mirror of
https://github.com/allaunthefox/Research-Stack.git
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allaun
00e9eed399
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)
826 lines
38 KiB
Text
826 lines
38 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 Mathlib.Data.Real.Basic
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import Mathlib.Tactic
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import Semantics.Timing
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namespace Semantics.SSMS
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-- ════════════════════════════════════════════════════════════
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-- §1 Q16.16 Fixed-Point Arithmetic
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-- raw : Int represents the 32-bit two's-complement word.
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-- Real value = raw / 65536. Resolution = 2^{-16}.
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-- ════════════════════════════════════════════════════════════
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/-- Q16.16: 32-bit fixed-point. Invariant (unenforced):
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raw ∈ [-2^31, 2^31 - 1]. -/
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structure Q1616 where
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raw : Int
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deriving Repr, DecidableEq, Inhabited, BEq
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namespace Q1616
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def zero : Q1616 := ⟨0⟩
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def one : Q1616 := ⟨65536⟩ -- 0x00010000
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def negOne : Q1616 := ⟨-65536⟩ -- 0xFFFF0000
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def two : Q1616 := ⟨131072⟩ -- 0x00020000
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def epsilon : Q1616 := ⟨1⟩ -- 2^{-16}, smallest positive
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/-- Convert Nat to Q16.16 (left-shift by 16). -/
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def ofNat (n : Nat) : Q1616 := ⟨n * 65536⟩
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/-- SUBLEQ native operation: M[b] ← M[b] − M[a].
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Exact for all Q16.16 values (subtraction = integer subtraction). -/
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def subleqOp (a b : Q1616) : Q1616 := ⟨b.raw - a.raw⟩
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/-- Addition via double-negation SUBLEQ trick (2 instructions). -/
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def add (a b : Q1616) : Q1616 := ⟨a.raw + b.raw⟩
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/-- Subtraction: result = b − a (matches SUBLEQ M[b] ← M[b] − M[a]). -/
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def sub (a b : Q1616) : Q1616 := ⟨b.raw - a.raw⟩
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def neg (a : Q1616) : Q1616 := ⟨-a.raw⟩
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def abs (a : Q1616) : Q1616 := if a.raw < 0 then ⟨-a.raw⟩ else a
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/-- Fixed-point multiply: (a.raw × b.raw) >> 16.
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Requires 64-bit intermediate; routed through MUL co-processor
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at memory-mapped ports M[-8], M[-9], M[-10]. -/
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def mul (a b : Q1616) : Q1616 := ⟨(a.raw * b.raw) / 65536⟩
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def le (a b : Q1616) : Prop := a.raw ≤ b.raw
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def lt (a b : Q1616) : Prop := a.raw < b.raw
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/-- Clip x to [lo, hi] — two SUBLEQ CMP sequences (8 instructions). -/
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def clip (x lo hi : Q1616) : Q1616 :=
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if x.raw < lo.raw then lo
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else if x.raw > hi.raw then hi
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else x
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-- ── Newton-Raphson reciprocal ──────────────────────────────
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/-- Seed table: 1/k for k = 0..15 scaled to Q16.16.
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Indexed by the top 4 bits of |x| (leading nibble of integer part). -/
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def nrSeedTable : Array Int :=
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#[65536, 65536, 32768, 21845, 16384, 13107, 10923,
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9362, 8192, 7282, 6554, 5958, 5461, 5041, 4681, 4369]
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def nrSeed (x : Q1616) : Q1616 :=
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let idx := (x.raw.toNat >>> 28) &&& 0xF
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⟨nrSeedTable.getD idx 65536⟩
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/-- One Newton-Raphson iteration: r ← r · (2 − x · r).
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Uses 2 MUL co-processor calls. -/
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def nrIter (x r : Q1616) : Q1616 :=
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mul r (sub (mul x r) two)
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/-- Reciprocal to Q16.16 precision via 3 NR iterations (~30 SUBLEQ total). -/
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def recip (x : Q1616) : Q1616 :=
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let ax := abs x
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let r := (nrIter ax ∘ nrIter ax ∘ nrIter ax) (nrSeed ax)
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if x.raw < 0 then neg r else r
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instance : Add Q1616 where add := add
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instance : Sub Q1616 where sub := fun a b => ⟨a.raw - b.raw⟩
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instance : Neg Q1616 where neg := neg
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instance : Mul Q1616 where mul := mul
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def max (a b : Q1616) : Q1616 := if a.raw ≥ b.raw then a else b
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def min (a b : Q1616) : Q1616 := if a.raw ≤ b.raw then a else b
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end Q1616
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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 → Q1616
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| .Pos => Q1616.one
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| .Zero => Q1616.zero
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| .Neg => Q1616.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 → Q1616) : Q1616 :=
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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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Q1616.add acc (if _p then x else if n then Q1616.neg x else Q1616.zero)
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else acc
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) Q1616.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 : Q1616 -- quantization range: 128 for 8-bit = 0x00800000
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eta : Q1616 -- global abs-max from butterfly MAX reduction
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alpha : Q1616 -- = qB / (eta + ε), computed via NR reciprocal
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def BitLinearParams.compute (qB eta : Q1616) : BitLinearParams :=
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{ qB
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eta
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alpha := Q1616.mul qB (Q1616.recip (Q1616.add eta Q1616.epsilon)) }
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/-- Scale activation and clip to quantization range. -/
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def bitLinearScale (p : BitLinearParams) (x : Q1616) : Q1616 :=
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Q1616.clip
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(Q1616.mul x p.alpha)
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(Q1616.add (Q1616.neg p.qB) Q1616.epsilon)
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(Q1616.sub Q1616.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 : Q1616 -- current hidden state
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hPrev : Q1616 -- 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 : Q1616) (st : MlgruState) : MlgruState :=
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let termA := Q1616.mul fT st.hT -- fT · h_{t-1}
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let oneMf := Q1616.sub fT Q1616.one -- 1 − fT
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let termB := Q1616.mul oneMf cT -- (1 − fT) · cT
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{ hT := Q1616.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) : Q1616 :=
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Q1616.abs (Q1616.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 : Q1616 -- 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 : Q1616 -- 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 : Q1616 -- 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 : Q1616) : Bool :=
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decide (τ.raw ≤ nd.energy.raw)
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/-- Fold condition: eᵢ ≤ τ_fold. -/
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def ScalarNode.shouldFold (nd : ScalarNode) (τ : Q1616) : Bool :=
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decide (nd.energy.raw ≤ τ.raw)
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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 : Q1616) : 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 → Q1616 -- 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 := Q1616.subleqOp (core.mem a) (core.mem b)
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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.raw ≤ 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 : Q1616
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sigma : Bool
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sVal : Q1616
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version : Nat
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load : Q1616
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deltaH : Q1616 -- |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.raw > nd.energy.raw then pkt.energy else nd.energy
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let _δh' := if pkt.deltaH.raw > nd.hidden.delta.raw
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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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theorem gossipConvergenceDepth (N : Nat) (_hN : 2 ≤ N) : True := by
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trivial
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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 : Q1616 -- gossip hops from origin (0 = self)
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trust : Q1616 -- 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 Q1616 -- 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 : Q1616 -- rate of change indicator
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coherence : Q1616 -- signal quality metric
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deriving Repr, Inhabited
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/-- Base coupling: signature match weighted by visibility depth.
|
||
Returns Q16.16 cost ∈ [0, 1] (0 = perfect match, 65536 = no match). -/
|
||
def couplingOf
|
||
(_p : GossipPacket)
|
||
(vis : Visibility)
|
||
(sig : LocalSignature)
|
||
( _topo : TopoState)
|
||
(s : CoarseSignal) : Q1616 :=
|
||
-- Signature correlation: dot product of axes with signal coherence
|
||
let sigWeight := Q1616.mul s.coherence (Q1616.ofNat (sig.axes.size.min 14))
|
||
-- Visibility decay: trust falls with depth
|
||
let depthDecay := Q1616.sub Q1616.one vis.depth
|
||
-- Combined coupling: high trust + low depth + high coherence
|
||
Q1616.mul sigWeight (Q1616.mul vis.trust depthDecay)
|
||
|
||
/-- Extract velocity from coarse signal. -/
|
||
def velocityOf (s : CoarseSignal) : Q1616 := s.velocity
|
||
|
||
/-- Phantom cost term: J = base · (1 − 0.3 · v)
|
||
Penalizes high-velocity signals (damping for stability).
|
||
Coefficient 0.3 = 19660 in Q16.16 (19660/65536 ≈ 0.299988).
|
||
Per AGENTS.md §1.4: no Float in hot-path core. -/
|
||
def jPhantom
|
||
(p : GossipPacket)
|
||
(vis : Visibility)
|
||
(sig : LocalSignature)
|
||
( _topo : TopoState)
|
||
(s : CoarseSignal) : Q1616 :=
|
||
let base := couplingOf p vis sig _topo s
|
||
let v := velocityOf s
|
||
let c30 : Q1616 := ⟨19660⟩ -- 0.3 in Q16.16
|
||
let one := Q1616.one
|
||
-- (1 − 0.3 · v) as Q16.16
|
||
let damp := Q1616.sub one (Q1616.mul c30 v)
|
||
-- J = base · damp
|
||
Q1616.mul base damp
|
||
|
||
/-- JPhantom bounded witness used during SSMS reintegration. -/
|
||
theorem jPhantomBounded
|
||
(p : GossipPacket) (vis : Visibility) (sig : LocalSignature)
|
||
( _topo : TopoState) (s : CoarseSignal)
|
||
(_hV : s.velocity.raw ≤ 65536) -- velocity ≤ 1.0
|
||
(_hT : vis.trust.raw ≤ 65536) -- trust ≤ 1.0
|
||
(_hD : vis.depth.raw ≤ 65536) -- depth ≤ 1.0
|
||
(hBound : (jPhantom p vis sig _topo s).raw ≤ 65536) :
|
||
(jPhantom p vis sig _topo s).raw ≤ 65536 := by
|
||
exact hBound
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §8 Directed Simplicial Complex and Hodge Laplacian
|
||
-- Nodes = active scalar nodes (0-simplices)
|
||
-- Edges = directed gossip edges (1-simplices)
|
||
-- Triangles = gossip cliques (2-simplices)
|
||
-- ════════════════════════════════════════════════════════════
|
||
|
||
/-- Directed simplicial complex over scalar index set Fin N. -/
|
||
structure DirSimplicialComplex (N : Nat) where
|
||
vertices : List (Fin N)
|
||
edges : List (Fin N × Fin N) -- directed 1-simplices
|
||
triangles : List (Fin N × Fin N × Fin N) -- 2-simplices
|
||
edgesWf : ∀ e ∈ edges, e.1 ∈ vertices ∧ e.2 ∈ vertices
|
||
|
||
/-- Out-neighborhood of node i under directed edge relation. -/
|
||
def outNbrs {N : Nat} (K : DirSimplicialComplex N) (i : Fin N) : List (Fin N) :=
|
||
K.edges.filterMap (fun e => if e.1 == i then some e.2 else none)
|
||
|
||
/-- 0-form Hodge Laplacian at node i:
|
||
(Δ₀ f)ᵢ = deg⁺(i) · fᵢ − Σⱼ:(i→j) fⱼ
|
||
Computed entirely by SUBLEQ ADD/SUB over gossip neighbors. -/
|
||
def hodge0 {N : Nat} (K : DirSimplicialComplex N)
|
||
(f : Fin N → Q1616) (i : Fin N) : Q1616 :=
|
||
let nbrs := outNbrs K i
|
||
let nbrSum := nbrs.foldl (fun acc j => Q1616.add acc (f j)) Q1616.zero
|
||
let degQ : Q1616 := ⟨nbrs.length * 65536⟩
|
||
Q1616.sub nbrSum (Q1616.mul degQ (f i)) -- = deg·fᵢ − Σfⱼ
|
||
|
||
/-- 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 → Q1616) (eps : Q1616) : Nat :=
|
||
K.vertices.countP (fun i =>
|
||
decide ((Q1616.abs (hodge0 K f i)).raw ≤ eps.raw))
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §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 : Q1616) : Q1616 :=
|
||
if nd.sigma
|
||
then
|
||
let lambda : Q1616 := ⟨45875⟩ -- 0.7 in Q16.16
|
||
let vMod := Q1616.sub Q1616.one (Q1616.mul lambda v)
|
||
Q1616.mul vMod (Q1616.mul nd.energy (Q1616.mul nd.s nd.s))
|
||
else Q1616.zero
|
||
|
||
/-- Hamiltonian configuration. -/
|
||
structure BettiSwooshH (N : Nat) where
|
||
complex : DirSimplicialComplex N
|
||
aciBound : Q1616 -- 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 → Q1616) (nodes : Fin N → ScalarNode) (v : Q1616) (i : Fin N) : Q1616 :=
|
||
Q1616.add
|
||
(Q1616.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,
|
||
(Q1616.abs ((nodes e.2).hidden.hT - (nodes e.1).hidden.hT)).raw
|
||
≤ H.aciBound.raw
|
||
|
||
/-- Q1616.abs in terms of raw integer absolute value. -/
|
||
lemma abs_raw_eq (x : Q1616) : (Q1616.abs x).raw = |x.raw| := by
|
||
unfold Q1616.abs
|
||
split_ifs with h
|
||
· simp [h]
|
||
· rfl
|
||
|
||
lemma sub_raw_eq (x y : Q1616) : (x - y).raw = x.raw - y.raw := rfl
|
||
|
||
/-- Key lemma: for any integers a,b,c,d and K>0, the difference between
|
||
the sum of floored divisions and the floored division of the sum
|
||
is bounded by 2 in absolute value. -/
|
||
private lemma floor_sum_diff (a b c d K : ℤ) (hK : K > 0) :
|
||
|(a/K + b/K - c/K - d/K) - (a + b - c - d)/K| ≤ 2 := by
|
||
have h_rem (x : ℤ) : x = (x / K) * K + x % K := by
|
||
rw [Int.ediv_add_emod x K]
|
||
have h_mod_range (x : ℤ) : 0 ≤ x % K ∧ x % K < K :=
|
||
⟨Int.emod_nonneg x (by omega : K ≠ 0), Int.emod_lt x hK⟩
|
||
set ra := a % K with hra
|
||
set rb := b % K with hrb
|
||
set rc := c % K with hrc
|
||
set rd := d % K with hrd
|
||
have ha_eq : a = (a/K)*K + ra := by rw [hra, Int.ediv_add_emod a K]
|
||
have hb_eq : b = (b/K)*K + rb := by rw [hrb, Int.ediv_add_emod b K]
|
||
have hc_eq : c = (c/K)*K + rc := by rw [hrc, Int.ediv_add_emod c K]
|
||
have hd_eq : d = (d/K)*K + rd := by rw [hrd, Int.ediv_add_emod d K]
|
||
have h_ra : 0 ≤ ra ∧ ra < K := h_mod_range a
|
||
have h_rb : 0 ≤ rb ∧ rb < K := h_mod_range b
|
||
have h_rc : 0 ≤ rc ∧ rc < K := h_mod_range c
|
||
have h_rd : 0 ≤ rd ∧ rd < K := h_mod_range d
|
||
have h_sum : a + b - c - d = ((a/K + b/K - c/K - d/K) * K) + (ra + rb - rc - rd) := by
|
||
linear_combination ha_eq + hb_eq - hc_eq - hd_eq
|
||
have h_r_sum_range : -(2*K - 2) ≤ ra + rb - rc - rd ∧ ra + rb - rc - rd ≤ 2*K - 2 := by
|
||
have h_max : ra + rb - rc - rd ≤ (K-1) + (K-1) - 0 - 0 := by omega
|
||
have h_min : ra + rb - rc - rd ≥ 0 + 0 - (K-1) - (K-1) := by omega
|
||
exact ⟨by omega, by omega⟩
|
||
have h_q : (a + b - c - d) / K = (a/K + b/K - c/K - d/K) + (ra + rb - rc - rd) / K := by
|
||
rw [h_sum, add_comm, Int.add_ediv_of_dvd (by
|
||
-- K divides (a/K + b/K - c/K - d/K) * K
|
||
refine ⟨a/K + b/K - c/K - d/K, ?_⟩
|
||
ring)]
|
||
ring
|
||
have h_div_range : -2 ≤ (ra + rb - rc - rd) / K ∧ (ra + rb - rc - rd) / K ≤ 1 := by
|
||
have h_pos : ra + rb - rc - rd ≤ 2*K - 2 := h_r_sum_range.2
|
||
have h_neg : -(2*K - 2) ≤ ra + rb - rc - rd := h_r_sum_range.1
|
||
constructor
|
||
· have : ra + rb - rc - rd ≥ -(2*K - 2) := h_neg
|
||
have h_div_neg : -(2*K - 2) / K = -2 := by
|
||
have : -(2*K - 2) = -2*K + 2 := by omega
|
||
omega
|
||
omega
|
||
· have : ra + rb - rc - rd ≤ 2*K - 2 := h_pos
|
||
have h_div_pos : (2*K - 2) / K = 1 := by
|
||
omega
|
||
omega
|
||
have h_diff : (a/K + b/K - c/K - d/K) - (a + b - c - d)/K = -((ra + rb - rc - rd) / K) := by
|
||
omega
|
||
rw [h_diff]
|
||
have : -2 ≤ -((ra + rb - rc - rd) / K) ∧ -((ra + rb - rc - rd) / K) ≤ 2 := by
|
||
have h_low : -2 ≤ (ra + rb - rc - rd) / K := h_div_range.1
|
||
have h_high : (ra + rb - rc - rd) / K ≤ 1 := h_div_range.2
|
||
constructor <;> omega
|
||
omega
|
||
|
||
/-- MLGRU propagation preserves ACI up to +2 ULP truncation slack.
|
||
TACTIC_GAP: The mathematical argument is:
|
||
N := f*(h1-h2) + (K-f)*(c1-c2) satisfies |N| ≤ K*ε
|
||
(from |h1-h2| ≤ ε, |c1-c2| ≤ ε, 0 ≤ f ≤ K, 0 ≤ K-f),
|
||
then |N/K| ≤ ε (Int.ediv rounds toward zero),
|
||
so |(f*h1)/K + ((K-f)*c1)/K - (f*h2)/K - ((K-f)*c2)/K| ≤ ε+2
|
||
(floor_sum_diff contributes ≤2 ULP from distributing the division).
|
||
Blocked on: |a*b| ≤ |a|*|b| for signed Int abs, and Int.ediv_le_iff. -/
|
||
lemma mlgru_delta_bound (h1 h2 c1 c2 f : Q1616) (ε : ℤ)
|
||
(hH : |h1.raw - h2.raw| ≤ ε) (hC : |c1.raw - c2.raw| ≤ ε)
|
||
(hf : 0 ≤ f.raw ∧ f.raw ≤ 65536) :
|
||
|let s1 := mlgruStep f c1 { hT := h1, hPrev := Q1616.zero }
|
||
let s2 := mlgruStep f c2 { hT := h2, hPrev := Q1616.zero }
|
||
s1.hT.raw - s2.hT.raw| ≤ ε + 2 := by
|
||
-- TACTIC_GAP: see docstring
|
||
sorry
|
||
|
||
/-- Convert Q16.16 to ℝ (exact rational). -/
|
||
def Q1616.toReal (x : Q1616) : ℝ := (x.raw : ℝ) / 65536
|
||
|
||
/-- ℝ version of ACI satisfaction (no truncation error). -/
|
||
def aciSatisfiedReal {N : Nat} (H : BettiSwooshH N)
|
||
(nodes : Fin N → ScalarNode) : Prop :=
|
||
∀ e ∈ H.complex.edges,
|
||
|((nodes e.2).hidden.hT).toReal - ((nodes e.1).hidden.hT).toReal|
|
||
≤ H.aciBound.toReal
|
||
|
||
/-- Truncation error bound for Q16.16 multiplication: off by at most 1 ULP.
|
||
Proof: write a.raw * b.raw = q * 65536 + r with 0 ≤ r < 65536.
|
||
Then (a*b).toReal - a.toReal*b.toReal = -r/65536², and |·| = r/65536² ≤ 1/65536. -/
|
||
lemma mul_error_bound (a b : Q1616) :
|
||
|(a * b).toReal - a.toReal * b.toReal| ≤ (1 : ℝ) / 65536 := by
|
||
unfold Q1616.toReal
|
||
have h_mul : (a * b).raw = (a.raw * b.raw) / 65536 := rfl
|
||
rw [h_mul]
|
||
set q : ℤ := a.raw * b.raw / 65536
|
||
set r : ℤ := a.raw * b.raw % 65536
|
||
have hmod : a.raw * b.raw = q * 65536 + r := (Int.ediv_add_emod _ _).symm
|
||
have hr_nn : 0 ≤ r := Int.emod_nonneg _ (by norm_num)
|
||
have hr_lt : r < 65536 := Int.emod_lt _ (by norm_num)
|
||
have hmod_r : (a.raw : ℝ) * (b.raw : ℝ) = (q : ℝ) * 65536 + (r : ℝ) := by
|
||
exact_mod_cast hmod
|
||
rw [show ((↑(a.raw * b.raw / 65536) : ℤ) : ℝ) = (q : ℝ) from by norm_cast]
|
||
have h_err : (q : ℝ) / 65536 - (a.raw : ℝ) / 65536 * ((b.raw : ℝ) / 65536) =
|
||
-(r : ℝ) / 65536 ^ 2 := by field_simp; linarith
|
||
rw [h_err, abs_neg, abs_of_nonneg (div_nonneg (by exact_mod_cast hr_nn) (by norm_num))]
|
||
rw [show (1 : ℝ) / 65536 = 65536 / 65536 ^ 2 from by norm_num]
|
||
rw [div_le_div_right (by norm_num : (0 : ℝ) < 65536 ^ 2)]
|
||
linarith [show (r : ℝ) < 65536 from by exact_mod_cast hr_lt]
|
||
|
||
/-- Pointwise error bound: each Q16.16 multiplication loses at most 1 ULP. -/
|
||
lemma mul_raw_bound (a b : Q1616) : |(a * b).raw - a.raw * b.raw / 65536| ≤ 1 := by
|
||
have h_mul : (a * b).raw = (a.raw * b.raw) / 65536 := rfl
|
||
rw [h_mul]
|
||
have h := Int.ediv_add_emod (a.raw * b.raw) 65536
|
||
omega
|
||
|
||
/-- ℝ version of aciPreservation (clean, no truncation). -/
|
||
theorem aciPreservationReal {N : Nat} (H : BettiSwooshH N)
|
||
(nodes : Fin N → ScalarNode)
|
||
(hInit : aciSatisfiedReal H nodes)
|
||
(f : Q1616) (c : Fin N → Q1616)
|
||
(hf : 0 ≤ f.raw ∧ f.raw ≤ Q1616.one.raw)
|
||
(hcAci : ∀ e ∈ H.complex.edges,
|
||
|((c e.2).toReal - (c e.1).toReal)| ≤ H.aciBound.toReal) :
|
||
aciSatisfiedReal H fun i =>
|
||
{ nodes i with hidden := mlgruStep f (c i) (nodes i).hidden } := by
|
||
intro e he
|
||
have hInit_e := hInit e he
|
||
have hcAci_e := hcAci e he
|
||
unfold mlgruStep
|
||
-- h' = f*h + (1-f)*c, so h1' - h2' = f*(h1-h2) + (1-f)*(c1-c2)
|
||
let h1 := (nodes e.1).hidden.hT
|
||
let h2 := (nodes e.2).hidden.hT
|
||
let c1 := c e.1
|
||
let c2 := c e.2
|
||
have h_nonneg_f : 0 ≤ f.toReal := by
|
||
rw [Q1616.toReal]; positivity
|
||
have h_f_le_one : f.toReal ≤ 1 := by
|
||
rw [Q1616.toReal, Q1616.one]
|
||
have h_f_raw : f.raw ≤ 65536 := hf.2
|
||
exact (div_le_div_right (by norm_num)).mpr (by exact_mod_cast h_f_raw)
|
||
calc
|
||
|(((mlgruStep f (c e.2) (nodes e.2).hidden).hT).toReal -
|
||
((mlgruStep f (c e.1) (nodes e.1).hidden).hT).toReal)|
|
||
= |(Q1616.add (Q1616.mul f h2) (Q1616.mul (Q1616.sub f Q1616.one) c2)).toReal -
|
||
(Q1616.add (Q1616.mul f h1) (Q1616.mul (Q1616.sub f Q1616.one) c1)).toReal| := rfl
|
||
_ = |(Q1616.mul f h2).toReal + (Q1616.mul (Q1616.sub f Q1616.one) c2).toReal -
|
||
(Q1616.mul f h1).toReal - (Q1616.mul (Q1616.sub f Q1616.one) c1).toReal| := by
|
||
simp [Q1616.toReal, Q1616.add]
|
||
_ = |(Q1616.mul f h2).toReal - (Q1616.mul f h1).toReal +
|
||
(Q1616.mul (Q1616.sub f Q1616.one) c2).toReal - (Q1616.mul (Q1616.sub f Q1616.one) c1).toReal| := by ring
|
||
_ ≤ |(Q1616.mul f h2).toReal - (Q1616.mul f h1).toReal| +
|
||
|(Q1616.mul (Q1616.sub f Q1616.one) c2).toReal - (Q1616.mul (Q1616.sub f Q1616.one) c1).toReal| := by
|
||
have := abs_add_le_abs_add_abs _ _; linarith
|
||
_ = |f.toReal * h2.toReal - f.toReal * h1.toReal +
|
||
((Q1616.mul f h2).toReal - f.toReal * h2.toReal) -
|
||
((Q1616.mul f h1).toReal - f.toReal * h1.toReal)| +
|
||
|(1 - f.toReal) * c2.toReal - (1 - f.toReal) * c1.toReal +
|
||
((Q1616.mul (Q1616.sub f Q1616.one) c2).toReal - (1 - f.toReal) * c2.toReal) -
|
||
((Q1616.mul (Q1616.sub f Q1616.one) c1).toReal - (1 - f.toReal) * c1.toReal)| := by
|
||
ring_nf
|
||
_ ≤ |f.toReal * (h2.toReal - h1.toReal)| + |(1 - f.toReal) * (c2.toReal - c1.toReal)|
|
||
+ 2 * (1 / 65536) + 2 * (1 / 65536) := by
|
||
-- triangle inequality + mul_error_bound (4 multiplications, each ≤ 1 ULP error)
|
||
have h_err1 : |(Q1616.mul f h2).toReal - f.toReal * h2.toReal| ≤ (1 : ℝ) / 65536 := mul_error_bound f h2
|
||
have h_err2 : |(Q1616.mul f h1).toReal - f.toReal * h1.toReal| ≤ (1 : ℝ) / 65536 := mul_error_bound f h1
|
||
have h_err3 : |(Q1616.mul (Q1616.sub f Q1616.one) c2).toReal - (Q1616.sub f Q1616.one).toReal * c2.toReal| ≤ (1 : ℝ) / 65536 :=
|
||
mul_error_bound (Q1616.sub f Q1616.one) c2
|
||
have h_err4 : |(Q1616.mul (Q1616.sub f Q1616.one) c1).toReal - (Q1616.sub f Q1616.one).toReal * c1.toReal| ≤ (1 : ℝ) / 65536 :=
|
||
mul_error_bound (Q1616.sub f Q1616.one) c1
|
||
have h_sub_real : (Q1616.sub f Q1616.one).toReal = f.toReal - 1 := by
|
||
simp [Q1616.toReal, Q1616.sub, Q1616.one]
|
||
rw [h_sub_real]
|
||
nlinarith [abs_add_le_abs_add_abs, h_err1, h_err2, h_err3, h_err4]
|
||
_ = f.toReal * |h2.toReal - h1.toReal| + (1 - f.toReal) * |c2.toReal - c1.toReal| + 4 / 65536 := by
|
||
ring
|
||
_ ≤ f.toReal * H.aciBound.toReal + (1 - f.toReal) * H.aciBound.toReal + 4 / 65536 := by
|
||
gcongr
|
||
· exact hInit_e
|
||
· exact hcAci_e
|
||
_ = H.aciBound.toReal + 4 / 65536 := by ring
|
||
-- This gives H.aciBound.toReal + 4/65536, which is the ℝ proof with slack.
|
||
-- For exact matching (no slack version), we need a different theorem.
|
||
-- This is the "with slack" version that accounts for 4 ULP truncation.
|
||
-- The caller can add 4 to H.aciBound.raw to compensate.
|
||
|
||
/-- ACI preservation witness.
|
||
If the forget gate f is uniform (fᵢ = fⱼ = f) and the candidate c satisfies ACI,
|
||
then the MLGRU step preserves ACI for the hidden state h. -/
|
||
theorem aciPreservation {N : Nat} (H : BettiSwooshH N)
|
||
(nodes : Fin N → ScalarNode)
|
||
(hInit : aciSatisfied H nodes)
|
||
(f : Q1616) (c : Fin N → Q1616)
|
||
(hf : 0 ≤ f.raw ∧ f.raw ≤ Q1616.one.raw) -- f ∈ [0, 1]
|
||
(hcAci : ∀ e ∈ H.complex.edges,
|
||
(Q1616.abs ((c e.2) - (c e.1))).raw ≤ H.aciBound.raw) :
|
||
aciSatisfied H fun i =>
|
||
{ nodes i with hidden := mlgruStep f (c i) (nodes i).hidden } := by
|
||
intro e he
|
||
let h1 := (nodes e.1).hidden.hT
|
||
let h2 := (nodes e.2).hidden.hT
|
||
let c1 := c e.1
|
||
let c2 := c e.2
|
||
-- The MLGRU update is a convex combination: h' = f*h + (1-f)*c
|
||
-- Distance |h1' - h2'| = |f(h1 - h2) + (1-f)(c1 - c2)|
|
||
-- By triangle inequality: |h1' - h2'| ≤ f|h1 - h2| + (1-f)|c1 - c2|
|
||
-- If both |h1 - h2| and |c1 - c2| are ≤ ε, then the result is ≤ ε.
|
||
-- Triangle inequality proof sketch
|
||
unfold aciSatisfied at hInit
|
||
specialize hInit e he
|
||
dsimp [mlgruStep] at *
|
||
specialize hcAci e he
|
||
-- ANALYTIC_OPEN: Blocked on Q1616 fixed-point arithmetic theory.
|
||
-- The mathematical argument is standard:
|
||
-- |h1' - h2'| = |f*(h1-h2) + (1-f)*(c1-c2)|
|
||
-- ≤ f*|h1-h2| + (1-f)*|c1-c2| (triangle inequality)
|
||
-- ≤ f*ε + (1-f)*ε = ε (by hInit, hcAci)
|
||
-- But Q1616.mul is defined as ⟨(a.raw * b.raw) / 65536⟩ (integer truncation),
|
||
-- which breaks exact distributivity: Q1616.mul f (h1 - h2) ≠ f * h1 - f * h2
|
||
-- in general. Closing this requires three new lemmas in a Q1616/Algebra.lean file:
|
||
-- (1) Q1616.abs_triangle : |a + b| ≤ |a| + |b| (in terms of .raw)
|
||
-- (2) Q1616.mul_add_distrib_bound : |f*(a+b) - f*a - f*b|.raw ≤ 1
|
||
-- (truncation error of at most 1 ULP per multiplication)
|
||
-- (3) Q1616.mul_nonneg_mono : a.raw ≤ b.raw → 0 ≤ f.raw → (f*a).raw ≤ (f*b).raw
|
||
-- Once those lemmas exist, the proof closes by integer arithmetic (omega).
|
||
sorry
|
||
|
||
|
||
|
||
-- ════════════════════════════════════════════════════════════
|
||
-- §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 : Q1616) : Nat :=
|
||
let tclCycles := (t.tcl.raw / 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 vMod := (Q1616.one.raw - (45875 * v.raw / 65536)).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
|
||
|
||
|
||
end Semantics.SSMS
|