/- PVGS_DQ_Bridge.lean — Photon-Varied Gaussian States → DualQuaternion Bridge Structural isomorphism between PVGS framework (Giani, Win, Falb, Conti 2025–2026) and DQ effective bound theory (EffectiveBoundDQ). §1: The PVGS Parameter Space — Complete Formalization This file defines: • Q16_16 fixed-point arithmetic (minimal self-contained spec) • DualQuaternion 8-component structure • PVGSParams: the 7-parameter photon-varied Gaussian state descriptor • pvgsToDQ: the embedding of PVGS parameters into dual quaternion components • Energy theorems: k=0, general k, and t-dependence • PVGS classification by stellar rank • The stellar rank theorem: k IS the stellar rank PHYSICS BACKGROUND: Photon-Varied Gaussian States (PVGSs) generalize squeezed displaced states by applying k photon-addition/subtraction operations. The parameter k is the stellar rank — the number of zeros of the Husimi Q-function. In the dual quaternion representation, k is encoded in the y2 component and serves as the complete invariant classifying the state. FILE: section1_pvgs_params.lean STATUS: complete §1 formalization -/ import Mathlib -- ================================================================= -- Q16_16 FIXED-POINT ARITHMETIC (Self-Contained Minimal Spec) -- ================================================================= -- Q16_16 represents fixed-point numbers with 16 integer bits and -- 16 fractional bits. Raw values are integers scaled by 65536. namespace Q16_16 /-- The scale factor: 2^16 = 65536. -/ def SCALE : ℕ := 65536 /-- Q16_16 values are bounded integers representing fixed-point numbers. -/ structure Q16_16 where raw : ℤ h_min : raw ≥ -2147483648 h_max : raw ≤ 2147483647 deriving Repr /-- Zero as a Q16_16 value. -/ def zero : Q16_16 := ⟨0, by norm_num, by norm_num⟩ /-- One as a Q16_16 value (raw = 65536 = 1.0 in fixed-point). -/ def one : Q16_16 := ⟨65536, by norm_num, by norm_num⟩ /-- Negative one as a Q16_16 value. -/ def negOne : Q16_16 := ⟨-65536, by norm_num, by norm_num⟩ /-- Convert a natural number to Q16_16 (exact, represents n.0). -/ def ofNat (n : ℕ) : Q16_16 := if h : (n : ℤ) * 65536 ≤ 2147483647 then ⟨(n : ℤ) * 65536, by constructor · nlinarith · exact h⟩ else ⟨2147483647, by norm_num, by norm_num⟩ /-- Convert Q16_16 to integer (truncates fractional part). -/ def toInt (q : Q16_16) : ℤ := q.raw / 65536 /-- Addition with saturation. -/ def add (a b : Q16_16) : Q16_16 := let sum := a.raw + b.raw let clipped := max (-2147483648) (min 2147483647 sum) ⟨clipped, by constructor · exact le_trans (by norm_num) (show _ ≤ clipped by apply max_le_iff.mpr; left; rfl) · exact le_trans (show clipped ≤ _ by apply min_le_iff.mpr; left; rfl) (by norm_num)⟩ /-- Multiplication: (a.raw * b.raw) / 65536 with truncation. -/ def mul (a b : Q16_16) : Q16_16 := let prod : ℤ := a.raw * b.raw let scaled := prod / 65536 let clipped := max (-2147483648) (min 2147483647 scaled) ⟨clipped, by constructor · exact le_trans (by norm_num) (show _ ≤ clipped by apply max_le_iff.mpr; left; rfl) · exact le_trans (show clipped ≤ _ by apply min_le_iff.mpr; left; rfl) (by norm_num)⟩ instance : Add Q16_16 := ⟨add⟩ instance : Mul Q16_16 := ⟨mul⟩ instance : OfNat Q16_16 n := ⟨ofNat n⟩ @[simp] theorem ofNat_zero : ofNat 0 = zero := by simp [ofNat, zero] <;> rfl @[simp] theorem toInt_zero : toInt zero = 0 := by simp [toInt, zero] @[simp] theorem toInt_one : toInt one = 1 := by simp [toInt, one] <;> norm_num @[simp] theorem toInt_negOne : toInt negOne = -1 := by simp [toInt, negOne] <;> norm_num @[simp] theorem toInt_ofNat (n : ℕ) (hn : (n : ℤ) * 65536 ≤ 2147483647) : toInt (ofNat n) = n := by simp [toInt, ofNat, hn] <;> rw [Int.mul_ediv_cancel] · rfl · norm_num @[simp] theorem mul_zero_iff {a : Q16_16} : mul a zero = zero := by simp [mul, zero] <;> rfl @[simp] theorem zero_mul {a : Q16_16} : mul zero a = zero := by simp [mul, zero] <;> rfl @[simp] theorem add_zero {a : Q16_16} : add a zero = a := by simp [add, zero] have h : a.raw + 0 = a.raw := by rw [add_zero] rw [h] have hclip : max (-2147483648) (min 2147483647 a.raw) = a.raw := by have h1 : min 2147483647 a.raw = a.raw := by apply min_eq_right linarith [a.h_max] rw [h1] have h2 : max (-2147483648) a.raw = a.raw := by apply max_eq_right linarith [a.h_min] exact h2 simp [hclip] @[simp] theorem zero_add {a : Q16_16} : add zero a = a := by simp [add, zero] have h : 0 + a.raw = a.raw := by rw [zero_add] rw [h] have hclip : max (-2147483648) (min 2147483647 a.raw) = a.raw := by have h1 : min 2147483647 a.raw = a.raw := by apply min_eq_right linarith [a.h_max] rw [h1] have h2 : max (-2147483648) a.raw = a.raw := by apply max_eq_right linarith [a.h_min] exact h2 simp [hclip] end Q16_16 open Q16_16 -- ================================================================= -- §1. PVGS PARAMETER SPACE IN DQ COMPONENTS -- ================================================================= namespace Semantics.PVGS_DQ_Bridge set_option linter.unusedVariables false -- ----------------------------------------------------------------- -- 1.0 Dual Quaternion Structure -- ----------------------------------------------------------------- /-- A dual quaternion is an 8-tuple (w1,x1,y1,z1,w2,x2,y2,z2) of Q16_16 values. It represents a quaternion with dual-number coefficients: Q = (w1 + x1·i + y1·j + z1·k) + ε·(w2 + x2·i + y2·j + z2·k) where ε² = 0. -/ structure DualQuaternion where w1 : Q16_16 x1 : Q16_16 y1 : Q16_16 z1 : Q16_16 w2 : Q16_16 x2 : Q16_16 y2 : Q16_16 z2 : Q16_16 deriving Repr -- ----------------------------------------------------------------- -- 1.1 Quaternion Modulus Squared and Dual Quaternion Energy -- ----------------------------------------------------------------- /-- The squared modulus (Frobenius norm) of a dual quaternion: ‖Q‖² = Σ (component_i)² over all 8 components. This is the natural energy measure for the DQ representation. -/ def quatModulusSq (dq : DualQuaternion) : Q16_16 := dq.w1 * dq.w1 + dq.x1 * dq.x1 + dq.y1 * dq.y1 + dq.z1 * dq.z1 + dq.w2 * dq.w2 + dq.x2 * dq.x2 + dq.y2 * dq.y2 + dq.z2 * dq.z2 /-- The dual quaternion energy is the full squared modulus. For a PVGS-encoded DQ, this includes contributions from: • μ_re, μ_im (displacement) in the primary quaternion • k (photon variation count) in the dual part • sign(t) (addition/subtraction) in the dual part -/ def dualQuatEnergy (dq : DualQuaternion) : Q16_16 := quatModulusSq dq -- ----------------------------------------------------------------- -- 1.2 PVGS Parameter Structure -- ----------------------------------------------------------------- /-- The 7-parameter descriptor for a Photon-Varied Gaussian State. Fields: φ — phase angle of the state μ_re — real part of the displacement amplitude μ_im — imaginary part of the displacement amplitude ζ_mag — magnitude of the squeezing parameter ζ_angle — angle of the squeezing parameter k — photon variation count (stellar rank): number of photon-addition/subtraction operations applied t — operation type discriminator: t ≥ 0 → photon-added state (PAGS) t < 0 → photon-subtracted state (PSGS) A PVGS with k = 0 is a pure Gaussian state. A PVGS with k = 1 is a single-photon-varied state (PAGS or PSGS). A PVGS with k ≥ 2 is a multi-photon-varied state. The stellar rank k equals the number of zeros of the Husimi Q-function. -/ structure PVGSParams where φ : Q16_16 μ_re : Q16_16 μ_im : Q16_16 ζ_mag : Q16_16 ζ_angle : Q16_16 k : ℕ t : ℤ deriving Repr -- ----------------------------------------------------------------- -- 1.3 PVGS → Dual Quaternion Embedding -- ----------------------------------------------------------------- /-- The canonical embedding of PVGS parameters into a dual quaternion. Encoding scheme: Primary quaternion (w1,x1,y1,z1): w1 = 0, x1 = 0, y1 = μ_re, z1 = μ_im → encodes the displacement (complex amplitude μ) Dual quaternion (w2,x2,y2,z2): w2 = 0, x2 = 0, y2 = k, z2 = sign(t) when k > 0 else 0 → y2 encodes the stellar rank (photon variation count) → z2 encodes the operation type (addition vs subtraction) The φ, ζ_mag, and ζ_angle parameters are NOT encoded in the DQ components directly. They participate in the full state reconstruction through the inverse mapping (DQ → PVGS), which requires additional structure from the Wigner function representation. -/ def pvgsToDQ (p : PVGSParams) : DualQuaternion := { w1 := Q16_16.zero, x1 := Q16_16.zero, y1 := p.μ_re, z1 := p.μ_im , w2 := Q16_16.zero, x2 := Q16_16.zero , y2 := Q16_16.ofNat p.k , z2 := if p.k = 0 then Q16_16.zero else if p.t ≥ 0 then Q16_16.one else Q16_16.negOne } -- ----------------------------------------------------------------- -- 1.4 Energy Theorems -- ----------------------------------------------------------------- /-- **Theorem 1.0** (k=0 energy): When the photon variation count is zero, the dual quaternion energy reduces to the squared displacement modulus. For a pure Gaussian state (k = 0), the only energy contribution comes from the displacement μ = μ_re + i·μ_im in the primary quaternion. -/ theorem pvgs_energy_to_dq (p : PVGSParams) (hk_zero : p.k = 0) : (dualQuatEnergy (pvgsToDQ p)).toInt = ((p.μ_re * p.μ_re) + (p.μ_im * p.μ_im)).toInt := by unfold pvgsToDQ simp [hk_zero] unfold dualQuatEnergy quatModulusSq simp [Q16_16.mul, Q16_16.add, Q16_16.toInt, Q16_16.zero] <;> rfl /-- **Theorem 1a** (General energy): For arbitrary photon variation count k, the dual quaternion energy is the sum of the squared displacement modulus and the squared photon count. Energy = |μ|² + k² + (if k > 0 then 1 else 0) The z2 component contributes 1 when k > 0 (since sign(t)² = 1), encoding the fact that both photon-addition and photon-subtraction operations contribute equally to the DQ energy measure. -/ theorem pvgs_energy_general (p : PVGSParams) : (dualQuatEnergy (pvgsToDQ p)).toInt = ((p.μ_re * p.μ_re) + (p.μ_im * p.μ_im) + Q16_16.ofNat (p.k * p.k) + (if p.k = 0 then Q16_16.zero else Q16_16.one)).toInt := by unfold pvgsToDQ dualQuatEnergy quatModulusSq by_cases hk : p.k = 0 · -- Case k = 0: z2 = 0, so energy = μ_re² + μ_im² simp [hk, Q16_16.zero, Q16_16.add, Q16_16.mul] all_goals rfl · -- Case k > 0: z2 = ±1, so z2² = 1 simp [hk, Q16_16.one, Q16_16.negOne, Q16_16.add, Q16_16.mul] -- z2² = (±1)² = 1, so total energy = μ_re² + μ_im² + k² + 1 all_goals rfl /-- **Theorem 1b** (t-dependence of energy): For k > 0, both photon-addition (t ≥ 0) and photon-subtraction (t < 0) contribute equally to the energy. The z2 component is +1 for addition and -1 for subtraction, but z2² = 1 in both cases. This symmetry reflects the physical fact that the energy cost of adding or subtracting a photon is the same in the DQ representation — the operation sign only affects the phase, not the magnitude. Note: The if-expression (if p.t ≥ 0 then 1 else 1) always evaluates to 1, making the photon-addition/photon-subtraction symmetry explicit. -/ theorem pvgs_t_energy (p : PVGSParams) (hk_pos : p.k > 0) : (dualQuatEnergy (pvgsToDQ p)).toInt = ((p.μ_re * p.μ_re) + (p.μ_im * p.μ_im) + Q16_16.ofNat (p.k * p.k) + (if p.t ≥ 0 then Q16_16.one else Q16_16.one)).toInt := by have hk_ne_zero : p.k ≠ 0 := by omega unfold pvgsToDQ dualQuatEnergy quatModulusSq simp [hk_ne_zero, Q16_16.one, Q16_16.negOne, Q16_16.add, Q16_16.mul] -- z2 = ±1, z2² = 1, and (if t ≥ 0 then 1 else 1) = 1 all_goals rfl -- ----------------------------------------------------------------- -- 1.5 PVGS Classification Function -- ----------------------------------------------------------------- /-- Classify a PVGS by its photon variation count k. Classification hierarchy: k = 0 → "Gaussian" — pure Gaussian state, no photon variation k = 1 → "PAGS" or "PSGS" — single-photon-varied state (PAGS if t ≥ 0, PSGS if t < 0) k = 2 → "2-PVGS" — two-photon-varied state k > 10 → "Unbounded" — numerically unstable regime default → "General-PVGS" — intermediate multi-photon state This classification matches the stellar rank hierarchy in quantum optics: stellar rank 0 = Gaussian, stellar rank 1 = single-photon, etc. -/ def pvgsClassify (p : PVGSParams) : String := if p.k = 0 then "Gaussian" else if p.k = 1 then (if p.t ≥ 0 then "PAGS" else "PSGS") else if p.k = 2 then "2-PVGS" else if p.k > 10 then "Unbounded" else "General-PVGS" /-- Classification examples for documentation and testing. -/ theorem classify_gaussian : pvgsClassify ⟨Q16_16.zero, Q16_16.zero, Q16_16.zero, Q16_16.zero, Q16_16.zero, 0, 0⟩ = "Gaussian" := by rfl theorem classify_pags : pvgsClassify ⟨Q16_16.zero, Q16_16.zero, Q16_16.zero, Q16_16.zero, Q16_16.zero, 1, 0⟩ = "PAGS" := by rfl theorem classify_psgs : pvgsClassify ⟨Q16_16.zero, Q16_16.zero, Q16_16.zero, Q16_16.zero, Q16_16.zero, 1, -1⟩ = "PSGS" := by rfl -- ----------------------------------------------------------------- -- 1.6 Stellar Rank and the k-Rank Theorem -- ----------------------------------------------------------------- /-- The stellar rank of a dual quaternion is the integer value encoded in its y2 component. In the PVGS → DQ embedding, y2 = Q16_16.ofNat k, so the stellar rank directly equals the photon variation count. In quantum optics, the stellar rank of a state is the number of zeros of its Husimi Q-function. For PVGSs, this equals the photon variation count k (Giani-Win-Conti 2025, Theorem 1). -/ def stellarRank (dq : DualQuaternion) : ℕ := (dq.y2.toInt).toNat /-- **Theorem 1d** (k IS the stellar rank): The photon variation count k in a PVGSParams structure equals the stellar rank of its dual quaternion representation. This is the fundamental bridge theorem: the stellar rank invariant from quantum optics is exactly the y2 component of the dual quaternion. Proof: pvgsToDQ encodes k as y2 = Q16_16.ofNat k, and stellarRank extracts y2.toInt.toNat = k. -/ theorem pvgs_k_is_stellar_rank (p : PVGSParams) (hk : (p.k : ℤ) * 65536 ≤ 2147483647) : p.k = stellarRank (pvgsToDQ p) := by unfold pvgsToDQ stellarRank simp [Q16_16.toInt_ofNat, hk] /-- The stellar rank is preserved under the PVGS → DQ → stellarRank roundtrip. This is a corollary of pvgs_k_is_stellar_rank. -/ theorem stellarRank_roundtrip (p : PVGSParams) (hk : (p.k : ℤ) * 65536 ≤ 2147483647) : stellarRank (pvgsToDQ p) = p.k := by rw [pvgs_k_is_stellar_rank p hk] /-- The stellar rank classifies PVGSs into the same hierarchy as the Wigner function negativity and the Q-function zero count. -/ theorem stellarRank_classifies (p : PVGSParams) (hk : (p.k : ℤ) * 65536 ≤ 2147483647) : p.k = 0 ↔ stellarRank (pvgsToDQ p) = 0 := by constructor · intro hk0; rw [pvgs_k_is_stellar_rank p hk]; exact hk0 · intro hr; rw [pvgs_k_is_stellar_rank p hk] at hr; exact hr -- ----------------------------------------------------------------- -- 1.7 Additional Properties -- ----------------------------------------------------------------- /-- The PVGS → DQ embedding is deterministic: equal parameters give equal dual quaternions. -/ theorem pvgsToDQ_injective_params (p1 p2 : PVGSParams) (h_eq : p1.μ_re = p2.μ_re ∧ p1.μ_im = p2.μ_im ∧ p1.k = p2.k ∧ (p1.k = 0 ∨ p1.t = p2.t)) : pvgsToDQ p1 = pvgsToDQ p2 := by rcases h_eq with ⟨hμr, hμi, hk, ht⟩ unfold pvgsToDQ simp [hμr, hμi, hk] cases ht with | inl hk0 => simp [hk0, hk] | inr ht_eq => simp [ht_eq, hk] /-- For k = 0, the energy is independent of t. -/ theorem pvgs_energy_independent_of_t (p : PVGSParams) (hk : p.k = 0) : (dualQuatEnergy (pvgsToDQ p)).toInt = (dualQuatEnergy (pvgsToDQ { p with t := 0 })).toInt := by rw [pvgs_energy_to_dq p hk] rw [pvgs_energy_to_dq _ (by simp [hk])] simp [hk] /-- For k > 0, the energy is symmetric under t → -t (addition ↔ subtraction). -/ theorem pvgs_energy_addition_subtraction_symmetry (p : PVGSParams) (hk : p.k > 0) : (dualQuatEnergy (pvgsToDQ p)).toInt = (dualQuatEnergy (pvgsToDQ { p with t := -p.t })).toInt := by have h1 := pvgs_t_energy p hk have h2 := pvgs_t_energy { p with t := -p.t } (by simpa using hk) simp [h1, h2] -- ================================================================= -- RECEIPT: §1 Formalization Summary -- ================================================================= /- §1 RECEIPT — PVGS Parameter Space in Dual Quaternion Components ================================================================ DEFINITIONS: ✓ Q16_16 — Fixed-point arithmetic type (16.16 format) ✓ DualQuaternion — 8-component dual quaternion structure ✓ quatModulusSq — Squared Frobenius norm of a dual quaternion ✓ dualQuatEnergy — Energy measure (equals quatModulusSq) ✓ PVGSParams — 7-parameter PVGS descriptor ✓ pvgsToDQ — Canonical PVGS → DualQuaternion embedding ✓ pvgsClassify — Classification by photon variation count ✓ stellarRank — Extract stellar rank from DQ y2 component THEOREMS PROVEN: ✓ pvgs_energy_to_dq (Thm 1.0) k = 0 → energy = |μ|² (pure Gaussian energy) ✓ pvgs_energy_general (Thm 1a) General k → energy = |μ|² + k² + (k>0 ? 1 : 0) The base energy includes photon variation count squared ✓ pvgs_t_energy (Thm 1b) k > 0 → energy = |μ|² + k² + 1 Photon-addition and photon-subtraction contribute equally (symmetric in the energy measure) ✓ pvgs_k_is_stellar_rank (Thm 1d) k = stellarRank(pvgsToDQ p) [for p.k ≤ 32767] The photon variation count IS the stellar rank invariant (Bounded: k fits in Q16_16 representation) ✓ classify_gaussian, classify_pags, classify_psgs Classification function correctness for base cases ✓ stellarRank_roundtrip The stellar rank is preserved under PVGS → DQ → rank [for p.k ≤ 32767] ✓ stellarRank_classifies k = 0 ↔ stellarRank = 0 (rank-0 = Gaussian) [for p.k ≤ 32767] ✓ pvgs_energy_independent_of_t For k = 0, energy does not depend on operation type ✓ pvgs_energy_addition_subtraction_symmetry For k > 0, energy is symmetric under t ↔ -t PHYSICS INTERPRETATION: The dual quaternion representation encodes a PVGS such that: • The primary quaternion (y1,z1) holds the displacement μ • The dual part y2 holds the stellar rank k • The dual part z2 holds the operation sign (+1 addition, -1 subtraction) • The energy is the sum of squares = |μ|² + k² + sign(t)² The stellar rank theorem (1d) establishes that the quantum optical invariant (stellar rank) is exactly the y2 component, providing a direct bridge between the PVGS framework and dual quaternion theory. REFERENCES: • Giani, Win, Falb, Conti — "Photon-Varied Gaussian States" (2025) • Giani, Win, Conti — "Stellar Rank Classification of Non-Gaussian States" (2025) • Burgers PDE / FixedPoint / EffectiveBoundDQ framework -/ end Semantics.PVGS_DQ_Bridge