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220 lines
12 KiB
Text
220 lines
12 KiB
Text
/-!
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# Combinatorial Precision Options: Q0.8 / Q0.16 / Q0.32 / Q0.64
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**Status:** TEST BRANCH — Experimental multi-precision palette.
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**Purpose:** Provide four fixed-point pure-fraction widths so the pipeline
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chooses the minimum sufficient precision per datum, not per file.
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**Resolution ladder:**
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| Width | Size | ε (resolution) | Use case |
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|-------|------|----------------|----------|
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| Q0.8 | 1 B | 7.8×10⁻³ | Coarse probabilities, RGB, early layers |
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| Q0.16 | 2 B | 3.0×10⁻⁵ | Default dimensionless (AGENTS.md §1.4) |
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| Q0.32 | 4 B | 4.7×10⁻¹⁰ | Sub-percentile tails, 4σ–5σ events |
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| Q0.64 | 8 B | 1.1×10⁻¹⁹ | 6.5σ tails, high-precision invariants |
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**Combinatorial rule:** Start at Q0.8. Promote one step on overflow or
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precision demand. Demote when result fits lower tier.
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This file is standalone: zero imports.
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-/
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Precision Constants (all four tiers)
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-- ═══════════════════════════════════════════════════════════════════════════
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def q08_maxVal : Nat := 0x7F -- 127
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def q08_epsilonVal : Nat := 1 -- 1/128 ≈ 7.8×10⁻³
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def q08_sizeBytes : Nat := 1
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def q16_maxVal : Nat := 0x7FFF -- 32767
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def q16_epsilonVal : Nat := 1 -- 1/32767 ≈ 3.05×10⁻⁵
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def q16_sizeBytes : Nat := 2
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def q32_maxVal : Nat := 0x7FFF_FFFF -- 2,147,483,647
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def q32_epsilonVal : Nat := 1 -- 1/2³¹ ≈ 4.66×10⁻¹⁰
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def q32_sizeBytes : Nat := 4
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def q64_maxVal : Nat := 0x8000_0000_0000_0000 -- 2⁶³
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def q64_epsilonVal : Nat := 1 -- 1/2⁶³ ≈ 1.08×10⁻¹⁹
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def q64_sizeBytes : Nat := 8
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Combinatorial Scalar Type (four-tier adaptive)
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-- ═══════════════════════════════════════════════════════════════════════════
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inductive CombinatorialScalar where
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| q08 (val : UInt8)
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| q16 (val : UInt16)
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| q32 (val : UInt32)
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| q64 (val : UInt64)
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deriving Repr, BEq, Inhabited
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def CombinatorialScalar.sizeBytes (s : CombinatorialScalar) : Nat :=
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match s with
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| .q08 _ => q08_sizeBytes
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| .q16 _ => q16_sizeBytes
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| .q32 _ => q32_sizeBytes
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| .q64 _ => q64_sizeBytes
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Promotion Rules (upgrade on demand)
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-- ═══════════════════════════════════════════════════════════════════════════
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def promote08to16 (v : UInt8) : CombinatorialScalar :=
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CombinatorialScalar.q16 (v.toNat.toUInt16)
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def promote16to32 (v : UInt16) : CombinatorialScalar :=
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CombinatorialScalar.q32 (v.toNat.toUInt32)
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def promote32to64 (v : UInt32) : CombinatorialScalar :=
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CombinatorialScalar.q64 (v.toNat.toUInt64)
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/-- Cascade promotion: try q08→q16→q32→q64 until the raw value fits.
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This is the entry point for unknown-magnitude values. -/
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def promoteCascading (raw : Nat) : CombinatorialScalar :=
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if raw ≤ q08_maxVal then
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CombinatorialScalar.q08 raw.toUInt8
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else if raw ≤ q16_maxVal then
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CombinatorialScalar.q16 raw.toUInt16
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else if raw ≤ q32_maxVal then
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CombinatorialScalar.q32 raw.toUInt32
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else
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CombinatorialScalar.q64 raw.toUInt64
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Demotion Rules (downgrade when safe)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Demote q64→q32 if upper 32 bits are zero. -/
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def demote64to32 (v : UInt64) : CombinatorialScalar :=
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if v >>> 32 = 0 then CombinatorialScalar.q32 v.toNat.toUInt32
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else CombinatorialScalar.q64 v
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/-- Demote q32→q16 if upper 16 bits are zero. -/
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def demote32to16 (v : UInt32) : CombinatorialScalar :=
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if v >>> 16 = 0 then CombinatorialScalar.q16 v.toNat.toUInt16
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else CombinatorialScalar.q32 v
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/-- Demote q16→q08 if upper 8 bits are zero. -/
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def demote16to08 (v : UInt16) : CombinatorialScalar :=
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if v >>> 8 = 0 then CombinatorialScalar.q08 v.toNat.toUInt8
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else CombinatorialScalar.q16 v
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/-- Full cascade demote: q64→q32→q16→q08, stopping at first lossless step. -/
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def demoteCascading (v : UInt64) : CombinatorialScalar :=
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let d32 := demote64to32 v
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match d32 with
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| .q32 w =>
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let d16 := demote32to16 w
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match d16 with
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| .q16 x => demote16to08 x
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| other => other
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| other => other
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Tail-Event Precision Selector
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Select the minimum precision tier that can represent a given tail
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probability without rounding to zero.
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3σ tail ≈ 0.135% → Q0.8 (ε = 0.78%) ROUNDS TO ZERO
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4σ tail ≈ 0.0032% → Q0.16 (ε = 0.003%) ROUNDS TO ZERO
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5σ tail ≈ 2.9×10⁻⁷ → Q0.32 (ε = 4.7×10⁻¹⁰) representable
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6.5σ tail ≈ 4×10⁻¹¹ → Q0.64 required -/
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def selectPrecision (probability : Nat) (isTailEvent : Bool) : CombinatorialScalar :=
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if isTailEvent ∧ probability < q16_epsilonVal then
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-- Below Q0.16 resolution: need at least Q0.32, possibly Q0.64
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if probability < q32_epsilonVal then
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CombinatorialScalar.q64 (probability.toUInt64 <<< 48)
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else
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CombinatorialScalar.q32 (probability.toUInt32 <<< 16)
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else if probability ≤ q08_maxVal then
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CombinatorialScalar.q08 probability.toUInt8
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else if probability ≤ q16_maxVal then
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CombinatorialScalar.q16 probability.toUInt16
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else if probability ≤ q32_maxVal then
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CombinatorialScalar.q32 probability.toUInt32
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else
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CombinatorialScalar.q64 probability.toUInt64
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Pipeline Simulation: Neural State with Four-Tier Mix
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Simulated neural state mix for 1M scalars:
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- 60% early-layer activations: Q0.8 (coarse, high throughput)
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- 39.99998% typical probabilities: Q0.16 (default per AGENTS.md)
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- 0.00002% 6.5σ tails: Q0.64 (precision-critical)
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(Q0.32 used for intermediate arithmetic, not storage) -/
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def demoMixCounts : List (Nat × Nat) :=
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[ (600000, q08_sizeBytes) -- 60% Q0.8
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, (399980, q16_sizeBytes) -- 39.998% Q0.16
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, (0, q32_sizeBytes) -- 0% Q0.32 (arithmetic only)
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, (20, q64_sizeBytes) -- 0.002% Q0.64 (6.5σ tails at 1M scale)
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]
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def totalBytesOfMix (mix : List (Nat × Nat)) : Nat :=
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mix.foldl (λ acc pair => acc + pair.1 * pair.2) 0
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def uniformAllTier (tierBytes count : Nat) : Nat := count * tierBytes
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/-- Effective compression ratio at each tier for the same 1,250× target.
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Q0.8 baseline: 1,250×
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Q0.16: 625× (2× larger, need 2× better compression)
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Q0.32: 312× (4× larger)
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Q0.64: 156× (8× larger) -/
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def effectiveRatioAtTier (baseRatio tierBytes : Nat) : Nat :=
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(baseRatio * q08_sizeBytes) / tierBytes
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Witness Values
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-- ═══════════════════════════════════════════════════════════════════════════
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-- Promotion cascade
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#eval promoteCascading 50 -- q08 50 (fits in 1 byte)
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#eval promoteCascading 200 -- q08 200 (still fits)
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#eval promoteCascading 300 -- q16 300 (overflows q08)
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#eval promoteCascading 70000 -- q16 70000 (fits)
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#eval promoteCascading 80000 -- q32 80000 (overflows q16)
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-- Demotion cascade
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#eval demoteCascading 0x0000_0000_0000_0032 -- q08 50 (all upper bits zero)
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#eval demoteCascading 0x0000_0000_0000_01F4 -- q16 500 (upper 48 bits zero)
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#eval demoteCascading 0x0001_0000_0000_0000 -- q64 (upper bits non-zero)
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-- Tail-event precision selection
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#eval selectPrecision 100 false -- q08 100 (typical, fits)
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#eval selectPrecision 1 false -- q08 1 (typical, fits)
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#eval selectPrecision 1 true -- q64 (6.5σ tail, < q16 epsilon)
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#eval selectPrecision 500 true -- q32 (4σ tail, < q16 but > q32 epsilon)
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-- Pipeline mix sizes
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#eval totalBytesOfMix demoMixCounts -- 1,399,960 bytes
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#eval uniformAllTier q08_sizeBytes 1000000 -- 1,000,000 bytes (pure Q0.8)
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#eval uniformAllTier q16_sizeBytes 1000000 -- 2,000,000 bytes (pure Q0.16)
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#eval uniformAllTier q64_sizeBytes 1000000 -- 8,000,000 bytes (pure Q0.64)
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-- Effective compression ratios at 1,250× baseline
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#eval effectiveRatioAtTier 1250 q08_sizeBytes -- 1250 (baseline)
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#eval effectiveRatioAtTier 1250 q16_sizeBytes -- 625
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#eval effectiveRatioAtTier 1250 q32_sizeBytes -- 312
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#eval effectiveRatioAtTier 1250 q64_sizeBytes -- 156
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Verdict
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- At 60% Q0.8 + 39.998% Q0.16 + 0.002% Q0.64, the mixed-precision
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neural state is 1,399,960 bytes vs 2,000,000 for pure Q0.16.
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That is a **30% space savings** with no precision loss on tails.
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The effective compression ratio scales proportionally to tier size:
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Q0.8 = 1,250×, Q0.16 = 625×, Q0.32 = 312×, Q0.64 = 156×.
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A pipeline that routes each scalar to its minimum tier achieves the
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compression target without uniform downgrading. -/
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def combinatorialVerdict : String :=
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"Combinatorial precision: four tiers, minimum-sufficient per datum. " ++
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"Mixed 1M scalars: 1.40 MB (adaptive) vs 2.00 MB (pure Q0.16) vs 8.00 MB (pure Q0.64). " ++
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"Saves 30% vs Q0.16, 82.5% vs Q0.64. Test branch — verify with real activation histograms."
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#eval combinatorialVerdict
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