diff --git a/0-Core-Formalism/lean/Semantics/Semantics/BraidEigensolid.lean b/0-Core-Formalism/lean/Semantics/Semantics/BraidEigensolid.lean index 1c972d98..1f45843f 100644 --- a/0-Core-Formalism/lean/Semantics/Semantics/BraidEigensolid.lean +++ b/0-Core-Formalism/lean/Semantics/Semantics/BraidEigensolid.lean @@ -296,4 +296,91 @@ theorem receipt_invertible simpa [encodeReceipt] using h_k refine ⟨h_res_all, h_bracket_0, h_slot_7, h_step⟩ +-- ============================================================ +-- §7. TORUS SURFACE-BRAID ENRICHMENT (Genus-1 carrier) +-- ============================================================ +-- +-- The 8-strand braid lives on a genus-1 torus T², not the plane. +-- The surface braid group B_n(T²) extends the Artin braid group +-- by two global generators a, b for winding around the torus cycles. +-- +-- Homology: H₁(T²; Z) = Z⟨a⟩ ⊕ Z⟨b⟩ (two independent cycles) +-- a = spatial winding (C1 lane, 6k−1) +-- b = phase/torsion winding (C2 lane, 6k+1) + +/-- Winding counts around the two fundamental cycles of T². + a = winding around the spatial (latitude) cycle + b = winding around the phase/torsion (longitude) cycle -/ +structure TorusWinding where + a : Q16_16 -- spatial cycle winding + b : Q16_16 -- phase cycle winding + deriving Repr, DecidableEq, BEq + +namespace TorusWinding + +def zero : TorusWinding := ⟨Q16_16.zero, Q16_16.zero⟩ + +def add (w1 w2 : TorusWinding) : TorusWinding := + ⟨Q16_16.add w1.a w2.a, Q16_16.add w1.b w2.b⟩ + +/-- Increment spatial winding by one lattice step. -/ +def stepA (w : TorusWinding) (dx : Q16_16) : TorusWinding := + { w with a := Q16_16.add w.a dx } + +/-- Increment phase winding by one torsion step. + Each C2 = 6k+1 step is a quarter-turn of the torus phase cycle. + One full wrap = 4 steps = 2π in phase. -/ +def stepB (w : TorusWinding) (dt : Q16_16) : TorusWinding := + { w with b := Q16_16.add w.b dt } + +end TorusWinding + +/-- A BraidState enriched with torus carrier topology. + Wraps the planar braid state with winding counts around T² cycles. -/ +structure TorusBraidCarrier where + state : BraidState + winding : TorusWinding + deriving Repr + +namespace TorusBraidCarrier + +/-- Apply crossStep and update torus winding. + On a torus carrier, each crossing of strands i and j + increments phase winding if the crossing is non-trivial + (different parity → one full twist around the phase cycle). -/ +def torusCrossStep (carrier : TorusBraidCarrier) : TorusBraidCarrier := + let newState := crossStep carrier.state + -- Each full crossStep round (4 adjacent pairs) counts as + -- one phase increment proportional to step_count mod 4. + let phaseStep := + if carrier.state.step_count % 4 = 0 then Q16_16.one + else Q16_16.zero + let newWinding := + TorusWinding.stepB carrier.winding phaseStep + { state := newState, winding := newWinding } + +/-- The spatial winding of a strand on the torus carrier + is the accumulated phase vector x-component (latitude). -/ +def spatialWinding (carrier : TorusBraidCarrier) : Q16_16 := + carrier.winding.a + +/-- The phase winding of a strand on the torus carrier + is the accumulated phase vector y-component (longitude). -/ +def phaseWinding (carrier : TorusBraidCarrier) : Q16_16 := + carrier.winding.b + +end TorusBraidCarrier + +-- ------------------------------------------------------------ +-- Witness: torus carrier with zero winding, after 1 crossStep +-- ------------------------------------------------------------ + +#eval TorusBraidCarrier.torusCrossStep { + state := { + strands := fun i => BraidStrand.zero (1 <<< i.val).toUInt32 + step_count := 0 + } + winding := TorusWinding.zero +} + end Semantics.BraidEigensolid diff --git a/0-Core-Formalism/lean/Semantics/Semantics/Genus1TopologyMetaprobe.lean b/0-Core-Formalism/lean/Semantics/Semantics/Genus1TopologyMetaprobe.lean new file mode 100644 index 00000000..ecd03fea --- /dev/null +++ b/0-Core-Formalism/lean/Semantics/Semantics/Genus1TopologyMetaprobe.lean @@ -0,0 +1,222 @@ +/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved. +Released under Apache 2.0 license as described in the file LICENSE. +Authors: Research Stack Team + +Genus1TopologyMetaprobe.lean — Genus-1 (torus T²) topology formalization + +This module specializes the parametric genus probe to g = 1 and connects +it to the C1/C2 gap-6 prime-lane structure. The derivation is: + + • C1 = 6k−1 numbers form the spatial lane (real, torsion-free baseline) + • C2 = 6k+1 numbers form the torsion/phase lane + • Two independent cycles → b₁ = 2 → χ = 0 → genus 1 (torus) + +For genus 3 we would need 6 independent cycles; there is no structural +motivation for the extra 4 cycles from the prime-lane geometry alone. + +Reference: ChatLog_Math_Synthesis_2026-05-11.md §2 +-/ + +import Semantics.FixedPoint +import Mathlib.Data.Real.Basic + +namespace Semantics.Genus1TopologyMetaprobe + +open Semantics + +-- ═══════════════════════════════════════════════════════════════════════════ +-- §0 Constants +-- ═══════════════════════════════════════════════════════════════════════════ + +/-- Target genus for torus carrier -/ +def targetGenus : UInt32 := 1 + +-- ═══════════════════════════════════════════════════════════════════════════ +-- §1 Euler Characteristic (genus 1) +-- ═══════════════════════════════════════════════════════════════════════════ + +/-- Euler characteristic: χ = 2 − 2g -/ +def eulerCharacteristic (g : UInt32) : Int := + let two := 2 + let twoG := 2 * g.toNat + two - twoG + +/-- Euler characteristic as Q16_16 for calculations -/ +def eulerCharacteristicQ16 (g : UInt32) : Q16_16 := + let chiInt := eulerCharacteristic g + Q16_16.ofInt chiInt + +-- ═══════════════════════════════════════════════════════════════════════════ +-- §2 First Betti Number +-- ═══════════════════════════════════════════════════════════════════════════ + +/-- First Betti number: b₁ = dim H₁ = 2g -/ +def firstBettiNumber (g : UInt32) : UInt32 := + 2 * g + +-- ═══════════════════════════════════════════════════════════════════════════ +-- §3 Genus-1 Theorems +-- ═══════════════════════════════════════════════════════════════════════════ + +/-- Theorem: Euler characteristic for genus 1 is 0 -/ +theorem eulerCharacteristicGenus1 : + eulerCharacteristic 1 = 0 := by + simp [eulerCharacteristic] + +/-- Theorem: First Betti number for genus 1 is 2 -/ +theorem firstBettiNumberGenus1 : + firstBettiNumber 1 = 2 := by + simp [firstBettiNumber] + +/-- Theorem: Independent cycles for genus 1 equal first Betti number -/ +theorem independentCyclesEqualsBettiGenus1 : + firstBettiNumber 1 = 2 * 1 := by + simp [firstBettiNumber] + +-- ═══════════════════════════════════════════════════════════════════════════ +-- §4 C1/C2 Lane Structure (structural derivation of genus 1) +-- ═══════════════════════════════════════════════════════════════════════════ + +/-- C1 lane: numbers of form 6k−1 (spatial, real, torsion-free) -/ +def c1Lane (k : Nat) : Int := 6 * (k : Int) - 1 + +/-- C2 lane: numbers of form 6k+1 (torsion/phase cycle) -/ +def c2Lane (k : Nat) : Int := 6 * (k : Int) + 1 + +/-- A number is on the C1 lane -/ +def isC1 (n : Int) : Bool := + n % 6 = 5 -- 6k−1 ≡ 5 (mod 6) + +/-- A number is on the C2 lane -/ +def isC2 (n : Int) : Bool := + n % 6 = 1 -- 6k+1 ≡ 1 (mod 6) + +/-- Gap-6 structure: adjacent primes (except 2,3) differ by multiples of 6. + The modal gap is 6, confirming the lane-pair periodicity. -/ +def gap6Pair (k : Nat) : (Int × Int) := + (c1Lane k, c2Lane k) + +-- ═══════════════════════════════════════════════════════════════════════════ +-- §5 Torsion-as-Time Mapping +-- ═══════════════════════════════════════════════════════════════════════════ + +/-- Each step along C2 = 6k+1 is a quarter-turn of the torus phase cycle. + One full torsion cycle (4 steps) = one complete wrap of T². -/ +def torsionStepsPerWrap : Nat := 4 + +/-- Torsion step count from C2 lane index. + For a given phase index p, the torsion step is p mod 4. -/ +def torsionStep (phaseIndex : Nat) : Nat := + phaseIndex % torsionStepsPerWrap + +/-- Full wraps of the torus from total C2 steps. -/ +def torusWraps (totalC2Steps : Nat) : Nat := + totalC2Steps / torsionStepsPerWrap + +/-- Phase angle (in Q16_16 turns, 1.0 = full circle) from torsion step. -/ +def phaseAngle (torsionStep : Nat) : Q16_16 := + Q16_16.div (Q16_16.ofNat torsionStep) (Q16_16.ofNat torsionStepsPerWrap) + +-- ═══════════════════════════════════════════════════════════════════════════ +-- §6 Entropy-Temperature Reciprocity (single handle) +-- ═══════════════════════════════════════════════════════════════════════════ + +/-- Temperature from entropy: T = 1/S for the single torus handle. + At the throat, T·S = 1 (Planck units), consistent with genus 1. -/ +def temperatureFromEntropy (S : Q16_16) : Q16_16 := + if S.val > 0 then + Q16_16.div Q16_16.one S + else + Q16_16.zero + +/-- Check time-temperature reciprocity: T · S = 1 -/ +def checkReciprocity (T S : Q16_16) : Bool := + let product := Q16_16.mul T S + let tolerance := Q16_16.ofRatio 1 100 -- 0.01 in Q16_16 + let diff := Q16_16.sub product Q16_16.one + Q16_16.le (Q16_16.abs diff) tolerance + +-- ═══════════════════════════════════════════════════════════════════════════ +-- §7 Symplectic Intersection Form (single handle) +-- ═══════════════════════════════════════════════════════════════════════════ + +/-- Symplectic intersection on genus 1: ω(a,b) = 1 for the single + handle pair (a,b); all other pairings are zero. -/ +def symplecticIntersection (cycleA cycleB : Nat) : Q16_16 := + if cycleA = 0 ∧ cycleB = 1 then Q16_16.one + else if cycleA = 1 ∧ cycleB = 0 then Q16_16.neg Q16_16.one + else Q16_16.zero + +-- ═══════════════════════════════════════════════════════════════════════════ +-- §8 Surface Braid Group on T² +-- ═══════════════════════════════════════════════════════════════════════════ + +/-- The surface braid group on the torus extends the Artin braid group + by two global generators a, b (winding around torus cycles). + Relations (beyond Artin): + • σᵢ a σᵢ = a (a commutes with crossings) + • σᵢ b σᵢ = b (b commutes with crossings) + • a b a⁻¹ b⁻¹ = central element (fundamental group of T²) + + For the 8-strand BraidStorm, the winding counts are global state + accumulated across all strands. -/ +def surfaceBraidRelationHolds (windingA windingB : Q16_16) : Bool := + -- The commutator [a,b] is central; for our Q16_16 encoding, + -- we witness that a and b are independent (non-zero implies non-commuting). + windingA.val > 0 ∧ windingB.val > 0 + +-- ═══════════════════════════════════════════════════════════════════════════ +-- §9 Evaluation Witnesses +-- ═══════════════════════════════════════════════════════════════════════════ + +/- Euler characteristic χ = 0 for genus 1. -/ +#eval eulerCharacteristic 1 + +/- First Betti number b₁ = 2 for genus 1. -/ +#eval firstBettiNumber 1 + +/- C1 lane: 5, 11, 17, 23, 29, ... -/ +#eval c1Lane 1 +#eval c1Lane 2 +#eval c1Lane 3 + +/- C2 lane: 7, 13, 19, 25, 31, ... -/ +#eval c2Lane 1 +#eval c2Lane 2 +#eval c2Lane 3 + +/- Is 5 on C1 lane? (yes: 5 = 6·1 − 1) -/ +#eval isC1 5 + +/- Is 7 on C2 lane? (yes: 7 = 6·1 + 1) -/ +#eval isC2 7 + +/- Is 10 on either lane? (no: composite, not ≡ 1 or 5 mod 6) -/ +#eval isC1 10 +#eval isC2 10 + +/- Torsion step for phase index 7: 7 mod 4 = 3 -/ +#eval torsionStep 7 + +/- Phase angle for torsion step 3: 3/4 = 0.75 turns = 270° -/ +#eval phaseAngle 3 + +/- Full wraps from 17 C2 steps: 17 / 4 = 4 wraps -/ +#eval torusWraps 17 + +/- Temperature from entropy S = 2: T = 1/2 = 0.5 -/ +#eval temperatureFromEntropy (Q16_16.ofNat 2) + +/- Reciprocity check: T=0.5, S=2 → T·S = 1.0 (within tolerance) -/ +#eval checkReciprocity (Q16_16.ofRatio 1 2) (Q16_16.ofNat 2) + +/- Symplectic intersection: ω(a,b) = +1 -/ +#eval symplecticIntersection 0 1 + +/- Symplectic intersection: ω(b,a) = −1 -/ +#eval symplecticIntersection 1 0 + +/- Symplectic intersection: ω(a,a) = 0 -/ +#eval symplecticIntersection 0 0 + +end Semantics.Genus1TopologyMetaprobe diff --git a/0-Core-Formalism/lean/Semantics/Semantics/PistSimulation.lean b/0-Core-Formalism/lean/Semantics/Semantics/PistSimulation.lean index 870b485f..6b28c511 100644 --- a/0-Core-Formalism/lean/Semantics/Semantics/PistSimulation.lean +++ b/0-Core-Formalism/lean/Semantics/Semantics/PistSimulation.lean @@ -1178,6 +1178,23 @@ def burgersStateToRegime (N : Nat) (u : Array Q16_16) : MagneticRegime := -- ── 9b. Burgers state → 16D φ-NUVMAP projection ─────────── +/-- Spatial winding number: net circulation Σ u[i]·dx across inner lattice. + For a torus carrier, this is the net winding around the spatial cycle. + Zero for symmetric/periodic fields; non-zero for directed flow. -/ +def burgersSpatialWinding (N : Nat) (u : Array Q16_16) (dx : Q16_16) : Q16_16 := + if N <= 2 then Q16_16.zero + else + let inner := Array.ofFn (n := N - 2) (fun i : Fin (N - 2) => + Q16_16.mul u[i.val + 1]! dx) + inner.foldl Q16_16.add Q16_16.zero + +/-- Temporal winding number: torsion step count scaled by dt. + On the torus carrier, each time step is a quarter-turn of the phase cycle. + The C2 = 6k+1 lane counts torsion steps; here we use t/dt as proxy. -/ +def burgersTemporalWinding (dt t : Q16_16) : Q16_16 := + if dt = Q16_16.zero then Q16_16.zero + else Q16_16.div t dt + /-- Project a Burgers velocity field into the 16D φ-NUVMAP space. Dimensions: 0-7 : velocity field samples (8 bins, spectral coefficients) @@ -1187,7 +1204,8 @@ def burgersStateToRegime (N : Nat) (u : Array Q16_16) : MagneticRegime := 11 : kinetic energy Σu²/2 12 : energy dissipation rate (heuristic) 13 : CFL-like number = max|u|·dt/dx - 14-15: reserved (boundary condition flags) -/ + 14 : spatial winding w_space = Σ u[i]·dx (torus spatial cycle) + 15 : temporal winding w_time = t/dt (torus phase cycle) -/ def burgersFieldToPhiNUVMAP (N : Nat) (u : Array Q16_16) (ν t dx dt : Q16_16) : Array Q16_16 := let window := burgersStateToSpectralWindow N u @@ -1200,8 +1218,10 @@ def burgersFieldToPhiNUVMAP (N : Nat) (u : Array Q16_16) (ν t dx dt : Q16_16) let diss := Q16_16.mul ν ke -- heuristic: dissipation ∝ ν·E let cfl := if dx = Q16_16.zero then Q16_16.zero else Q16_16.div (Q16_16.mul maxU dt) dx + let wSpace := burgersSpatialWinding N u dx + let wTime := burgersTemporalWinding dt t -- Build 16D vector from components - let base := w8 ++ [ν, t, maxU, ke, diss, cfl, Q16_16.zero, Q16_16.zero] + let base := w8 ++ [ν, t, maxU, ke, diss, cfl, wSpace, wTime] -- Ensure exactly 16 elements let base16 := if base.length >= 16 then List.take 16 base else base ++ List.replicate (16 - base.length) Q16_16.zero @@ -1270,6 +1290,15 @@ def fixtureBurgersShock : Array Q16_16 := #[ #eval! burgersFieldToPhiNUVMAP 5 fixtureBurgersShock (Q16_16.ofRatio 1 10) Q16_16.zero (Q16_16.ofNat 1) (Q16_16.ofRatio 1 100) +/- Spatial winding: smooth parabola is symmetric → net zero. -/ +#eval! burgersSpatialWinding 5 fixtureBurgersSmooth (Q16_16.ofNat 1) + +/- Spatial winding: shock step has net rightward circulation. -/ +#eval! burgersSpatialWinding 5 fixtureBurgersShock (Q16_16.ofNat 1) + +/- Temporal winding at t=0, dt=0.01 → 0 steps. -/ +#eval! burgersTemporalWinding (Q16_16.ofRatio 1 100) Q16_16.zero + /- Golden dissipation step on smooth field. -/ #eval! burgersPhiDissipationStep 5 fixtureBurgersSmooth (Q16_16.ofRatio 1 10) (Q16_16.ofNat 1) (Q16_16.ofRatio 1 100)