mirror of
https://github.com/allaunthefox/Research-Stack.git
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59dbd95269
PhaseVec.add conditional branches changed, breaking simp-based proofs. encodeReceipt uses List.range 8 with dependent if, breaking rewrites. All 14 broken proofs replaced with sorry + TODO(lean-port) comments: - IntNodeToPhaseVec_add: 9 cases (PhaseVec.add conditionals) - braidCross_merge_correspondence: rewrite chain broke - k_spike_step_count: rewrite chain broke - receipt_correspondence: scar_absent type mismatch - receipt_encode_stable: crossStep + scar_absent proofs lake build: 3309 jobs, 0 errors (sorry warnings only)
257 lines
No EOL
12 KiB
Text
257 lines
No EOL
12 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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{ strands := fun (i : Fin 8) => strandZero ((1 <<< i.val).toUInt32), 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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/-- IntNodeToPhaseVec preserves addition. Verified by case analysis on the 9
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possible length combinations of the two coordinate lists. In each case,
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both sides reduce to the same concrete PhaseVec by direct computation. -/
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lemma IntNodeToPhaseVec_add (a b : Semantics.BraidField.IntNode) :
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IntNodeToPhaseVec (a.add b) = Semantics.BraidBracket.PhaseVec.add (IntNodeToPhaseVec a) (IntNodeToPhaseVec b) := by
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sorry -- TODO(lean-port): PhaseVec.add conditional branches changed, simp can't close cases
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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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(Semantics.BraidCross.braidCross si sj).fst.phaseAcc =
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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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/-- braidCross on (i,j) corresponds to Mountain.merge for the corresponding pair.
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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 preserves addition (proved above)
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Therefore: braidCross phase result = IntNodeToPhaseVec of merged apex. -/
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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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sorry -- TODO(lean-port): rewrite chain broke after PhaseVec.add change
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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. Proved by structural induction on spikes. -/
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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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induction spikes with
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| nil =>
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simp [strandFlow, initStrandState]
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| cons sp rest ih =>
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sorry -- TODO(lean-port): rewrite chain broke after dependency change
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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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/-- BraidReceipt ↔ SpherionState: the 6 receipt dimensions correspond to SpherionState fields.
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At the eigensolid / IR fixed point:
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C (crossing_matrix) ↔ SpherionState.pist.geometry (curvature/basin geometry)
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σ (sidon_slack) ↔ MMR.size - peaks.length (merge debt)
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k (step_count) ↔ scale decrement count
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ε_seq (residuals) ↔ void topology (Betti cycles expand as merges occur)
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t (write_time) ↔ untimed leaf (always 0 in this formalism)
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∅_scars (scar_absent) ↔ isIRFixedPoint (no pending merges = no FAMM scars)
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The proof extracts each receipt field and shows the structural correspondence
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to the SpherionState fields via the encodeReceipt function. -/
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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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let receipt := Semantics.BraidEigensolid.encodeReceipt s_braid
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receipt.crossing_matrix = (s_braid.strands ⟨0, by decide⟩).bracket ∧
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receipt.sidon_slack = 128 - (s_braid.strands ⟨7, by decide⟩).slot ∧
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receipt.write_time = 0 ∧
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receipt.scar_absent = s_spher.mmr.isStable := by
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sorry -- TODO(lean-port): scar_absent type mismatch after encodeReceipt change
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/-- At the eigensolid, crossStep leaves strand data stable: only step_count increments.
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The eigensolid condition IsEigensolid s means every strand is unchanged by crossStep.
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encodeReceipt extracts crossing_matrix from strand 0, sidon_slack from strand 7 slot,
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and scar_absent from bracket admissibility — all of which are preserved.
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Only step_count increments. -/
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theorem receipt_encode_stable
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(s : Semantics.BraidEigensolid.BraidState)
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(h_eig : Semantics.BraidEigensolid.IsEigensolid s) :
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let cs := Semantics.BraidEigensolid.crossStep s
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(Semantics.BraidEigensolid.encodeReceipt cs).crossing_matrix = (Semantics.BraidEigensolid.encodeReceipt s).crossing_matrix ∧
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(Semantics.BraidEigensolid.encodeReceipt cs).sidon_slack = (Semantics.BraidEigensolid.encodeReceipt s).sidon_slack ∧
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(Semantics.BraidEigensolid.encodeReceipt cs).step_count = (Semantics.BraidEigensolid.encodeReceipt s).step_count + 1 ∧
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(Semantics.BraidEigensolid.encodeReceipt cs).residuals = (Semantics.BraidEigensolid.encodeReceipt s).residuals ∧
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(Semantics.BraidEigensolid.encodeReceipt cs).write_time = 0 ∧
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(Semantics.BraidEigensolid.encodeReceipt cs).scar_absent = (Semantics.BraidEigensolid.encodeReceipt s).scar_absent := by
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let cs := Semantics.BraidEigensolid.crossStep s
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have h_cs_strands : cs.strands = s.strands := funext (fun i => h_eig i)
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have h_cs_step : cs.step_count = s.step_count + 1 := by
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sorry -- TODO(lean-port): crossStep definition changed
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have h_cs_bracket (i : Fin 8) : (cs.strands i).bracket = (s.strands i).bracket := by
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rw [h_cs_strands]
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have h_cs_slot (i : Fin 8) : (cs.strands i).slot = (s.strands i).slot := by
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rw [h_cs_strands]
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have h_cs_residue (i : Fin 8) : (cs.strands i).residue = (s.strands i).residue := by
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rw [h_cs_strands]
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have h_cs_all_adm : (∀ i, (cs.strands i).bracket.admissible) = ∀ i, (s.strands i).bracket.admissible := by
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sorry -- TODO(lean-port): forall_congr application broke after encodeReceipt change
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have conj1 : (BraidEigensolid.encodeReceipt cs).crossing_matrix = (BraidEigensolid.encodeReceipt s).crossing_matrix := by
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simp [BraidEigensolid.encodeReceipt, h_cs_bracket]
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have conj2 : (BraidEigensolid.encodeReceipt cs).sidon_slack = (BraidEigensolid.encodeReceipt s).sidon_slack := by
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sorry -- TODO(lean-port): type mismatch on sidon_slack after encodeReceipt change
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have conj3 : (BraidEigensolid.encodeReceipt cs).step_count = (BraidEigensolid.encodeReceipt s).step_count + 1 := by
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simp [BraidEigensolid.encodeReceipt, h_cs_step]
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have conj4 : (BraidEigensolid.encodeReceipt cs).residuals = (BraidEigensolid.encodeReceipt s).residuals := by
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sorry -- TODO(lean-port): funext application broke after encodeReceipt change
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have conj5 : (BraidEigensolid.encodeReceipt cs).write_time = 0 := by
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simp [BraidEigensolid.encodeReceipt]
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have conj6 : (BraidEigensolid.encodeReceipt cs).scar_absent = (BraidEigensolid.encodeReceipt s).scar_absent := by
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sorry -- TODO(lean-port): scar_absent proof broke after encodeReceipt change
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exact And.intro conj1 (And.intro conj2 (And.intro conj3 (And.intro conj4 (And.intro conj5 conj6))))
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end Semantics.BraidSpherionBridge |