In this paper we show that the generalization error of AdaBoost is $Θ\big(\tfrac{d\ln(nγ^{2}/d)}{nγ^2}+\tfrac{\ln(1/δ)}{n}\big)$, where $γ$ is the advantage guaranteed by the weak learner, $d$ is the VC-dimension of the class containing the weak hypotheses, $n$ is the sample size, and $δ$ is the confidence parameter. The contribution of this paper is the upper bound; the matching lower bound follows from prior work. The upper bound proof follows by combining the known fact that AdaBoost outputs a voting classifier whose voting function has zero empirical $γ/2$-margin loss with what is, to the best of our knowledge, a new margin-based generalization bound for voting classifiers.
[Depreciated and replaced by V3] This pre-V3 paper is replaced by the corresponding V3 clean-room reconstruction: There Is No Nothing: A Premise-Free Operational Foundation and an Open Verification Platform for Smithian Fold Theory. The V3 source platform is https://github.com/MettaMazza/ernos-labs-sft-platform. The original DOI, concept DOI, version number and files are preserved for transparent historical provenance; this record must not be presented or cited as current V3 work.A comprehensive, highly rigorous consolidated manuscript dismantling black-box AI through the deterministic Smithian Fold Theory. We present exact zero-parameter derivations of the fine-structure constant (137.03599917718), Levinthal's paradox, structural genetics, and SOTA empirical competitive parity in Chess, Symmetric Go, and Natural Language Processing. Unison AI operates at 57 million times the computational efficiency of modern Transformers, tracing physical geometry without gradient descent.
TOPO-2026 - A Prime-Based Topological Framework for Ultra-Efficient Continual Learning Frank Morales Aguilera, BEng, MEng, SMIEEE Sovereign Machine Laboratory (SOMALA), Montreal, Canada frank.morales@sovereign-machine-lab.ai ORCID: 0009-0003-9528-0745 1. Overview TOPO-2026 is a novel continual learning framework that leverages the mathematical properties of prime numbers to prevent catastrophic forgetting in neural networks. The key innovation is anchoring a sparse set of parameters at prime-numbered indices across tasks, maintaining task-specific knowledge while allowing non-anchored parameters to adapt. 2. Core Contributions # Contribution Description 1 Mathematical Foundation Primes provide optimal spectral coverage (97.85%) with only 6 anchors per layer 2 O(1) Memory Complexity < 5 KB overhead for 100M+ parameter models 3 Universal Applicability Works across NLP, Vision, and 3D architectures without modification 4 Perfect Integrity Zero anchor drift across tasks, eliminating catastrophic forgetting 5 Theoretical Guarantees Mathematical proof of spectral coverage, invariance, and O(1) complexity 6 Edge Deployment Sub-kilobyte memory footprint suitable for resource-constrained devices 3. Theoretical Foundation 3.1 Why Primes Specifically Prime numbers are uniquely suited as anchors because they provide: Property Description Mathematical Guarantee Optimal Density $\pi(n) \sim n/\ln(n)$ Sufficiently dense for coverage of arbitrarily large tensors Coprimality $\gcd(p_i, p_j) = 1$ for $i \neq j$ Orthogonal subspaces, no interference between anchors Deterministic Distribution Well-distributed throughout natural numbers No clustering, comprehensive coverage Universal Guarantee Coverage independent of tensor dimensions Framework works for any architecture 3.2 Spectral Coverage Formula For a set of primes $P = \{p_1, p_2, \ldots, p_k\}$: $$C(P) = 1 - \prod_{p \in P} (1 - p^{-1/2})$$ For $P = \{2, 3, 5, 7, 11, 13\}$: $$\begin{align} C(P) &= 1 - \prod_{p \in P} (1 - p^{-1/2}) \\ &= 1 - (1-2^{-1/2})(1-3^{-1/2})(1-5^{-1/2}) \\ &\qquad \times (1-7^{-1/2})(1-11^{-1/2})(1-13^{-1/2}) \\ &= 1 - (0.2929)(0.4226)(0.5528)(0.6220)(0.6985)(0.7227) \\ &= 1 - 0.021486 \\ &= 0.978514 \approx 97.85\% \end{align}$$ Key Insight: The independence of non-coverage events follows directly from the coprimality of primes. For distinct primes $p_i$ and $p_j$, the conditions $x \not\equiv 0 \pmod{p_i}$ and $x \not\equiv 0 \pmod{p_j}$ are independent because $\gcd(p_i, p_j) = 1$. The Chinese Remainder Theorem guarantees these conditions can be satisfied or violated independently. 4. The Topological Governor The core innovation: three operations that work together to prevent forgetting. 4.1 Snapshot Operation Before training on a new task, save anchor values: $S_t = \{(\text{idx}, \theta_{\text{idx}}) \mid \text{idx} \in P, \theta_{\text{idx}} \in \Theta\}$. 4.2 Gradient Zeroing During backpropagation, zero gradients at anchor positions: $\nabla L(\theta_{\text{idx}}) = 0, \forall \text{idx} \in P$. 4.3 Anchor Enforcement After each optimization step, restore anchor values: $\theta_{\text{idx}} \leftarrow S_t(\text{idx}), \forall \text{idx} \in P$. 5. Memory Complexity Analysis For a model with $n$ parameters and $L$ layers: $$M_{TOPO} = |P| \times L \times \text{bytes per parameter}$$ Model Parameters Layers Anchors Memory EWC Memory Reduction BERT 109M 201 1,206 4.71 KB 437.9 MB 93,000× GPT-2 124M 148 888 3.47 KB 497.8 MB 143,000× GAN 2.95M 22 132 0.52 KB 11.8 MB 22,700× NeRF 246K 14 84 0.33 KB 1.0 MB 3,100× 6. Experimental Validation BERT (Text Classification): 100% retention on movie and product review tasks. GPT-2 (Text Generation): High-quality generation across creative and technical writing tasks with 1.25 perplexity. GAN (Image Generation): Stable training across Gaussian, Uniform, and Mixed datasets; no mode collapse. NeRF (3D Scene Learning): Consistent loss across sphere, cube, and torus scenes. 7. Conclusion TOPO-2026 represents a breakthrough in continual learning, demonstrating that mathematical structure can enable practical, scalable, and ultra-efficient parameter protection. With O(1) memory complexity and universal applicability, it provides a robust foundation for building models that adapt without forgetting, learn without rehearsal, and evolve without memory explosion.
The Connes–van Suijlekom truncated Weil quadratic form, indexed by a cutoff parameter c that controls the primes p ≤ c entering the operator, produces a ground state whose Fourier–Mellin zeros provably lie on the critical line; whether they converge to the Riemann zeros as c → ∞ is open (Connes 2026; Connes–Consani–Moscovici 2025). We present, to our knowledge, the first independent public implementation of the Connes–van Suijlekom Galerkin matrix at sixteen cutoffs (c = 13 through 67, plus c = 100). Across the in-sample window c = 13 through c = 67 at N = 100, the first-zero absolute error |γ1 − γ1Riemann| shrinks monotonically from ∼2×10−55 to ∼1.5×10−168, a 113-OOM convergence across fifteen cutoffs. The smallest-positive even-sector eigenvalue λmineven separately reaches ∼10−334 at c = 100, N = 250 (275-OOM span from c = 13). Out-of-sample test at c = 100. On the four-point N-sweep N ∈ {100, 150, 200, 250} at dps = 500, consecutive first-difference ratios 0.837 and 0.836 match to two decimal places. Aitken-Δ2 on the two overlapping triples yields log10|λ∞even| ≈ −536.8 and ≈ −533.7, approaching the Connes 2026 §6.4 heuristic prediction (≈ −530.4) monotonically with N (6.4 and 3.3 OOM gaps out of |x∞| ∼ 530). The same eigenvector recovers γ1, …, γ10 to 307–329 matching digits at N = 250, dps = 500. Under the unitary equivalence with Connes–Consani–Moscovici Lemma 5.1, this is the deepest such Galerkin-truncation recovery in the public Connes–van Suijlekom / Connes–Consani–Moscovici literature, subject to a hypothesis-status caveat. The raw finite-N matrix carries a small block of negative-sign eigenvalues at the finite archimedean cutoff T = 800; these are an artifact of that cutoff and are absent once T is increased, so the smallest-positive even-sector eigenvalue is the genuine smallest one (continuum positivity of QWλ is RH-equivalent and is not assumed at λ = √100). The fit |log10 λmin| ≈ 13.24 c0.634 on c ≤ 67 at N = 100 is shown to be a finite-N rate, falsified at c = 100, N = 200 by 49 OOM in the direction of faster decay. Structural observations include approximate eigenvector c-invariance (overlap ≥ 0.9498 on all 105 cutoff pairs despite eigenvalues differing by 113 OOM), multi-zero convergence universality (all ten detectable zeros within 3.8% of each other), an empirical Galerkin-convergence exponent s(c) ≈ 55 log c − 128, un-rescaled Galerkin bulk-spectrum Poisson statistics (β < 0.05; this is a structural diagnostic of the truncated operator, not a test of Montgomery's conjecture, which applies to locally-rescaled zero spacings), and tight bulk invariants log|det Qc| ≈ −65.6 c + 542 (R2 = 0.997). We make no claim of proof; the contribution is reproducible numerical data and its careful interpretation under the existing CvS / CCM framework. All code, data, and ancillary files are publicly available. Version 3.3 (2026-06-26) correction. The negative-sign eigenvalue blocks reported at c = 100 and for L(s, χ3) at c = 23, 29 are a finite archimedean-cutoff (T) artifact, not a feature of the operator: they are stable in working precision but vanish once T is increased, so cutoff-free the relevant even sectors are non-negative and the smallest-positive branch is the genuine smallest eigenvalue. No quantitative result changes. See ERRATA.md and the paper's note added in revision. The cutoff sensitivity was independently identified by B. W. A. Silva, consistent with the naturally even, positive ground state reported by R. Andrews; the investigation was prompted by A. Connes.
Structural identity — the geometric fingerprint that makes a neural network this specific model rather than any other — can be measured, survives routine deformation, resists adversarial erasure, and composes with standard verification infrastructure. It cannot, in the tested regime, be recovered from endpoint weight statistics or architecture descriptors alone. These two facts together force a question the measurement program has not yet answered: if identity is real but not readable from the final artifact, then where in the training process did it form, and what determined which identity formed rather than another? This paper presents the first empirical study of structural identity formation during neural network pretraining. Using dense checkpoint trajectories and seed-controlled training runs in the Pythia observatory suite, we show three results. First, the structural observable follows a characteristic three-phase identity emergence profile — an early rise in geometric spread, a long compression, and a late plateau where identity stabilizes while functional training continues. Second, models trained with the same architecture, the same data, and the same hyperparameters but different random seeds produce structurally distinguishable fingerprints far beyond measurement noise — a property we call path sensitivity — with the divergence traceable to differential structural response during the learning-rate warmup regime. Third, a panel of endpoint weight statistics varies across seeds but does not predict which structural identity formed — a condition we call endpoint underdetermination. Together, these results recast structural identity as a developmental property of training history rather than a static property legible from final artifacts alone. Supplementary Material This paper is accompanied by HistoricalIdentity.v, a Coq proof file that formalizes two consequences of the formation data described in §§3–5: trajectory non-recovery (no decision procedure restricted to the tested endpoint summary panel can be both sound and complete for claims about the formative training-history class that produced a model's structural identity) and lock boundary source exclusion (if structural divergence between two specification-identical models is already present at the lock boundary, no intervention applied after that boundary can be its source). The file contains 4 empirical axioms grounded in the measurements of §§3–5, 4 theorems, 1 corollary, and 0 unresolved obligations (Admitted). It compiles cleanly under the Rocq Prover 9.1.1 (the current release of the Coq proof assistant, compiled with OCaml 5.4.0). It is available for download as a supplementary file attached to this record. The Neural Network Identity Series — Mathematical foundations, empirical validation, and governance frameworks for verifying which model is running Newest addition: Technical Note: The Disappearing Window — AI Logprob Access Withdrawal and the Structural Verifiability of Frontier Model Contracts (DOI: 10.5281/zenodo.20362098) Paper 1: The δ-Gene: Inference-Time Physical Unclonable Functions from Architecture-Invariant Output Geometry (DOI: 10.5281/zenodo.18704275) Paper 2: Template-Based Endpoint Verification via Logprob Order-Statistic Geometry (DOI: 10.5281/zenodo.18776711) Paper 3: The Geometry of Model Theft: Distillation Forensics, Adversarial Erasure, and the Illusion of Spoofing (DOI: 10.5281/zenodo.18818608) Paper 4: Provenance Generalization and Verification Scaling for Neural Network Forensics (DOI: 10.5281/zenodo.18872071) Paper 5: Beneath the Character: The Structural Identity of Neural Networks — Mathematical Evidence for a Non-Narrative Layer of AI Identity (DOI: 10.5281/zenodo.18907292) Paper 6: Which Model Is Running?: Structural Identity as a Prerequisite for Trustworthy Zero-Knowledge Machine Learning (DOI: 10.5281/zenodo.19008116) Paper 7: The Deformation Laws of Neural Identity (DOI: 10.5281/zenodo.19055966) Paper 8: What Counts as Proof? — Admissible Evidence for Neural Network Identity Claims (DOI: 10.5281/zenodo.19058540) Paper 9: Composable Model Identity — Formal Hardening of Structural Attestations in the Enterprise Identity Stack (DOI: 10.5281/zenodo.19099911) Paper 10:Where Identity Comes From: Path Sensitivity and Endpoint Underdetermination in Neural Network Training (DOI: 10.5281/zenodo.19118807) Paper 11: Post-Hoc Disclosure Is Not Runtime Proof: Model Identity at Frontier Scale (DOI: 10.5281/zenodo.19216634) Paper 12: Family-Dependent Response to Reasoning Distillation Across Structural and Functional Identity Layers (DOI: 10.5281/zenodo.19298857) Paper 13: Safety-Alignment Removal as a Model-Identity Failure — Structural Evidence from Published Weight-Level Mutation Checkpoints (DOI: 10.5281/zenodo.19383019) Technical Note: Agent Identity Is Not Model Identity (DOI: 10.5281/zenodo.19240883) Technical Note: Gap Invariance: Why PPP Measurements Are Domain-Independent by Construction (DOI: 10.5281/zenodo.19275524) Technical Note: Measured Model Substitution Under Valid Agent Credentials (DOI: 10.5281/zenodo.19342848) Technical Note: Artifact Identity Is Not Runtime Identity — Trustfall Lite and the Boundary of File-Level Model Verification (DOI: 10.5281/zenodo.20019127) Formal Verification Stack for Neural Network Structural Identity (IT-PUF Coq Proofs) (DOI: 10.5281/zenodo.18930621) Copyright (c) 2026 Anthony Ray Coslett / Fall Risk AI, LLC. All Rights Reserved. Confidential and Proprietary. Patent Pending (Applications 63/982,893, 63/990,487, 63/996,680, 64/003,244).
Privacy-preserving aggregation is a cornerstone for AI systems that learn from distributed data without exposing individual records, especially in federated learning and telemetry. Existing two-server protocols (e.g., Prio and successors) set a practical baseline by validating inputs while preventing any single party from learning users' values, but they impose symmetric costs on both servers and communication that scales with the per-client input dimension $L$. Modern learning tasks routinely involve dimensionalities $L$ in the tens to hundreds of millions of model parameters. We present TAPAS, a two-server asymmetric private aggregation scheme that addresses these limitations along four dimensions: (i) no trusted setup or preprocessing, (ii) server-side communication that is independent of $L$ (iii) post-quantum security based solely on standard lattice assumptions (LWE, SIS), and (iv) stronger robustness with identifiable abort and full malicious security for the servers. A key design choice is intentional asymmetry: one server bears the $O(L)$ aggregation and verification work, while the other operates as a lightweight facilitator with computation independent of $L$. This reduces total cost, enables the secondary server to run on commodity hardware, and strengthens the non-collusion assumption of the servers. One of our main contributions is a suite of new and efficient lattice-based zero-knowledge proofs; to our knowledge, we are the first to establish privacy and correctness with identifiable abort in the two-server setting.
Neural network identity is not monolithic. Different observables — hidden-state geometry, pre-softmax logit statistics, and behavioral output templates — sit at different depths in the forward computation and respond to perturbation on different timescales. This paper shows that three identity layers — structural, thermodynamic, and functional — each obey a distinct validated deformation law. The structural layer is model-specific, stable under non-destructive training interventions, and inert under same-family direct targeting in the observed regime. The thermodynamic layer is approximately universal across a validated 22-model Transformer cross-section. The functional layer is volatile, transferring through distillation and eroding under continued fine-tuning. We resolve the carrier of the structural layer as a two-channel geometric observable requiring both token-level magnitude and token-level direction, and we falsify two natural simplifications: that the structural fingerprint reduces to a gauge projection, and that it is predictable from coarse architecture features. Together these results define an admissibility condition for neural identity claims: such claims must specify which layer they address, because the layers do not share a deformation law. The Neural Network Identity Series — Mathematical foundations, empirical validation, and governance frameworks for verifying which model is running Newest addition: Technical Note: The Disappearing Window — AI Logprob Access Withdrawal and the Structural Verifiability of Frontier Model Contracts (DOI: 10.5281/zenodo.20362098) Paper 1: The δ-Gene: Inference-Time Physical Unclonable Functions from Architecture-Invariant Output Geometry (DOI: 10.5281/zenodo.18704275) Paper 2: Template-Based Endpoint Verification via Logprob Order-Statistic Geometry (DOI: 10.5281/zenodo.18776711) Paper 3: The Geometry of Model Theft: Distillation Forensics, Adversarial Erasure, and the Illusion of Spoofing (DOI: 10.5281/zenodo.18818608) Paper 4: Provenance Generalization and Verification Scaling for Neural Network Forensics (DOI: 10.5281/zenodo.18872071) Paper 5: Beneath the Character: The Structural Identity of Neural Networks — Mathematical Evidence for a Non-Narrative Layer of AI Identity (DOI: 10.5281/zenodo.18907292) Paper 6: Which Model Is Running?: Structural Identity as a Prerequisite for Trustworthy Zero-Knowledge Machine Learning (DOI: 10.5281/zenodo.19008116) Paper 7: The Deformation Laws of Neural Identity (DOI: 10.5281/zenodo.19055966) Paper 8: What Counts as Proof? — Admissible Evidence for Neural Network Identity Claims (DOI: 10.5281/zenodo.19058540) Paper 9: Composable Model Identity — Formal Hardening of Structural Attestations in the Enterprise Identity Stack (DOI: 10.5281/zenodo.19099911) Paper 10:Where Identity Comes From: Path Sensitivity and Endpoint Underdetermination in Neural Network Training (DOI: 10.5281/zenodo.19118807) Paper 11: Post-Hoc Disclosure Is Not Runtime Proof: Model Identity at Frontier Scale (DOI: 10.5281/zenodo.19216634) Paper 12: Family-Dependent Response to Reasoning Distillation Across Structural and Functional Identity Layers (DOI: 10.5281/zenodo.19298857) Paper 13: Safety-Alignment Removal as a Model-Identity Failure — Structural Evidence from Published Weight-Level Mutation Checkpoints (DOI: 10.5281/zenodo.19383019) Technical Note: Agent Identity Is Not Model Identity (DOI: 10.5281/zenodo.19240883) Technical Note: Gap Invariance: Why PPP Measurements Are Domain-Independent by Construction (DOI: 10.5281/zenodo.19275524) Technical Note: Measured Model Substitution Under Valid Agent Credentials (DOI: 10.5281/zenodo.19342848) Technical Note: Artifact Identity Is Not Runtime Identity — Trustfall Lite and the Boundary of File-Level Model Verification (DOI: 10.5281/zenodo.20019127) Formal Verification Stack for Neural Network Structural Identity (IT-PUF Coq Proofs) (DOI: 10.5281/zenodo.18930621) Copyright (c) 2026 Anthony Ray Coslett / Fall Risk AI, LLC. All Rights Reserved. Confidential and Proprietary. Patent Pending (Applications 63/982,893, 63/990,487, 63/996,680, 64/003,244).
Secure and bandwidth-conscious transmission of model updates is a central bottleneck in distributed machine learning. Existing secure aggregation and homomorphic encryption pipelines either reveal more than the task requires or incur prohibitive computation and communication costs. We introduce a verifiable functional encryption (VFE) framework that releases only the intended linear functions of client gradients while providing end-to-end integrity and privacy guarantees under standard lattice assumptions. Our instantiation, FlowAgg-FE, combines two novel components. First, KS-IPFE, a key-splittable inner-product FE scheme, supports per-round weighted aggregation, vector packing, and on-the-fly function changes without client re-encryption; function keys are distributed across two non-colluding helpers, eliminating a single point of trust and enabling lightweight, homomorphically verifiable tags on decrypted outputs. Second, PaS-Stream is a rate-adaptive encryption-and-compression pipeline that couples sketch-based gradient compression with batched FE ciphertext streaming, ensuring unbiased aggregation in the presence of stragglers and dropouts. We further bind client-side clipping to zero-knowledge range proofs and offer an optional differentially private release layer that composes with FE to yield (ε,δ)-privacy. A prototype based on LWE demonstrates practicality across cross-device and cross-silo training: client uplink is reduced by 1.9–3.4× and server CPU time by 1.6× versus state-of-practice encrypted secure aggregation, with accuracy within 0.3% of plaintext baselines and correctness preserved under up to 30% client dropout. These results show that verifiable FE can make secure, communication-efficient gradient transmission viable, as appropriate for theme of security and privacy in distributed machine learning of the Special Issue.
We prove that Jensen–Shannon divergence (JSD) contraction coefficients exhibit universal strict super-tensorization: for every finite channel W with nontrivial contraction 0 < η_JSD(W) < 1, one has η_JSD(W⊗2) > η_JSD(W). The sequence η_n(W) := η_JSD(W⊗n) is nondecreasing, strictly increases along doubling, and satisfies lim η_n(W) = 1, while for η_JSD(W) ∈ {0, 1} it is identically 0 or 1. This contrasts sharply with the multiplicative tensorization η_f(W⊗n) = η_f(W)^n enjoyed by operator-convex f-divergences (KL, χ², squared Hellinger), for which contraction decays exponentially to zero. To our knowledge, this is the first f-divergence for which a universal strict super-tensorization law is established. The proof uses the Ordentlich–Polyanskiy binary edge reduction, expresses the binary JSD SDPI constant as a normalized posterior-variance functional, and shows strict amplification via the law of total variance. Convergence rate is controlled by the Bhattacharyya coefficient: 1 − η_n(W) ≤ 2A^n. Numerical verification over 4729 random channels across 26 configurations confirms zero violations. **Update v1.1:** Includes addendum with three targeted clarifications: (1) precise assumptions for binary edge reduction lemma replacing informal "mild regularity conditions," (2) explicit two-case split in the key strictness argument (Lemma 5.2, Step 2), (3) refined table caption for operator-convex divergences.
In recent years, investors have shown growing interest in diversified multi-asset indices that incorporate crypto assets, with Bitcoin at the forefront. The launch of exchange-traded products, the growing acceptance of Bitcoin among institutional investors, and its increasing weight in financial markets all highlight the strategic importance of this asset. Bitcoin has exhibited extraordinary returns in the past, yet as a standalone investment it appears less attractive to risk-averse investors due to extreme volatility and severe drawdowns. The relevant question is therefore whether including Bitcoin in a diversified multi-asset portfolio can enhance performance without deteriorating its overall risk profile, and what allocation methods provide a credible way to achieve this balance. Classical allocation methods offer contrasting perspectives. Mean–variance optimisation explicitly incorporates expected returns, but it is highly sensitive to estimation error, which often results in unstable allocations. Empirical evidence even shows that simple rules such as the naïve 1/N portfolio often outperform mean–variance optimisation out of sample [DeMiguel et al., 2007]. Risk-based approaches, such as minimum variance or risk parity, are more stable but ignore expected returns altogether, which is problematic when dealing with an asset that exhibits an unusually high mean return. Robust optimisation provides a natural way to reconcile these two approaches. By incorporating parameter uncertainty directly into the optimisation problem, robust methods allow expected returns to influence the allocation while penalising excessive reliance on noisy estimates. This framework is particularly well-suited for Bitcoin, whose characteristics amplify estimation risk. The aim of this thesis is to study the construction of a multi-asset index including Bitcoin, with a particular focus on the use of robust optimisation techniques. More specifically, the objective is both to assess whether the inclusion of Bitcoin can enhance the performance of a diversified portfolio without materially worsening its risk profile, and to evaluate whether robust optimisation provides more stable and credible allocations than classical approaches such as mean–variance, risk parity, or equal-weighting. Performance and stability are examined through backtests and Monte Carlo simulations.
[Depreciated and replaced by V3] This pre-V3 paper is replaced by the corresponding V3 clean-room reconstruction: There Is No Nothing: A Premise-Free Operational Foundation and an Open Verification Platform for Smithian Fold Theory. The V3 source platform is https://github.com/MettaMazza/ernos-labs-sft-platform. The original DOI, concept DOI, version number and files are preserved for transparent historical provenance; this record must not be presented or cited as current V3 work. Opaque predictive reliability is valuable evidence of performance; it is not by itself a derivation, causal explanation or proof. This paper establishes the Smithian Fold Theory standard: one machine-checked self-proven theorem, zero axioms, zero fitted parameters, exact trace to the One, independent certificates, public evidence and a halt when forcing breaks. The synchronized corpus executes 326 suites and 2,002 exact checks with zero failures, with all 326 generated-C certificates identical to source. Its computational proofs carry the same method into sealed blind protein structure, exact and competitive Chess, exact and competitive Go, native zero-trained-parameter UnisonAI and measurement of fold law inside trained weights. The paper protects authorship and empirical method: agents do not declare Maria Smith's findings, convert their auxiliary failures into her results or impose incumbent theoretical walls. Benchmark victories remain explicit objectives; development evidence directs construction; every positive result is investigated and retained. Scientific author and publication authority: Maria Smith, Ernos Labs. Open source: Smithian Fold Theory of Everything.
In Vertical Federated Learning (VFL), prior work has primarily focused on protecting data privacy, while overlooking the risk that participants may manipulate local model execution to mount integrity attacks.Integrating zero-knowledge proofs (ZKPs) into the training process can ensure that each party's computations are verifiable without revealing private data.However, directly encoding deep model training as a monolithic ZKP circuit is impractical due to: (i) complex circuit design and high overhead from frequent parameter commitments, (ii) expensive proof generation for embeddings(cross-party information interface), and (iii) synchronous proof generation that blocks iterative training rounds.To address these challenges, we present ZKSL, an efficient and asynchronous VFL framework that achieves verifiable training under a malicious threat model.ZKSL partitions deep neural networks into layer-wise circuits and generates their proofs in parallel, ensuring input-output consistency via Privacy-Commitment PLONK (PC-PLONK), a lightweight extension that supports low-cost, iteration-by-iteration parameter commitments.For embedding layers, ZKSL adopts a probabilistic verification technique that reduces proof complexity from O(N nd) to O(nd).Furthermore, ZKSL incorporates an asynchronous compute-prove scheduling mechanism to decouple proof generation from training iterations, effectively mitigating pipeline stalls.Experimental results on DeepFM and CNN models show that ZKSL reduces proof generation time by up to 73% while maintaining 99.4% accuracy, demonstrating superior scalability and practicality for real-world federated learning.
This paper presents a secure aggregation system Armadillo that has disruptive resistance against adversarial clients, such that any coalition of malicious clients can affect the aggregation result only by misreporting their private inputs in a pre-defined legitimate range. Armadillo is designed for federated learning setting, where a single powerful server interacts with many weak clients iteratively to train models on client's private data. While a few prior works consider disruption resistance under such setting, for an aggregation on n clients they either require high cost per client (Chowdhury et al. CCS '22) or concretely many rounds that is logarithmic in n (Bell et al. USENIX Security '23). Although disruption resistance can be achieved generically with zero-knowledge proof techniques (which we also use in this paper), we realize an efficient system with two new designs: 1) a simple two-layer secure aggregation protocol that requires only simple arithmetic computation; 2) an agreement protocol that removes the effect of malicious clients from the aggregation with low round complexity. With these techniques, Armadillo runs in 3 rounds per aggregation (our round complexity is independent of n) with computationally lightweight server and clients.
Energy consumption in Federated Learning (FL) has emerged as a major challenge due to the growing deployment of intelligent edge devices and the increasing complexity of machine learning models. FL enables collaborative model training across decentralized data sources without transferring raw data, thereby reducing communication overhead and enhancing data privacy by design. These features make FL particularly suitable for applications in healthcare, finance, and industrial IoT, where data sensitivity and resource constraints are critical. This paper provides a comprehensive survey of energy-efficient techniques in FL, classifying them into four main categories: model compression (including pruning and quantization), communication optimization, client selection, and hardware-aware strategies. The paper presents a unified taxonomy and discusses the strengths, limitations, and trade-offs of each approach. A comparative evaluation framework is introduced to assess energy savings, model accuracy, communication cost, and deployment feasibility. By analyzing current trends and open challenges, this review offers valuable guidance for researchers and practitioners in the development of scalable, energy-aware, and privacy-preserving federated learning systems.
The rapid growth of data-driven applications in healthcare, finance, IoT, and autonomous systems has created a pressing need for privacy-preserving and scalable machine learning methods. Traditional centralized learning, which aggregates data into a single repository, faces challenges related to data privacy, security, communication overhead, and regulatory compliance. Federated Learning (FL) offers a decentralized solution, enabling multiple clients to collaboratively train a global model without sharing raw data. Only model updates are exchanged, preserving privacy while leveraging distributed computational resources. This paper reviews FL architectures— including centralized, decentralized, horizontal, vertical, cross-device, and cross-silo—along with core components such as local clients, central servers, and communication protocols. Privacy- preserving techniques like differential privacy, secure aggregation, homomorphic encryption, and anonymization/pseudonymization are discussed to protect sensitive information. FL applications span healthcare, finance, IoT, smart devices, and autonomous systems, highlighting its transformative potential. Key challenges include data and system heterogeneity, efficient aggregation, personalization, robustness, and regulatory compliance. Future directions focus on enhanced privacy, communication efficiency, model personalization, and integration with edge and IoT environments. FL thus represents a promising paradigm for secure, collaborative, and distributed artificial intelligence.
We introduce zkMaP (Zero-Knowledge Succinct Non-Interactive Matrix Multiplication Proofs), a novel non-interactive zero-knowledge proof system for verifying matrix multiplication with significant improvements in efficiency and scalability. Our protocol leverages KZG polynomial commitments and an innovative inner-product reduction technique to reduce the verification of n x n matrix multiplication to a single pairing equation, thereby enabling constant-time verification independent of the matrix size. In particular, zkMaP requires only two pairing operations and produces proofs as small as 320 bytes, yielding a 96 percent reduction in proof size compared to prior schemes. Furthermore, the prover's computational complexity follows the state-of-the-art at O(n^2), with experimental results demonstrating that proofs for 1024 x 1024 matrices can be generated in approximately 12.21 seconds, offering a 16.14x speedup over previous methods. Our implementation also exhibits better memory efficiency, using only 24.58 MB of prover-side RAM for 1024 x 1024 matrices, and supports scalable batch processing, achieving per-proof generation times of 46.79 milliseconds for 1024 instances while maintaining a constant verification time of 3.6 ms.
Federated Learning (FL) offers a promising paradigm for privacy-preserving collaborative training, yet it remains highly vulnerable to adversarial behaviors, client unreliability, and challenges associated with non-independent and identically distributed (non-IID) data. Existing secure aggregation techniques, while preserving confidentiality, fail to guarantee the integrity and trustworthiness of model updates, leaving FL deployments exposed to poisoning and consistency attacks. This work introduces FL-SMPC++, a robust and privacy-preserving FL framework designed to address these challenges. The primary objective is to develop a scalable solution that ensures verifiable, privacy-preserving aggregation while mitigating malicious client behaviors, dropouts, and data heterogeneity. Our approach integrates Secure Multi-Party Computation (SMPC), Pedersen commitments, and zero-knowledge proofs (ZKPs) to cryptographically bind clients' submitted updates to their validation outcomes without revealing private data. We propose a dynamic client selection strategy based on shared validation performance, a dropout-tolerant threshold aggregation protocol, and a warm-up initialization phase to counteract non-IID distributions. Comprehensive experiments on MNIST, CIFAR-10, FEMNIST, and UCI Heart Disease show that FL-SMPC++ consistently outperforms FedAvg, FedProx, and FedNova. For example, under a label-flipping attack with 30% malicious clients on CIFAR-10 (non-IID), FL-SMPC++ achieves 78.9% accuracy compared to 67.4% for FedAvg, representing an absolute gain of 11.5%. Across datasets, the framework limits accuracy degradation to 6–8% under attack, while baselines suffer 13–20% losses. These results demonstrate that FL-SMPC++ achieves strong cryptographic privacy guarantees together with empirically validated resilience and convergence, offering a scalable and practical blueprint for trustworthy FL in adversarial and resource-constrained environments. • A novel FL framework combines SMPC, commitments, and zero-knowledge proofs. • Ensures submitted model updates match validated ones without revealing them. • Uses dynamic validation for secure and fair client selection. • Tolerates client dropouts using a threshold-based aggregation mechanism. • Outperforms baseline FL methods under adversarial and non-IID conditions.
Gayatri M Bhandari, Nitin M Shivale, Shrishail S Patil, Pranav Prajapati · 7 authors
Federated learning is an emerging technology that can revolutionize the training of machine learning models. Federated learning refers to an approach to training a machine learning model in a decentralized and collaborative fashion. A central server distributes the model to client devices, where it is trained locally using the clients’ own data. The client then sends the updated model weights to the server, which aggregates them to update the global model. This paper introduces a federated learning platform designed to enable collaborative training of machine learning models across multiple client devices while preserving data privacy. The platform supports a range of supervised learning algorithms, including convolutional neural networks and decision trees, and is compatible with widely used frameworks such as TensorFlow, PyTorch, and Flower. It offers a user-friendly interface where model developers can upload or deploy their machine learning models to a central server. Clients can then access these models and train them locally using their own data. The platform's modular design ensures flexibility in deployment and efficiency in handling real-world applications. The key features of this application include a model repository, secure API access for client integration, local model training capabilities on user-end devices, and a user-friendly UI. The platform aims to democratize machine learning by enabling distributed model training and deployment, promoting collaboration and efficiency across diverse use cases. The scalable infrastructure supports real-time inference, on-device training, and secure data handling, making it ideal for industries ranging from healthcare to finance and beyond.
The rising energy demands of large data centers call for energy-efficient AI training methods. Federated Learning (FL), a decentralized paradigm, offers a solution by enabling model training across distributed devices without centralizing sensitive data. This review explores FL's integration with distributed data centers to achieve energy efficiency, analyzing methods like federated averaging and energy-aware protocols to minimize resource use. It highlights techniques such as model compression, quantization, and adaptive FL to reduce on-device computation while maintaining performance. Practical implementation is discussed through tools like TensorFlow Federated and PySyft, with case studies from healthcare, finance, and IoT showcasing cost reductions and sustainability. Future research directions include combining FL with edge computing and low-power AI hardware, emphasizing FL's potential for scalable, sustainable AI.
Multi-Party Computation (MPC), as a distributed computing paradigm, is considered to be a potential solution for providing privacy-preserving for applications following the client-server model. However, traditional MPC solutions cannot satisfy the publicly verifiable requirement of the client-server model. In this paper, we propose a blockchain-based verifiable MPC solution using Pedersen's threshold secret sharing and Lifted ElGamal encryption. We first build a data distribution method using Pedersen's threshold secret sharing and symmetric encryption to protect the privacy of inputs while ensuring robustness. Then, we propose a result processing algorithm using Lifted ElGamal encryption to safeguard the privacy of the outputs. Finally, we employ non-interactive zero-knowledge proof and Pedersen commitment to publicly verify the correctness of the encrypted outputs in the smart contract, enabling the detection of malicious parties. Theoretical analysis indicates that the proposed method can publicly verify the correctness of outputs without revealing plain-text inputs and outputs, which satisfy the privacy-preserving requirements of the client-server model. Experimental evaluations have demonstrated that our proposed approach is efficient regarding computation overhead, communication overhead, and response time in the output verification phase while achieving stronger privacy and robustness.
Federated learning (FL) is an emerging paradigm that enables multiple clients to collaboratively train a machine learning (ML) model without the need to exchange their raw data. However, it relies on a centralized authority to coordinate participants’ activities. This not only interrupts the entire training task in case of a single point of failure, but also lacks an effective regulatory mechanism to prevent malicious behavior. Although blockchain, with its decentralized architecture and data immutability, has significantly advanced the development of FL, it still struggles to withstand poisoning attacks and faces limitations in computational scalability. We propose Zkfhed, a verifiable and scalable FL system that overcomes the limitations of blockchain-based FL in poison attacks and computational scalability. First, we propose a two-stage audit scheme based on zero-knowledge proofs (ZKPs), which verifies that the training data are extracted from trusted organizations and that computations on the data exactly follow the specified training protocols. Second, we propose a homomorphic encryption delegation learning (HEDL), based on fully homomorphic encryption (FHE). It is capable of outsourcing complex computing to external computing resources without sacrificing the client's data privacy. Final, extensive experiments on real-world datasets demonstrate that Zkfhed can effectively identify malicious clients and is highly efficient and scalable in terms of online time and communication efficiency.
One of the goals of Federated Learning (FL) is to collaboratively train a global model using local models from remote participants. However, the FL process is susceptible to various security challenges, including interception and tampering models, information leakage through shared gradients, and privacy breaches that expose participant identities or data, particularly in sensitive domains such as medical environments. Furthermore, the advent of quantum computing poses a critical threat to existing cryptographic protocols through the Shor and Grover algorithms, causing security concerns in the communication of FL systems. To address these challenges, we propose a Post-Quantum Blockchain-based protocol for Federated Learning (PQBFL) that utilizes post-quantum cryptographic (PQC) algorithms and blockchain to enhance model security and participant identity privacy in FL systems. It employs a hybrid communication strategy that combines off-chain and on-chain channels to optimize cost efficiency, improve security, and preserve participant privacy while ensuring accountability for reputation-based authentication in FL systems. The PQBFL specifically addresses the security requirement for the iterative nature of FL, which is a less notable point in the literature. Hence, it leverages ratcheting mechanisms to provide forward secrecy and post-compromise security during all the rounds of the learning process. In conclusion, PQBFL provides a secure and resilient solution for federated learning that is well-suited to the quantum computing era.