/-!
# Q0_64 Experimental — Ultra-Precision Dimensionless Scalars

**Status:** TEST BRANCH — Experimental, do not commit to main.
**Purpose:** Explore 64-bit pure fraction for dimensionless quantities
where Q0_16 (≈ 0.00003 resolution) is insufficient.

**Q0_16 resolution:** 1/32767 ≈ 3.05×10⁻⁵
**Q0_64 resolution:** 1/2⁶³ ≈ 1.08×10⁻¹⁹

**Trade-off:** Q0_64 is 4× larger than Q0_16 (8 bytes vs 2 bytes).
Per AGENTS.md §1.4 default: Q0_16. Q0_64 is for contexts where
confidence thresholds demand sub-ppm precision (e.g., 6.5σ tail events).

This file is standalone: zero imports.
-/

-- ═══════════════════════════════════════════════════════════════════════════
-- §0  Q0_64 Structure
-- ═══════════════════════════════════════════════════════════════════════════

/-- Q0.64 pure fraction: 64-bit unsigned interpreted as signed fixed-point.
    Range: [-1.0, 1.0 - 2⁻⁶⁴] ≈ [-1.0, 0.99999999999999999989]
    Resolution: 2⁻⁶³ ≈ 1.08 × 10⁻¹⁹
    0x8000_0000_0000_0000 = 1.0 (max positive)
    0x0000_0000_0000_0000 = 0.0 -/
structure Q0_64 where
  val : UInt64
  deriving Repr, BEq, Inhabited

-- ═══════════════════════════════════════════════════════════════════════════
-- §1  Constants
-- ═══════════════════════════════════════════════════════════════════════════

def Q0_64.zero : Q0_64 := ⟨0x0000_0000_0000_0000⟩
def Q0_64.one  : Q0_64 := ⟨0x8000_0000_0000_0000⟩  -- ~1.0
def Q0_64.half : Q0_64 := ⟨0x4000_0000_0000_0000⟩ -- 0.5

-- ═══════════════════════════════════════════════════════════════════════════
-- §2  Basic Arithmetic
-- ═══════════════════════════════════════════════════════════════════════════

def Q0_64.neg (x : Q0_64) : Q0_64 := ⟨-x.val⟩

def Q0_64.add (a b : Q0_64) : Q0_64 := ⟨a.val + b.val⟩

def Q0_64.sub (a b : Q0_64) : Q0_64 := ⟨a.val - b.val⟩

/-- Multiplication: (a/2⁶³) × (b/2⁶³) × 2⁶³ = (a×b)/2⁶³.
    Uses unbounded Nat intermediate to avoid 64-bit overflow. -/
def Q0_64.mul (a b : Q0_64) : Q0_64 :=
  let prod : Nat := a.val.toNat * b.val.toNat
  ⟨(prod >>> 63).toUInt64⟩

/-- Division: a/b = (a × 2⁶³) / b. Saturate on div-by-zero. -/
def Q0_64.div (a b : Q0_64) : Q0_64 :=
  if b.val = 0 then ⟨0x8000_0000_0000_0000⟩
  else
    let num : Nat := a.val.toNat <<< 63
    ⟨(num / b.val.toNat).toUInt64⟩

-- ═══════════════════════════════════════════════════════════════════════════
-- §3  Precision Comparison (Q0_16 vs Q0_64)
-- ═══════════════════════════════════════════════════════════════════════════

/-- Smallest representable positive value in Q0_16: 1/32767 ≈ 3.05×10⁻⁵ -/
def q0_16Epsilon : Nat := 1
  -- Q0_16.val = 1 corresponds to ≈ 3.05×10⁻⁵ in float space

/-- Smallest representable positive value in Q0_64: 1/2⁶³ ≈ 1.08×10⁻¹⁹ -/
def q0_64Epsilon : Nat := 1
  -- Q0_64.val = 1 corresponds to ≈ 1.08×10⁻¹⁹ in float space

/-- Ratio of precisions: Q0_64 is 2⁴⁸× finer than Q0_16.
    2⁶³ / 2¹⁵ = 2⁴⁸ ≈ 2.8 × 10¹⁴ -/
def precisionRatio : Nat :=
  let q16Max : Nat := 0x8000    -- 32768
  let q64Max : Nat := 0x8000_0000_0000_0000  -- 2^63
  q64Max / q16Max

-- ═══════════════════════════════════════════════════════════════════════════
-- §4  6.5σ Tail Probability in Q0_64
-- ═══════════════════════════════════════════════════════════════════════════

/-- Standard normal CDF at 6.5σ: Φ(6.5) ≈ 4.02 × 10⁻¹¹.
    In Q0_16: this rounds to 0 (below epsilon).
    In Q0_64: representable as val ≈ 346 (6.5σ in Q0_64 units, approximate). -/
def sigma65TailProbabilityQ0_16Representable : Bool := false
  -- 4.02×10⁻¹¹ << Q0_16 epsilon (3.05×10⁻⁵). Rounds to zero.

def sigma65TailProbabilityQ0_64Representable : Bool := true
  -- 4.02×10⁻¹¹ >> Q0_64 epsilon (1.08×10⁻¹⁹). Easily representable.

-- ═══════════════════════════════════════════════════════════════════════════
-- §5  Witness Values
-- ═══════════════════════════════════════════════════════════════════════════

#eval Q0_64.zero  -- { val := 0 }
#eval Q0_64.one   -- { val := 9223372036854775808 }
#eval Q0_64.half  -- { val := 4611686018427387904 }

#eval Q0_64.add Q0_64.half Q0_64.half  -- { val := 9223372036854775808 } = one
#eval Q0_64.sub Q0_64.one Q0_64.half   -- { val := 4611686018427387904 } = half
#eval Q0_64.mul Q0_64.half Q0_64.half  -- 0.25: { val := 2305843009213693952 }
#eval Q0_64.div Q0_64.one Q0_64.half    -- 2.0: { val := 0 }  -- wraps (2.0 exceeds [-1,1) range)

#eval precisionRatio  -- 281474976710656 (2^48)

-- ═══════════════════════════════════════════════════════════════════════════
-- §6  Size Impact on Neural Compression (Back-of-Envelope)
-- ═══════════════════════════════════════════════════════════════════════════

/-- Q0_16 neural state: 1 PB = 10¹⁵ bytes.
    Q0_64 neural state: 4 PB = 4×10¹⁵ bytes (4× larger). -/
def q0_16StateBytes : Nat := 1000000000000000
def q0_64StateBytes : Nat := q0_16StateBytes * 4

/-- At 1,250× compression:
    Q0_16 compressed: 800 GB
    Q0_64 compressed: 3,200 GB (exceeds 800 GB target) -/
def q0_16CompressedGb : Nat := q0_16StateBytes / 1250 / 1000000000
def q0_64CompressedGb : Nat := q0_64StateBytes / 1250 / 1000000000

/-- To hit 800 GB with Q0_64 raw state, compression ratio must be:
    C = 4×10¹⁵ / 8×10¹¹ = 5,000× -/
def requiredRatioForQ0_64 : Nat := q0_64StateBytes / 800000000000

#eval q0_16CompressedGb        -- 800
#eval q0_64CompressedGb        -- 3200 (fails 800 GB target)
#eval requiredRatioForQ0_64    -- 5000 (need 4× higher compression)

-- ═══════════════════════════════════════════════════════════════════════════
-- §7  Verdict
-- ═══════════════════════════════════════════════════════════════════════════

/-- Q0_64 buys 2^48× precision at a 4× size cost.
    For 6.5σ tail probabilities (4×10⁻¹¹), Q0_16 rounds to zero.
    Q0_64 captures it with ~37 bits of headroom.

    Trade-off: if the neural state is stored as Q0_64, the compression
    pipeline must achieve 5,000× instead of 1,250× to hit the same
    storage target. This is feasible with deeper quantization-aware
    training but must be proven in the pipeline. -/
def testBranchVerdict : String :=
  "Q0_64: 2^48 precision gain, 4x size cost. " ++
  "6.5σ tails representable. " ++
  "Compression target increases 1,250x -> 5,000x. " ++
  "Test branch only — do not merge without pipeline proof."

#eval testBranchVerdict