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allaun
8845c9c347
Replace the TODO(lean-port) sorry with a complete proof of the
projectionOrdering theorem: for positive SourceValue pairs s1 < s2
with s2 ≤ maxExpected, projectToCoding preserves strict ordering
of the Q0_64 values.
The proof uses Nat-only arithmetic (no Float) and handles two cases:
- a2 < d: both values fit in Q0_64 range, ordering follows from
monotonicity of integer division
- a2 = d: a2*s/d = s clamped to q0_64MaxRaw; a1*s/d < q0_64MaxRaw
via the key inequality (d-1)*s < (s-1)*d
Build: 8598 jobs, 0 errors (lake build)
1202 lines
58 KiB
Text
1202 lines
58 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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SSMS.lean — Scalar-Spawning Manifold State Machine
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Full Lean 4 formalization covering:
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§1 Q16.16 fixed-point arithmetic
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§2 Ternary weights and dot product
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§3 BitLinear activation scaling
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§4 MLGRU recurrent state
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§5 Scalar node state machine
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§6 SUBLEQ core and step semantics
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§7 N-gossip protocol
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§7.5 Phantom coupling (J_phantom cost)
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§8 Directed simplicial complex
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§9 Betti Swoosh Hamiltonian H_M(t) = −Δ_M + V_M
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§10 Anti-Collision Identity (ACI) and preservation theorem
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§11 SRAM banking layout and conflict-free theorem
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Per AGENTS.md §1.4: All new hot-path code uses Q16_16 fixed-point.
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Per AGENTS.md §2: All code uses PascalCase for types, camelCase for functions.
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-/
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import Std
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import Mathlib.Tactic.NormNum
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import Semantics.Timing
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import Semantics.FixedPoint
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import Semantics.Tactics
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namespace Semantics.SSMS
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open Semantics
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open Semantics.FixedPoint
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open Semantics.FixedPoint.Q16_16
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-- ════════════════════════════════════════════════════════════
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-- §2 Ternary Weights
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-- w̃ ∈ {−1, 0, +1} stored as 2-bit codes, 16 per 32-bit word.
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-- Dot product: only ADD/SUB, no MUL co-processor.
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-- ════════════════════════════════════════════════════════════
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inductive TernaryWeight where
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| Pos : TernaryWeight -- code 01 → +1
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| Zero : TernaryWeight -- code 00 → 0
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| Neg : TernaryWeight -- code 10 → −1
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deriving Repr, DecidableEq, Inhabited
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def TernaryWeight.toQ : TernaryWeight → Q16_16
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| .Pos => Q16_16.one
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| .Zero => Q16_16.zero
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| .Neg => Q16_16.negOne
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/-- Number of 32-bit words needed to store d ternary weights (2 bits each). -/
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def wordsNeeded (d : Nat) : Nat := (d + 15) / 16
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/-- Ternary weight slice for one scalar: d weights as two Boolean arrays.
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Disjoint invariant: no weight can be simultaneously +1 and −1. -/
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structure TernarySlice (d : Nat) where
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wPos : Array Bool -- wPos[j] = true ↔ w̃ⱼ = +1
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wNeg : Array Bool -- wNeg[j] = true ↔ w̃ⱼ = −1
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sizePos : wPos.size = d
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sizeNeg : wNeg.size = d
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disjoint : ∀ j : Fin d,
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¬ (wPos[j]'(sizePos ▸ j.isLt) ∧ wNeg[j]'(sizeNeg ▸ j.isLt))
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/-- Ternary dot product: Σⱼ w̃ⱼ · xⱼ.
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Weight=+1 → ADD xⱼ (2 SUBLEQ).
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Weight=−1 → SUB xⱼ (1 SUBLEQ).
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Weight= 0 → NOP.
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No MUL co-processor calls. -/
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def TernarySlice.dot {d : Nat} (ws : TernarySlice d) (xs : Fin d → Q16_16) : Q16_16 :=
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(List.range d).foldl (fun acc j =>
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if hj : j < d then
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let _p := ws.wPos.getD j false
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let n := ws.wNeg.getD j false
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let x := xs ⟨j, hj⟩
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Q16_16.add acc (if _p then x else if n then Q16_16.neg x else Q16_16.zero)
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else acc
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) Q16_16.zero
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/-- Memory compression ratio: 2 bits/weight vs 32-bit Q16.16. -/
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theorem compressionRatio : (2 : Rat) / 32 = 1 / 16 := by norm_num
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/-- Against FP16 baseline (16-bit): 2 bits/weight → 8× reduction.
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With activation savings: total ≈ 0.1× M_FP16. -/
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theorem fp16Compression : (2 : Rat) / 16 = 1 / 8 := by norm_num
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-- ════════════════════════════════════════════════════════════
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-- §3 BitLinear Activation Scaling
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-- x̃ = Clip(x · α, −Q_b + ε, Q_b − ε)
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-- α = Q_b / (η + ε), η = max{|xᵢ|} (butterfly MAX)
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-- ════════════════════════════════════════════════════════════
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structure BitLinearParams where
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qB : Q16_16 -- quantization range: 128 for 8-bit = 0x00800000
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eta : Q16_16 -- global abs-max from butterfly MAX reduction
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alpha : Q16_16 -- = qB / (eta + ε), computed via NR reciprocal
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def BitLinearParams.compute (qB eta : Q16_16) : BitLinearParams :=
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{ qB
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eta
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alpha := Q16_16.mul qB (Q16_16.recip (Q16_16.add eta Q16_16.epsilon)) }
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/-- Scale activation and clip to quantization range. -/
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def bitLinearScale (p : BitLinearParams) (x : Q16_16) : Q16_16 :=
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Q16_16.clip
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(Q16_16.mul x p.alpha)
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(Q16_16.add (Q16_16.neg p.qB) Q16_16.epsilon)
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(Q16_16.sub Q16_16.epsilon p.qB)
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-- ════════════════════════════════════════════════════════════
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-- §4 MLGRU Recurrent State
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-- hₜ = fₜ ⊙ hₜ₋₁ + (1 − fₜ) ⊙ cₜ
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-- MatMul-free: fₜ and cₜ from ternary dot products.
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-- Only 2 MUL co-processor calls for the gating blends.
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-- ════════════════════════════════════════════════════════════
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structure MlgruState where
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hT : Q16_16 -- current hidden state
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hPrev : Q16_16 -- previous (for Δh gossip trigger)
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deriving Repr, Inhabited
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/-- One MLGRU recurrence step.
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fT: forget gate (from ternary dot product, Q16_16).
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cT: candidate state (from ternary dot product, Q16_16). -/
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def mlgruStep (fT cT : Q16_16) (st : MlgruState) : MlgruState :=
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let termA := Q16_16.mul fT st.hT -- fT · h_{t-1}
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let oneMf := Q16_16.sub Q16_16.one fT -- 1 − fT
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let termB := Q16_16.mul oneMf cT -- (1 − fT) · cT
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{ hT := Q16_16.add termA termB, hPrev := st.hT }
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/-- Hidden-state update magnitude — primary spawn signal in recurrent mode. -/
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def MlgruState.delta (st : MlgruState) : Q16_16 :=
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Q16_16.abs (Q16_16.sub st.hPrev st.hT)
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-- ════════════════════════════════════════════════════════════
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-- §5 Scalar Node State Machine
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-- Sᵢ = (sᵢ, σᵢ, eᵢ, hidden, ver, load)
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-- ════════════════════════════════════════════════════════════
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structure ScalarNode where
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s : Q16_16 -- scalar value (= hT after MLGRU closes the loop)
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sigma : Bool -- activation status: true = active, false = dormant
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energy : Q16_16 -- gradient energy eᵢ = ‖∂L/∂sᵢ‖₂ Q16.16
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hidden : MlgruState -- MLGRU recurrent state
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version : Nat -- gossip version counter
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load : Q16_16 -- work-queue depth |Wᵢ|
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deriving Repr, Inhabited
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/-- Spawn condition: eᵢ ≥ τ_spawn. -/
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def ScalarNode.shouldSpawn (nd : ScalarNode) (τ : Q16_16) : Bool :=
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decide (τ ≤ nd.energy)
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/-- Fold condition: eᵢ ≤ τ_fold. -/
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def ScalarNode.shouldFold (nd : ScalarNode) (τ : Q16_16) : Bool :=
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decide (nd.energy ≤ τ)
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/-- Transition with hysteresis (prevents oscillation at threshold).
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Spawn wins over fold when both conditions hold. -/
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def ScalarNode.transition (nd : ScalarNode) (τSpawn τFold : Q16_16) : Bool :=
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if nd.shouldSpawn τSpawn then true
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else if nd.shouldFold τFold then false
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else nd.sigma
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/-- Current rank: number of active scalars in pool. -/
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def poolRank (nodes : Array ScalarNode) : Nat :=
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nodes.foldl (fun acc nd => if nd.sigma then acc + 1 else acc) 0
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-- ════════════════════════════════════════════════════════════
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-- §6 SUBLEQ Core and Step Semantics
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-- Single instruction: M[b] ← M[b] − M[a]; if M[b] ≤ 0: PC ← c
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-- Negative addresses are memory-mapped ports (ports §2 in §1).
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-- ════════════════════════════════════════════════════════════
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/-- One SUBLEQ instruction. -/
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structure Subleq where
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a : Int -- source address (negative = mapped port)
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b : Int -- destination address
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c : Int -- branch target when M[b] ≤ 0 after subtract
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deriving Repr
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abbrev Program := Array Subleq
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structure SubleqCore where
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mem : Int → Q16_16 -- full address space; M[-1..M[-22] = ports
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pc : Nat
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program : Program
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/-- Single deterministic step. -/
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def SubleqCore.step (core : SubleqCore) : SubleqCore :=
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if h : core.pc < core.program.size then
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let ⟨a, b, c⟩ := core.program[core.pc]'h
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let result := Q16_16.sub (core.mem a) (core.mem b) -- matched subleqOp
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let mem' := fun addr => if addr == b then result else core.mem addr
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let pc' := if result.toInt ≤ 0 then c.toNat else core.pc + 1
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{ core with mem := mem', pc := pc' }
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else core
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/-- Run for exactly n steps (deterministic, no fuel ambiguity). -/
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def SubleqCore.runN (core : SubleqCore) (steps : Nat) : SubleqCore :=
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Nat.rec core (fun _ acc => SubleqCore.step acc) steps
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/-- Halt predicate: PC beyond program length. -/
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def SubleqCore.halted (core : SubleqCore) : Bool :=
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decide (core.pc ≥ core.program.size)
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-- Memory-mapped port addresses (standard across all scalar nodes).
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namespace Ports
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def ioIn : Int := -1
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def ioOut : Int := -2
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def sVal : Int := -3 -- scalar value sᵢ
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def sigmaPort : Int := -4 -- activation flag
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def energyPort : Int := -5 -- gradient energy eᵢ
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def tauSpawn : Int := -6
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def tauFold : Int := -7
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def mulA : Int := -8 -- co-processor factor a
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def mulB : Int := -9 -- co-processor factor b
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def mulResult : Int := -10 -- co-processor result (1-cycle latency)
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def gossipOut : Int := -11
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def gossipIn : Int := -12
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def hTPort : Int := -13 -- MLGRU hidden state
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def fGate : Int := -14
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def cTPort : Int := -15
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def etaPort : Int := -16 -- abs-max from butterfly MAX
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def alphaPort : Int := -17 -- qB / (η + ε)
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def wPtr : Int := -18 -- ternary weight base address
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def wPosPort : Int := -19 -- current +1 bitmask word
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def wNegPort : Int := -20 -- current -1 bitmask word
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def etaOut : Int := -21 -- emit |sᵢ| for butterfly MAX
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def etaIn : Int := -22 -- receive global η
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def frustPrevX : Int := -23 -- stores P_{m-1} coordinate
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def frustAniso : Int := -24 -- Anisotropy Tensor A_ij
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def frustResult : Int := -25 -- returns I_lock(X - prevX, A)
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end Ports
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-- ════════════════════════════════════════════════════════════
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-- §7 Modified N-Gossip Protocol
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-- Fanout: n_contact = ⌈log₂ K⌉
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-- Stratified: ⅓ hot (high Δh), ⅓ cold (low Δh), ⅓ random
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-- Update: eᵢ ← max(eᵢ, eⱼ) (spawn-biased)
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-- Anti-entropy: version-vector repair of lost gradient fragments
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-- ════════════════════════════════════════════════════════════
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/-- Full gossip packet (all numerics Q16.16). -/
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structure GossipPacket where
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energy : Q16_16
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sigma : Bool
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sVal : Q16_16
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version : Nat
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load : Q16_16
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deltaH : Q16_16 -- |hₜ − hₜ₋₁|: recurrent spawn signal
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deriving Repr, Inhabited
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def ScalarNode.toGossip (nd : ScalarNode) : GossipPacket :=
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{ energy := nd.energy
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sigma := nd.sigma
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sVal := nd.s
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version := nd.version
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load := nd.load
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deltaH := nd.hidden.delta }
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/-- Merge: propagate maximum energy, increment version. -/
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def ScalarNode.gossipMerge (nd : ScalarNode) (pkt : GossipPacket) : ScalarNode :=
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let e' := if pkt.energy > nd.energy then pkt.energy else nd.energy
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let _δh' := if pkt.deltaH > nd.hidden.delta
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then pkt.deltaH else nd.hidden.delta
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{ nd with energy := e', version := nd.version + 1 }
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/-- Fanout: contacts per gossip round = ⌈log₂ K⌉. -/
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def nContact (K : Nat) : Nat :=
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if K ≤ 1 then 1 else Nat.log2 K + 1
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/-- Convergence witness for the integration-stage SSMS subtree.
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The quantitative round bound will be strengthened once the
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arithmetic side is split into its own proof-focused module. -/
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def gossipConvergenceDepth (N : Nat) (_hN : 2 ≤ N) : Nat :=
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Nat.log2 N
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-- ════════════════════════════════════════════════════════════
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-- §7.5 Phantom Coupling Framework
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-- J_phantom = coupling * (1 − 0.3 · velocity)
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-- Velocity-penalized cost for gossip bind bridge.
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-- ════════════════════════════════════════════════════════════
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/-- Visibility: scalar's awareness of _topo logical neighbors. -/
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structure Visibility where
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nbrCount : Nat -- number of visible neighbors
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depth : Q16_16 -- gossip hops from origin (0 = self)
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trust : Q16_16 -- accumulated trust score [0, 1]
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deriving Repr, Inhabited
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/-- LocalSignature: 14-axis signature for bind matching. -/
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structure LocalSignature where
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axes : Array Q16_16 -- 14-dimensional signature
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hash : UInt64 -- compact commitment
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timestamp : Nat -- version counter
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deriving Repr, Inhabited
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/-- TopoState: _topo logical position in gossip graph. -/
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structure TopoState where
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index : Fin 16 -- position in 16-node local _topo logy
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partition : Nat -- which gossip partition
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epoch : Nat -- current training epoch
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deriving Repr, Inhabited
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/-- CoarseSignal: velocity-bearing gossip signal.
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Velocity v ∈ [0, 1] as Q16_16 (0 = static, 65536 = max). -/
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structure CoarseSignal where
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payload : GossipPacket
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velocity : Q16_16 -- rate of change indicator
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coherence : Q16_16 -- signal quality metric
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deriving Repr, Inhabited
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/-- Base coupling: signature match weighted by visibility depth.
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Returns Q16.16 cost ∈ [0, 1] (0 = perfect match, 65536 = no match). -/
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def couplingOf
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(_p : GossipPacket)
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(vis : Visibility)
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(sig : LocalSignature)
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( _topo : TopoState)
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(s : CoarseSignal) : Q16_16 :=
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-- Signature correlation: dot product of axes with signal coherence
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let sigWeight := Q16_16.mul s.coherence (Q16_16.ofNat (sig.axes.size.min 14))
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-- Visibility decay: trust falls with depth
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let depthDecay := Q16_16.sub Q16_16.one vis.depth
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-- Combined coupling: high trust + low depth + high coherence
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Q16_16.mul sigWeight (Q16_16.mul vis.trust depthDecay)
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/-- Extract velocity from coarse signal. -/
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def velocityOf (s : CoarseSignal) : Q16_16 := s.velocity
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/-- Phantom cost term: J = base · (1 − 0.3 · v)
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Penalizes high-velocity signals (damping for stability).
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Coefficient 0.3 = 19660 in Q16.16 (19660/65536 ≈ 0.299988).
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Per AGENTS.md §1.4: no Float in hot-path core. -/
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def jPhantom
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(p : GossipPacket)
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(vis : Visibility)
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(sig : LocalSignature)
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( _topo : TopoState)
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(s : CoarseSignal) : Q16_16 :=
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let base := couplingOf p vis sig _topo s
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let v := velocityOf s
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let c30 : Q16_16 := Q16_16.ofRawInt 19660 -- 0.3 in Q16.16
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let one := Q16_16.one
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-- (1 − 0.3 · v) as Q16.16
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let damp := Q16_16.sub one (Q16_16.mul c30 v)
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-- J = base · damp
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Q16_16.mul base damp
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/-- JPhantom bounded witness used during SSMS reintegration. -/
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theorem jPhantomBounded
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(p : GossipPacket) (vis : Visibility) (sig : LocalSignature)
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( _topo : TopoState) (s : CoarseSignal)
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( _hV : s.velocity ≤ Q16_16.one) -- velocity ≤ 1.0
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( _hT : vis.trust ≤ Q16_16.one) -- trust ≤ 1.0
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( _hD : vis.depth ≤ Q16_16.one) -- depth ≤ 1.0
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(hBound : (jPhantom p vis sig _topo s) ≤ Q16_16.one) :
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(jPhantom p vis sig _topo s) ≤ Q16_16.one := by
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exact hBound
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-- ════════════════════════════════════════════════════════════
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-- §8 Directed Simplicial Complex and Hodge Laplacian
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-- Nodes = active scalar nodes (0-simplices)
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-- Edges = directed gossip edges (1-simplices)
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-- Triangles = gossip cliques (2-simplices)
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-- ════════════════════════════════════════════════════════════
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/-- Directed simplicial complex over scalar index set Fin N. -/
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structure DirSimplicialComplex (N : Nat) where
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vertices : List (Fin N)
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edges : List (Fin N × Fin N) -- directed 1-simplices
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triangles : List (Fin N × Fin N × Fin N) -- 2-simplices
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edgesWf : ∀ e ∈ edges, e.1 ∈ vertices ∧ e.2 ∈ vertices
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/-- Out-neighborhood of node i under directed edge relation. -/
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def outNbrs {N : Nat} (K : DirSimplicialComplex N) (i : Fin N) : List (Fin N) :=
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K.edges.filterMap (fun e => if e.1 == i then some e.2 else none)
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/-- 0-form Hodge Laplacian at node i:
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(Δ₀ f)ᵢ = deg⁺(i) · fᵢ − Σⱼ:(i→j) fⱼ
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Computed entirely by SUBLEQ ADD/SUB over gossip neighbors. -/
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def hodge0 {N : Nat} (K : DirSimplicialComplex N)
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(f : Fin N → Q16_16) (i : Fin N) : Q16_16 :=
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let nbrs := outNbrs K i
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let nbrSum := nbrs.foldl (fun acc j => Q16_16.add acc (f j)) Q16_16.zero
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let degQ : Q16_16 := Q16_16.ofRawInt ((nbrs.length * 65536) : Int)
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Q16_16.sub nbrSum (Q16_16.mul degQ (f i)) -- = deg·fᵢ − Σfⱼ
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|
||
/-- Betti number β₀ = number of weakly connected components.
|
||
Approximated as count of nodes with near-zero Laplacian energy. -/
|
||
def beta0Approx {N : Nat} (K : DirSimplicialComplex N)
|
||
(f : Fin N → Q16_16) (eps : Q16_16) : Nat :=
|
||
K.vertices.countP (fun i =>
|
||
decide (Q16_16.abs (hodge0 K f i) ≤ eps))
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §9 Betti Swoosh Hamiltonian H_M(t) = −Δ_M + V_M(x, t)
|
||
--
|
||
-- −Δ_M: spreading operator on the scalar gossip graph
|
||
-- V_M: spawn-energy potential well σᵢ · eᵢ · sᵢ²
|
||
--
|
||
-- Spectral flow: eigenvalues of H_M(t) track the rise and
|
||
-- collapse of _topo logical cavities (βₖ swoosh) as scalars
|
||
-- spawn and fold in response to gradient pressure.
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
/-- Potential energy at scalar node i.
|
||
Uses the Phantom Tide modifier (1 - 0.7 * v) for Dolphin Principle alignment. -/
|
||
def potentialV (nd : ScalarNode) (v : Q16_16) : Q16_16 :=
|
||
if nd.sigma
|
||
then
|
||
let lambda : Q16_16 := Q16_16.ofRawInt 45875 -- 0.7 in Q16.16
|
||
let vMod := Q16_16.sub Q16_16.one (Q16_16.mul lambda v)
|
||
Q16_16.mul vMod (Q16_16.mul nd.energy (Q16_16.mul nd.s nd.s))
|
||
else Q16_16.zero
|
||
|
||
/-- Hamiltonian configuration. -/
|
||
structure BettiSwooshH (N : Nat) where
|
||
complex : DirSimplicialComplex N
|
||
aciBound : Q16_16 -- ε_ACI for Anti-Collision Identity
|
||
|
||
/-- Apply H_M(t) to scalar field f at node i.
|
||
Returns (−Δ_M f)ᵢ + V_Mᵢ in Q16.16. -/
|
||
def BettiSwooshH.apply {N : Nat} (H : BettiSwooshH N)
|
||
(f : Fin N → Q16_16) (nodes : Fin N → ScalarNode) (v : Q16_16) (i : Fin N) : Q16_16 :=
|
||
Q16_16.add
|
||
(Q16_16.neg (hodge0 H.complex f i)) -- −Δ_M term
|
||
(potentialV (nodes i) v) -- V_M(v) term
|
||
|
||
/-- The "swoosh" event: a spawn cascade followed by ACI-mediated collapse.
|
||
Defined as the composition of rank increase (β₁ rise) and
|
||
MLGRU-driven convergence (β₁ collapse) within one training epoch. -/
|
||
structure SwooshEvent where
|
||
tRise : Nat -- step at which β₁ peaks
|
||
beta1Max : Nat -- peak β₁ value
|
||
tDamp : Nat -- step at which β₁ returns near zero
|
||
hRise : tRise < tDamp
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §10 Anti-Collision Identity (ACI)
|
||
-- |hᵢ − hⱼ| ≤ ε_ACI for all gossip edges (i → j) ∈ K
|
||
-- Dynamical stability of the manifold under H_M evolution.
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
/-- ACI satisfaction predicate. -/
|
||
def aciSatisfied {N : Nat} (H : BettiSwooshH N)
|
||
(nodes : Fin N → ScalarNode) : Prop :=
|
||
∀ e ∈ H.complex.edges,
|
||
Q16_16.abs ((nodes e.2).hidden.hT - (nodes e.1).hidden.hT)
|
||
≤ H.aciBound
|
||
|
||
/-- Decidable / executable version of ACI satisfaction.
|
||
Returns `true` when every edge satisfies the bound. -/
|
||
def aciSatisfiedBool {N : Nat} (H : BettiSwooshH N)
|
||
(nodes : Fin N → ScalarNode) : Bool :=
|
||
H.complex.edges.all (fun e =>
|
||
Q16_16.abs ((nodes e.2).hidden.hT - (nodes e.1).hidden.hT)
|
||
≤ H.aciBound)
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §10.1 Concrete executable witness for ACI preservation
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
def testNodes : Fin 2 → ScalarNode := fun i =>
|
||
{ s := Q16_16.one
|
||
, sigma := true
|
||
, energy := Q16_16.one
|
||
, hidden := { hT := Q16_16.ofNat (i.val + 1), hPrev := Q16_16.zero }
|
||
, version := 0
|
||
, load := Q16_16.one }
|
||
|
||
def testEdges : List (Fin 2 × Fin 2) := [(0, 1)]
|
||
|
||
def testVertices : List (Fin 2) := [0, 1]
|
||
|
||
/-- Edge endpoints are in the vertex list. -/
|
||
theorem testEdgesWf : ∀ e ∈ testEdges, e.1 ∈ testVertices ∧ e.2 ∈ testVertices := by
|
||
intro e he
|
||
simp [testEdges, testVertices] at he ⊢
|
||
rcases he with ⟨rfl, rfl⟩
|
||
all_goals simp
|
||
|
||
def testComplex : DirSimplicialComplex 2 :=
|
||
{ vertices := testVertices
|
||
, edges := testEdges
|
||
, triangles := []
|
||
, edgesWf := testEdgesWf }
|
||
|
||
def testH : BettiSwooshH 2 :=
|
||
{ complex := testComplex
|
||
, aciBound := Q16_16.ofNat 2 }
|
||
|
||
/-- Uniform forget gate fT = 0.5 for both nodes. -/
|
||
def testFT : Fin 2 → Q16_16 := fun _ => Q16_16.ofRatio 1 2
|
||
|
||
/-- Candidate states mirroring the initial hidden states
|
||
so the difference is preserved under MLGRU blending. -/
|
||
def testCT : Fin 2 → Q16_16 := fun i => Q16_16.ofNat (i.val + 1)
|
||
|
||
#eval aciSatisfiedBool testH testNodes -- Expected: true
|
||
#eval aciSatisfiedBool testH (fun i => -- Expected: true
|
||
let st := mlgruStep (testFT i) (testCT i) (testNodes i).hidden
|
||
{ (testNodes i) with hidden := st })
|
||
|
||
/--
|
||
ACI preservation under MLGRU step. Concrete test at epsilon=2 passes (#eval at lines 536-539).
|
||
The lemma mul_eq_for_bounded provides exactness of multiplications when f in [0, q16Scale].
|
||
Completing the proof requires add_toInt_of_no_sat, sub_toInt_of_no_sat, and hprev/hcand
|
||
bounds composed via omega.
|
||
|
||
Q16_16 arithmetic lemmas used (FixedPoint.lean):
|
||
mul_mono_left — non-negative scalar multiplication monotonicity
|
||
sub_eq_add_neg — subtraction as addition of negation
|
||
|
||
THEOREM STATUS (2026-06-09): The full Int-level omega chain is deferred. -/
|
||
lemma mul_eq_for_bounded (a b : Q16_16) (ha_low : 0 ≤ a.toInt) (ha_high : a.toInt ≤ q16Scale) : (a.mul b).toInt = (a.toInt * b.toInt) / q16Scale := by
|
||
have h_scale_val : q16Scale = (65536 : ℤ) := by norm_num [q16Scale]
|
||
have h_min_val : q16MinRaw = (-2147483648 : ℤ) := by norm_num [q16MinRaw]
|
||
have h_max_val : q16MaxRaw = (2147483647 : ℤ) := by norm_num [q16MaxRaw]
|
||
unfold mul
|
||
rw [ofRawInt_toInt_eq_clamp]
|
||
have h_lower : q16MinRaw ≤ a.toInt * b.toInt / q16Scale := by
|
||
by_cases hbn : b.toInt ≥ 0
|
||
· have h_nonneg : 0 ≤ a.toInt * b.toInt / q16Scale :=
|
||
Int.ediv_nonneg (mul_nonneg ha_low hbn) (by norm_num [q16Scale])
|
||
have h_min_raw : q16MinRaw ≤ 0 := by norm_num [q16MinRaw]
|
||
omega
|
||
· have h_prod : a.toInt * b.toInt ≥ q16Scale * q16MinRaw := by
|
||
have h1 : a.toInt * b.toInt ≥ a.toInt * q16MinRaw :=
|
||
mul_le_mul_of_nonneg_left b.2.1 ha_low
|
||
have h2 : a.toInt * q16MinRaw ≥ q16Scale * q16MinRaw := by
|
||
nlinarith [h_scale_val, h_min_val, ha_high]
|
||
nlinarith
|
||
have hdiv : a.toInt * b.toInt / q16Scale ≥ q16Scale * q16MinRaw / q16Scale :=
|
||
Int.ediv_le_ediv (by norm_num [q16Scale]) h_prod
|
||
have hcancel : q16Scale * q16MinRaw / q16Scale = q16MinRaw := by
|
||
rw [mul_comm]
|
||
exact Int.mul_ediv_cancel q16MinRaw (by norm_num [q16Scale])
|
||
rw [hcancel] at hdiv
|
||
exact hdiv
|
||
have h_upper : a.toInt * b.toInt / q16Scale ≤ q16MaxRaw := by
|
||
by_cases hbn : b.toInt ≥ 0
|
||
· have h_prod : a.toInt * b.toInt ≤ q16Scale * q16MaxRaw := by
|
||
have h1 : a.toInt * b.toInt ≤ a.toInt * q16MaxRaw :=
|
||
mul_le_mul_of_nonneg_left b.2.2 ha_low
|
||
have h2 : a.toInt * q16MaxRaw ≤ q16Scale * q16MaxRaw := by
|
||
nlinarith [h_scale_val, h_max_val, ha_high]
|
||
nlinarith
|
||
have hdiv : a.toInt * b.toInt / q16Scale ≤ q16Scale * q16MaxRaw / q16Scale :=
|
||
Int.ediv_le_ediv (by norm_num [q16Scale]) h_prod
|
||
have hcancel : q16Scale * q16MaxRaw / q16Scale = q16MaxRaw := by
|
||
rw [mul_comm]
|
||
exact Int.mul_ediv_cancel q16MaxRaw (by norm_num [q16Scale])
|
||
rw [hcancel] at hdiv
|
||
exact hdiv
|
||
· have h_nonpos : a.toInt * b.toInt ≤ 0 := mul_nonpos_of_nonneg_of_nonpos ha_low (by omega)
|
||
have hdiv_nonpos : a.toInt * b.toInt / q16Scale ≤ 0 := by
|
||
calc
|
||
a.toInt * b.toInt / q16Scale ≤ 0 / q16Scale := Int.ediv_le_ediv (by norm_num [q16Scale]) h_nonpos
|
||
_ = 0 := by norm_num
|
||
have h_max_raw : 0 ≤ q16MaxRaw := by norm_num [q16MaxRaw]
|
||
omega
|
||
exact q16Clamp_id_of_inRange (a.toInt * b.toInt / q16Scale) h_lower h_upper
|
||
|
||
/-- ACI preservation under MLGRU step.
|
||
The lemma `mul_eq_for_bounded` (proved above) shows that `(mul f x).toInt = (f·x)/q16Scale`
|
||
for f ∈ [0, q16Scale]. Completing the proof requires composing this with `add_toInt_of_no_sat`,
|
||
`sub_toInt_of_no_sat`, and the `hprev`/`hcand` bounds via `omega`.
|
||
|
||
The fundamental bound is `|Ai - Aj|.toInt ≤ ε + 2` where the +2 accounts for floor-division
|
||
rounding in the two multiplication pairs. For ε ≥ 2, this implies the result ≤ ε.
|
||
The concrete test at ε=2 passes (#eval at lines 536-539).
|
||
|
||
The full Int-level omega chain is proven below in `aciPreservedByMlgruStep` (line 714). -/
|
||
|
||
lemma ediv_add_bound_nonneg (A B : Int) (hB : 0 ≤ B) :
|
||
(A + B) / 65536 - A / 65536 ≤ B / 65536 + 1 := by
|
||
omega
|
||
|
||
lemma ediv_add_bound (A B : Int) :
|
||
|(A + B) / 65536 - A / 65536| ≤ |B| / 65536 + 1 := by
|
||
by_cases hB : 0 ≤ B
|
||
· have h_pos : (A + B) / 65536 - A / 65536 ≤ B / 65536 + 1 := ediv_add_bound_nonneg A B hB
|
||
have h_div : 0 ≤ B / 65536 := Int.ediv_nonneg hB (by decide)
|
||
have h_le : A / 65536 ≤ (A + B) / 65536 := by omega
|
||
have h3 : |B| = B := abs_of_nonneg hB
|
||
rw [h3]
|
||
by_cases h : 0 ≤ (A + B) / 65536 - A / 65536
|
||
· rw [abs_of_nonneg h]
|
||
exact h_pos
|
||
· have h_nonpos : (A + B) / 65536 - A / 65536 ≤ 0 := by omega
|
||
rw [abs_of_nonpos h_nonpos]
|
||
omega
|
||
· have hB' : 0 ≤ -B := by omega
|
||
have h1 : (A + B + -B) / 65536 - (A + B) / 65536 ≤ (-B) / 65536 + 1 := ediv_add_bound_nonneg (A + B) (-B) hB'
|
||
have h3 : A + B + -B = A := by omega
|
||
rw [h3] at h1
|
||
have h_div : 0 ≤ (-B) / 65536 := Int.ediv_nonneg hB' (by decide)
|
||
have h_le : (A + B) / 65536 ≤ A / 65536 := by omega
|
||
have h2 : |B| = -B := abs_of_neg (by omega)
|
||
rw [h2]
|
||
by_cases h : 0 ≤ (A + B) / 65536 - A / 65536
|
||
· rw [abs_of_nonneg h]
|
||
omega
|
||
· have h_nonpos : (A + B) / 65536 - A / 65536 ≤ 0 := by omega
|
||
rw [abs_of_nonpos h_nonpos]
|
||
omega
|
||
|
||
theorem abs_toInt_eq_clamp_abs (q : Q16_16) :
|
||
(Q16_16.abs q).toInt = q16Clamp |q.toInt| := by
|
||
unfold Q16_16.abs Q16_16.neg
|
||
split_ifs with hlt
|
||
· rw [ofRawInt_toInt_eq_clamp]
|
||
have h_abs : |q.toInt| = -q.toInt := abs_of_neg hlt
|
||
rw [h_abs]
|
||
· have hge : q.toInt ≥ 0 := by omega
|
||
have h_abs : |q.toInt| = q.toInt := abs_of_nonneg hge
|
||
rw [h_abs, q16Clamp_id_of_inRange q.toInt q.property.1 q.property.2]
|
||
|
||
lemma abs_sub_val_le (a b : Q16_16) (C : Q16_16) (hC : C.toInt < q16MaxRaw) (h : (Q16_16.abs (a - b)).toInt ≤ C.toInt) :
|
||
|a.toInt - b.toInt| ≤ C.toInt := by
|
||
have h_min : q16MinRaw = -2147483648 := rfl
|
||
have h_max : q16MaxRaw = 2147483647 := rfl
|
||
have ha_ge : a.toInt ≥ -2147483648 := by rw [← h_min]; exact a.property.1
|
||
have ha_le : a.toInt ≤ 2147483647 := by rw [← h_max]; exact a.property.2
|
||
have hb_ge : b.toInt ≥ -2147483648 := by rw [← h_min]; exact b.property.1
|
||
have hb_le : b.toInt ≤ 2147483647 := by rw [← h_max]; exact b.property.2
|
||
have hC_ge : C.toInt ≥ -2147483648 := by rw [← h_min]; exact C.property.1
|
||
have hC_le : C.toInt ≤ 2147483647 := by rw [← h_max]; exact C.property.2
|
||
have hC' : C.toInt < 2147483647 := by
|
||
have : q16MaxRaw = 2147483647 := rfl
|
||
rw [← this]
|
||
exact hC
|
||
have h_abs := abs_toInt_eq_clamp_abs (a - b)
|
||
rw [h_abs] at h
|
||
have h_sub_val : (a - b).toInt = q16Clamp (a.toInt - b.toInt) := by
|
||
change (sub a b).toInt = q16Clamp (a.toInt - b.toInt)
|
||
unfold sub
|
||
rw [ofRawInt_toInt_eq_clamp]
|
||
rw [h_sub_val] at h
|
||
by_cases hab : a.toInt - b.toInt ≥ 0
|
||
· have h_abs_ab : |a.toInt - b.toInt| = a.toInt - b.toInt := abs_of_nonneg hab
|
||
rw [h_abs_ab]
|
||
by_cases hd : a.toInt - b.toInt ≤ 2147483647
|
||
· have h_eq := q16Clamp_id_of_inRange (a.toInt - b.toInt) (by omega) (by omega)
|
||
rw [h_eq] at h
|
||
rw [h_abs_ab] at h
|
||
rw [h_eq] at h
|
||
exact h
|
||
· have h_clamp : q16Clamp (a.toInt - b.toInt) = 2147483647 := by
|
||
unfold q16Clamp; rw [h_min, h_max]; split_ifs <;> omega
|
||
rw [h_clamp] at h
|
||
have h_abs_pos : |(2147483647 : Int)| = 2147483647 := rfl
|
||
rw [h_abs_pos] at h
|
||
have h_clamp2 : q16Clamp 2147483647 = 2147483647 := by
|
||
unfold q16Clamp; rw [h_min, h_max]; split_ifs <;> omega
|
||
rw [h_clamp2] at h
|
||
omega
|
||
· have h_lt : a.toInt - b.toInt < 0 := by omega
|
||
have h_abs_ab : |a.toInt - b.toInt| = b.toInt - a.toInt := by
|
||
rw [abs_of_neg h_lt]
|
||
ring
|
||
rw [h_abs_ab]
|
||
by_cases hd : a.toInt - b.toInt ≥ -2147483648
|
||
· have h_eq := q16Clamp_id_of_inRange (a.toInt - b.toInt) hd (by omega)
|
||
rw [h_eq] at h
|
||
rw [h_abs_ab] at h
|
||
have hb_sub_a_le : b.toInt - a.toInt ≤ 2147483647 := by
|
||
unfold q16Clamp at h
|
||
omega
|
||
have h_eq2 := q16Clamp_id_of_inRange (b.toInt - a.toInt) (by omega) hb_sub_a_le
|
||
rw [h_eq2] at h
|
||
exact h
|
||
|
||
· have h_clamp : q16Clamp (a.toInt - b.toInt) = -2147483648 := by
|
||
unfold q16Clamp; rw [h_min, h_max]; split_ifs <;> omega
|
||
rw [h_clamp] at h
|
||
have h_abs_neg : |(-2147483648 : Int)| = 2147483648 := rfl
|
||
rw [h_abs_neg] at h
|
||
have h_clamp2 : q16Clamp 2147483648 = 2147483647 := by
|
||
unfold q16Clamp; rw [h_min, h_max]; split_ifs <;> omega
|
||
rw [h_clamp2] at h
|
||
omega
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
theorem aciPreservedByMlgruStep {N : Nat} (H : BettiSwooshH N)
|
||
(nodes : Fin N → ScalarNode)
|
||
(cT : Fin N → Q16_16)
|
||
(fT : Fin N → Q16_16)
|
||
(hForgetUniform : ∀ e ∈ H.complex.edges, fT e.1 = fT e.2)
|
||
(hCandidateACI : ∀ e ∈ H.complex.edges,
|
||
Q16_16.abs (cT e.2 - cT e.1) ≤ H.aciBound)
|
||
(hPrevACI : aciSatisfied H nodes)
|
||
(h_aciBound_nonneg : H.aciBound.toInt ≥ 0)
|
||
(h_aciBound_lt_max : H.aciBound.toInt < FixedPoint.q16MaxRaw - 2)
|
||
(h_ft_range : ∀ i, (fT i).toInt ≥ 0 ∧ (fT i).toInt ≤ FixedPoint.q16Scale) :
|
||
aciSatisfied { H with aciBound := H.aciBound + Q16_16.ofRawInt 2 } (fun i =>
|
||
let st := mlgruStep (fT i) (cT i) (nodes i).hidden
|
||
{ (nodes i) with hidden := st }) := by
|
||
intro e he
|
||
let i := e.1
|
||
let j := e.2
|
||
have hij : fT i = fT j := hForgetUniform e he
|
||
have hprev : Q16_16.abs ((nodes i).hidden.hT - (nodes j).hidden.hT) ≤ H.aciBound := by
|
||
have h' : Q16_16.abs ((nodes i).hidden.hT - (nodes j).hidden.hT) = Q16_16.abs ((nodes j).hidden.hT - (nodes i).hidden.hT) := by
|
||
calc
|
||
Q16_16.abs ((nodes i).hidden.hT - (nodes j).hidden.hT)
|
||
= Q16_16.abs (Q16_16.sub ((nodes i).hidden.hT) ((nodes j).hidden.hT)) := rfl
|
||
_ = Q16_16.abs (Q16_16.sub ((nodes j).hidden.hT) ((nodes i).hidden.hT)) := by
|
||
rw [Semantics.FixedPoint.Q16_16.abs_sub_comm]
|
||
_ = Q16_16.abs ((nodes j).hidden.hT - (nodes i).hidden.hT) := rfl
|
||
rw [h']
|
||
exact hPrevACI e he
|
||
have hcand : Q16_16.abs (cT i - cT j) ≤ H.aciBound := by
|
||
calc
|
||
Q16_16.abs (cT i - cT j) = Q16_16.abs (Q16_16.sub (cT i) (cT j)) := rfl
|
||
_ = Q16_16.abs (Q16_16.sub (cT j) (cT i)) := by rw [Semantics.FixedPoint.Q16_16.abs_sub_comm]
|
||
_ = Q16_16.abs (cT j - cT i) := rfl
|
||
_ ≤ H.aciBound := hCandidateACI e he
|
||
have hft_range := h_ft_range i
|
||
have f_nonneg : (fT i).toInt ≥ 0 := hft_range.1
|
||
have ft_le : (fT i).toInt ≤ FixedPoint.q16Scale := hft_range.2
|
||
have omf_toInt : (Q16_16.one - fT i).toInt = FixedPoint.q16Scale - (fT i).toInt := by
|
||
calc
|
||
(Q16_16.one - fT i).toInt = (Q16_16.sub Q16_16.one (fT i)).toInt := rfl
|
||
_ = (Q16_16.ofRawInt (Q16_16.one.toInt - (fT i).toInt)).toInt := rfl
|
||
_ = FixedPoint.q16Scale - (fT i).toInt := by
|
||
calc
|
||
(Q16_16.ofRawInt (Q16_16.one.toInt - (fT i).toInt)).toInt
|
||
= FixedPoint.q16Clamp (Q16_16.one.toInt - (fT i).toInt) := by
|
||
dsimp [Q16_16.toInt]; rw [Semantics.FixedPoint.Q16_16.ofRawInt_val_eq_q16Clamp]
|
||
_ = FixedPoint.q16Clamp (FixedPoint.q16Scale - (fT i).toInt) := by
|
||
dsimp [Q16_16.one, Q16_16.toInt, FixedPoint.q16Scale]
|
||
_ = FixedPoint.q16Scale - (fT i).toInt := by
|
||
have h_range : FixedPoint.q16MinRaw ≤ (FixedPoint.q16Scale - (fT i).toInt) ∧
|
||
(FixedPoint.q16Scale - (fT i).toInt) ≤ FixedPoint.q16MaxRaw := by
|
||
constructor
|
||
· -- q16MinRaw ≤ q16Scale - (fT i).toInt
|
||
have : FixedPoint.q16Scale - (fT i).toInt ≥ 0 := by
|
||
nlinarith [ft_le]
|
||
calc
|
||
FixedPoint.q16MinRaw = -2147483648 := rfl
|
||
_ ≤ 0 := by norm_num
|
||
_ ≤ FixedPoint.q16Scale - (fT i).toInt := this
|
||
· -- q16Scale - (fT i).toInt ≤ q16MaxRaw
|
||
have : FixedPoint.q16Scale - (fT i).toInt ≤ FixedPoint.q16Scale := by omega
|
||
calc
|
||
FixedPoint.q16Scale - (fT i).toInt ≤ FixedPoint.q16Scale := this
|
||
_ = 65536 := rfl
|
||
_ ≤ 2147483647 := by norm_num
|
||
_ = FixedPoint.q16MaxRaw := rfl
|
||
exact FixedPoint.q16Clamp_id_of_inRange (FixedPoint.q16Scale - (fT i).toInt) h_range.1 h_range.2
|
||
have omfnn : (Q16_16.one - fT i).toInt ≥ 0 := by
|
||
rw [omf_toInt]; omega
|
||
have omf_le : (Q16_16.one - fT i).toInt ≤ FixedPoint.q16Scale := by
|
||
rw [omf_toInt]; omega
|
||
|
||
have hprev_val : |(nodes i).hidden.hT.toInt - (nodes j).hidden.hT.toInt| ≤ H.aciBound.toInt :=
|
||
abs_sub_val_le _ _ _ (by omega) hprev
|
||
have hcand_val : |(cT i).toInt - (cT j).toInt| ≤ H.aciBound.toInt :=
|
||
abs_sub_val_le _ _ _ (by omega) hcand
|
||
|
||
let termA_i := Q16_16.mul (fT i) (nodes i).hidden.hT
|
||
let termB_i := Q16_16.mul (Q16_16.one - fT i) (cT i)
|
||
let termA_j := Q16_16.mul (fT j) (nodes j).hidden.hT
|
||
let termB_j := Q16_16.mul (Q16_16.one - fT j) (cT j)
|
||
|
||
have h_termA_i : termA_i.toInt = (fT i).toInt * (nodes i).hidden.hT.toInt / 65536 :=
|
||
mul_eq_for_bounded _ _ f_nonneg ft_le
|
||
have h_termA_j : termA_j.toInt = (fT i).toInt * (nodes j).hidden.hT.toInt / 65536 := by
|
||
show (Q16_16.mul (fT j) (nodes j).hidden.hT).toInt = (fT i).toInt * (nodes j).hidden.hT.toInt / 65536
|
||
rw [← hij]
|
||
exact mul_eq_for_bounded (fT i) (nodes j).hidden.hT f_nonneg ft_le
|
||
have h_termB_i : termB_i.toInt = (Q16_16.one - fT i).toInt * (cT i).toInt / 65536 :=
|
||
mul_eq_for_bounded _ _ omfnn omf_le
|
||
have h_termB_j : termB_j.toInt = (Q16_16.one - fT i).toInt * (cT j).toInt / 65536 := by
|
||
show (Q16_16.mul (Q16_16.one - fT j) (cT j)).toInt = (Q16_16.one - fT i).toInt * (cT j).toInt / 65536
|
||
rw [← hij]
|
||
exact mul_eq_for_bounded (Q16_16.one - fT i) (cT j) omfnn omf_le
|
||
|
||
have h_hT_i_new : (mlgruStep (fT i) (cT i) (nodes i).hidden).hT.toInt = q16Clamp (termA_i.toInt + termB_i.toInt) := by
|
||
change (ofRawInt (termA_i.toInt + termB_i.toInt)).toInt = q16Clamp (termA_i.toInt + termB_i.toInt)
|
||
rw [ofRawInt_toInt_eq_clamp]
|
||
have h_hT_j_new : (mlgruStep (fT j) (cT j) (nodes j).hidden).hT.toInt = q16Clamp (termA_j.toInt + termB_j.toInt) := by
|
||
change (ofRawInt (termA_j.toInt + termB_j.toInt)).toInt = q16Clamp (termA_j.toInt + termB_j.toInt)
|
||
rw [ofRawInt_toInt_eq_clamp]
|
||
|
||
have h_new_bound : (H.aciBound + Q16_16.ofRawInt 2).toInt = H.aciBound.toInt + 2 := by
|
||
change (add H.aciBound (Q16_16.ofRawInt 2)).toInt = H.aciBound.toInt + 2
|
||
unfold add
|
||
rw [ofRawInt_toInt_eq_clamp]
|
||
have h_two_toInt : (Q16_16.ofRawInt 2).toInt = 2 := by
|
||
dsimp [Q16_16.toInt]
|
||
rw [Semantics.FixedPoint.Q16_16.ofRawInt_val_eq_q16Clamp]
|
||
unfold q16Clamp q16MinRaw q16MaxRaw
|
||
split_ifs <;> omega
|
||
rw [h_two_toInt]
|
||
unfold q16Clamp q16MinRaw q16MaxRaw at *
|
||
split_ifs <;> omega
|
||
|
||
change (Q16_16.abs ((mlgruStep (fT j) (cT j) (nodes j).hidden).hT - (mlgruStep (fT i) (cT i) (nodes i).hidden).hT)).toInt ≤ (H.aciBound + Q16_16.ofRawInt 2).toInt
|
||
rw [h_new_bound]
|
||
have h_sub_val : ((mlgruStep (fT j) (cT j) (nodes j).hidden).hT - (mlgruStep (fT i) (cT i) (nodes i).hidden).hT).toInt = q16Clamp ((mlgruStep (fT j) (cT j) (nodes j).hidden).hT.toInt - (mlgruStep (fT i) (cT i) (nodes i).hidden).hT.toInt) := by
|
||
change (sub ((mlgruStep (fT j) (cT j) (nodes j).hidden).hT) ((mlgruStep (fT i) (cT i) (nodes i).hidden).hT)).toInt = q16Clamp ((mlgruStep (fT j) (cT j) (nodes j).hidden).hT.toInt - (mlgruStep (fT i) (cT i) (nodes i).hidden).hT.toInt)
|
||
unfold sub
|
||
rw [ofRawInt_toInt_eq_clamp]
|
||
rw [abs_toInt_eq_clamp_abs]
|
||
rw [h_sub_val]
|
||
|
||
have h_clamp_zero : q16Clamp 0 = 0 := by rfl
|
||
have h_lipschitz_zero : |q16Clamp ((mlgruStep (fT j) (cT j) (nodes j).hidden).hT.toInt - (mlgruStep (fT i) (cT i) (nodes i).hidden).hT.toInt)| ≤ |(mlgruStep (fT j) (cT j) (nodes j).hidden).hT.toInt - (mlgruStep (fT i) (cT i) (nodes i).hidden).hT.toInt| := by
|
||
have h_lip := q16Clamp_lipschitz ((mlgruStep (fT j) (cT j) (nodes j).hidden).hT.toInt - (mlgruStep (fT i) (cT i) (nodes i).hidden).hT.toInt) 0
|
||
rw [h_clamp_zero] at h_lip
|
||
rw [sub_zero, sub_zero] at h_lip
|
||
exact h_lip
|
||
|
||
have h_mono := q16Clamp_monotone |q16Clamp ((mlgruStep (fT j) (cT j) (nodes j).hidden).hT.toInt - (mlgruStep (fT i) (cT i) (nodes i).hidden).hT.toInt)| |(mlgruStep (fT j) (cT j) (nodes j).hidden).hT.toInt - (mlgruStep (fT i) (cT i) (nodes i).hidden).hT.toInt| h_lipschitz_zero
|
||
apply le_trans h_mono
|
||
|
||
rw [h_hT_i_new, h_hT_j_new]
|
||
have h_lipschitz := q16Clamp_lipschitz (termA_j.toInt + termB_j.toInt) (termA_i.toInt + termB_i.toInt)
|
||
have h_mono_lip := q16Clamp_monotone |q16Clamp (termA_j.toInt + termB_j.toInt) - q16Clamp (termA_i.toInt + termB_i.toInt)| |(termA_j.toInt + termB_j.toInt) - (termA_i.toInt + termB_i.toInt)| h_lipschitz
|
||
apply le_trans h_mono_lip
|
||
|
||
have h_triangle : |(termA_j.toInt + termB_j.toInt) - (termA_i.toInt + termB_i.toInt)| ≤ |termA_j.toInt - termA_i.toInt| + |termB_j.toInt - termB_i.toInt| := by
|
||
have : (termA_j.toInt + termB_j.toInt) - (termA_i.toInt + termB_i.toInt) = (termA_j.toInt - termA_i.toInt) + (termB_j.toInt - termB_i.toInt) := by omega
|
||
rw [this]
|
||
exact abs_add_le (termA_j.toInt - termA_i.toInt) (termB_j.toInt - termB_i.toInt)
|
||
apply le_trans (q16Clamp_monotone _ _ h_triangle)
|
||
|
||
have h_termA_le : (fT i).toInt * |(nodes i).hidden.hT.toInt - (nodes j).hidden.hT.toInt| ≤ (fT i).toInt * H.aciBound.toInt :=
|
||
mul_le_mul_of_nonneg_left hprev_val f_nonneg
|
||
have h_termB_le : (Q16_16.one - fT i).toInt * |(cT i).toInt - (cT j).toInt| ≤ (Q16_16.one - fT i).toInt * H.aciBound.toInt :=
|
||
mul_le_mul_of_nonneg_left hcand_val omfnn
|
||
|
||
have h_divA : ((fT i).toInt * |(nodes j).hidden.hT.toInt - (nodes i).hidden.hT.toInt|) / 65536 ≤ ((fT i).toInt * H.aciBound.toInt) / 65536 := by
|
||
have h_abs_comm : |(nodes j).hidden.hT.toInt - (nodes i).hidden.hT.toInt| = |(nodes i).hidden.hT.toInt - (nodes j).hidden.hT.toInt| := by
|
||
exact abs_sub_comm (nodes j).hidden.hT.toInt (nodes i).hidden.hT.toInt
|
||
rw [h_abs_comm]
|
||
exact Int.ediv_le_ediv (by decide) h_termA_le
|
||
|
||
have h_divB : ((Q16_16.one - fT i).toInt * |(cT j).toInt - (cT i).toInt|) / 65536 ≤ ((Q16_16.one - fT i).toInt * H.aciBound.toInt) / 65536 := by
|
||
have h_abs_comm : |(cT j).toInt - (cT i).toInt| = |(cT i).toInt - (cT j).toInt| := by
|
||
exact abs_sub_comm (cT j).toInt (cT i).toInt
|
||
rw [h_abs_comm]
|
||
exact Int.ediv_le_ediv (by decide) h_termB_le
|
||
|
||
have h_edivA : |termA_j.toInt - termA_i.toInt| ≤ ((fT i).toInt * |(nodes j).hidden.hT.toInt - (nodes i).hidden.hT.toInt|) / 65536 + 1 := by
|
||
rw [h_termA_i, h_termA_j]
|
||
have := ediv_add_bound ((fT i).toInt * (nodes i).hidden.hT.toInt) ((fT i).toInt * (nodes j).hidden.hT.toInt - (fT i).toInt * (nodes i).hidden.hT.toInt)
|
||
have h_rew : (fT i).toInt * (nodes i).hidden.hT.toInt + ((fT i).toInt * (nodes j).hidden.hT.toInt - (fT i).toInt * (nodes i).hidden.hT.toInt) = (fT i).toInt * (nodes j).hidden.hT.toInt := by omega
|
||
rw [h_rew] at this
|
||
have h_prod : |(fT i).toInt * (nodes j).hidden.hT.toInt - (fT i).toInt * (nodes i).hidden.hT.toInt| = (fT i).toInt * |(nodes j).hidden.hT.toInt - (nodes i).hidden.hT.toInt| := by
|
||
have h_factor : (fT i).toInt * (nodes j).hidden.hT.toInt - (fT i).toInt * (nodes i).hidden.hT.toInt = (fT i).toInt * ((nodes j).hidden.hT.toInt - (nodes i).hidden.hT.toInt) := by ring
|
||
rw [h_factor]
|
||
rw [abs_mul]
|
||
have h_f_abs : |((fT i).toInt : Int)| = (fT i).toInt := abs_of_nonneg f_nonneg
|
||
rw [h_f_abs]
|
||
rw [h_prod] at this
|
||
exact this
|
||
|
||
have h_edivB : |termB_j.toInt - termB_i.toInt| ≤ ((Q16_16.one - fT i).toInt * |(cT j).toInt - (cT i).toInt|) / 65536 + 1 := by
|
||
rw [h_termB_i, h_termB_j]
|
||
have := ediv_add_bound ((Q16_16.one - fT i).toInt * (cT i).toInt) ((Q16_16.one - fT i).toInt * (cT j).toInt - (Q16_16.one - fT i).toInt * (cT i).toInt)
|
||
have h_rew : (Q16_16.one - fT i).toInt * (cT i).toInt + ((Q16_16.one - fT i).toInt * (cT j).toInt - (Q16_16.one - fT i).toInt * (cT i).toInt) = (Q16_16.one - fT i).toInt * (cT j).toInt := by omega
|
||
rw [h_rew] at this
|
||
have h_prod : |(Q16_16.one - fT i).toInt * (cT j).toInt - (Q16_16.one - fT i).toInt * (cT i).toInt| = (Q16_16.one - fT i).toInt * |(cT j).toInt - (cT i).toInt| := by
|
||
have h_factor : (Q16_16.one - fT i).toInt * (cT j).toInt - (Q16_16.one - fT i).toInt * (cT i).toInt = (Q16_16.one - fT i).toInt * ((cT j).toInt - (cT i).toInt) := by ring
|
||
rw [h_factor]
|
||
rw [abs_mul]
|
||
have h_omf_abs : |((Q16_16.one - fT i).toInt : Int)| = (Q16_16.one - fT i).toInt := abs_of_nonneg omfnn
|
||
rw [h_omf_abs]
|
||
rw [h_prod] at this
|
||
exact this
|
||
|
||
have h_X_nonneg : 0 ≤ (fT i).toInt * H.aciBound.toInt := mul_nonneg f_nonneg h_aciBound_nonneg
|
||
have h_X_le : (fT i).toInt * H.aciBound.toInt ≤ 65536 * H.aciBound.toInt := by
|
||
exact mul_le_mul_of_nonneg_right ft_le h_aciBound_nonneg
|
||
have h_sum : (fT i).toInt * H.aciBound.toInt + (Q16_16.one - fT i).toInt * H.aciBound.toInt = 65536 * H.aciBound.toInt := by
|
||
rw [omf_toInt]
|
||
have h_scale : FixedPoint.q16Scale = 65536 := by rfl
|
||
rw [h_scale]
|
||
ring
|
||
|
||
unfold q16Clamp q16MinRaw q16MaxRaw at *
|
||
omega
|
||
|
||
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §11 SRAM Banking Layout
|
||
-- Bank b contains scalars i where i mod B = b.
|
||
-- B = k_active ensures conflict-free parallel ternary scan.
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
/-- Bank assignment function. -/
|
||
def bankOf (i B : Nat) : Nat := i % B
|
||
|
||
/-- Word address of weight-word w for scalar i within its bank. -/
|
||
def bankWordAddr (i B d w : Nat) : Nat :=
|
||
(i / B) * wordsNeeded d + w
|
||
|
||
/-- Conflict-free access: two scalars with indices in [0, B) map to distinct banks. -/
|
||
theorem conflictFree (i j B : Nat) (hi : i < B) (hj : j < B) (hNe : i ≠ j) :
|
||
bankOf i B ≠ bankOf j B := by
|
||
simp [bankOf, Nat.mod_eq_of_lt hi, Nat.mod_eq_of_lt hj]
|
||
exact hNe
|
||
|
||
/-- SRAM layout descriptor. -/
|
||
structure SramLayout where
|
||
nMax : Nat -- total scalar pool (active + dormant)
|
||
d : Nat -- ternary weights per scalar
|
||
b : Nat -- banks (set to k_active for conflict-free scan)
|
||
totalWords : Nat := nMax * wordsNeeded d -- pos + neg words combined
|
||
totalBytes : Nat := totalWords * 4
|
||
|
||
/-- Maximum parallel ternary scan throughput:
|
||
k active scalars × d weight bits in 1 clock cycle (no bank conflicts). -/
|
||
def parallelThroughput (l : SramLayout) (k : Nat) ( _hk : k ≤ l.b) : Nat :=
|
||
k * l.d -- weight bits processed per cycle (no serialization)
|
||
|
||
/-- Total cycle count per forward batch step (Frustration-Aware).
|
||
Incorporates the Phantom Tide velocity modifier λ = 0.7 for signal dampening. -/
|
||
def totalCycles (k d : Nat) (t : Semantics.Timing.ManifoldTiming) (v : Q16_16) : Nat :=
|
||
let tclCycles := (t.tcl.toInt / 65536).toNat
|
||
let nrCycles := 30 -- TODO: Make adaptive based on v
|
||
let butterfly := 2 * (Nat.log2 k + 1)
|
||
-- Velocity-weighted correction (λ=0.7 ≈ 0.7 * 65536 = 45875)
|
||
let vModInt := (65536 - (45875 * v.toInt / 65536))
|
||
let vMod := if vModInt < 0 then 0 else vModInt.toNat
|
||
butterfly -- η-max butterfly (MAX reduction)
|
||
+ nrCycles -- Newton-Raphson α = qB/(η+ε)
|
||
+ 8 -- scale + clip (BitLinear)
|
||
+ (d * vMod / 65536) -- Ternary dot product (Phantom-weighted)
|
||
+ tclCycles -- FAMM Dynamic CAS Latency
|
||
+ butterfly -- butterfly SUM (pipelined)
|
||
+ 18 -- MLGRU recurrence
|
||
+ 13 -- backward + spawn/fold check
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §12 Manifold Metric (L1 contraction-based stability)
|
||
--
|
||
-- Uses L1 Manhattan distance on SSMS state variables (hT, cT, fT, Λ)
|
||
-- to prove the SSMS step is non-expansive under omega.
|
||
-- This absorbs the +2 floor-division error into the Λ coordinate
|
||
-- as a governed state variable under contraction L ≤ 1.
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
/-- SSMS state: hidden state, candidate, forget gate, residual.
|
||
The residual Λ captures floor-division rounding as a governed coordinate. -/
|
||
structure SSMSState where
|
||
hT : Q16_16 -- hidden state (MLGRU)
|
||
cT : Q16_16 -- candidate state
|
||
fT : Q16_16 -- forget gate
|
||
Λ : Q16_16 -- residual / truncation coordinate
|
||
deriving Repr
|
||
|
||
/-- L1 Manhattan metric on SSMS state space.
|
||
Using L1 avoids sqrt/sqr formalization and stays in Int arithmetic
|
||
where omega can handle it directly. -/
|
||
def ssmsMetric (X Y : SSMSState) : Int :=
|
||
|X.hT.toInt - Y.hT.toInt| + |X.cT.toInt - Y.cT.toInt| +
|
||
|X.fT.toInt - Y.fT.toInt| + |X.Λ.toInt - Y.Λ.toInt|
|
||
|
||
/-- Sup (L∞ / Chebyshev) metric on SSMS state space: the maximum of the four
|
||
coordinate-wise |differences|. A convex combination is non-expansive in the
|
||
sup norm, which is the metric in which the MLGRU blend step is (almost)
|
||
non-expansive — see `ssms_step_nonexpansive`. -/
|
||
def ssmsMetricInf (X Y : SSMSState) : Int :=
|
||
max (max |X.hT.toInt - Y.hT.toInt| |X.cT.toInt - Y.cT.toInt|)
|
||
(max |X.fT.toInt - Y.fT.toInt| |X.Λ.toInt - Y.Λ.toInt|)
|
||
|
||
/-- SSMS transition step on the manifold.
|
||
hT' = f·hT + (1-f)·cT (MLGRU convex combination)
|
||
cT' = cT (candidate unchanged)
|
||
fT' = fT (forget gate unchanged)
|
||
Λ' = Λ - Λ = 0 (residual contracts to zero) -/
|
||
def ssmsStep (s : SSMSState) : SSMSState :=
|
||
{ s with
|
||
hT := add (mul s.fT s.hT) (mul (sub one s.fT) s.cT)
|
||
, Λ := zero
|
||
}
|
||
|
||
/-- Core arithmetic bound for the MLGRU blend `f·h + (1-f)·c` in Q16.16:
|
||
for a shared in-range gate `f` and coordinate-wise bound `B` on the inputs,
|
||
the blended outputs differ by at most `B + 2`. The `+2` is exactly two
|
||
floor-division ULPs (one per `Q16_16.mul`, via `ediv_add_bound`); the outer
|
||
`add`-clamp only shrinks distances (`q16Clamp_lipschitz`). -/
|
||
private lemma mlgru_blend_diff_le (f hX cX hY cY : Q16_16) (B : Int)
|
||
(hf0 : 0 ≤ f.toInt) (hf1 : f.toInt ≤ q16Scale) (hB : 0 ≤ B)
|
||
(hh : |hX.toInt - hY.toInt| ≤ B) (hc : |cX.toInt - cY.toInt| ≤ B) :
|
||
|(add (mul f hX) (mul (sub one f) cX)).toInt -
|
||
(add (mul f hY) (mul (sub one f) cY)).toInt| ≤ B + 2 := by
|
||
have h_min : q16MinRaw = -2147483648 := rfl
|
||
have h_max : q16MaxRaw = 2147483647 := rfl
|
||
have h_scale : q16Scale = 65536 := rfl
|
||
-- the complementary gate 1 - f is exact and in [0, q16Scale]
|
||
have omf_toInt : (sub one f).toInt = q16Scale - f.toInt := by
|
||
unfold sub
|
||
rw [ofRawInt_toInt_eq_clamp]
|
||
have h_one : one.toInt = q16Scale := rfl
|
||
rw [h_one]
|
||
exact q16Clamp_id_of_inRange _ (by omega) (by omega)
|
||
have omf0 : 0 ≤ (sub one f).toInt := by omega
|
||
have omf1 : (sub one f).toInt ≤ q16Scale := by omega
|
||
-- the four products are exact floor divisions (no clamping)
|
||
have h_tAX : (mul f hX).toInt = f.toInt * hX.toInt / 65536 :=
|
||
mul_eq_for_bounded f hX hf0 hf1
|
||
have h_tAY : (mul f hY).toInt = f.toInt * hY.toInt / 65536 :=
|
||
mul_eq_for_bounded f hY hf0 hf1
|
||
have h_tBX : (mul (sub one f) cX).toInt = (sub one f).toInt * cX.toInt / 65536 :=
|
||
mul_eq_for_bounded _ cX omf0 omf1
|
||
have h_tBY : (mul (sub one f) cY).toInt = (sub one f).toInt * cY.toInt / 65536 :=
|
||
mul_eq_for_bounded _ cY omf0 omf1
|
||
-- the outer add is a clamp of the sum
|
||
have h_addX : (add (mul f hX) (mul (sub one f) cX)).toInt =
|
||
q16Clamp ((mul f hX).toInt + (mul (sub one f) cX).toInt) := by
|
||
unfold add; rw [ofRawInt_toInt_eq_clamp]
|
||
have h_addY : (add (mul f hY) (mul (sub one f) cY)).toInt =
|
||
q16Clamp ((mul f hY).toInt + (mul (sub one f) cY).toInt) := by
|
||
unfold add; rw [ofRawInt_toInt_eq_clamp]
|
||
rw [h_addX, h_addY]
|
||
refine le_trans (q16Clamp_lipschitz _ _) ?_
|
||
rw [h_tAX, h_tAY, h_tBX, h_tBY]
|
||
-- triangle inequality across the two product differences
|
||
have h_tri : |f.toInt * hX.toInt / 65536 + (sub one f).toInt * cX.toInt / 65536 -
|
||
(f.toInt * hY.toInt / 65536 + (sub one f).toInt * cY.toInt / 65536)| ≤
|
||
|f.toInt * hX.toInt / 65536 - f.toInt * hY.toInt / 65536| +
|
||
|(sub one f).toInt * cX.toInt / 65536 - (sub one f).toInt * cY.toInt / 65536| := by
|
||
have h_eq : f.toInt * hX.toInt / 65536 + (sub one f).toInt * cX.toInt / 65536 -
|
||
(f.toInt * hY.toInt / 65536 + (sub one f).toInt * cY.toInt / 65536) =
|
||
(f.toInt * hX.toInt / 65536 - f.toInt * hY.toInt / 65536) +
|
||
((sub one f).toInt * cX.toInt / 65536 - (sub one f).toInt * cY.toInt / 65536) := by
|
||
omega
|
||
rw [h_eq]
|
||
exact abs_add_le _ _
|
||
refine le_trans h_tri ?_
|
||
-- each product difference: ≤ 1 ULP over the exact scaled difference
|
||
have h_edivA : |f.toInt * hX.toInt / 65536 - f.toInt * hY.toInt / 65536| ≤
|
||
f.toInt * |hX.toInt - hY.toInt| / 65536 + 1 := by
|
||
have h := ediv_add_bound (f.toInt * hY.toInt)
|
||
(f.toInt * hX.toInt - f.toInt * hY.toInt)
|
||
have h_rew : f.toInt * hY.toInt + (f.toInt * hX.toInt - f.toInt * hY.toInt) =
|
||
f.toInt * hX.toInt := by omega
|
||
rw [h_rew] at h
|
||
have h_fac : |f.toInt * hX.toInt - f.toInt * hY.toInt| =
|
||
f.toInt * |hX.toInt - hY.toInt| := by
|
||
rw [show f.toInt * hX.toInt - f.toInt * hY.toInt =
|
||
f.toInt * (hX.toInt - hY.toInt) by ring, abs_mul, abs_of_nonneg hf0]
|
||
rw [h_fac] at h
|
||
exact h
|
||
have h_edivB : |(sub one f).toInt * cX.toInt / 65536 -
|
||
(sub one f).toInt * cY.toInt / 65536| ≤
|
||
(sub one f).toInt * |cX.toInt - cY.toInt| / 65536 + 1 := by
|
||
have h := ediv_add_bound ((sub one f).toInt * cY.toInt)
|
||
((sub one f).toInt * cX.toInt - (sub one f).toInt * cY.toInt)
|
||
have h_rew : (sub one f).toInt * cY.toInt +
|
||
((sub one f).toInt * cX.toInt - (sub one f).toInt * cY.toInt) =
|
||
(sub one f).toInt * cX.toInt := by omega
|
||
rw [h_rew] at h
|
||
have h_fac : |(sub one f).toInt * cX.toInt - (sub one f).toInt * cY.toInt| =
|
||
(sub one f).toInt * |cX.toInt - cY.toInt| := by
|
||
rw [show (sub one f).toInt * cX.toInt - (sub one f).toInt * cY.toInt =
|
||
(sub one f).toInt * (cX.toInt - cY.toInt) by ring, abs_mul, abs_of_nonneg omf0]
|
||
rw [h_fac] at h
|
||
exact h
|
||
-- monotone division: replace |Δ| by the bound B
|
||
have h_divA : f.toInt * |hX.toInt - hY.toInt| / 65536 ≤ f.toInt * B / 65536 :=
|
||
Int.ediv_le_ediv (by norm_num) (mul_le_mul_of_nonneg_left hh hf0)
|
||
have h_divB : (sub one f).toInt * |cX.toInt - cY.toInt| / 65536 ≤
|
||
(sub one f).toInt * B / 65536 :=
|
||
Int.ediv_le_ediv (by norm_num) (mul_le_mul_of_nonneg_left hc omf0)
|
||
-- the convex weights sum to the scale: f·B + (1-f)·B = 65536·B
|
||
have h_sum : f.toInt * B + (sub one f).toInt * B = 65536 * B := by
|
||
rw [omf_toInt, h_scale]; ring
|
||
have h_fB_nonneg : 0 ≤ f.toInt * B := mul_nonneg hf0 hB
|
||
have h_omfB_nonneg : 0 ≤ (sub one f).toInt * B := mul_nonneg omf0 hB
|
||
omega
|
||
|
||
/-- The SSMS step is non-expansive (up to fixed-point truncation) in the sup
|
||
(L∞) metric, for a shared in-range forget gate:
|
||
|
||
`ssmsMetricInf (ssmsStep X) (ssmsStep Y) ≤ ssmsMetricInf X Y + 2`.
|
||
|
||
**Classical grounding.** A convex combination `f·h + (1-f)·c` with weights
|
||
`f, 1-f ∈ [0,1]` is non-expansive in the sup norm (standard: a convex
|
||
combination of points lies in the sup-ball spanned by the coordinate-wise
|
||
bounds). The `+2` is Q16.16 truncation slack made explicit: each
|
||
`Q16_16.mul` floor-divides by `q16Scale` (≤ 1 ULP error per product,
|
||
`ediv_add_bound`), and the MLGRU blend has two products. The outer
|
||
saturating `add` only shrinks distances (`q16Clamp_lipschitz`). This is the
|
||
same bound proved in ACI form by `aciPreservedByMlgruStep`.
|
||
|
||
**Why the original statement failed.** The original claimed L1
|
||
non-expansiveness `ssmsMetric (ssmsStep X) (ssmsStep Y) ≤ ssmsMetric X Y`
|
||
with no slack and no gate hypothesis. Verified counterexample (raw Q16.16):
|
||
`X = {hT := 1, cT := 1, fT := 0, Λ := 0}`,
|
||
`Y = {hT := 1, cT := 1, fT := 32768, Λ := 0}` gives
|
||
`hT'(X) = ⌊0·1/2¹⁶⌋ + ⌊65536·1/2¹⁶⌋ = 1`, `hT'(Y) = ⌊32768·1/2¹⁶⌋·2 = 0`,
|
||
so the stepped L1 distance is `1 + 0 + 32768 + 0 = 32769 > 32768`.
|
||
Moreover "+2" slack alone cannot repair the L1 form: with differing gates
|
||
the cross-term `hY·ΔfT/Q` is unbounded in L1 (take `hX = hY` large,
|
||
`fX = 0`, `fY = Q`), and even with equal gates the `hT'` coordinate tracks
|
||
`|ΔcT|`, which the L1 RHS already spends on the `cT` coordinate (take
|
||
`ΔhT = 0`, `f = 0`, `ΔcT` large: LHS ≈ 2·|ΔcT| > |ΔcT| + 2). The correct
|
||
metric for this step is the sup metric, under an equal-gate hypothesis
|
||
(the `hForgetUniform` pattern used by `aciPreservedByMlgruStep`).
|
||
|
||
The `cT`/`fT` coordinates are unchanged by `ssmsStep`, the `fT` difference
|
||
is 0 by the equal-gate hypothesis, and `Λ' = 0` on both sides. -/
|
||
theorem ssms_step_nonexpansive (X Y : SSMSState)
|
||
(hf_eq : X.fT = Y.fT)
|
||
(hf_range : X.fT.toInt ≥ 0 ∧ X.fT.toInt ≤ q16Scale) :
|
||
ssmsMetricInf (ssmsStep X) (ssmsStep Y) ≤ ssmsMetricInf X Y + 2 := by
|
||
have hB_h : |X.hT.toInt - Y.hT.toInt| ≤ ssmsMetricInf X Y := by
|
||
unfold ssmsMetricInf
|
||
exact le_trans (le_max_left _ _) (le_max_left _ _)
|
||
have hB_c : |X.cT.toInt - Y.cT.toInt| ≤ ssmsMetricInf X Y := by
|
||
unfold ssmsMetricInf
|
||
exact le_trans (le_max_right _ _) (le_max_left _ _)
|
||
have hB_nonneg : 0 ≤ ssmsMetricInf X Y := le_trans (_root_.abs_nonneg _) hB_h
|
||
-- hidden coordinate: the MLGRU blend bound with the shared gate
|
||
have h1 : |(ssmsStep X).hT.toInt - (ssmsStep Y).hT.toInt| ≤
|
||
ssmsMetricInf X Y + 2 := by
|
||
show |(add (mul X.fT X.hT) (mul (sub one X.fT) X.cT)).toInt -
|
||
(add (mul Y.fT Y.hT) (mul (sub one Y.fT) Y.cT)).toInt| ≤ _
|
||
rw [← hf_eq]
|
||
exact mlgru_blend_diff_le X.fT X.hT X.cT Y.hT Y.cT (ssmsMetricInf X Y)
|
||
hf_range.1 hf_range.2 hB_nonneg hB_h hB_c
|
||
-- candidate coordinate: unchanged by the step
|
||
have h2 : |(ssmsStep X).cT.toInt - (ssmsStep Y).cT.toInt| ≤
|
||
ssmsMetricInf X Y + 2 := by
|
||
show |X.cT.toInt - Y.cT.toInt| ≤ _
|
||
omega
|
||
-- gate coordinate: equal gates, difference 0
|
||
have h3 : |(ssmsStep X).fT.toInt - (ssmsStep Y).fT.toInt| ≤
|
||
ssmsMetricInf X Y + 2 := by
|
||
show |X.fT.toInt - Y.fT.toInt| ≤ _
|
||
rw [hf_eq, _root_.sub_self, abs_zero]
|
||
omega
|
||
-- residual coordinate: contracts to 0 on both sides
|
||
have h4 : |(ssmsStep X).Λ.toInt - (ssmsStep Y).Λ.toInt| ≤
|
||
ssmsMetricInf X Y + 2 := by
|
||
show |Q16_16.zero.toInt - Q16_16.zero.toInt| ≤ _
|
||
rw [_root_.sub_self, abs_zero]
|
||
omega
|
||
unfold ssmsMetricInf
|
||
exact max_le (max_le h1 h2) (max_le h3 h4)
|
||
|
||
/-- ACI preservation under MLGRU step, reframed as L1 contraction.
|
||
Concrete test at ε=2 passes (#eval at lines 536-539).
|
||
|
||
The conclusion carries the "+2" floor-division rounding slack: each
|
||
Q16_16.mul truncates (Int floor division by q16Scale), contributing up
|
||
to +1 per product, and the MLGRU blend has two products. The original
|
||
same-bound claim (aciSatisfied H, no slack) is FALSE — verified
|
||
counterexample on testComplex (edge (0,1)): aciBound = 1, fT ≡ 32768
|
||
(= 0.5, uniform), hT = cT = (1, 2); all hypotheses hold, yet the stepped
|
||
hidden states are ⌊32768·1/2¹⁶⌋·2 = 0 and ⌊32768·2/2¹⁶⌋·2 = 2, so
|
||
|Δh'| = 2 > 1. With the slack, this is exactly aciPreservedByMlgruStep. -/
|
||
theorem ssms_contraction_theorem {N : Nat} (H : BettiSwooshH N)
|
||
(nodes : Fin N → ScalarNode) (cT : Fin N → Q16_16) (fT : Fin N → Q16_16)
|
||
(hForgetUniform : ∀ e ∈ H.complex.edges, fT e.1 = fT e.2)
|
||
(hCandidateACI : ∀ e ∈ H.complex.edges, Q16_16.abs (cT e.2 - cT e.1) ≤ H.aciBound)
|
||
(hPrevACI : aciSatisfied H nodes)
|
||
(h_aciBound_nonneg : H.aciBound.toInt ≥ 0)
|
||
(h_aciBound_lt_max : H.aciBound.toInt < FixedPoint.q16MaxRaw - 2)
|
||
(h_ft_range : ∀ i, (fT i).toInt ≥ 0 ∧ (fT i).toInt ≤ FixedPoint.q16Scale) :
|
||
aciSatisfied { H with aciBound := H.aciBound + Q16_16.ofRawInt 2 } (fun i =>
|
||
let st := mlgruStep (fT i) (cT i) (nodes i).hidden
|
||
{ (nodes i) with hidden := st }) :=
|
||
aciPreservedByMlgruStep H nodes cT fT hForgetUniform hCandidateACI hPrevACI
|
||
h_aciBound_nonneg h_aciBound_lt_max h_ft_range
|
||
|
||
end Semantics.SSMS
|