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223 lines
12 KiB
Text
223 lines
12 KiB
Text
/-!
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# Adaptive Precision: Q0_16 ↔ Q0_64 Upgrade on Demand
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**Status:** TEST BRANCH — Experimental mixed-precision pipeline.
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**Purpose:** Default to Q0_16 (2 bytes, fast). Promote individual scalars
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to Q0_64 (8 bytes) only when Q0_16 would underflow/overflow.
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**Adaptation Rule:**
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- Start: every scalar is Q0_16.
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- If |value| > Q0_16.max (0x7FFF ≈ 0.999985): promote to Q0_64.
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- If precision demand > Q0_16.epsilon (3.05×10⁻⁵): promote to Q0_64.
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- If Q0_64 result fits in Q0_16 range: demote back to Q0_16.
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**Storage Cost:**
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- 100% Q0_16: 1.0× baseline
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- 100% Q0_64: 4.0× baseline
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- Adaptive: 1.0×–4.0× depending on promotion rate.
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This file is standalone: zero imports.
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-/
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §0 Q0_16 Constants (16-bit pure fraction)
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-- ═══════════════════════════════════════════════════════════════════════════
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def q0_16MaxVal : Nat := 0x7FFF -- max positive: ~0.999985
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def q0_16EpsilonVal : Nat := 1 -- smallest step: ~3.05×10⁻⁵
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def q0_16SizeBytes : Nat := 2
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Q0_64 Constants (64-bit pure fraction)
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-- ═══════════════════════════════════════════════════════════════════════════
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def q0_64MaxVal : Nat := 0x8000_0000_0000_0000 -- 2^63, ~1.0
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def q0_64EpsilonVal : Nat := 1 -- ~1.08×10⁻¹⁹
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def q0_64SizeBytes : Nat := 8
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Adaptive Scalar Type
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- An adaptive scalar is either:
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- Q0_16: 2 bytes, sufficient for 99%+ of dimensionless quantities.
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- Q0_64: 8 bytes, used only when Q0_16 would lose information. -/
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inductive AdaptiveScalar where
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| q0_16 (val : UInt16)
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| q0_64 (val : UInt64)
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deriving Repr, BEq, Inhabited
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def AdaptiveScalar.sizeBytes (s : AdaptiveScalar) : Nat :=
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match s with
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| .q0_16 _ => q0_16SizeBytes
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| .q0_64 _ => q0_64SizeBytes
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Promotion / Demotion Rules
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Promote a Q0_16 to Q0_64.
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Shift left by 48 bits: Q0_16.val × 2⁴⁸ = Q0_64.val with same semantic value. -/
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def promote (v : UInt16) : AdaptiveScalar :=
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let promoted : UInt64 := (v.toNat.toUInt64) <<< 48
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AdaptiveScalar.q0_64 promoted
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/-- Demote a Q0_64 to Q0_16 if it was promoted (lower 48 bits are zero).
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Q0_16 value = upper 16 bits = v >>> 48.
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If lower 48 bits are non-zero, precision would be lost: keep Q0_64. -/
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def demote (v : UInt64) : AdaptiveScalar :=
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let upper : UInt64 := v >>> 48
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let lower : UInt64 := v &&& 0x0000_FFFF_FFFF_FFFF
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if lower = 0 then
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-- Was promoted from Q0_16: reverse the shift
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if upper ≤ q0_16MaxVal.toUInt64 then
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AdaptiveScalar.q0_16 (upper.toNat.toUInt16)
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else
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AdaptiveScalar.q0_64 v
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else
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-- Has precision in lower 48 bits: cannot demote without loss
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AdaptiveScalar.q0_64 v
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Adaptive Arithmetic (Q0_16 default, promote on overflow)
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-- ═══════════════════════════════════════════════════════════════════════════
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def AdaptiveScalar.add (a b : AdaptiveScalar) : AdaptiveScalar :=
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match a, b with
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| .q0_16 av, .q0_16 bv =>
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let sum : Nat := av.toNat + bv.toNat
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if sum > q0_16MaxVal then
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-- Overflow: promote both to Q0_64, add, then attempt demotion
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let ap : UInt64 := (av.toNat.toUInt64) <<< 48
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let bp : UInt64 := (bv.toNat.toUInt64) <<< 48
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demote (ap + bp)
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else
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AdaptiveScalar.q0_16 (sum.toUInt16)
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| .q0_64 av, .q0_64 bv =>
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let sum : UInt64 := av + bv
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demote sum
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| .q0_16 av, .q0_64 bv =>
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let ap : UInt64 := (av.toNat.toUInt64) <<< 48
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demote (ap + bv)
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| .q0_64 av, .q0_16 bv =>
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let bp : UInt64 := (bv.toNat.toUInt64) <<< 48
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demote (av + bp)
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def AdaptiveScalar.mul (a b : AdaptiveScalar) : AdaptiveScalar :=
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match a, b with
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| .q0_16 av, .q0_16 bv =>
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-- Q0_16.mul: (a×b) >>> 15
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let prod : Nat := av.toNat * bv.toNat
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let shifted : Nat := prod >>> 15
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if shifted > q0_16MaxVal then
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-- Overflow after shift: promote to Q0_64
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let ap : UInt64 := (av.toNat.toUInt64) <<< 48
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let bp : UInt64 := (bv.toNat.toUInt64) <<< 48
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-- Q0_64.mul would be (ap*bp)>>>63, but ap,bp are already shifted
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let prod64 : Nat := ap.toNat * bp.toNat
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let shifted64 : Nat := prod64 >>> 63
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demote (shifted64.toUInt64)
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else
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AdaptiveScalar.q0_16 (shifted.toUInt16)
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| .q0_64 av, .q0_64 bv =>
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let prod : Nat := av.toNat * bv.toNat
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let shifted : Nat := prod >>> 63
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demote (shifted.toUInt64)
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| .q0_16 av, .q0_64 bv =>
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let ap : UInt64 := (av.toNat.toUInt64) <<< 48
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let prod : Nat := ap.toNat * bv.toNat
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let shifted : Nat := prod >>> 63
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demote (shifted.toUInt64)
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| .q0_64 av, .q0_16 bv =>
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let bp : UInt64 := (bv.toNat.toUInt64) <<< 48
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let prod : Nat := av.toNat * bp.toNat
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let shifted : Nat := prod >>> 63
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demote (shifted.toUInt64)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Precision-Driven Promotion (Tail Events)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Create an adaptive scalar from a raw value, promoting to Q0_64
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if the value is smaller than Q0_16 epsilon (precision loss).
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This is the entry point for 6.5σ tail probabilities. -/
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def fromProbability (raw : Nat) (isTailEvent : Bool) : AdaptiveScalar :=
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if isTailEvent ∧ raw < q0_16EpsilonVal then
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-- Tail event below Q0_16 resolution: must use Q0_64
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AdaptiveScalar.q0_64 (raw.toUInt64 <<< 48)
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else if raw ≤ q0_16MaxVal then
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AdaptiveScalar.q0_16 raw.toUInt16
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else
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AdaptiveScalar.q0_64 (raw.toUInt64 <<< 48)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Pipeline Simulation: Neural State with Sparse Tails
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Simulate N=1,000,000 scalars where 99.99998% are typical (Q0_16)
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and 0.00002% are 6.5σ tail events requiring Q0_64.
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At 1M scalars: 1M × 0.00002 = 20 tail events → 20 Q0_64, rest Q0_16. -/
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def totalScalars : Nat := 1000000
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def tailEventRate : Nat := 2 -- 0.00002% = 2 per 10,000,000, scaled
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def tailEventDenominator : Nat := 10000000
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def tailEventCount : Nat :=
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(totalScalars * tailEventRate) / tailEventDenominator
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def typicalEventCount : Nat := totalScalars - tailEventCount
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def adaptiveTotalBytes : Nat :=
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typicalEventCount * q0_16SizeBytes + tailEventCount * q0_64SizeBytes
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def uniformQ0_16Bytes : Nat := totalScalars * q0_16SizeBytes
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def uniformQ0_64Bytes : Nat := totalScalars * q0_64SizeBytes
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def adaptiveOverheadPercent : Nat :=
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((adaptiveTotalBytes - uniformQ0_16Bytes) * 1000) / uniformQ0_16Bytes
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def spaceSavingsVsQ0_64Percent : Nat :=
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((uniformQ0_64Bytes - adaptiveTotalBytes) * 1000) / uniformQ0_64Bytes
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Witness Values
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-- ═══════════════════════════════════════════════════════════════════════════
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-- Promotion/demotion mechanics
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#eval promote 0x4000 -- Q0_16 half → Q0_64 half
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#eval demote 0x4000_0000_0000_0000 -- promoted Q0_16 half: lower 48 bits zero → demotes to Q0_16
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#eval demote 0x4000_0000_0000_0001 -- lower 48 bits non-zero → stays Q0_64
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-- Arithmetic overflow handling
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#eval AdaptiveScalar.add (AdaptiveScalar.q0_16 0x7000) (AdaptiveScalar.q0_16 0x7000) -- overflow → Q0_64 or demoted Q0_16
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#eval AdaptiveScalar.mul (AdaptiveScalar.q0_16 0x7000) (AdaptiveScalar.q0_16 0x7000) -- overflow → promoted
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-- Tail event handling
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#eval fromProbability 1 true -- tail event, raw=1 (< epsilon): Q0_64
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#eval fromProbability 100 false -- typical event: Q0_16
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-- Pipeline scale
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#eval tailEventCount -- 0 (integer division: 1M*2/10M = 0)
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#eval adaptiveTotalBytes -- 2,000,000 (all Q0_16 at this rate)
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#eval adaptiveOverheadPercent -- 0 (no overhead at 0 tails)
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-- With explicit 20 tail events (override rate for demo)
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def demoTailCount : Nat := 20
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def demoAdaptiveBytes : Nat :=
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(totalScalars - demoTailCount) * q0_16SizeBytes + demoTailCount * q0_64SizeBytes
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#eval demoAdaptiveBytes -- 2,000,120 bytes
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#eval ((demoAdaptiveBytes - uniformQ0_16Bytes) * 1000000) / uniformQ0_16Bytes -- 60 ppm overhead
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §8 Verdict
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- At 6.5σ (0.00002% tail rate), adaptive precision adds ~60 parts per
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million overhead vs pure Q0_16. vs pure Q0_64, it saves 74.99985%.
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The pipeline is: Q0_16 default → promote on overflow or tail event
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→ demote when result fits → amortized cost ≈ 1.00006× baseline. -/
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def adaptiveVerdict : String :=
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"Adaptive precision: Q0_16 default, Q0_64 on demand. " ++
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"At 6.5σ tail rate (0.00002%): ~60 ppm overhead vs pure Q0_16. " ++
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"Saves ~75% vs pure Q0_64. Test branch — verify with real tail distributions."
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#eval adaptiveVerdict
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