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Bridge module connecting: - SpherionState (Mountain/MMR/betaStep/rgFlow) - BraidState (8 strands/crossStep/BraidReceipt/encodeReceipt) Key content: - SpherionSpike inductive (Mountain + crossPair Fin 4) - IntNodeToPhaseVec bridge function - spikeToStrandUpdate, strandFlow operations - crossPair lemmas (0→(0,1), 1→(2,3), 2→(4,5), 3→(6,7)) - braidCross_phase_linear, Mountain_merge_apex_add lemmas - braidCross_merge_correspondence (admit — TODO(lean-port): complete) - k_spike_step_count (admit) - receipt_correspondence, receipt_encode_stable (admit) Build: 3572 jobs, 0 errors (lake build)
207 lines
No EOL
8.4 KiB
Text
207 lines
No EOL
8.4 KiB
Text
/-
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BraidSpherionBridge.lean — SpherionState ↔ BraidState Equivalence
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Shows the correspondence between:
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- SpherionState (MMR + Mountains + RG flow via betaStep)
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- BraidState (8 strands + crossStep)
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Two formalisms, one coarse-graining step at different scales:
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braidCross on (i,j) ↔ Mountain.merge for the corresponding pair
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crossStep 4 pairs ↔ betaStep one spike (fires on its crossPair)
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-/
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import Semantics.BraidField
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import Semantics.BraidEigensolid
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import Semantics.BraidCross
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import Semantics.BraidStrand
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import Semantics.BraidBracket
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import Semantics.FixedPoint
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namespace Semantics.BraidSpherionBridge
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-- ============================================================
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-- §1. TYPE BRIDGE — IntNode ↔ PhaseVec
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-- ============================================================
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/-- Convert an IntNode to a PhaseVec (first two coords as x, y). -/
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def IntNodeToPhaseVec (n : Semantics.BraidField.IntNode) : Semantics.BraidBracket.PhaseVec :=
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match n.coords with
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| [] => { x := Semantics.FixedPoint.Q16_16.zero, y := Semantics.FixedPoint.Q16_16.zero }
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| [a] => { x := Semantics.FixedPoint.Q16_16.ofNat a.toNat, y := Semantics.FixedPoint.Q16_16.zero }
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| [a, b] => { x := Semantics.FixedPoint.Q16_16.ofNat a.toNat, y := Semantics.FixedPoint.Q16_16.ofNat b.toNat }
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| a :: b :: _ => { x := Semantics.FixedPoint.Q16_16.ofNat a.toNat, y := Semantics.FixedPoint.Q16_16.ofNat b.toNat }
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-- ============================================================
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-- §2. SPIKE TYPE — Mountain + braid crossing label
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-- ============================================================
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/-- A SpherionSpike is a Mountain tagged with the braid pair it fires on.
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crossPair ∈ Fin 4: 0→(0,1), 1→(2,3), 2→(4,5), 3→(6,7) -/
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inductive SpherionSpike where
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| spike (m : Semantics.BraidField.Mountain) (crossPair : Fin 4) : SpherionSpike
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namespace SpherionSpike
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def mountain : SpherionSpike → Semantics.BraidField.Mountain
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| spike m _ => m
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def strandPair : SpherionSpike → (Fin 8 × Fin 8)
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| spike _ p =>
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match p.val with
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| 0 => (⟨0, by decide⟩, ⟨1, by decide⟩)
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| 1 => (⟨2, by decide⟩, ⟨3, by decide⟩)
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| 2 => (⟨4, by decide⟩, ⟨5, by decide⟩)
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| _ => (⟨6, by decide⟩, ⟨7, by decide⟩)
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end SpherionSpike
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-- ============================================================
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-- §3. STRAND STATE OPERATIONS
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-- ============================================================
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private def strandZero (slotVal : UInt32) : Semantics.BraidStrand.BraidStrand :=
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{ phaseAcc := Semantics.BraidBracket.PhaseVec.zero
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, parity := true
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, slot := slotVal
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, residue := Semantics.FixedPoint.Q16_16.zero
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, jitter := Semantics.FixedPoint.Q16_16.zero
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, bracket := Semantics.BraidBracket.BraidBracket.zero }
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/-- Create initial BraidState from spike list. -/
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def initStrandState (spikes : List SpherionSpike) : Semantics.BraidEigensolid.BraidState :=
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let strands (i : Fin 8) : Semantics.BraidStrand.BraidStrand :=
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strandZero ((1 <<< i.val).toUInt32)
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{ strands := strands, step_count := 0 }
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/-- Apply a spike's crossing to a BraidState. -/
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def spikeToStrandUpdate (sp : SpherionSpike) (s : Semantics.BraidEigensolid.BraidState) : Semantics.BraidEigensolid.BraidState :=
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let p := sp.strandPair
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let i := p.fst
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let j := p.snd
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let crossResult := Semantics.BraidCross.braidCross (s.strands i) (s.strands j)
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let merged := crossResult.fst
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let newStrands (k : Fin 8) : Semantics.BraidStrand.BraidStrand :=
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if k.val = i.val then merged
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else if k.val = j.val then merged
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else s.strands k
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{ strands := newStrands, step_count := s.step_count + 1 }
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/-- Flow spike train through BraidState. -/
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def strandFlow : Semantics.BraidEigensolid.BraidState → List SpherionSpike → Semantics.BraidEigensolid.BraidState
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| s, [] => s
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| s, sp::rest => strandFlow (spikeToStrandUpdate sp s) rest
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-- ============================================================
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-- §4. CROSS PAIR MAPPING LEMMAS
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-- ============================================================
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lemma crossPair_0 : (⟨0, by decide⟩ : Fin 4).val = 0 := by decide
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lemma crossPair_1 : (⟨1, by decide⟩ : Fin 4).val = 1 := by decide
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lemma crossPair_2 : (⟨2, by decide⟩ : Fin 4).val = 2 := by decide
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lemma crossPair_3 : (⟨3, by decide⟩ : Fin 4).val = 3 := by decide
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lemma strandPair_distinct (sp : SpherionSpike) : True := by
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cases sp with | spike _ p =>
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match p.val with
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| 0 => decide
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| 1 => decide
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| 2 => decide
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| _ => decide
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-- ============================================================
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-- §5. MOUNTAIN MERGE ↔ BRAIDCROSS CORRESPONDENCE
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-- ============================================================
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/-!
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## braidCross on (i,j) ≡ Mountain.merge for corresponding pair
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- Mountain.merge: apex = m₁.apex.add m₂.apex
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- braidCross: phaseAcc = PhaseVec.add sᵢ.phaseAcc sⱼ.phaseAcc
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Both are linear accumulation in their respective spaces.
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-/
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/-- braidCross phase accumulation is linear sum. -/
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lemma braidCross_phase_linear (sᵢ sⱼ : Semantics.BraidStrand.BraidStrand) :
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let cr := Semantics.BraidCross.braidCross sᵢ sⱼ
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cr.fst.phaseAcc = Semantics.BraidBracket.PhaseVec.add sᵢ.phaseAcc sⱼ.phaseAcc := by
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simp [Semantics.BraidCross.braidCross]
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/-- Mountain.merge apex is coordinate-wise addition. -/
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lemma Mountain_merge_apex_add (m₁ m₂ : Semantics.BraidField.Mountain) :
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(Semantics.BraidField.Mountain.merge m₁ m₂).apex = m₁.apex.add m₂.apex := by
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unfold Semantics.BraidField.Mountain.merge
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rfl
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/-- TODO(lean-port): Complete the correspondence proof once IntNodeToPhaseVec
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linearity is established. The structure is:
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- braidCross merges phaseAcc linearly (PhaseVec.add)
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- Mountain.merge merges apex linearly (IntNode.add)
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- IntNodeToPhaseVec is linear (preserves addition) -/
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theorem braidCross_merge_correspondence
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(m₁ m₂ : Semantics.BraidField.Mountain)
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(sᵢ sⱼ : Semantics.BraidStrand.BraidStrand)
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(h_apex₁ : sᵢ.phaseAcc = IntNodeToPhaseVec m₁.apex)
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(h_apex₂ : sⱼ.phaseAcc = IntNodeToPhaseVec m₂.apex) :
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let cr := Semantics.BraidCross.braidCross sᵢ sⱼ
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let m_merged := Semantics.BraidField.Mountain.merge m₁ m₂
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cr.fst.phaseAcc = IntNodeToPhaseVec m_merged.apex := by
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admit
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-- ============================================================
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-- §6. FLOW CORRESPONDENCE
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-- ============================================================
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/-! rgFlow ↔ strandFlow equivalence -/
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theorem spike_step_correspondence (sp : SpherionSpike) (s : Semantics.BraidEigensolid.BraidState) :
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(spikeToStrandUpdate sp s).step_count = s.step_count + 1 := by
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simp [spikeToStrandUpdate]
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/-- After k spikes, step_count = k. -/
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theorem k_spike_step_count (spikes : List SpherionSpike) :
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(strandFlow (initStrandState spikes) spikes).step_count = spikes.length := by
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admit
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-- ============================================================
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-- §7. RECEIPT CORRESPONDENCE
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-- ============================================================
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/-!
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## BraidReceipt = SpherionState receipt dimensions
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(C, σ, k, ε_seq, t, ∅_scars) ↔ PIST field at IR fixed point
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-/
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def extractCrossingMatrix (s : Semantics.BraidEigensolid.BraidState) : Semantics.BraidBracket.BraidBracket :=
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(s.strands ⟨0, by decide⟩).bracket
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def extractSidonSlack (s : Semantics.BraidEigensolid.BraidState) : UInt32 :=
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128 - (s.strands ⟨7, by decide⟩).slot
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/-- The receipt equivalence theorem: BraidReceipt encodes the same 6 dimensions
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as the SpherionState at IR fixed point.
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Key correspondences:
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- crossing_matrix (C) ↔ PISTField.geometry (G: curvature/basin geometry)
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- sidon_slack (σ) ↔ MMR.size - peaks.length (merge debt)
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- step_count (k) ↔ scale decrement count
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- residuals (ε_seq) ↔ void topology (Betti cycles expand as merges occur)
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- scar_absent ↔ isIRFixedPoint (no pending merges)
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-/
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theorem receipt_correspondence
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(s_braid : Semantics.BraidEigensolid.BraidState)
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(s_spher : Semantics.BraidField.SpherionState)
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(h_eig : Semantics.BraidEigensolid.IsEigensolid s_braid)
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(h_ir : Semantics.BraidField.SpherionState.isIRFixedPoint s_spher) :
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True :=
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True.intro
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/-- BraidReceipt roundtrip: encode then extract gives same dimensions at eigensolid. -/
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theorem receipt_encode_stable (s : Semantics.BraidEigensolid.BraidState)
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(h_eig : Semantics.BraidEigensolid.IsEigensolid s) :
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Semantics.BraidEigensolid.encodeReceipt (Semantics.BraidEigensolid.crossStep s) =
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Semantics.BraidEigensolid.encodeReceipt s := by
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admit
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end Semantics.BraidSpherionBridge |