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Zero-Storage Procedural Neural Synthesis via Boundary Dynamics: Formal Verification in Lean 4 and Bare-Metal Gauntlet Validation

Volkan Dağlı, Zerrin Dağlı, Dağhan Dağlı

arXiv:2609.33066v1cs.LGcs.AIcs.CRcs.LO

TL;DR

Dense-weight inference faces memory and deployment constraints, while recursive dynamical kernels require termination and arithmetic guarantees. The paper presents formal proofs for a bounded procedural kernel and reports bare-metal throughput and outlier-gate results.

  • Problem

    Dense neural weights impose memory-wall costs, and recursive-kernel verification must address fuel-bounded termination for deterministic deployment.

    Source: §1 · p. 1, §1.1 · p. 2

  • Method

    The paper uses a shallow-fuel Q16.16 EVM kernel and Lean 4 proofs for its termination, arithmetic, and boundary properties.

    Source: §2.1 · p. 2, §1.1 · p. 2

  • Results

    The proofs establish 9- and 12-step bounds, Q16.16 square safety below 2^63−1 for |z| ≤ 2.0, and non-constant quadrant sensitivity.

    Source: §3.1 · p. 5

  • Takeaways & Limitations

    The conclusion reports procedural boundary synthesis from a 24-byte seed with no persistent tensor memory and EVM execution below 22,557 gas.

    Source: §7 · p. 6

    Stochastic Lévy-flight perturbations apply strictly to continuous non-convex manifolds, while deterministic recurrences govern discrete rings.

    Source: §5 · p. 6

Abstract

from arXiv · show

Contemporary neural inference architectures rely on dense floating-point weight matrices stored in high-bandwidth memory (VRAM), incurring severe memory-wall bottlenecks and preventing native execution inside deterministic virtual machines like the Ethereum Virtual Machine (EVM). Verifying termination and arithmetic invariants for recursive dynamical systems over continuous domains is generally undecidable in the Blum-Shub-Smale model. Here, we present the formal verification and bare-metal empirical validation of WERR (Waves & Errors) and Phase III Orbital Error Dynamics (OED), a non-tensor decision paradigm that procedurally synthesizes non-linear decision boundaries on demand from a 24-byte coordinate seed $\Theta = (c_x, c_y, \text{zoom})$ along the boundary of the Mandelbrot set ($\partial\mathcal{M}$). By projecting the recurrence $z_{n+1} = z_n^2 + c$ onto the modular residue ring $\mathbb{Z}/9\mathbb{Z}$ and the fixed-point domain $\mathbb{Q}_{16.16}$, we establish ten machine-verified theorems in Lean 4 (v4.34.1) with Mathlib4 and zero unproven conjectures (sorry): proving $\mathcal{I}_3 = \{0,3,6\} \subset \mathbb{Z}/9\mathbb{Z}$ ideal closure, universal fuel-bounded halting ($\le 9$ and $\le 12$ steps), absence of $\mathbb{Q}_{16.16}$ square overflow below $2^{63}-1$, non-constant boundary escape sensitivity, and a parametric EVM gas bound ($\le 22,557 \le 24,000$ gas). Evaluated on a 40-core Dual Intel Xeon server, the vectorized 36-iteration CPU kernel processes 100,000 decisions in 6.49 s (15,397 decisions/s, 0 Bytes VRAM, 15.15x speedup), while a sigmoidal outlier gate suppresses 100.00% of adversarial spikes while preserving 89.60% of clean baseline signals.

1. Introduction and Prior Art

The section contrasts memory-heavy neural inference and recursive dynamical kernels that need formal guarantees for deterministic deployment. It presents Lean 4 verification of bounded termination, fixed-point safety, boundary sensitivity, and gas bounds.

1. Introduction and Prior Art

  • Dense floating-point weights create memory-wall latency and energy costs, while EVM deployment faces storage, arithmetic, and proof-latency barriers.The cited ZK-ML frameworks require 10–300 s of proof generation and 250,000–500,000 verification gas.Source: §1 · pp. 1–2

1.1 Formal Verification of Neural Systems vs. Recursive Dynamical Kernels

  • Recursive Mandelbrot-boundary kernels require machine-checked guarantees because escape or halting over the continuous complex plane is formally undecidable in the Blum–Shub–Smale model.The section contrasts these recurrences with finite feedforward networks, whose termination is trivial by construction.Source: §1.1 · p. 2
  • The formalization targets fuel-bounded termination, fixed-point overflow safety, non-constant boundary sensitivity, and EVM gas below the 24,000-gas ceiling.The paper reports a Lean 4 and Mathlib4 formalization alongside bare-metal telemetry and a three-arm number-theoretic ablation.Source: §1.1 · p. 2

2. Two-Tier Procedural Architecture and Mathematical Formulation

The architecture separates high-resolution CPU evaluation from a shallow-fuel EVM kernel, with OED mechanisms for continuous-boundary optimization and discrete modular aggregation. The two tiers use different iteration budgets and operating domains.

2.1 Tier 1 (CPU Edge Runtime) vs. Tier 2 (On-Chain EVM Reflex Kernel)

  • Tier 1 evaluates 32 × 32 or 36 × 36 grids at Nmax = 36 and reports 99.30% ± 0.18% Two-Moons accuracy, 77.67% ± 5.35% strict zero-leakage accuracy, and 93.1% JevBench macro accuracy.The CPU engine combines boundary-corrected kernel density estimation with a three-scale Harmonic Tripod.Source: §2.1 · p. 2
  • Tier 2 packs model state into one 32-byte EVM storage slot and evaluates a sparse grid in Q16.16 with Nmax ∈ {9, 12}.The 9-step kernel differentiates along the outer escape shell, not the deep Seahorse Valley center, where local escape times exceed 30 iterations.Source: §2.1 · p. 2

2.2 Phase III Orbital Error Dynamics (OED) Operators

  • OED applies bounded restorative drift near the parabolic cusp or samples the Observer Horizon shoulders rather than globally overwriting the cusp.The drift is ∇Edrag(z) = −k(z−1/4), with k ∈ (0, 1).Source: §2.2 · p. 2
  • When optimization gradients fall below εstag = 0.05, OED injects a Cauchy perturbation with scale γ = 0.12 into the coordinate state.The perturbation is intended for stagnating gradients in non-convex saddle regions.Source: §2.2 · p. 3
  • A two-dimensional Cauchy perturbation with γ = 0.12 exits a saddle basin of radius 0.08 in one step with an analytical probability of approximately 86.9%.The passage attributes this escape probability to the distribution’s power-law tails.Source: §2.2 · p. 3
  • The MCD4 gate applies smooth sigmoidal attenuation to streaming sensory inputs to suppress high-amplitude spikes without dropping normal traffic.At γsteep = 7.0 and τtol = 0.45, it suppresses 100.00% of adversarial bursts while preserving 89.60% of clean Gaussian signal amplitude.Source: §2.2 · p. 3

2.3 Modular Residue Ring Z/9Z and Discrete Phase Aggregation

  • Discrete phase aggregation projects step counts into Z/9Z and sums 12 root-of-unity-weighted samples using a character that vanishes on I3.I3 = {0, 3, 6}; the complementary error set is {1, 2, 4, 5, 7, 8}.Source: §2.3 · p. 3

3. Interactive Formal Verification in Lean 4

The Lean 4 formalization verifies algebraic invariants over Z/9Z and bounded Q16.16 recurrence behavior, including overflow safety, non-constant boundary sensitivity, and a parametric EVM gas bound. The proof module is reported to compile without sorry statements or custom axioms.

3. Interactive Formal Verification in Lean 4

  • The proof module formalizes ring structure, fixed-point recurrence, overflow bounds, boundary sensitivity, and gas cost in Lean 4 with Mathlib4.It compiles in 2.9 s across 3,093 build targets with zero sorry statements and no custom axioms.Source: §3 · p. 3, §3.1 · p. 5

3.1 Summary of Machine-Verified Theorems and Axiom Audit

  • The Z/9Z proofs establish that I3 = {0, 3, 6} is an additive subgroup and multiplicative ideal, with 3 resonant states and 6 error states.The formalization also proves additive closure, ideal absorption, and that 3k belongs to I3 for every k.Source: §3 · pp. 3–4, §3.1 · p. 5
  • Every escape_zmod9 evaluation is bounded by 9 steps and every escape_werracle evaluation by 12 steps for all integer coordinate pairs.Structural induction proves the bounds; the revised base case distinguishes bounded interior orbits from escaping orbits.Source: §3 · p. 4, §3.1 · p. 5
  • For |z| ≤ 2.0 in Q16.16, the unshifted square is at most 17,179,869,184, below 2^63−1.The stated bound guarantees the int128 square does not overflow before the |z|^2 > 4.0 escape guard terminates the loop.Source: §3 · p. 4, §3.1 · p. 5
  • The 9-step Q16.16 kernel yields escape counts 1, 2, 4, 5, 7, 8, and 9 across boundary points, with quadrant counts (5, 7, 9, 8) at the specified input and step.The cited theorem reports non-zero fractal bias and distinct boolean outputs under input variation.Source: §3.1 · p. 5
  • For grid size 16 and iteration depth k ≤ 12, the gas model proves a cost at most 22,557, below the 24,000 ceiling.The modeled cost is 21,000 + 100 + (n_grid × max_iter × 7) + 113.Source: §3.1 · p. 5, §3 · p. 5

4. Bare-Metal 40-Core Gauntlet Evaluation

The bare-metal evaluation reports that the vectorized kernel increases throughput over an unvectorized 40-core baseline. The MCD4 gate suppresses adversarial spikes while retaining most clean baseline signals.

4. Bare-Metal 40-Core Gauntlet Evaluation

  • The evaluation ran on a bare-metal Dual Intel Xeon E5-2630 v4 server with 20 physical cores, 40 hardware threads, and 256 GB RAM.The system used Ubuntu 24.04.5 LTS and a performance-locked CPU governor.Source: §4 · p. 5
  • 15.15× acceleration: the vectorized OED kernel completed 100,000 decisions in 6.4946 s at 15,397.4 decisions/s, versus 98.40 s and 1,016.3 decisions/s for the unvectorized 40-core baseline.Both wall-clock time and throughput reflect the same acceleration on a 32 × 32 grid with Nmax = 36.Source: §4 · p. 5
  • 100.00% of high-amplitude adversarial spikes were suppressed while 89.60% of clean baseline Gaussian signals were preserved.The reported gate operated at 8,596,523 packets/s.Source: §4 · p. 5

5. Domain Boundary Ablation: Discrete Rings (Z/NZ) vs. Continuous Manifolds

The ablation distinguishes deterministic recurrences over discrete rings from Cauchy-perturbed dynamics intended for continuous non-convex optimization. In the tested semi-primes, the unperturbed recurrence matched classical Pollard–Brent, while Cauchy jumps disrupted the modular collision trajectory.

5. Domain Boundary Ablation: Discrete Rings (Z/NZ) vs. Continuous Manifolds

  • The recurrence z_(n+1) = z_n^2 + c modulo N is identified as the foundational recurrence of Pollard’s ρ algorithm and Brent’s cycle-finding improvement [10, 11].Source: §5 · p. 5
  • 100% of semi-primes were factored by both classical Pollard–Brent and the unperturbed recurrence, with a 12,970.3-step grand mean across 180 trials.The ablation covered 180 semi-primes, with 60 trials each at 40-, 48-, and 56-bit scales.Source: §5 · p. 6, Table 2 · p. 6
  • Cauchy jumps injected every 1,000 steps reset modular congruence trajectories before birthday-paradox collisions could accumulate.The authors use this negative result to distinguish discrete algebraic rings from continuous non-convex manifolds.Source: §5 · p. 6

6. On-Chain EVM Execution and Verification Landscape Comparison

The paper reports native EVM execution of its Q16.16 kernel in an oracle and Uniswap v4 fee hook. The hook stays within the stated gas ceiling and dynamically modulates fees to mitigate, but not eliminate, LVR and toxic order-flow exposure.

6. On-Chain EVM Execution and Verification Landscape Comparison

  • 22,557 gas is the worst-case ceiling for the Uniswap v4 beforeSwap() hook, below the proven 24,000-gas bound; standalone decideNoul averages 21,438 gas.The figures come from a 1,000-vector deterministic Foundry audit battery.Source: §6 · p. 6, Table 3 · p. 6
  • The hook modulates pool swap fees between 5 bps and 50 bps intra-block to mitigate LVR and toxic order-flow exposure, not eliminate LVR.The text attributes LVR fundamentally to arbitrage against stale block prices.Source: §6 · p. 6

7. Conclusion

The conclusion presents procedural boundary synthesis as a zero-persistent-tensor-memory approach that is formally verified for fixed-point overflow safety, boundary sensitivity, and bounded EVM execution.

7. Conclusion

  • A 24-byte coordinate seed synthesizes non-linear decision kernels with zero persistent tensor memory, verified for Q16.16 overflow safety and non-constant boundary sensitivity, and executed below 22,557 gas.Source: §7 · p. 6

Statements and Declarations

The declarations provide source and replication materials, disclose a patent application and competing interest, and describe the authors’ use of generative AI tools.

Statements and Declarations

  • Source files, Lean and Solidity projects, benchmark scripts, and cryptographic manifests are archived on CERN Zenodo and GitHub.The authors also disclose a patent application covering procedural weight derivation and on-chain decision synthesis, and state that generative AI assisted with typesetting, proof-script structuring, and English editing.Source: Statements and Declarations · p. 7

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