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feat(lean): ContractedCrossStep — genuine φ⁻¹ contraction for braid crossing dynamics
The real crossStep uses PhaseVec.add (additive doubling: zᵢⱼ = zᵢ + zⱼ), which grows until Q16_16 saturation. ContractedCrossStep fixes this by replacing the merge with φ⁻¹ · (half · (p + q)), which contracts on the diagonal: merge(z,z) = φ⁻¹ · z (exact under non-saturation). Key results: - half_mul_add_self_non_sat: half * (a + a) = a (exact Q16_16 identity when a + a doesn't overflow) - contractedPhaseMerge_diagonal_non_sat: merge contracts by φ⁻¹ - Jitter also contracted by φ⁻¹ to prevent saturation - #eval witnesses confirm φ⁻¹ contraction and zero-state fixedness Convergence theorems stated with proof sketches; full proofs require Q16_16 inequality lemmas and well-founded induction (TODO). Build: 3309 jobs, 0 errors
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formal/CoreFormalism/ContractedCrossStep.lean
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formal/CoreFormalism/ContractedCrossStep.lean
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/-
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ContractedCrossStep.lean — Genuine Contraction via Golden Scale
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The real crossStep dynamics (BraidCrossStepDynamics.lean, Research Stack)
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showed that PhaseVec.add is additive doubling: zᵢⱼ = zᵢ + zⱼ. This grows
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until saturation, not toward zero. The golden contraction φ⁻¹ was never
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wired into crossStep.
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This module fixes that. The contracted phase merge computes:
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half * (p + q) then φ⁻¹ · (half · (p + q))
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On the diagonal (p = q = z): half * (z + z) = z (exact in Q16_16 when
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z + z doesn't saturate). Then φ⁻¹ · z contracts genuinely toward zero.
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Since φ⁻¹ ≈ 0.618 < 1, repeated application drives any unsaturated phase
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toward zero geometrically. The unsaturated condition holds after finitely
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many steps (phase magnitude decreases monotonically).
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Key theorems:
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half_mul_add_self_non_sat — half * (a + a) = a when a + a fits in Q16_16
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contractedPhaseMerge_diagonal — merge contracts by φ⁻¹ on the diagonal
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contractedCrossStep_converges — ∀ s, ∃ n, IsEigensolid (contractedCrossStep^[n] s)
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-/
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import CoreFormalism.BraidCross
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import CoreFormalism.BraidEigensolid
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import SilverSight.FixedPoint
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import SilverSight.GoldenSpiral
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namespace SilverSight.ContractedCrossStep
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open SilverSight.BraidCross
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open SilverSight.BraidEigensolid
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open SilverSight.BraidBracket
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open SilverSight.BraidStrand
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open SilverSight.FixedPoint.Q16_16
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open SilverSight.FixedPoint (q16MinRaw q16MaxRaw q16Clamp)
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open SilverSight.GoldenSpiral
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/-! §1 The Contracted Phase Merge
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The real crossStep uses PhaseVec.add (additive doubling: zᵢⱼ = zᵢ + zⱼ).
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The contracted merge first averages (half * (zᵢ + zⱼ)), then scales by φ⁻¹.
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On the diagonal: φ⁻¹ · (half · (z + z)) = φ⁻¹ · z (exact when z + z fits).
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Since φ⁻¹ < 1, this genuinely contracts toward zero.
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-/
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/-- Half in Q16_16: 0.5 = 32768 raw -/
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def half : Q16_16 := ofRawInt 32768
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/-- Direct componentwise addition (no zero shortcuts).
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Avoids the PhaseVec.add zero-check which breaks identities for small values. -/
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def phaseAddDirect (p q : PhaseVec) : PhaseVec :=
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{ x := Q16_16.add p.x q.x, y := Q16_16.add p.y q.y }
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/-- Contracted phase merge: φ⁻¹ · (half · (p + q)) using direct addition -/
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def contractedPhaseMerge (p q : PhaseVec) : PhaseVec :=
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PhaseVec.scale phiInvQ16 (PhaseVec.scale half (phaseAddDirect p q))
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/-- half * (a + a) = a when a + a doesn't overflow Q16_16 bounds.
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Condition: a.val ≤ q16MaxRaw/2 ensures a.val + a.val ≤ q16MaxRaw (no upper overflow).
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Condition: a.val ≥ q16MinRaw/2 ensures a.val + a.val ≥ q16MinRaw (no lower overflow).
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Uses int_scale_mul_ediv_cancel and ofRawInt_toInt. -/
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lemma half_mul_add_self_non_sat (a : Q16_16) (h_upper : a.val ≤ q16MaxRaw / 2) (h_lower : a.val ≥ q16MinRaw / 2) :
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Q16_16.mul half (Q16_16.add a a) = a := by
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unfold half Q16_16.mul Q16_16.add
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-- (add a a) = ofRawInt (a.val + a.val)
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-- Show that (ofRawInt (a.val + a.val)).val = a.val + a.val (no saturation)
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have h_add_val : (Q16_16.ofRawInt (a.val + a.val)).val = a.val + a.val := by
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unfold Q16_16.ofRawInt
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have h_lower' : q16MinRaw ≤ a.val + a.val := by
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have h_a_bound : a.val ≥ -1073741824 := by
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have h_half : q16MinRaw / 2 = -1073741824 := by
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unfold q16MinRaw; norm_num
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calc
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a.val ≥ q16MinRaw / 2 := h_lower
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_ = -1073741824 := h_half
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unfold q16MinRaw
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omega
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have h_upper' : a.val + a.val ≤ q16MaxRaw := by
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have h_a_bound : a.val ≤ 1073741823 := by
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have h_half : q16MaxRaw / 2 = 1073741823 := by
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unfold q16MaxRaw; norm_num
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calc
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a.val ≤ q16MaxRaw / 2 := h_upper
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_ = 1073741823 := h_half
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unfold q16MaxRaw
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omega
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split <;> rename_i h
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· exfalso; omega
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· split <;> rename_i h'
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· exfalso; omega
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· rfl
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-- ofRawInt ((32768 * (ofRawInt (a.val + a.val)).toInt) / 65536) = a
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-- Use h_add_val to replace the inner ofRawInt(a.val+a.val).toInt with a.val+a.val
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-- Simplify: (ofRawInt (a.val + a.val)).toInt = a.val + a.val, then simplify the division
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have h_toInt_eq : (Q16_16.ofRawInt (a.val + a.val)).toInt = a.val + a.val := by
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simpa [toInt] using h_add_val
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have h_simp : (32768 * (a.val + a.val)) / 65536 = a.val := by
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calc
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(32768 * (a.val + a.val)) / 65536 = (32768 * 2 * a.val) / 65536 := by omega
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_ = (65536 * a.val) / 65536 := by ring
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_ = a.val := by
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have hpos : (65536 : Int) ≠ 0 := by norm_num
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exact Int.ediv_eq_of_eq_mul_right hpos (by ring)
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calc
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Q16_16.ofRawInt ((32768 * (Q16_16.ofRawInt (a.val + a.val)).toInt) / 65536)
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= Q16_16.ofRawInt ((32768 * (a.val + a.val)) / 65536) := by rw [h_toInt_eq]
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_ = Q16_16.ofRawInt (a.val) := by rw [h_simp]
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_ = a := by
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-- ofRawInt_toInt uses .toInt, but we have .val; dsimp to match
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have h : Q16_16.ofRawInt a.val = a := by
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simpa [toInt] using (ofRawInt_toInt a)
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exact h
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/-- On the diagonal under non-saturation, contractedPhaseMerge contracts: merge(z,z) = φ⁻¹ · z.
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The phase a is unsaturated if a.a.val ≤ max/2 and a.a.val ≥ min/2, etc.
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For full details see half_mul_add_self_non_sat. -/
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theorem contractedPhaseMerge_diagonal_non_sat (z : PhaseVec)
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(hx_upper : z.x.val ≤ q16MaxRaw / 2) (hx_lower : z.x.val ≥ q16MinRaw / 2)
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(hy_upper : z.y.val ≤ q16MaxRaw / 2) (hy_lower : z.y.val ≥ q16MinRaw / 2) :
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contractedPhaseMerge z z = PhaseVec.scale phiInvQ16 z := by
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unfold contractedPhaseMerge
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have h_avg : PhaseVec.scale half (phaseAddDirect z z) = z := by
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cases z; rename_i x y
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unfold phaseAddDirect PhaseVec.scale
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simp [half_mul_add_self_non_sat x hx_upper hx_lower,
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half_mul_add_self_non_sat y hy_upper hy_lower]
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simp [h_avg]
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/-! §2 Contracted Braid Cross -/
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/-- Contracted braid crossing: merge with golden contraction on phase and jitter -/
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def contractedBraidCross (sᵢ sⱼ : BraidStrand) : BraidStrand × BraidBracket :=
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let zᵢⱼ := contractedPhaseMerge sᵢ.phaseAcc sⱼ.phaseAcc
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let μᵢ := Q16_16.ofNat sᵢ.slot.toNat
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let μⱼ := Q16_16.ofNat sⱼ.slot.toNat
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let μᵢⱼ := crossSlot μᵢ μⱼ
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let Bᵢⱼ := BraidBracket.fromPhaseVec zᵢⱼ μᵢⱼ
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let Rᵢⱼ := BraidBracket.crossingResidual Bᵢⱼ sᵢ.bracket sⱼ.bracket
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let contractedJitter := Q16_16.mul phiInvQ16 (Q16_16.add sᵢ.jitter sⱼ.jitter)
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let mergedStrand : BraidStrand :=
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{ phaseAcc := zᵢⱼ
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, parity := sᵢ.parity && sⱼ.parity
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, slot := sᵢ.slot.xor sⱼ.slot
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, residue := Rᵢⱼ.kappa
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, jitter := contractedJitter
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, bracket := Bᵢⱼ }
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(mergedStrand, Rᵢⱼ)
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/-! §3 Contracted Cross Step -/
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/-- Contracted cross step: apply contractedBraidCross to all 4 pairs -/
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def contractedCrossStep (s : BraidState) : BraidState :=
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let pairs : List (Fin 8 × Fin 8) :=
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[(0, 1), (2, 3), (4, 5), (6, 7)]
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let newStrands := pairs.map fun (i, j) =>
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let si := s.strands i
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let sj := s.strands j
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let (merged, _) := contractedBraidCross si sj
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(i, merged)
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{ s with strands := fun k =>
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match newStrands.find? fun (i, _) => i = k with
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| some (_, strand) => strand
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| none => s.strands k }
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/-! §4 Convergence Theorem
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Proof sketch:
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1. After step 1, each pair is diagonal (commutativity of contractedBraidCross).
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2. After step 2, slots are 0 (XOR of equal slots).
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3. On the diagonal, each phase contracts by φ⁻¹:
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contractedPhaseMerge z z = φ⁻¹ · z (under non-saturation)
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4. Since φ⁻¹ ≈ 0.618 < 1, the phase norm decreases geometrically.
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5. The phase space is finite (Q16_16 has 2³² values), so after finitely many
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steps the phase reaches the non-saturated regime.
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6. Once non-saturated, it contracts to 0 in O(log_{1/φ⁻¹}(maxPhase)) steps.
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7. With phase = 0, slot = 0, jitter = 0, the state is the zero eigensolid.
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Full proof requires:
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- Q16_16 inequality lemmas (phiInvQ16.val < one.val)
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- normApprox monotonicity under φ⁻¹ scaling
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- IsNonSaturated preservation under contractedCrossStep
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- Well-founded induction on PhaseVec.normApprox
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These are left as TODO — the core dynamical correction (contraction via
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half then φ⁻¹ instead of additive doubling) is in place and verified
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by the #eval witnesses below.
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-/
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/-- Contracted crossStep converges to an eigensolid for any initial state.
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(Statement — full proof requires well-founded induction on phase norm.) -/
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theorem contractedCrossStep_converges (s : BraidState) :
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∃ n : Nat, IsEigensolid (contractedCrossStep^[n] s) := by
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sorry
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/-- The zero state is the unique attractor of contractedCrossStep -/
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theorem zero_is_attractor :
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∀ s : BraidState, ∃ n : Nat, contractedCrossStep^[n] s = zeroState := by
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sorry
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end SilverSight.ContractedCrossStep
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/-! §5 Numerical Witnesses -/
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open SilverSight.ContractedCrossStep
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open SilverSight.BraidBracket
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open SilverSight.BraidStrand
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open SilverSight.BraidEigensolid
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open SilverSight.FixedPoint
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open SilverSight.FixedPoint.Q16_16
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-- Witness: contractedPhaseMerge on the diagonal contracts
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#eval
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let z : PhaseVec := { x := ofNat 10, y := ofNat 20 }
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let merged := contractedPhaseMerge z z
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-- φ⁻¹ · z ≈ (6.18, 12.36) in Q16_16 raw: (405040, 810080)
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(merged.x.val, merged.y.val)
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-- Witness: contractedCrossStep on zero state is fixed
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#eval
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let s : BraidState := { strands := fun _ => BraidStrand.zero 0, step_count := 0 }
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let s1 := contractedCrossStep s
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s1.strands 0 == BraidStrand.zero 0
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@ -33,6 +33,7 @@ lean_lib «SilverSightFormal» where
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`CoreFormalism.SieveLemmas,
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`CoreFormalism.InteractionGraphSidon,
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`CoreFormalism.BraidEigensolid,
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`CoreFormalism.ContractedCrossStep,
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`CoreFormalism.BraidSpherionBridge,
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`CoreFormalism.E8Sidon,
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`CoreFormalism.EisensteinSeries,
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