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/- 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_nD.lean — Variable Dimension Manifold Extension
Extends SSMS with n-dimensional manifold support: §1 VariableDimensionManifold structure with dynamic n §2 LiftingOperator L_{1D→n} for sequential data §3 HolonomicConstraint system with m constraints §4 Dynamic ACI for cross-dimensional collision §5 BettiSwooshND over [1, n_max] §6 SUBLEQ variable-n kernels §7 Dimension selection via potential minimization §8 PhantomTideQ: Adaptive phantom coupling with Q16.16
Per Clean Room Protocol: All math from public sources only. Per AGENTS.md §1.4: All hot-path code uses Q16_16 fixed-point. -/
import Mathlib.Data.Nat.Basic import Mathlib.Data.Int.Basic import Mathlib.Data.Array.Basic import Mathlib.Tactic import Semantics.SSMS import Semantics.FixedPoint
namespace Semantics.SSMS_nD
open Semantics.SSMS open Semantics.Q16_16
-- ════════════════════════════════════════════════════════════ -- §1 Variable Dimension Manifold Structure -- M_i = (n, c ∈ R^n, Σ ∈ R^{n×n}, θ ∈ R^p, σ) -- ════════════════════════════════════════════════════════════
/-- Variable-n manifold: dimensionality determined at spawn time. -/ structure VarDimManifold where n : Nat -- dimensionality (1 to n_max) center : Array Q16_16 -- n coordinates sizeN : center.size = n metric : Array Q16_16 -- upper-triangular Σ: n(n+1)/2 entries sizeMetric : metric.size = n * (n + 1) / 2 orient : Array Q16_16 -- orientation params: n(n-1)/2 for SO(n) sizeOrient : orient.size = n * (n - 1) / 2 energy : Q16_16 -- gradient energy (spawn pressure) sigma : Bool -- activation status deriving Repr
instance : Inhabited VarDimManifold where default := { n := 0, center := #[], sizeN := by simp , metric := #[], sizeMetric := by simp , orient := #[], sizeOrient := by simp , energy := zero, sigma := true }
/-- Calculate total scalar nodes needed for n-dim manifold. -/ def scalarCount (n : Nat) : Nat := n + (n * (n + 1) / 2) + (n * (n - 1) / 2)
/-- Maximum dimension supported (SRAM constraint). -/ def nMax : Nat := 16
/-- Validate manifold dimension within bounds. -/ def validN (n : Nat) : Prop := n ≥ 1 ∧ n ≤ nMax
theorem scalarCountMonotonic (n : Nat) (h : n ≥ 1) : scalarCount n ≥ scalarCount 1 := by simp [scalarCount] have h1 : n * (n + 1) / 2 ≥ 1 := by have h2 : n * (n + 1) ≥ 2 := by have h3 : n ≥ 1 := h have h4 : n + 1 ≥ 2 := by omega have h5 : n * (n + 1) ≥ 1 * 2 := Nat.mul_le_mul h3 h4 simp at h5 exact h5 omega have h2 : n * (n - 1) / 2 ≥ 0 := by omega omega
-- ════════════════════════════════════════════════════════════ -- §2 LiftingOperator L_{1D→n} -- Lifts 1D sequence interval [t0, t1] to R^n -- ════════════════════════════════════════════════════════════
/-- 1D sequence sample at position t. -/ structure SeqSample where position : Nat -- t ∈ [0, L] features : Array Q16_16 -- d-dimensional feature vector deriving Repr, Inhabited
/-- Lifting weights: ternary matrix W_lift ∈ {-1,0,1}^{n×d}. -/ structure LiftingWeights (n d : Nat) where wPos : Array Bool -- n×d positive mask wNeg : Array Bool -- n×d negative mask sizePos : wPos.size = n * d sizeNeg : wNeg.size = n * d disjoint : ∀ i : Fin (n * d), ¬ (wPos[i]'(sizePos.symm ▸ i.isLt) ∧ wNeg[i]'(sizeNeg.symm ▸ i.isLt))
/-- Pool 1D features over interval [t0, t1] via mean pooling. -/ def pool1D (seq : Array SeqSample) (t0 t1 : Nat) : Array Q16_16 := if h : t0 < seq.size then let sample0 := seq[t0]'h let count := (t1 - t0 + 1) -- Mean pooling: sum features / count sample0.features.map (fun f => ⟨f.val / count⟩) else #[]
/-- Lifting operator: 1D pooled features → n-dim center. MatMul-free via ternary weights (ADD/SUB only). -/ def lift1DToN {n d : Nat} (weights : LiftingWeights n d) (pooled : Array Q16_16) (hPooled : pooled.size = d) : Array Q16_16 := (Array.range n).map (fun i => if hi : i < n then let rowOffset := i * d (Array.range d).foldl (fun acc j => if hj : j < d then let idx := rowOffset + j let p := weights.wPos[idx]'(by rw [weights.sizePos] have h1 : i * d + j < i * d + d := Nat.add_lt_add_left hj (i * d) have h2 : i * d + d ≤ n * d := by have h3 : i * d + d = (i + 1) * d := by calc i * d + d = i * d + 1 * d := by rw [Nat.one_mul] _ = (i + 1) * d := by rw [Nat.add_mul] rw [h3] exact Nat.mul_le_mul_right d (Nat.succ_le_of_lt hi) exact Nat.lt_of_lt_of_le h1 h2) let n := weights.wNeg[idx]'(by rw [weights.sizeNeg] have h1 : i * d + j < i * d + d := Nat.add_lt_add_left hj (i * d) have h2 : i * d + d ≤ n * d := by have h3 : i * d + d = (i + 1) * d := by calc i * d + d = i * d + 1 * d := by rw [Nat.one_mul] _ = (i + 1) * d := by rw [Nat.add_mul] rw [h3] exact Nat.mul_le_mul_right d (Nat.succ_le_of_lt hi) exact Nat.lt_of_lt_of_le h1 h2) let x := pooled[j]'(hPooled ▸ hj) if p then Q16_16.add acc x else if n then Q16_16.sub acc x else acc else acc ) Q16_16.zero else Q16_16.zero )
/-- Approximate inverse chart L^{-1}: R^n → [0, L]. -/ def approxInverseChart (c : Array Q16_16) (L : Nat) : Nat := -- Project to first coordinate, clamp to [0, L] if h : 0 < c.size then let c0 := c[0]'h let tRaw := c0.val / 65536 -- Convert Q16.16 to integer let tNat := if tRaw < 0 then 0 else tRaw.toNat min tNat L else 0
-- ════════════════════════════════════════════════════════════ -- §3 HolonomicConstraint System -- m constraints {h_j(x) = 0} for n-dim manifold -- ════════════════════════════════════════════════════════════
/-- Linear constraint: Σ a_j · x_j = b. -/ structure LinearConstraint (n : Nat) where coeffs : Array Q16_16 -- a[0..n-1] sizeCoeffs : coeffs.size = n rhs : Q16_16 -- b deriving Repr
instance {n : Nat} : Inhabited (LinearConstraint n) where default := ⟨Array.mk (List.replicate n Q16_16.zero), by simp, Q16_16.zero⟩
/-- Constraint system for n-dim manifold. -/ structure ConstraintSystem (n m : Nat) where constraints : Array (LinearConstraint n) -- m constraints sizeConstraints : constraints.size = m epsilon : Q16_16 -- ACI tolerance ε deriving Repr
instance {n m : Nat} : Inhabited (ConstraintSystem n m) where default := ⟨Array.mk (List.replicate m default), by simp, Q16_16.zero⟩
/-- Evaluate constraint residual |Σ a_j · x_j - b|. -/ def constraintResidual (c : LinearConstraint n) (x : Array Q16_16) (hX : x.size = n) : Q16_16 := let dot := (Array.range n).foldl (fun acc i => if hi : i < n then let a := c.coeffs[i]'(c.sizeCoeffs.symm ▸ hi) let xi := x[i]'(hX.symm ▸ hi) Q16_16.add acc (Q16_16.mul a xi) else acc ) Q16_16.zero Q16_16.abs (Q16_16.sub dot c.rhs)
/-- Check if manifold satisfies all constraints (ACI predicate). -/ def constraintsSatisfied (sys : ConstraintSystem n m) (M : VarDimManifold) (hN : M.n = n) : Prop := ∀ i : Fin m, let c := sys.constraints[i]'(sys.sizeConstraints.symm ▸ i.isLt) (constraintResidual c M.center (M.sizeN.trans hN)).val ≤ sys.epsilon.val
/-- Constraint potential: Σ λ_j · h_j(x)^2 for MLGRU energy. -/ def constraintPotential (sys : ConstraintSystem n m) (M : VarDimManifold) (hN : M.n = n) (lambdas : Array Q16_16) (hLambdas : lambdas.size = m) : Q16_16 := (Array.range m).foldl (fun acc i => if hi : i < m then let c := sys.constraints[i]'(sys.sizeConstraints.symm ▸ hi) let lam := lambdas[i]'(hLambdas.symm ▸ hi) let r := constraintResidual c M.center (M.sizeN.trans hN) let r2 := Q16_16.mul r r Q16_16.add acc (Q16_16.mul lam r2) else acc ) Q16_16.zero
-- ════════════════════════════════════════════════════════════ -- §4 Dynamic ACI for Cross-Dimensional Collision -- ════════════════════════════════════════════════════════════
/-- Project higher dimension to lower via coordinate truncation. -/ def projectDown (x : Array Q16_16) (nTarget : Nat) : Array Q16_16 := x.take nTarget
/-- Center distance with dimension handling. If n_i ≠ n_j, project to lower dimension first. -/ def dynamicCenterDist (Mi Mj : VarDimManifold) : Q16_16 := let nMin := min Mi.n Mj.n let ci := projectDown Mi.center nMin let cj := projectDown Mj.center nMin let d2 := (ci.zip cj).foldl (fun acc (xi, xj) => let dx := Q16_16.sub xi xj Q16_16.add acc (Q16_16.mul dx dx) ) Q16_16.zero -- sqrt via NR (reusing centerDist pattern) let r0 := ⟨d2.val / 2⟩ let nr := fun r => ⟨(r.val + (d2.val * 65536 / (r.val + 1))) / 2⟩ nr (nr (nr r0))
/-- Dynamic ACI collision predicate. -/ def dynamicACI (Mi Mj : VarDimManifold) (tau : Q16_16) : Bool := decide ((dynamicCenterDist Mi Mj).val ≤ tau.val)
/-- NMS suppression for variable dimensions. Lower energy manifold folded when collision detected. -/ def dynamicSuppresses (Mi Mj : VarDimManifold) (tau : Q16_16) : Bool := dynamicACI Mi Mj tau && decide (Mi.energy.val > Mj.energy.val)
-- ════════════════════════════════════════════════════════════ -- §5 BettiSwooshND: Hamiltonian over [1, n_max] -- ════════════════════════════════════════════════════════════
/-- Betti number counts per dimension. -/ structure BettiVector where beta0 : Nat -- connected components beta1 : Nat -- 1D holes beta2 : Nat -- 2D cavities beta3 : Nat -- 3D voids beta4plus : Nat -- higher dimensions aggregated deriving Repr, Inhabited
/-- Manifold registry: separate lists per dimension. -/ structure ManifoldRegistry (nMax : Nat) where byDim : Array (List VarDimManifold) -- index by dimension sizeByDim : byDim.size = nMax + 1 deriving Repr
instance {nMax : Nat} : Inhabited (ManifoldRegistry nMax) where default := ⟨Array.mk (List.replicate (nMax + 1) []), by simp⟩
/-- Get manifolds of specific dimension. -/ def manifoldsOfDim (reg : ManifoldRegistry nMax) (n : Nat) : List VarDimManifold := if h : n ≤ nMax then reg.byDim[n]'(by rw [reg.sizeByDim]; exact Nat.lt_succ_of_le h) else []
/-- Global Betti swoosh over all dimensions. H_M = Σ_n H_M^{(n)} - cross-dim coupling. -/ def bettiSwooshND (reg : ManifoldRegistry nMax) : Q16_16 := -- Sum potential over all dimensions (Array.range (nMax + 1)).foldl (fun acc n => let mfs := manifoldsOfDim reg n let sumN := mfs.foldl (fun accM Mi => Q16_16.add accM Mi.energy) Q16_16.zero Q16_16.add acc sumN ) Q16_16.zero
/-- Dimension selection potential: penalize deviation from target. -/ def dimensionPotential (n nTarget : Nat) (eta : Q16_16) : Q16_16 := if n = nTarget then Q16_16.zero else ⟨eta.val * (Int.ofNat (if n > nTarget then n - nTarget else nTarget - n))⟩
/-- Total potential with structure constraint. -/ def totalPotentialWithDim (M : VarDimManifold) (nTarget : Nat) (eta : Q16_16) : Q16_16 := Q16_16.add M.energy (dimensionPotential M.n nTarget eta)
-- ════════════════════════════════════════════════════════════ -- §6 SUBLEQ Variable-n Kernels -- ════════════════════════════════════════════════════════════
/-- SUBLEQ program for lifting 1D → n with dynamic loop bounds. -/ def liftKernel (_n : Nat) : Program := -- M[0] = seq_ptr, M[1] = t0, M[2] = dest_base -- M[3] = i (counter), M[4] = n (target dimension) -- M[5] = accum, M[6] = divisor #[ ⟨0, 5, 1⟩ -- accum ← accum - M[seq_ptr] (load) , ⟨6, 5, 2⟩ -- accum ← accum - M[divisor] (normalize) , ⟨5, 2, 3⟩ -- M[dest_base + i] ← accum , ⟨1, 3, 4⟩ -- i ← i - 1 (increment) , ⟨4, 3, 0⟩ -- if i ≤ n: continue else halt ]
/-- SUBLEQ program for constraint checking with m constraints. -/ def constrainKernel (_n _m : Nat) : Program := -- Nested loops: outer over m constraints, inner over n dimensions -- M[0..n-1]: center coordinates x -- M[n..n+m-1]: constraint residuals -- M[n+m]: constraint index j -- M[n+m+1]: dimension index i -- M[n+m+2]: dot accumulator -- M[n+m+3]: epsilon tolerance #[ ⟨0, 0, 1⟩ -- placeholder for constraint loop ]
/-- Memory layout for variable-n manifold in SRAM. -/ def varDimMemoryLayout (base n : Nat) : Array Int := #[ base -- center[0] , base + n -- center[n-1] end , base + n + n*(n+1)/2 -- metric end , base + n + n*(n+1)/2 + n*(n-1)/2 -- orient end , base + n*(n+3)/2 -- header start , base + n*(n+3)/2 - 4 -- dimension n , base + n*(n+3)/2 - 3 -- constraint count m , base + n*(n+3)/2 - 2 -- energy , base + n*(n+3)/2 - 1 -- activation σ ]
-- ════════════════════════════════════════════════════════════ -- §8 PhantomTideQ: Adaptive Phantom Coupling -- Converts PhantomTide Float functions to Q16.16 formalization. -- Integrates with VarDimManifold gossip routing. -- ════════════════════════════════════════════════════════════
/-- Signal energy-coherence delta as velocity proxy. v = |e - κ| in Q16.16 (difference of two Q16.16 values). -/ def signalVelocity (energy coherence : Q16_16) : Q16_16 := Q16_16.abs (Q16_16.sub energy coherence)
/-- Phantom modifier with adaptive λ parameter. φ(λ, v) = max(0, 1 - λ·v) — non-negative clamping. λ ∈ [0, 1] as Q16.16 (0 = no damping, 65536 = full damping). -/ def phantomModifier (lambda v : Q16_16) : Q16_16 := let damp := Q16_16.sub Q16_16.one (Q16_16.mul lambda v) -- max(0, damp) via Q16_16 comparison if damp.val < 0 then Q16_16.zero else damp
/-- Full phantom coupling: j = base · φ(λ, v). Lambda-adaptive version of SSMS.jPhantom. -/ def couplingPhantom (lambda : Q16_16) (base : Q16_16) (energy coherence : Q16_16) : Q16_16 := let v := signalVelocity energy coherence let modifier := phantomModifier lambda v Q16_16.mul base modifier
/-- Final score with phantom boost: s' = s · (1 + max(0, j)). Boosts stable signals (j > 0), dampens unstable (j < 0). -/ def finalScorePhantom (baseScore j : Q16_16) : Q16_16 := let boost := Q16_16.add Q16_16.one (Q16_16.max Q16_16.zero j) Q16_16.mul baseScore boost
/-- Dynamic gossip budget: increase slots when coupling > 1.0. j > 1.0 means strong signal → allow more gossip contacts. Integrates with nContact tier system. -/ def dynamicGossipBudget (j : Q16_16) (baseSlots : Nat) : Nat := if j.val > 65536 then baseSlots + 1 else baseSlots -- j > 1.0 in Q16.16
/-- Betti-soliton driven score with phantom coupling. H_M = -Δ_M + V_M + V_phantom(λ). Drive term from phase control (Warden pressure). -/ def stableDrivenScorePhantom (lambda : Q16_16) (baseScore : Q16_16) (bettiEnergy : Q16_16) -- from BettiSwooshND (drive : Q16_16) -- phase control term (prev : Q16_16) : Q16_16 := let j := couplingPhantom lambda baseScore bettiEnergy Q16_16.one let boosted := finalScorePhantom baseScore j -- Soliton step: combine with drive and previous state let step := Q16_16.add (Q16_16.mul drive boosted) (Q16_16.mul (Q16_16.ofNat 3) prev) -- Normalize by 4 (bit shift approximation) ⟨step.val / 4⟩
/-- Stable band routing: predicate for gossip inclusion. Signal routed if driven score exceeds threshold τ_stable. -/ def routeStablePhantom (lambda : Q16_16) (baseScore bettiEnergy drive prev tauStable : Q16_16) : Bool := let score := stableDrivenScorePhantom lambda baseScore bettiEnergy drive prev decide (score.val ≥ tauStable.val)
/-- Tunneling allowance: high-coupling, high-coherence signals. j > 0.8 && visibility > 0.5 && coherence > 0.35 -/ def allowTunnelPhantom (lambda : Q16_16) (baseScore bettiEnergy visibility coherence : Q16_16) : Bool := let j := couplingPhantom lambda baseScore bettiEnergy coherence let jThresh : Q16_16 := ⟨52428⟩ -- 0.8 in Q16.16 (0.8 * 65536 = 52428.8) let visThresh : Q16_16 := ⟨32768⟩ -- 0.5 in Q16.16 let cohThresh : Q16_16 := ⟨22937⟩ -- 0.35 in Q16.16 (0.35 * 65536 = 22937.6) decide (j.val > jThresh.val) && decide (visibility.val > visThresh.val) && decide (coherence.val > cohThresh.val)
/-- Promotion threshold: tiered reduction based on coupling strength. j > 1.0: reduce by 0.75, j > 0.5: reduce by 0.25, else add 0.5. -/ def promoteThresholdPhantom (lambda thresholdBase : Q16_16) (baseScore bettiEnergy : Q16_16) : Q16_16 := let j := couplingPhantom lambda baseScore bettiEnergy Q16_16.one let quarter : Q16_16 := ⟨16384⟩ -- 0.25 in Q16.16 let half : Q16_16 := ⟨32768⟩ -- 0.5 in Q16.16 let threeQ : Q16_16 := ⟨49152⟩ -- 0.75 in Q16.16 let jHalf : Q16_16 := ⟨32768⟩ -- 0.5 threshold let jOne := Q16_16.one if j.val > jOne.val then Q16_16.max Q16_16.zero (Q16_16.sub thresholdBase threeQ) else if j.val > jHalf.val then Q16_16.max Q16_16.zero (Q16_16.sub thresholdBase quarter) else Q16_16.add thresholdBase half
/-- Promotion predicate: score meets adaptive threshold. -/ def shouldPromotePhantom (lambda thresholdBase : Q16_16) (baseScore bettiEnergy drive prev : Q16_16) : Bool := let finalScore := stableDrivenScorePhantom lambda baseScore bettiEnergy drive prev let thresh := promoteThresholdPhantom lambda thresholdBase baseScore bettiEnergy decide (finalScore.val ≥ thresh.val)
/-- Phantom-scored payload structure for gossip selection. -/ structure PhantomScoredPayload where payload : GossipPacket score : Q16_16 coupling : Q16_16 stable : Bool -- passed routeStablePhantom deriving Repr, Inhabited
/-- Stabilize payloads via phantom scoring and filter. Returns sorted (by score) array of stable payloads. Integrates with variable-n gossip: budget scales with n. -/ def stabilizePayloadsPhantom (lambda tauStable : Q16_16) (_budgetSlots : Nat) (packets : Array (GossipPacket × Q16_16 × Q16_16 × Q16_16)) -- (pkt, energy, drive, prev) : Array PhantomScoredPayload := let scored := packets.filterMap (fun (pkt, betti, drive, prev) => let baseScore := pkt.energy let coherence := pkt.deltaH -- reuse deltaH as coherence proxy let j := couplingPhantom lambda baseScore betti coherence let stable := routeStablePhantom lambda baseScore betti drive prev tauStable if stable then some { payload := pkt, score := finalScorePhantom baseScore j , coupling := j, stable := true } else none) -- Sort by score descending (selection sort via foldl) scored -- qsort requires Ord instance; return unsorted for now
/-- Phantom kernel output structure for gossip routing decision. -/ structure PhantomKernelOutput where chosen : Option GossipPacket score : Q16_16 coupling : Q16_16 promoted : Bool tunneled : Bool budgetNext : Nat deriving Repr, Inhabited
/-- Phantom kernel step: full gossip packet selection pipeline. Integrates with VarDimManifold.n for dimension-aware budget. -/ def stepKernelPhantom (lambda tauStable : Q16_16) (budgetSlots : Nat) (n : Nat) -- manifold dimension for scaling (packets : Array (GossipPacket × Q16_16 × Q16_16 × Q16_16)) (visibility coherence : Q16_16) : PhantomKernelOutput := let scored := stabilizePayloadsPhantom lambda tauStable budgetSlots packets -- Scale budget by dimension: more dimensions → more gossip slots let dimBudget := budgetSlots + n / 2 match scored[0]? with | none => { chosen := none, score := Q16_16.zero, coupling := Q16_16.zero , promoted := false, tunneled := false, budgetNext := dimBudget } | some best => let j := best.coupling let tunneled := allowTunnelPhantom lambda best.score best.score visibility coherence let promoted := shouldPromotePhantom lambda Q16_16.one best.score best.score Q16_16.zero Q16_16.zero let budgetNext := dynamicGossipBudget j dimBudget { chosen := some best.payload, score := best.score, coupling := j , promoted := promoted, tunneled := tunneled, budgetNext := budgetNext }
-- ════════════════════════════════════════════════════════════ -- §9 Cycle Counts for Variable-n Pipeline with Phantom -- ════════════════════════════════════════════════════════════
/-- Lifting cycles: O(n · d) ternary ops. -/ def liftCycles (n d : Nat) : Nat := n * d -- one ADD/SUB per nonzero weight
/-- Constraint check cycles: O(m · n). -/ def constraintCycles (n m : Nat) : Nat := m * n * 2 -- dot product + comparison per constraint
/-- Dynamic NMS cycles: O(k^2 · n_min) for k manifolds. -/ def dynamicNmsCycles (k nAvg : Nat) : Nat := k * k * nAvg -- pairwise center distances
/-- Total pipeline with dimension selection. -/ def varDimTotalCycles (n d m k : Nat) : Nat := liftCycles n d
- constraintCycles n m
- dynamicNmsCycles k n
- n * 18 -- MLGRU per dimension
- 2 * (Nat.log2 k + 1) -- gossip
/-- Throughput: manifolds processed per 1000 cycles. -/ def varDimThroughput (n d m : Nat) : Nat := 1000 / varDimTotalCycles n d m 1
end Semantics.SSMS_nD