Keys/Interactions.lean — FrcCore

1import FrcCore.Theme.Interactions23/-!4# FrcCore.Keys.Interactions — the keyed theorems of the interactions theme (ledger migration, task LM19)56One declaration per key, `FRC.Ledger.p<key>`, generated by `make_keys.py` from the ledgers' Lean bindings: the theorem7that every row carrying the key asserts, proved from the themes alone. The paper's module keeps its declaration8`FRC.<Module>.p<key>` as an alias. Every declaration has no axioms (`check_core_axioms.py`).9-/1011namespace FRC.Ledger1213-- Keys of the interactions theme (generated by make_keys.py from the ledgers' Lean bindings; edit the ledgers, not this file)14/-- p00081 — 00:G16. \textbf{Flavour} (Koide): the charged-lepton ratio $Q=2/3$ is the cube-root generation orbit at the self-dual quarter-turn, $Q=\tfrac13+\tfrac23\rho^2$ exact, the value $\rho^2=2^{-1}$ the quarter-turn realisation (28-flav B5; falsifier: $Q_\ell$ off $\tfrac23$ beyond the $\tau$-mass tolerance $\pm7\times10^{-6}$); the winding-phase selection onto that orientation is bedrock, $\Om$-hard in magnitude, its $\Z_2$ sign the derived drive orientation (C7); the sub-horizon reading is $\delta_0=Q/3=2/9$ (radians). -/15theorem p00081 : (∀ {p : Nat} [FRC.Pos p] (a₀ a₁ a₂ ω : FRC.Shell p), ω * ω + ω + (1 : FRC.Shell p) = (0 : FRC.Shell p) → (3 : FRC.Shell p) * (a₀ * a₀ + a₁ * a₁ + a₂ * a₂) = (a₀ + a₁ + a₂) * (a₀ + a₁ + a₂) + (2 : FRC.Shell p) * ((a₀ + a₁ * ω + a₂ * (ω * ω)) * (a₀ + a₁ * (ω * ω) + a₂ * ω))) ∧ (∀ {p : Nat} [FRC.Pos p] (a₀ a₁ a₂ ω : FRC.Shell p), ω * ω + ω + (1 : FRC.Shell p) = (0 : FRC.Shell p) → ((3 : FRC.Shell p) * (a₀ * a₀ + a₁ * a₁ + a₂ * a₂) = (2 : FRC.Shell p) * ((a₀ + a₁ + a₂) * (a₀ + a₁ + a₂)) ↔ (2 : FRC.Shell p) * ((a₀ + a₁ * ω + a₂ * (ω * ω)) * (a₀ + a₁ * (ω * ω) + a₂ * ω)) = (a₀ + a₁ + a₂) * (a₀ + a₁ + a₂))) ∧ (∀ {p : Nat} [FRC.Pos p] {ν h : FRC.Shell p}, h + h = (1 : FRC.Shell p) → ν + (1 : FRC.Shell p) + (1 : FRC.Shell p) + (1 : FRC.Shell p) = (0 : FRC.Shell p) → ∀ (a₀ a₁ a₂ : FRC.Shell p), FRC.Interactions.omega3 ν h * FRC.Interactions.omega3 ν h + FRC.Interactions.omega3 ν h + (1 : FRC.Extension.Ext p ν) = (0 : FRC.Extension.Ext p ν) ∧ FRC.Interactions.hat2 ν h a₀ a₁ a₂ = (FRC.Interactions.hat1 ν h a₀ a₁ a₂).conj ∧ FRC.Interactions.hat1 ν h a₀ a₁ a₂ * FRC.Interactions.hat2 ν h a₀ a₁ a₂ = FRC.Extension.Ext.ofShell (FRC.Interactions.hat1 ν h a₀ a₁ a₂).norm ∧ (3 : FRC.Shell p) * (a₀ * a₀ + a₁ * a₁ + a₂ * a₂) = (a₀ + a₁ + a₂) * (a₀ + a₁ + a₂) + (2 : FRC.Shell p) * (FRC.Interactions.hat1 ν h a₀ a₁ a₂).norm ∧ ((3 : FRC.Shell p) * (a₀ * a₀ + a₁ * a₁ + a₂ * a₂) = (2 : FRC.Shell p) * ((a₀ + a₁ + a₂) * (a₀ + a₁ + a₂)) ↔ (2 : FRC.Shell p) * (FRC.Interactions.hat1 ν h a₀ a₁ a₂).norm = (a₀ + a₁ + a₂) * (a₀ + a₁ + a₂))) ∧ ∀ {p : Nat} [FRC.Pos p] {κ : Nat}, p = (4 : Nat) * κ + (1 : Nat) → FRC.Shell.ofNat ((2 : Nat) * κ + (1 : Nat)) + FRC.Shell.ofNat ((2 : Nat) * κ + (1 : Nat)) = (1 : FRC.Shell p) ∧ (-3 : FRC.Shell p) + (1 : FRC.Shell p) + (1 : FRC.Shell p) + (1 : FRC.Shell p) = (0 : FRC.Shell p) :=16  And.intro @FRC.Interactions.koide_parseval (And.intro @FRC.Interactions.koide_two_thirds (And.intro @FRC.Interactions.koide_extension (@FRC.Interactions.koide_half)))17/-- p00155 — 00:G12. Read with the quarter-turn ($4\mid\Om-1$), the Carrier's four representation charts are four directions; their reflection algebra has basis $e_S$, $S\subseteq\{1,2,3,4\}$, $16$ elements, $e_Se_T=\pm e_{S\cup T}$ for disjoint $S,T$, else $0$. The quarter-turn as fifth direction maps $e_S\mapsto e_S$ ($|S|$ even), $e_Se_5$ ($|S|$ odd) onto the even basis of five directions: the chiral spinor $\mathbf{16}$. On the realisation 27:B5 matter is this spinor. -/18theorem p00155 : (((List.range (16 : Nat)).all fun S => (List.range (16 : Nat)).all fun T => decide ((FRC.Interactions.emul S T).snd < (16 : Nat))) = true ∧ ((List.range (16 : Nat)).all fun S => (List.range (16 : Nat)).all fun T => (List.range (16 : Nat)).all fun U => FRC.Interactions.assocL S T U == FRC.Interactions.assocR S T U) = true ∧ ((List.range (4 : Nat)).all fun i => (List.range (4 : Nat)).all fun j => if (i == j) = true then FRC.Interactions.emul ((2 : Nat) ^ i) ((2 : Nat) ^ j) == ((0 : Int), (0 : Nat)) else FRC.Interactions.emul ((2 : Nat) ^ i) ((2 : Nat) ^ j) == (-(FRC.Interactions.emul ((2 : Nat) ^ j) ((2 : Nat) ^ i)).fst, (FRC.Interactions.emul ((2 : Nat) ^ j) ((2 : Nat) ^ i)).snd)) = true) ∧ FRC.Interactions.gen16.length = (16 : Nat) ∧ ((List.range (16 : Nat)).all fun S => FRC.Interactions.gen16.contains (FRC.Interactions.fifth S)) = true ∧ ((List.range (16 : Nat)).all fun S => (List.range (16 : Nat)).all fun T => decide ((FRC.Interactions.fifth S == FRC.Interactions.fifth T) = true → (S == T) = true)) = true ∧ (FRC.Interactions.gen16.all fun R => (List.range (16 : Nat)).any fun S => FRC.Interactions.fifth S == R) = true ∧ ((List.range (16 : Nat)).all fun S => FRC.Interactions.cnt S (0 : Nat) (4 : Nat) % (2 : Nat) == (0 : Nat) || FRC.Interactions.emul S (16 : Nat) == ((1 : Int), FRC.Interactions.fifth S)) = true :=19  And.intro @FRC.Interactions.reflection_algebra (@FRC.Interactions.fifth_direction)20/-- p27020 — 27:C8, 00:G11. \textbf{One generation} (content): $=$ the spinor $\mathbf{16}$ of the rank-five frame --- all gauge reps, hypercharges, charges, anomaly-free, $\nu_R$ --- recovered in the compact theory, where the frame is $\C^3\oplus\C^2$, via the correspondence: the imported $\mathrm{SO}(10)$ template evaluated on the realisation (27-fld B5). -/21theorem p27020 : (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 :=22  @FRC.Interactions.one_generation23-- end keys2425end FRC.Ledger26