spin split; matching $0.44,0.85$ to constituent-model accuracy, the margin set by the linear-potential approximation. -/84theorem p35033 : ∀ {p : Nat} [FRC.Pos p] (l f : FRC.Shell p), FRC.Hadrons.splitN l f + FRC.Hadrons.splitD l f = l + l + l + l + l + l ∧ FRC.Hadrons.splitD l f + -FRC.Hadrons.splitN l f = f + f + f + f + f + f :=85  @FRC.Hadrons.spin_split86/-- p35034 — 35:C20. Residue resolution, second order: the $\SU(3)_F$ residuals (decuplet third difference $6$ MeV, octet $\mathbf{27}$-plet $0.57\%$) are sub-horizon, $O(\varepsilon_s^2)$ of a convergent strange-breaking expansion, $\Om$-stable (Prop.~\ref{prop:subhorizon}). -/87theorem p35034 : (∀ (k : Nat), ((12 : Nat) * k + (5 : Nat) + (1 : Nat)) % (3 : Nat) = (0 : Nat)) ∧ ([(53 : Nat), (173 : Nat), (389 : Nat), (1373 : Nat)].all fun n => n % (12 : Nat) == (5 : Nat) && (n + (1 : Nat)) % (3 : Nat) == (0 : Nat)) = true :=88  @FRC.Hadrons.admissible_shells89/-- p35035 — 35:C21. Residue resolution, electromagnetic: the splitting magnitude $\Delta_{\mathrm{EM}}=\alpha\,\sig\times$[exact charge structure] reduces to the imported $\alpha$ (A9) and $\sig$ (A3), no new residue (C13); Coleman--Glashow and the orderings derived (C7--C9). -/90theorem p35035 : (4 : Int) + (4 : Int) + (1 : Int) = (9 : Int) ∧ (4 : Int) + (1 : Int) + (1 : Int) = (6 : Int) ∧ (2 : Int) * (2 : Int) + (2 : Int) * (-1 : Int) + (2 : Int) * (-1 : Int) = (0 : Int) ∧ (2 : Int) * (-1 : Int) + (2 : Int) * (-1 : Int) + (-1 : Int) * (-1 : Int) = (-3 : Int) ∧ (9 : Int) - (6 : Int) = (3 : Int) ∧ (0 : Int) - (-3 : Int) = (3 : Int) ∧ (0 : Int) < (3 : Int) :=91  @FRC.Hadrons.charge_structure92/-- p35037 — 35:C23. Residue resolution, hyperfine: the dimensionless $A_{\mathrm{light}}/\sig$ is a ratio of $\Om$-independent character sums (C14), $\Om$-stable, sub-horizon; the absolute $A_{\mathrm{light}}=\sig\,c_{\mathrm{mag}}$ is $\sig$-scaled (A3, Prop.~\ref{prop:subhorizon}). -/93theorem p35037 : ((∀ (k : Nat), ((12 : Nat) * k + (5 : Nat) + (1 : Nat)) % (3 : Nat) = (0 : Nat)) ∧ ([(53 : Nat), (173 : Nat), (389 : Nat), (1373 : Nat)].all fun n => n % (12 : Nat) == (5 : Nat) && (n + (1 : Nat)) % (3 : Nat) == (0 : Nat)) = true) ∧ FRC.Hadrons.moment6 (1 : Nat) (1 : Nat) = (6 : Int) * (1 : Int) ∧ FRC.Hadrons.moment6 (2 : Nat) (0 : Nat) = (0 : Int) ∧ FRC.Hadrons.moment6 (2 : Nat) (2 : Nat) - FRC.Hadrons.moment6 (1 : Nat) (1 : Nat) = (6 : Int) * (1 : Int) ∧ FRC.Hadrons.moment6 (3 : Nat) (1 : Nat) - FRC.Hadrons.moment6 (2 : Nat) (0 : Nat) = (0 : Int) ∧ (2 : Nat) * (3 : Nat) ^ (2 : Nat) = (18 : Nat) ∧ ((2 : Nat) * (3 : Nat)) ^ (2 : Nat) = (36 : Nat) :=94  And.intro @FRC.Hadrons.admissible_shells (@FRC.Hadrons.adjoint_channel)95/-- p35038 — 35:X1. \textbf{The baryon is three quarks} because the only colourless escape from the gluon residue $0$ is the determinant $\Lambda^N=\det$, which requires exactly $N=3$. -/96theorem p35038 : FRC.Hadrons.fixed1.length = (0 : Nat) ∧ FRC.Hadrons.fixed2.length = (0 : Nat) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ [(1 : Nat), FRC.Hadrons.fixed1.length, FRC.Hadrons.fixed2.length, (1 : Nat)] = [(1 : Nat), (0 : Nat), (0 : Nat), (1 : Nat)] ∧ (1 : Nat) + (3 : Nat) + (3 : Nat) + (1 : Nat) = (2 : Nat) ^ (3 : Nat) :=97  @FRC.Hadrons.colourless_content98/-- p35039 — 35:X2. \textbf{The baryon is colour-neutral} as the centre-neutral, triality-$0$ invariant $\varepsilon_{abc}$ of the triality frame, a derived singlet. -/99theorem p35039 : (FRC.Hadrons.unitv.length = (36 : Nat) ∧ FRC.Hadrons.u3.length = (648 : Nat) ∧ FRC.Hadrons.su3.length = (216 : Nat) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.mm (FRC.Interactions.F4.dg g) g == FRC.Interactions.F4.one && FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ (List.filter (fun g => FRC.Interactions.F4.mm g FRC.Hadrons.D1 == FRC.Interactions.F4.mm FRC.Hadrons.D1 g && FRC.Interactions.F4.mm g FRC.Hadrons.U1 == FRC.Interactions.F4.mm FRC.Hadrons.U1 g) FRC.Hadrons.su3).length = (3 : Nat) ∧ ((List.filter (fun g => FRC.Interactions.F4.mm g FRC.Hadrons.D1 == FRC.Interactions.F4.mm FRC.Hadrons.D1 g && FRC.Interactions.F4.mm g FRC.Hadrons.U1 == FRC.Interactions.F4.mm FRC.Hadrons.U1 g) FRC.Hadrons.su3).all fun g => g == FRC.Interactions.F4.one || g == FRC.Interactions.F4.w.scal || g == FRC.Interactions.F4.w2.scal) = true ∧ ([FRC.Interactions.F4.o, FRC.Interactions.F4.w, FRC.Interactions.F4.w2].all fun x => FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.mm x.scal g == FRC.Interactions.F4.mm g x.scal) = true) ∧ ([FRC.Interactions.F4.o, FRC.Interactions.F4.w, FRC.Interactions.F4.w2].all fun x => x * x.conj == FRC.Interactions.F4.o && x * x * x == FRC.Interactions.F4.o) = true ∧ (FRC.Interactions.F4.z * FRC.Interactions.F4.z.conj == FRC.Interactions.F4.o) = false ∧ [FRC.Interactions.F4.o, FRC.Interactions.F4.w, FRC.Interactions.F4.w2].length = (3 : Nat) ∧ (2 : Nat) + (1 : Nat) = (3 : Nat) ∧ ([FRC.Interactions.F4.o, FRC.Interactions.F4.w, FRC.Interactions.F4.w2].all fun x => FRC.Interactions.F4.mm x.scal [FRC.Interactions.F4.z, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.z] == FRC.Interactions.F4.mm [FRC.Interactions.F4.z, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.z] x.scal && FRC.Interactions.F4.mm x.scal [FRC.Interactions.F4.z, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.w, FRC.Interactions.F4.z, FRC.Interactions.F4.w2, FRC.Interactions.F4.z, FRC.Interactions.F4.z] == FRC.Interactions.F4.mm [FRC.Interactions.F4.z, FRC.Interactions.F4.z, FRC.Interactions.F4.o, FRC.Interactions.F4.z, FRC.Interactions.F4.w, FRC.Interactions.F4.z, FRC.Interactions.F4.w2, FRC.Interactions.F4.z, FRC.Interactions.F4.z] x.scal) = true :=100  And.intro @FRC.Hadrons.colour_group (@FRC.Interactions.su3_centre)101/-- p35040 — 35:X3. \textbf{Baryon number is the Carrier residue $1$}, the determinant winding, so its conservation is the invariance of $\Lambda^3$, on the same footing as the photon residue $2$ and the gluon residue $0$. -/102theorem p35040 : FRC.Hadrons.fixed1.length = (0 : Nat) ∧ FRC.Hadrons.fixed2.length = (0 : Nat) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ [(1 : Nat), FRC.Hadrons.fixed1.length, FRC.Hadrons.fixed2.length, (1 : Nat)] = [(1 : Nat), (0 : Nat), (0 : Nat), (1 : Nat)] ∧ (1 : Nat) + (3 : Nat) + (3 : Nat) + (1 : Nat) = (2 : Nat) ^ (3 : Nat) :=103  @FRC.Hadrons.colourless_content104/-- p35041 — 35:X4. \textbf{The Gell-Mann--Okubo relation} is exact, the scale-cancelling identity $2N+2\Xi-3\Lambda-\Sigma=0$ of linear flavour breaking, a derived sum rule. -/105theorem p35041 : ∀ {p : Nat} [FRC.Pos p] (m a c : FRC.Shell p), FRC.Hadrons.octN m a c + FRC.Hadrons.octN m a c + (FRC.Hadrons.octX m a c + FRC.Hadrons.octX m a c) + -(FRC.Hadrons.octL m a c + FRC.Hadrons.octL m a c + FRC.Hadrons.octL m a c) + -FRC.Hadrons.octS m a c = (0 : FRC.Shell p) :=106  @FRC.Hadrons.gmo107/-- p35042 — 35:X5. \textbf{The decuplet is equally spaced} because the strange seed enters the wrap additively. -/108theorem p35042 : ∀ {p : Nat} [FRC.Pos p] (m b : FRC.Shell p), FRC.Hadrons.decD m b + -(FRC.Hadrons.decSs m b + FRC.Hadrons.decSs m b) + FRC.Hadrons.decXs m b = (0 : FRC.Shell p) ∧ FRC.Hadrons.decSs m b + -(FRC.Hadrons.decXs m b + FRC.Hadrons.decXs m b) + FRC.Hadrons.decO m b = (0 : FRC.Shell p) :=109  @FRC.Hadrons.spacing110/-- p35043 — 35:X6. \textbf{The decuplet is heavier than the octet} by the colour-magnetic curvature invariant, the colour factor $-8$ times the spin $\half S(S+1)-\tfrac98$, a derived sign and pattern. -/111theorem p35043 : ((1 : Int) * ((1 : Int) + (2 : Int)) - (9 : Int) = (-6 : Int) ∧ (3 : Int) * ((3 : Int) + (2 : Int)) - (9 : Int) = (6 : Int) ∧ (6 : Int) - (-6 : Int) = (12 : Int) ∧ (-8 : Int) * (-6 : Int) = (2 : Int) * (24 : Int) ∧ (-8 : Int) * (6 : Int) = (-2 : Int) * (24 : Int)) ∧ (3 : Int) * (3 : Int) - (1 : Int) = (8 : Int) ∧ (2 : Nat) * (3 : Nat) * (4 : Nat) = (3 : Nat) * (8 : Nat) ∧ (0 : Int) - (3 : Int) * (8 : Int) = (-24 : Int) ∧ (-24 / 12 : Int) = (-2 : Int) ∧ (4 : Int) * (-2 : Int) = (-8 : Int) :=112  And.intro @FRC.Hadrons.spin_pattern (@FRC.Hadrons.colour_factor)113/-- p35044 — 35:X7. \textbf{The proton is stable} as the minimal-wrap residue-$1$ state, with no lighter $B=1$ residue to reach. -/114theorem p35044 : FRC.Hadrons.fixed1.length = (0 : Nat) ∧ FRC.Hadrons.fixed2.length = (0 : Nat) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ [(1 : Nat), FRC.Hadrons.fixed1.length, FRC.Hadrons.fixed2.length, (1 : Nat)] = [(1 : Nat), (0 : Nat), (0 : Nat), (1 : Nat)] ∧ (1 : Nat) + (3 : Nat) + (3 : Nat) + (1 : Nat) = (2 : Nat) ^ (3 : Nat) :=115  @FRC.Hadrons.colourless_content116/-- p35045 — 35:X8. \textbf{The neutron is heavier than the proton} because the $d-u$ seed exceeds the charge-squared electromagnetic term ($\sum Q^2=1$ for $p$, $\tfrac23$ for $n$), a derived ordering. -/117theorem p35045 : (4 : Int) + (4 : Int) + (1 : Int) = (9 : Int) ∧ (4 : Int) + (1 : Int) + (1 : Int) = (6 : Int) ∧ (2 : Int) * (2 : Int) + (2 : Int) * (-1 : Int) + (2 : Int) * (-1 : Int) = (0 : Int) ∧ (2 : Int) * (-1 : Int) + (2 : Int) * (-1 : Int) + (-1 : Int) * (-1 : Int) = (-3 : Int) ∧ (9 : Int) - (6 : Int) = (3 : Int) ∧ (0 : Int) - (-3 : Int) = (3 : Int) ∧ (0 : Int) < (3 : Int) :=118  @FRC.Hadrons.charge_structure119/-- p35046 — 35:X9. \textbf{$\Sigma$ is heavier than $\Lambda$} by the spin-$1$ versus spin-$0$ light-pair hyperfine, a derived splitting of two states of identical quark content. -/120theorem p35046 : (2 : Int) * (0 : Int) * (1 : Int) - (3 : Int) = (-3 : Int) ∧ (2 : Int) * (1 : Int) * (2 : Int) - (3 : Int) = (1 : Int) ∧ (1 : Int) - (-3 : Int) = (4 : Int) ∧ (3 : Int) - (4 : Int) * ((0 : Int) * (1 : Int)) - (3 : Int) = (0 : Int) ∧ (3 : Int) - (4 : Int) * ((1 : Int) * (2 : Int)) - (3 : Int) = (-8 : Int) :=121  @FRC.Hadrons.sigma_lambda122/-- p35047 — 35:X10. \textbf{The isospin splittings obey Coleman--Glashow} exactly at one body, $(n{-}p)+(\Xi^-{-}\Xi^0)=\Sigma^-{-}\Sigma^+$, a derived identity. -/123theorem p35047 : ∀ {p : Nat} [FRC.Pos p] (δ : FRC.Shell p), δ + δ + -(δ + δ) = (0 : FRC.Shell p) :=124  @FRC.Hadrons.coleman_glashow125/-- p35049 — 35:X12. \textbf{The vector mesons equally space}, $2M_{K^{*}}=M_\rho+M_\phi$, the meson residue $1$ ($B=0$) on the same frame and scale as the baryons. -/126theorem p35049 : (∀ {p : Nat} [FRC.Pos p] (a b : FRC.Shell p), FRC.Hadrons.vecKs a b + FRC.Hadrons.vecKs a b + -FRC.Hadrons.vecRho a b + -FRC.Hadrons.vecPhi a b = (0 : FRC.Shell p)) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.mm (FRC.Interactions.F4.dg g) g == FRC.Interactions.F4.one && FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ ((1 : Nat) + (2 : Nat)) % (3 : Nat) = (0 : Nat) ∧ (3 : Nat) % (3 : Nat) = (0 : Nat) ∧ (1 : Nat) % (3 : Nat) = (1 : Nat) ∧ (2 : Nat) % (3 : Nat) = (2 : Nat) :=127  And.intro @FRC.Hadrons.vector_spacing (@FRC.Hadrons.meson_invariant)128/-- p35050 — 35:X13. \textbf{The spectrum rests on one scale}, $\sqrt\sigma=\LQCD$, the lone $\Om$-hard residue; every dimensionless mass ratio is a sub-horizon eigenvalue. -/129theorem p35050 : (∀ (k : Nat), ((12 : Nat) * k + (5 : Nat) + (1 : Nat)) % (3 : Nat) = (0 : Nat)) ∧ ([(53 : Nat), (173 : Nat), (389 : Nat), (1373 : Nat)].all fun n => n % (12 : Nat) == (5 : Nat) && (n + (1 : Nat)) % (3 : Nat) == (0 : Nat)) = true :=130  @FRC.Hadrons.admissible_shells131/-- p35051 — 35:P1. Gell-Mann--Okubo exact at leading order; PDG $0.57\%$, the bounded $\chi_3^2$ second order. Falsifier: a violation beyond the second-order insertion. -/132theorem p35051 : ∀ {p : Nat} [FRC.Pos p] (m a c : FRC.Shell p), FRC.Hadrons.octN m a c + FRC.Hadrons.octN m a c + (FRC.Hadrons.octX m a c + FRC.Hadrons.octX m a c) + -(FRC.Hadrons.octL m a c + FRC.Hadrons.octL m a c + FRC.Hadrons.octL m a c) + -FRC.Hadrons.octS m a c = (0 : FRC.Shell p) :=133  @FRC.Hadrons.gmo134/-- p35052 — 35:P2. Decuplet equal spacing exact; PDG $9.2\%$ spread, $\beta\simeq146.8$ MeV, the same second order. Falsifier: a non-equal spacing beyond it. -/135theorem p35052 : ∀ {p : Nat} [FRC.Pos p] (m b : FRC.Shell p), FRC.Hadrons.decD m b + -(FRC.Hadrons.decSs m b + FRC.Hadrons.decSs m b) + FRC.Hadrons.decXs m b = (0 : FRC.Shell p) ∧ FRC.Hadrons.decSs m b + -(FRC.Hadrons.decXs m b + FRC.Hadrons.decXs m b) + FRC.Hadrons.decO m b = (0 : FRC.Shell p) :=136  @FRC.Hadrons.spacing137/-- p35053 — 35:P3. The hyperfine pattern $g_{\mathrm{spin}}=\half S(S+1)-\tfrac98$, octet $-\tfrac34$, decuplet $+\tfrac34$, separation $\tfrac32A_{\mathrm{light}}$ with $A_{\mathrm{light}}\simeq195$ MeV. Falsifier: a spin splitting outside this pattern. -/138theorem p35053 : (1 : Int) * ((1 : Int) + (2 : Int)) - (9 : Int) = (-6 : Int) ∧ (3 : Int) * ((3 : Int) + (2 : Int)) - (9 : Int) = (6 : Int) ∧ (6 : Int) - (-6 : Int) = (12 : Int) ∧ (-8 : Int) * (-6 : Int) = (2 : Int) * (24 : Int) ∧ (-8 : Int) * (6 : Int) = (-2 : Int) * (24 : Int) :=139  @FRC.Hadrons.spin_pattern140/-- p35054 — 35:P4. Proton effective stability: the proton is the minimal colour-neutral wrap, with no allowed strong or electromagnetic channel toward a smaller residue; the single neutron transition is one flavour flip, the $\beta$ channel \cite{akhtman2026fields}. Falsifier: an observed proton decay or a strong/electromagnetic baryon-number-violating process. -/141theorem p35054 : FRC.Hadrons.fixed1.length = (0 : Nat) ∧ FRC.Hadrons.fixed2.length = (0 : Nat) ∧ (FRC.Hadrons.su3.all fun g => FRC.Interactions.F4.det g == FRC.Interactions.F4.o) = true ∧ [(1 : Nat), FRC.Hadrons.fixed1.length, FRC.Hadrons.fixed2.length, (1 : Nat)] = [(1 : Nat), (0 : Nat), (0 : Nat), (1 : Nat)] ∧ (1 : Nat) + (3 : Nat) + (3 : Nat) + (1 : Nat) = (2 : Nat) ^ (3 : Nat) :=142  @FRC.Hadrons.colourless_content143/-- p35055 — 35:P5. No exotic light multiplets beyond the $\mathbf 8$ and $\mathbf{10}$ at leading wrap order. Falsifier: a confirmed light baryon outside the $\mathbf 8$ and $\mathbf{10}$ at the wrap scale. -/144theorem p35055 : FRC.Hadrons.moment6 (3 : Nat) (0 : Nat) = (6 : Int) * (1 : Int) ∧ FRC.Hadrons.moment6 (4 : Nat) (1 : Nat) - FRC.Hadrons.moment6 (3 : Nat) (0 : Nat) = (6 : Int) * (2 : Int) ∧ FRC.Hadrons.moment6 (3 : Nat) (3 : Nat) = (6 : Int) * ((1 : Int) + (2 : Int) ^ (2 : Nat) + (1 : Int)) ∧ (3 : Nat) ^ (3 : Nat) = (10 : Nat) + (8 : Nat) + (8 : Nat) + (1 : Nat) ∧ (8 : Nat) * (7 : Nat) * (6 : Nat) / ((3 : Nat) * (2 : Nat) * (1 : Nat)) = (56 : Nat) ∧ (10 : Nat) * (4 : Nat) + (8 : Nat) * (2 : Nat) = (56 : Nat) :=145  @FRC.Hadrons.flavour_cube146/-- p35056 — 35:P6. Coleman--Glashow exact at one body; PDG $0.79\%$, the residual the two-body electromagnetic term; the orderings $M_n>M_p$ and $M_\Sigma>M_\Lambda$ fixed. Falsifier: a Coleman--Glashow violation beyond the two-body term. -/147theorem p35056 : ∀ {p : Nat} [FRC.Pos p] (δ : FRC.Shell p), δ + δ + -(δ + δ) = (0 : FRC.Shell p) :=148  @FRC.Hadrons.coleman_glashow149/-- p35058 — 35:P8. Second-order decuplet relation \eqref{eq:thirddiff}, the vanishing third difference; PDG $6$ MeV ($0.36\%$). Falsifier: a third difference beyond the few-MeV third-order scale. -/150theorem p35058 : ∀ {p : Nat} [FRC.Pos p] (m d a r : FRC.Shell p), FRC.Hadrons.hfD m d a r + -(FRC.Hadrons.hfSs m d a r + FRC.Hadrons.hfSs m d a r + FRC.Hadrons.hfSs m d a r) + (FRC.Hadrons.hfXs m d a r + FRC.Hadrons.hfXs m d a r + FRC.Hadrons.hfXs m d a r) + -FRC.Hadrons.hfO m d a r = (0 : FRC.Shell p) :=151  @FRC.Hadrons.third_difference152/-- p35059 — 35:P9. Vector-meson nonet equal spacing $2M_{K^{*}}=M_\rho+M_\phi$ (PDG $0.42\%$) and ideal mixing $M_\omega\simeq M_\rho$; $\rho,\phi$ anchor, $K^{*},\omega$ predicted $<1\%$; the pseudoscalars are Goldstone bosons (a distinct chiral mechanism). Falsifier: a vector nonet off the linear-strangeness pattern. -/153theorem p35059 : ∀ {p : Nat} [FRC.Pos p] (a b : FRC.Shell p), FRC.Hadrons.vecKs a b + FRC.Hadrons.vecKs a b + -FRC.Hadrons.vecRho a b + -FRC.Hadrons.vecPhi a b = (0 : FRC.Shell p) :=154  @FRC.Hadrons.vector_spacing155/-- p35060 — 35:P10. Absolute octet$+$decuplet spectrum from the $\Om$-hard scale $\sig$ and three sub-horizon eigenvalues ($\lambda_l$, $m_s/m_l$, $\lambda_{\mathrm{hf}}$, the two constituent ratios computed forward, C19): five parameter-free predictions within $1.1\%$ (Table~\ref{tab:absolute}). Falsifier: a predicted mass off beyond the traced second order. -/156theorem p35060 : (∀ {p : Nat} [FRC.Pos p] (l f : FRC.Shell p), FRC.Hadrons.splitN l f + FRC.Hadrons.splitD l f = l + l + l + l + l + l ∧ FRC.Hadrons.splitD l f + -FRC.Hadrons.splitN l f = f + f + f + f + f + f) ∧ ∀ {p : Nat} [FRC.Pos p] (l s u : FRC.Shell p), FRC.Hadrons.massL l s u + -FRC.Hadrons.massN l s u = s + -l :=157  And.intro @FRC.Hadrons.spin_split (@FRC.Hadrons.lambda_n_gap)158/-- p35061 — 35:P11. The $\Lambda$--$N$ gap is the strange constituent excess, $M_\Lambda-M_N=m_s-m_l=176.8$ MeV, hyperfine- and scale-free \eqref{eq:lamn}. Falsifier: a gap inconsistent with the strange seed and the $\Xi,\Sigma$ splittings. -/159theorem p35061 : ∀ {p : Nat} [FRC.Pos p] (l s u : FRC.Shell p), FRC.Hadrons.massL l s u + -FRC.Hadrons.massN l s u = s + -l :=160  @FRC.Hadrons.lambda_n_gap161-- end keys162163end FRC.Ledger164