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- cupfox-config.nix: add Open WebUI container with chat.researchstack.info proxy, gather-metrics service/timer, rclone, and tmpfiles for persistent storage - Lean semantics: reduce axiom count from 109 to 18 across 10 files; FixedPoint now 0 axioms, 0 sorries with 12 theorems - Documentation: update AGENTS.md with current axiom/sorry counts and FixedPoint status; refine bind signature - Add topology scripts, CGA/FAMM/GeneticOptimizer/MMRFAMM Lean modules, devcontainer config, MEMORY.md, and Modelfile
108 lines
3.6 KiB
Text
108 lines
3.6 KiB
Text
import Semantics.DynamicCanal
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namespace Semantics.Quaternion
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open DynamicCanal
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/--
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Quaternion: 4D Non-Commutative State for PIST (Propagated Informatic Sere Topology).
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Elements: q = w + xi + yj + zk
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-/
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structure Quaternion where
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w : Fix16
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x : Fix16
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y : Fix16
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z : Fix16
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deriving Repr, DecidableEq, BEq, Inhabited
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namespace Quaternion
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theorem ext {q1 q2 : Quaternion} (hw : q1.w = q2.w) (hx : q1.x = q2.x) (hy : q1.y = q2.y) (hz : q1.z = q2.z) : q1 = q2 := by
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cases q1; cases q2
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simp only at hw hx hy hz ⊢
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simp [hw, hx, hy, hz]
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def zero : Quaternion :=
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{ w := Fix16.zero, x := Fix16.zero, y := Fix16.zero, z := Fix16.zero }
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def one : Quaternion :=
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{ w := Fix16.one, x := Fix16.zero, y := Fix16.zero, z := Fix16.zero }
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def i : Quaternion :=
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{ w := Fix16.zero, x := Fix16.one, y := Fix16.zero, z := Fix16.zero }
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def j : Quaternion :=
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{ w := Fix16.zero, x := Fix16.zero, y := Fix16.one, z := Fix16.zero }
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def k : Quaternion :=
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{ w := Fix16.zero, x := Fix16.zero, y := Fix16.zero, z := Fix16.one }
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def add (p q : Quaternion) : Quaternion :=
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{ w := Fix16.add p.w q.w
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, x := Fix16.add p.x q.x
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, y := Fix16.add p.y q.y
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, z := Fix16.add p.z q.z }
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def sub (p q : Quaternion) : Quaternion :=
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{ w := Fix16.sub p.w q.w
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, x := Fix16.sub p.x q.x
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, y := Fix16.sub p.y q.y
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, z := Fix16.sub p.z q.z }
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def neg (q : Quaternion) : Quaternion :=
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{ w := Fix16.neg q.w, x := Fix16.neg q.x, y := Fix16.neg q.y, z := Fix16.neg q.z }
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/--
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Hamiltonian product: standard non-commutative multiplication.
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(w1 + x1i + y1j + z1k)(w2 + x2i + y2j + z2k)
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-/
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def mul (p q : Quaternion) : Quaternion :=
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let w := Fix16.sub (Fix16.sub (Fix16.sub (Fix16.mul p.w q.w) (Fix16.mul p.x q.x)) (Fix16.mul p.y q.y)) (Fix16.mul p.z q.z)
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let x := Fix16.add (Fix16.add (Fix16.add (Fix16.mul p.w q.x) (Fix16.mul p.x q.w)) (Fix16.mul p.y q.z)) (Fix16.neg (Fix16.mul p.z q.y))
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let y := Fix16.add (Fix16.add (Fix16.add (Fix16.mul p.w q.y) (Fix16.neg (Fix16.mul p.x q.z))) (Fix16.mul p.y q.w)) (Fix16.mul p.z q.x)
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let z := Fix16.add (Fix16.add (Fix16.add (Fix16.mul p.w q.z) (Fix16.mul p.x q.y)) (Fix16.neg (Fix16.mul p.y q.x))) (Fix16.mul p.z q.w)
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{ w := w, x := x, y := y, z := z }
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/-- Dot product for Quaternions. -/
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def dot (p q : Quaternion) : Fix16 :=
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Fix16.add (Fix16.mul p.w q.w) (Fix16.add (Fix16.mul p.x q.x) (Fix16.add (Fix16.mul p.y q.y) (Fix16.mul p.z q.z)))
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/--
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Approximation of the norm: max(|w|,|x|,|y|,|z|) + (3/8)·Σothers
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(A 4D extension of the octagonal norm).
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-/
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def normApprox (q : Quaternion) : Fix16 :=
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let abs_val (v : Fix16) := if v.val < 0x80000000 then v else Fix16.neg v
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let aw := abs_val q.w
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let ax := abs_val q.x
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let ay := abs_val q.y
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let az := abs_val q.z
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let m1 := if aw.val > ax.val then aw else ax
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let m2 := if ay.val > az.val then ay else az
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let hi := if m1.val > m2.val then m1 else m2
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-- Sum the others roughly
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let sum_others := Fix16.add aw (Fix16.add ax (Fix16.add ay az))
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let others := Fix16.sub sum_others hi
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let o38 := Fix16.mk ((others.val.toNat * 0x6000 / 0x10000).toUInt32)
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Fix16.add hi o38
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/-- Conjugate of a Quaternion: q* = w - xi - yj - zk -/
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def conj (q : Quaternion) : Quaternion :=
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{ w := q.w, x := Fix16.neg q.x, y := Fix16.neg q.y, z := Fix16.neg q.z }
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/-- Scalar multiplication. -/
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def smul (s : Fix16) (q : Quaternion) : Quaternion :=
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{ w := Fix16.mul s q.w, x := Fix16.mul s q.x, y := Fix16.mul s q.y, z := Fix16.mul s q.z }
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/-- Map a color index (0-3) to a Quaternion basis vector. -/
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def fromColor (c : Fin 4) : Quaternion :=
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match c.val with
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| 0 => one
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| 1 => i
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| 2 => j
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| 3 => k
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| _ => zero
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end Quaternion
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end Semantics.Quaternion
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