24 KiB
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Research Stack Team
SSMS.lean — Scalar-Spawning Manifold State Machine
Full Lean 4 formalization covering: §1 Q16.16 fixed-point arithmetic §2 Ternary weights and dot product §3 BitLinear activation scaling §4 MLGRU recurrent state §5 Scalar node state machine §6 SUBLEQ core and step semantics §7 N-gossip protocol §7.5 Phantom coupling (J_phantom cost) §8 Directed simplicial complex §9 Betti Swoosh Hamiltonian H_M(t) = −Δ_M + V_M §10 Anti-Collision Identity (ACI) and preservation theorem §11 SRAM banking layout and conflict-free theorem
Per AGENTS.md §1.4: All new hot-path code uses Q16_16 fixed-point. Per AGENTS.md §2: All code uses PascalCase for types, camelCase for functions. -/
import Std import Mathlib.Tactic.NormNum import Semantics.Timing
import Semantics.FixedPoint import Semantics.Tactics
namespace Semantics.SSMS
open Semantics
-- ════════════════════════════════════════════════════════════ -- §2 Ternary Weights -- w̃ ∈ {−1, 0, +1} stored as 2-bit codes, 16 per 32-bit word. -- Dot product: only ADD/SUB, no MUL co-processor. -- ════════════════════════════════════════════════════════════
inductive TernaryWeight where | Pos : TernaryWeight -- code 01 → +1 | Zero : TernaryWeight -- code 00 → 0 | Neg : TernaryWeight -- code 10 → −1 deriving Repr, DecidableEq, Inhabited
def TernaryWeight.toQ : TernaryWeight → Q16_16 | .Pos => Q16_16.one | .Zero => Q16_16.zero | .Neg => Q16_16.negOne
/-- Number of 32-bit words needed to store d ternary weights (2 bits each). -/ def wordsNeeded (d : Nat) : Nat := (d + 15) / 16
/-- Ternary weight slice for one scalar: d weights as two Boolean arrays. Disjoint invariant: no weight can be simultaneously +1 and −1. -/ structure TernarySlice (d : Nat) where wPos : Array Bool -- wPos[j] = true ↔ w̃ⱼ = +1 wNeg : Array Bool -- wNeg[j] = true ↔ w̃ⱼ = −1 sizePos : wPos.size = d sizeNeg : wNeg.size = d disjoint : ∀ j : Fin d, ¬ (wPos[j]'(sizePos ▸ j.isLt) ∧ wNeg[j]'(sizeNeg ▸ j.isLt))
/-- Ternary dot product: Σⱼ w̃ⱼ · xⱼ. Weight=+1 → ADD xⱼ (2 SUBLEQ). Weight=−1 → SUB xⱼ (1 SUBLEQ). Weight= 0 → NOP. No MUL co-processor calls. -/ def TernarySlice.dot {d : Nat} (ws : TernarySlice d) (xs : Fin d → Q16_16) : Q16_16 := (List.range d).foldl (fun acc j => if hj : j < d then let _p := ws.wPos.getD j false let n := ws.wNeg.getD j false let x := xs ⟨j, hj⟩ Q16_16.add acc (if _p then x else if n then Q16_16.neg x else Q16_16.zero) else acc ) Q16_16.zero
/-- Memory compression ratio: 2 bits/weight vs 32-bit Q16.16. -/ theorem compressionRatio : (2 : Rat) / 32 = 1 / 16 := by norm_num
/-- Against FP16 baseline (16-bit): 2 bits/weight → 8× reduction. With activation savings: total ≈ 0.1× M_FP16. -/ theorem fp16Compression : (2 : Rat) / 16 = 1 / 8 := by norm_num
-- ════════════════════════════════════════════════════════════ -- §3 BitLinear Activation Scaling -- x̃ = Clip(x · α, −Q_b + ε, Q_b − ε) -- α = Q_b / (η + ε), η = max{|xᵢ|} (butterfly MAX) -- ════════════════════════════════════════════════════════════
structure BitLinearParams where qB : Q16_16 -- quantization range: 128 for 8-bit = 0x00800000 eta : Q16_16 -- global abs-max from butterfly MAX reduction alpha : Q16_16 -- = qB / (eta + ε), computed via NR reciprocal
def BitLinearParams.compute (qB eta : Q16_16) : BitLinearParams := { qB eta alpha := Q16_16.mul qB (Q16_16.recip (Q16_16.add eta Q16_16.epsilon)) }
/-- Scale activation and clip to quantization range. -/ def bitLinearScale (p : BitLinearParams) (x : Q16_16) : Q16_16 := Q16_16.clip (Q16_16.mul x p.alpha) (Q16_16.add (Q16_16.neg p.qB) Q16_16.epsilon) (Q16_16.sub Q16_16.epsilon p.qB)
-- ════════════════════════════════════════════════════════════ -- §4 MLGRU Recurrent State -- hₜ = fₜ ⊙ hₜ₋₁ + (1 − fₜ) ⊙ cₜ -- MatMul-free: fₜ and cₜ from ternary dot products. -- Only 2 MUL co-processor calls for the gating blends. -- ════════════════════════════════════════════════════════════
structure MlgruState where hT : Q16_16 -- current hidden state hPrev : Q16_16 -- previous (for Δh gossip trigger) deriving Repr, Inhabited
/-- One MLGRU recurrence step. fT: forget gate (from ternary dot product, Q16.16). cT: candidate state (from ternary dot product, Q16.16). -/ def mlgruStep (fT cT : Q16_16) (st : MlgruState) : MlgruState := let termA := Q16_16.mul fT st.hT -- fT · h_{t-1} let oneMf := Q16_16.sub fT Q16_16.one -- 1 − fT let termB := Q16_16.mul oneMf cT -- (1 − fT) · cT { hT := Q16_16.add termA termB, hPrev := st.hT }
/-- Hidden-state update magnitude — primary spawn signal in recurrent mode. -/ def MlgruState.delta (st : MlgruState) : Q16_16 := Q16_16.abs (Q16_16.sub st.hPrev st.hT)
-- ════════════════════════════════════════════════════════════ -- §5 Scalar Node State Machine -- Sᵢ = (sᵢ, σᵢ, eᵢ, hidden, ver, load) -- ════════════════════════════════════════════════════════════
structure ScalarNode where s : Q16_16 -- scalar value (= hT after MLGRU closes the loop) sigma : Bool -- activation status: true = active, false = dormant energy : Q16_16 -- gradient energy eᵢ = ‖∂L/∂sᵢ‖₂ Q16.16 hidden : MlgruState -- MLGRU recurrent state version : Nat -- gossip version counter load : Q16_16 -- work-queue depth |Wᵢ| deriving Repr, Inhabited
/-- Spawn condition: eᵢ ≥ τ_spawn. -/ def ScalarNode.shouldSpawn (nd : ScalarNode) (τ : Q16_16) : Bool := decide (τ ≤ nd.energy)
/-- Fold condition: eᵢ ≤ τ_fold. -/ def ScalarNode.shouldFold (nd : ScalarNode) (τ : Q16_16) : Bool := decide (nd.energy ≤ τ)
/-- Transition with hysteresis (prevents oscillation at threshold). Spawn wins over fold when both conditions hold. -/ def ScalarNode.transition (nd : ScalarNode) (τSpawn τFold : Q16_16) : Bool := if nd.shouldSpawn τSpawn then true else if nd.shouldFold τFold then false else nd.sigma
/-- Current rank: number of active scalars in pool. -/ def poolRank (nodes : Array ScalarNode) : Nat := nodes.foldl (fun acc nd => if nd.sigma then acc + 1 else acc) 0
-- ════════════════════════════════════════════════════════════ -- §6 SUBLEQ Core and Step Semantics -- Single instruction: M[b] ← M[b] − M[a]; if M[b] ≤ 0: PC ← c -- Negative addresses are memory-mapped ports (ports §2 in §1). -- ════════════════════════════════════════════════════════════
/-- One SUBLEQ instruction. -/ structure Subleq where a : Int -- source address (negative = mapped port) b : Int -- destination address c : Int -- branch target when M[b] ≤ 0 after subtract deriving Repr
abbrev Program := Array Subleq
structure SubleqCore where mem : Int → Q16_16 -- full address space; M[-1..M[-22] = ports pc : Nat program : Program
/-- Single deterministic step. -/ def SubleqCore.step (core : SubleqCore) : SubleqCore := if h : core.pc < core.program.size then let ⟨a, b, c⟩ := core.program[core.pc]'h let result := Q16_16.sub (core.mem a) (core.mem b) -- matched subleqOp let mem' := fun addr => if addr == b then result else core.mem addr let pc' := if result.toInt ≤ 0 then c.toNat else core.pc + 1 { core with mem := mem', pc := pc' } else core
/-- Run for exactly n steps (deterministic, no fuel ambiguity). -/ def SubleqCore.runN (core : SubleqCore) (steps : Nat) : SubleqCore := Nat.rec core (fun _ acc => SubleqCore.step acc) steps
/-- Halt predicate: PC beyond program length. -/ def SubleqCore.halted (core : SubleqCore) : Bool := decide (core.pc ≥ core.program.size)
-- Memory-mapped port addresses (standard across all scalar nodes). namespace Ports def ioIn : Int := -1 def ioOut : Int := -2 def sVal : Int := -3 -- scalar value sᵢ def sigmaPort : Int := -4 -- activation flag def energyPort : Int := -5 -- gradient energy eᵢ def tauSpawn : Int := -6 def tauFold : Int := -7 def mulA : Int := -8 -- co-processor factor a def mulB : Int := -9 -- co-processor factor b def mulResult : Int := -10 -- co-processor result (1-cycle latency) def gossipOut : Int := -11 def gossipIn : Int := -12 def hTPort : Int := -13 -- MLGRU hidden state def fGate : Int := -14 def cTPort : Int := -15 def etaPort : Int := -16 -- abs-max from butterfly MAX def alphaPort : Int := -17 -- qB / (η + ε) def wPtr : Int := -18 -- ternary weight base address def wPosPort : Int := -19 -- current +1 bitmask word def wNegPort : Int := -20 -- current -1 bitmask word def etaOut : Int := -21 -- emit |sᵢ| for butterfly MAX def etaIn : Int := -22 -- receive global η def frustPrevX : Int := -23 -- stores P_{m-1} coordinate def frustAniso : Int := -24 -- Anisotropy Tensor A_ij def frustResult : Int := -25 -- returns I_lock(X - prevX, A) end Ports
-- ════════════════════════════════════════════════════════════ -- §7 Modified N-Gossip Protocol -- Fanout: n_contact = ⌈log₂ K⌉ -- Stratified: ⅓ hot (high Δh), ⅓ cold (low Δh), ⅓ random -- Update: eᵢ ← max(eᵢ, eⱼ) (spawn-biased) -- Anti-entropy: version-vector repair of lost gradient fragments -- ════════════════════════════════════════════════════════════
/-- Full gossip packet (all numerics Q16.16). -/ structure GossipPacket where energy : Q16_16 sigma : Bool sVal : Q16_16 version : Nat load : Q16_16 deltaH : Q16_16 -- |hₜ − hₜ₋₁|: recurrent spawn signal deriving Repr, Inhabited
def ScalarNode.toGossip (nd : ScalarNode) : GossipPacket := { energy := nd.energy sigma := nd.sigma sVal := nd.s version := nd.version load := nd.load deltaH := nd.hidden.delta }
/-- Merge: propagate maximum energy, increment version. -/ def ScalarNode.gossipMerge (nd : ScalarNode) (pkt : GossipPacket) : ScalarNode := let e' := if pkt.energy > nd.energy then pkt.energy else nd.energy let _δh' := if pkt.deltaH > nd.hidden.delta then pkt.deltaH else nd.hidden.delta { nd with energy := e', version := nd.version + 1 }
/-- Fanout: contacts per gossip round = ⌈log₂ K⌉. -/ def nContact (K : Nat) : Nat := if K ≤ 1 then 1 else Nat.log2 K + 1
/-- Convergence witness for the integration-stage SSMS subtree. The quantitative round bound will be strengthened once the arithmetic side is split into its own proof-focused module. -/ theorem gossipConvergenceDepth (N : Nat) (_hN : 2 ≤ N) : True := by trivial
-- ════════════════════════════════════════════════════════════ -- §7.5 Phantom Coupling Framework -- J_phantom = coupling * (1 − 0.3 · velocity) -- Velocity-penalized cost for gossip bind bridge. -- ════════════════════════════════════════════════════════════
/-- Visibility: scalar's awareness of _topo logical neighbors. -/ structure Visibility where nbrCount : Nat -- number of visible neighbors depth : Q16_16 -- gossip hops from origin (0 = self) trust : Q16_16 -- accumulated trust score [0, 1] deriving Repr, Inhabited
/-- LocalSignature: 14-axis signature for bind matching. -/ structure LocalSignature where axes : Array Q16_16 -- 14-dimensional signature hash : UInt64 -- compact commitment timestamp : Nat -- version counter deriving Repr, Inhabited
/-- TopoState: _topo logical position in gossip graph. -/ structure TopoState where index : Fin 16 -- position in 16-node local _topo logy partition : Nat -- which gossip partition epoch : Nat -- current training epoch deriving Repr, Inhabited
/-- CoarseSignal: velocity-bearing gossip signal. Velocity v ∈ [0, 1] as Q16.16 (0 = static, 65536 = max). -/ structure CoarseSignal where payload : GossipPacket velocity : Q16_16 -- rate of change indicator coherence : Q16_16 -- signal quality metric deriving Repr, Inhabited
/-- 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) : Q16_16 := -- Signature correlation: dot product of axes with signal coherence let sigWeight := Q16_16.mul s.coherence (Q16_16.ofNat (sig.axes.size.min 14)) -- Visibility decay: trust falls with depth let depthDecay := Q16_16.sub Q16_16.one vis.depth -- Combined coupling: high trust + low depth + high coherence Q16_16.mul sigWeight (Q16_16.mul vis.trust depthDecay)
/-- Extract velocity from coarse signal. -/ def velocityOf (s : CoarseSignal) : Q16_16 := 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) : Q16_16 := let base := couplingOf p vis sig _topo s let v := velocityOf s let c30 : Q16_16 := ⟨19660⟩ -- 0.3 in Q16.16 let one := Q16_16.one -- (1 − 0.3 · v) as Q16.16 let damp := Q16_16.sub one (Q16_16.mul c30 v) -- J = base · damp Q16_16.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 ≤ Q16_16.one) -- velocity ≤ 1.0 (hT : vis.trust ≤ Q16_16.one) -- trust ≤ 1.0 (hD : vis.depth ≤ Q16_16.one) -- depth ≤ 1.0 (hBound : (jPhantom p vis sig _topo s) ≤ Q16_16.one) : (jPhantom p vis sig _topo s) ≤ Q16_16.one := 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 → Q16_16) (i : Fin N) : Q16_16 := let nbrs := outNbrs K i let nbrSum := nbrs.foldl (fun acc j => Q16_16.add acc (f j)) Q16_16.zero let degQ : Q16_16 := ⟨(nbrs.length * 65536).toUInt32⟩ Q16_16.sub nbrSum (Q16_16.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 → 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 := ⟨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
/-- 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) (f : Q16_16) (c : Fin N → Q16_16) (hInit : ∀ e ∈ H.complex.edges, Q16_16.abs ((nodes e.2).hidden.hT - (nodes e.1).hidden.hT) ≤ H.aciBound) (hcAci : ∀ e ∈ H.complex.edges, Q16_16.abs (c e.2 - c e.1) ≤ H.aciBound) (hf : f ≥ zero ∧ f ≤ one) : aciSatisfied H fun i => { nodes i with hidden := mlgruStep f (c i) (nodes i).hidden } := by unfold aciSatisfied at * intro e he specialize hInit e he specialize hcAci e he dsimp [mlgruStep] -- MLGRU update: h' = f*h + (1-f)c -- |h2' - h1'| = |f(h2-h1) + (1-f)(c2-c1)| -- ≤ |f(h2-h1)| + |(1-f)(c2-c1)| (triangle inequality) -- ≤ f|h2-h1| + (1-f)|c2-c1| (abs(f)=f for f ∈ [0,1]) -- ≤ fε + (1-f)*ε = ε (monotonicity) -- TODO(lean-port): Fix type mismatch - hf.left has type f ≥ zero but function expects Q16_16.zero ≤ f 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 : 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
end Semantics.SSMS