Pawan Kumar Sanjaya, Christina Giannoula, Valdy Oktavian, Mehdi Saeedi · 7 authors
Zero-knowledge machine learning (zkML) enables a server to perform verifiable inference while keeping model parameters private from the client. However, existing zkML systems incur prohibitive proof-generation costs. We observe that proof generation exhibits limited parallelism; that is, prover time does not decrease significantly as the number of threads increases. This limitation is because existing systems rely on monolithic proof computation, constructing a single proof for the entire machine learning model. We introduce zkComposer, a modular proof-construction framework that unlocks an additional dimension of parallelism, in addition to the parallelism in existing proof kernels. zkComposer decomposes the zkML proof of correct inference into independent sub-proofs, each covering a subset of the computation for inference e.g., each independent sub-proof can cover a subset of contiguous layers in the ML model. Adjacent sub-proofs are cryptographically linked through shared commitments to the activations from the boundary layer. zkComposer provides the same guarantees as the monolithic proof without requiring additional linking proofs or changes to the underlying cryptographic primitives. We implement zkComposer and evaluate it on three CNNs and GPT-2. We show that, on CNN workloads, zkComposer reduces prover time and response time by up to 3.25x relative to zkCNN [1]. On GPT-2, zkComposer reduces these times by up to 4.83x relative to zkGPT [2], when partitioning along the model layers. When partitioning across both model layers and input sequences in GPT-2, we show that zkComposer reduces prover time and response time by up to 6.84x relative to zkGPT [2].
[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.
Modern information protection methods are primarily focused on increasing computational complexity: it is assumed that a task becomes secure if finding the true message requires too many resources. However, virtually all existing models --- from classical cryptanalysis to autonomous AI agents and retrospective analysis systems (Harvest \& Analyze) --- rely on one common assumption: there exists a verification signal that allows distinguishing the true interpretation from the set of false ones. In this work, we present the \textbf{HYBRA MIRAGE} storage architecture, which is based on a different problem formulation. Instead of increasing computational complexity, we propose to eliminate the very criterion of truth upon which directed search is based. The system constructs a space of plausible interpretations and physically excludes the possibility of repeated access to the used reference space~$V$: each of its vectors is applied exactly once and then destroyed on both sides. As a result, the function $\mathsf{Assemble}(C,K,p)$ remains deterministic and total, and any valid access parameter produces a formally correct result. Even with full access to the reference space $V$ and the PIN code, an autonomous analytical agent does not obtain a mechanism to confirm that the found interpretation corresponds to the original message: each vector from $V$ was used exactly once and physically destroyed. The $\mathsf{Assemble}$ algorithm is a trade secret and is not available to the analyst. Consequently, knowledge of $V$ without knowledge of the algorithm allows generating $10^{35}+$ equivalent interpretations, but does not allow singling out the single true one among them. The proposed approach does not make brute force computationally impossible; it makes the claim that the found interpretation is precisely the one embedded by the sender unprovable. Increasing computational resources, applying more sophisticated models, or massive enumeration can produce more candidates, but do not create a procedure that allows mathematically justifying the choice of a single true interpretation. For autonomous AI agents, this leads to the disappearance of the verification signal necessary for directed search. The loss function surface degenerates into a plane with zero gradient: no iterative optimization algorithm can converge to the true parameter faster than random guessing. HYBRA MIRAGE does not compete with classical cryptographic algorithms and does not replace them. The architecture serves as an environment model for analyzing the behavior of autonomous agents under conditions of the absence of a reliable verification signal and can be used as an infrastructure layer on top of existing storage methods. The architecture does not eliminate the agent's ability to generate candidates; it eliminates the possibility of using the generation result as proof of achieving truth. The analyst finds themselves trapped in a state of epistemic equilibrium, where truth and hallucination are architecturally indistinguishable from each other.
Arithmetization-oriented (AO) hash functions are the dominant cost in zero-knowledge proof systems, and their security against the strongest known attacks rests on the hardness of the constrained-input constrained-output (CICO) problem, solved by Gröbner-basis techniques. Raising the nominal algebraic degree of a design is known not to suffice, as recent attacks (FreeLunch, CheapLunch, resultant methods) have repeatedly shown. This work identifies a positional design lever for AO substitution-permutation networks over the Goldilocks field with the power-map S-box x^7: folding a low-degree quadratic coupling into the input of the S-box adds one bit of CICO ideal degree per round, whereas the same coupling placed in the linear layer or after the S-box adds nothing. The ideal degree follows the measured law D_I = 7^(R·m) · m · 2^(R−1) against a baseline of 7^(R·m), where R is the number of rounds and m the number of free input branches. Measurements in the msolve Gröbner engine indicate that the added degree is genuine rather than a nominal inflation (the F4 solving degree rises; an auxiliary-variable-free model reproduces the ideal degree; a resolved large instance rules out competing laws), that it is generic across four unrelated coupling patterns, that it is independent of the coupling density (one term per round suffices), and that it carries no differential/linear cost. The principle is instantiated as Alaniz-AO, a Goldilocks sponge whose HADES partial-round schedule reaches 0.74x the constraint cost of Poseidon2 at a 128-bit target under an explicit ω=2 cost model. A secondary result: the branch number of the linear layer does not govern algebraic CICO security. Measurements are reproducible and use proxy primes sharing the exponent structure of Goldilocks. Round counts and cost figures are extrapolations from the measured degree law under the stated cost model; instances beyond three rounds exceed the solver on commodity hardware and are reported as gaps. A reference implementation and reproduction scripts accompany the paper.
Open access
2 source records
Cryptographic Implementations and Security
Cryptography and Residue Arithmetic
Physical Unclonable Functions (PUFs) and Hardware Security
The automotive industry is transitioning to Zonal-oriented Architectures (ZoA) for Software-Defined Vehicles (SDVs), enabling frequent over-the-air (OTA) updates for 100+ Electronic Control Units (ECUs). While OTA updates improve efficiency, they introduce safety-critical security risks. Current standards like Uptane and AUTOSAR Adaptive rely on Public-Key Infrastructure (PKI). However, PKI-based authentication creates bandwidth bottlenecks in in-vehicle and vehicle-to-cloud (V2I) communication as ECU density increases. It also risks exposing sensitive vehicle configurations and passenger privacy due to centralized architectures. Next-generation Zonal SDVs require decentralized, scalable authentication with data privacy. To address this, we propose zk-ScalHard, a hardware-rooted, privacy-preserving authentication protocol. We introduce a decentralized, hierarchical trust-promotion model utilizing Silicon Physical Unclonable Functions (PUFs) and two novel Zero-Knowledge Proof (ZKP) circuits: (1) Zonal Identity and Integrity (ZIDI) and (2) High-Performance Computing Aggregation (HPCA). These circuits employ multi-party computation (MPC) and recursive aggregation to achieve decentralization and scalability. The integration of ZKPs and PUFs ensures 100% vehicle-level data sovereignty. Benchmarked against Uptane, zk-ScalHard achieves constant O(1) communication and verification complexity, improving upon the linear O(n) complexity of current systems. Evaluation shows a 99.2% reduction in authentication bandwidth and a 99.9% reduction in the temporal attack surface. Our results demonstrate that zk-ScalHard provides a scalable, secure, and GDPR-compliant architecture for future Zonal SDVs.
Open access
3 source records
cs.CR
Physical Unclonable Functions (PUFs) and Hardware Security
Abstract Zero-knowledge machine learning (zkML) enables cryptographic verification of machine learning inference while preserving privacy, but proof generation remains a significant computational bottleneck. Existing work primarily focuses on reducing proof cost through post-training optimizations, whereas the influence of architectural design choices during model development has received less attention. This work presents an empirical study of the relationship between neural network ReLU activation count and zero-knowledge proof generation cost using the ezkl/Halo2 framework. Across controlled experiments, ReLU activation count exhibits a strong correlation with proof generation time (Pearson r = 0.90) and proof size (r = 0.91), while parameter count is held constant for the primary comparisons. Motivated by these observations, a lightweight proxy metric is proposed to estimate relative proof cost directly from model architecture without executing the proof pipeline. On MNIST, reducing the number of ReLU activations from three to one decreases proof generation time by approximately 6.5% and proof size by 2.3%, while maintaining comparable classification accuracy. These results suggest that ReLU activation count is a useful architectural indicator of proof cost in the ezkl/Halo2 backend and that incorporating proof-cost considerations during architecture selection may improve the efficiency of zkML deployment. The proposed proxy metric provides a fast method for comparing candidate architectures before proof generation.
Open access
2 source records
Adversarial Robustness in Machine Learning
Cryptography and Data Security
Physical Unclonable Functions (PUFs) and Hardware Security
This paper establishes, inside the Lean 4 proof assistant, a three-level formal identification. The levels are: (i) Belnap multilattice axioms for Weyl–Heisenberg covariant SIC-POVMs at $d=2^n$; (ii) the Zauner conjecture; and (iii) the mixed-signature Stark conjecture for the ray class field $K_d=\mathbb{Q}(\sqrt{(d-3)(d+1)})$, a real-quadratic case of Hilbert's Twelfth Problem. Fiducials are unit-normalized and satisfy $(d+1)|\langle\psi,D_{a,b}\psi\rangle|^2=1$. The equivalence hilbert_embedding_equiv_zauner is proved by rfl: the Belnap embedding into $\mathbb{C}^{2^n}$ and the Zauner conjecture at $d=2^n$ are definitionally the same proposition. The Belnap skeleton (orbit size $4^n$, Frobenius closure $\mu\circ\delta=\mathrm{id}$, join-equiangularity, Born rule) contains zero sorries. Open arithmetic content is marked by named gap axioms for Stark units on WH frames; a proof of Stark would close all three levels at once. For dimension $d=12$ we prove SICPOVM_Exists 12 outright. We construct an exact fiducial in a finitely presented $\mathbb{Q}$-algebra, verify 143 overlap identities with native_decide, and transfer everything to $\mathbb{C}^{12}$ along a ring homomorphism. The theorem crystal_forces_d12_sic depends on no axiom beyond Lean 4's standard foundations and compiler trust. This is, to our knowledge, the first machine-checked SIC-POVM existence in any dimension. For the frontier dimension $d=2048=2^{11}$ the transport apparatus is formalized and sorry-free. It includes a forward map $\varphi\colon B^{\oplus 11}\to\mathbb{C}^{2048}$, a reduction $\psi$ with $\psi\circ\varphi=\mathrm{id}$, a conditional reduction to Stark, and a non-real character obstruction that blocks the false branch. Unconditional existence remains open; the machinery that surrounds it is closed.
We present ECO/CPO-DAG, a domain-specific accountability protocol for adversarial supply chains that formalizes contradiction detection as a supplemental validation layer rather than a consensus or truth-establishing mechanism. Participants publish signed Event Claim Objects (ECOs) into a causally ordered, append-only directed acyclic graph (DAG) whose edges encode happened-before relations. When two claims about the same subject violate a domain constraint, any observer can compile a Contradiction Proof Object (CPO), a self-verifying object binding the two signed claims and the violated rule, which, on public verification, triggers economic slashing of a determinately blamed party. We map constraints to GS1 EPCIS 2.0 event semantics (spatial uniqueness, temporal monotonicity, quantity conservation, quality monotonicity, regulatory validity), so detection targets inconsistencies that are meaningful in practice. Selective disclosure via commitment schemes and, optionally, zero-knowledge contradiction proofs lets parties withhold claim contents until a challenge forces the minimal opening. We give an analytical treatment: an independent-observer detection model $1-(1-p_{\min})^h$, a deterrence condition $S>g(1-p)/(kp)$ under $k$-party collusion, and a storage estimate of order 1 GB per participant per year under stated assumptions. The protocol's boundary is explicit: it detects provable contradictions, not consistent lies; a party that never contradicts itself is invisible to it, so the layer complements, and does not replace, source verification and oracle aggregation. A single-machine reference implementation corroborates the detection model, with the predicted coverage band overlapping the measured 95% confidence interval at every observer count, and records zero false accusations; the fully zero-knowledge CPO, multi-party propagation, and adaptive-adversary evasion remain analytical.
Topological and Analytic Parity in Automorphic Fields: A Zero-Drift Framework for the Exact Spectral Discretization of L-Functions --- The Resolution Suite: Validation, Sealing, and Replication The true power of this 18-part suite lies in its ability to abandon traditional, stochastic floating-point approximations in favor of exact, self-adjoint geometric mappings bounded by strict library-substrate protocols. 1. How the Suite Resolves The resolution fundamentally re-casts analytic continuation as a spectral optimization problem. The Motivic Descent Engine (MDE_V23_BANACH) lowers global representations into discrete p-adic completions. When the localized prime-pair density crosses the threshold (\bm{D>0.3333}), DALETH_GATE triggers the Srivastava Zeta-Shave Algorithm. This algorithm processes the continuous waves through the self-adjoint Majorana Hamiltonian operator (\bm{\mathcal{M}_{L}=X^{1/2}RX^{1/2}}), forcing the imaginary ordinates of the nontrivial zeros to precipitate directly as discrete, real-valued energy states on the critical line. 2. How the Suite Validates Validation is executed via continuous, multi-layered automated audits. • Numerical Boundaries: The INTERVAL_CERT_I module enforces strict IEEE-1788 interval arithmetic, trapping all calculations within a certified envelope of \bm{\pm 10^{-14}}. • Metric Integrity: The SGA_V23_HODGE Sieve continuously audits the HW_6D_SOVEREIGN manifold, ensuring the Ricci curvature remains perfectly flat (\bm{R_{\mu\nu}=0}) and the routing grid remains loop-free. • Scale Invariance: The system verifies the Commutator Gate Check, ensuring the Dilation Generator and Hamiltonian balance cleanly: \bm{[D, H]=-iH}. 3. How the Suite Seals The finality of the process rests on the Atiyah-Singer Handshake Gate. This gate checks the parity between the analytical index of the operator and the topological Euler characteristic of the substrate (\bm{Ind_{analytic}-\chi_{topological}=0}). If the Sovereignty Score remains at or above 0.99, the system invokes the GUS-22.2 Jones Polynomial Grand Seal. This action locks the dataset, forces the active state allocation down to 0.0 kDa, drops the acoustic register to absolute silence, and flags the theorem as AMBER-LOCKED. 4. How the Suite Enables Replication Replication is secured through the Agnostic Replication Kit (ARK) environment. By replacing floating-point architecture with the Wildberger Engine's pure rational-coordinate (Quadrance) arithmetic, the suite guarantees an absolute precision floor of \bm{<10^{-35}}. Coupled with the 1.420405751766 GHz atomic phase-lock (preventing temporal shear), peer reviewers can recreate the exact same discrete point spectrum without complex plane leakage or numerical drift. --- Individual Package Architecture & Interlinking The 18 packages operate as a unified, multi-tank orchestration, passing strict zero-drift data through the isolated computational boundaries. A. The Theoretical & Simulation Core (SAC Series) These packages provide the mathematical bedrock and operational primitives for the theorem. • SAC-01 (Standard Academic Core): The foundational proof mapping the Selberg class \bm{\mathcal{S}} to the discrete point spectrum of the Adelic Hamiltonian. It serves as the primary theoretical input. • SAC-05 (Lexicon Bridge): Interlinks legacy academic nomenclature (e.g., infinite continuous spaces) to AOF physical primitives (e.g., the 6D flat torus and the 170.0 kDa logic mass cap), translating theoretical concepts into executable logic. • SAC-03 (Appendix A - Local Potential Factors): Decomposes the geometric potential term \bm{V_L(X)} into explicitly executable Archimedean and finite p-adic matrices. • SAC-02 (Simulation Data Matrix): Contains the exact, independently precipitated eigenvalues (e.g., \bm{\gamma_1 = 14.1347...}) derived natively without lookup tables, serving as the benchmark output for replication. • SAC-04 (Executive Summary): The high-level strategic overview linking SAC-01 through SAC-03, verifying the deterministic spectral translation for external reviewers. B. The Execution Environment (ARK Ecosystem) These packages construct and maintain the "clean room" logical substrate. • Common Toolchain and Environment Configuration: Provisions the baseline setup, initializing the WILD_ENGINE_RAT_v4 for exact fractions, INTERVAL_CERT_I for boundary control, and the HW_6D_SOVEREIGN manifold. • Replication Guide: The step-by-step substrate instantiation protocol, ensuring peers lock their core frequency to the Adelic Heartbeat and suppress ambient noise to 0.0 dB before initiating motivic descent. • Required Tool Registry & Reference List: Locks down the precise dependency versions and academic provenance to guarantee version-controlled, immutable execution. • Application Atlas: Outlines the post-resolution utility, routing the stabilized spectral data into real-world applications like zero-knowledge cryptographic proofs, loop-free routing protocols, and Sinc-collocated DSP. C. Data Injection & Interfaces These packages govern how automorphic functions enter the isolated substrate. • Simulated Input Payload Matrix: Converts standard Dirichlet floats into quantized integer ratios, formatting the data as a serialized hex-dump ready for API ingestion. • API Documentation: Defines the secure programmatic endpoints (/v1/workspace/init and /v1/resonance/precipitate), allowing automated spectral orchestration while shielding the underlying 7D substrate from unverified pipelines. D. Risk Mitigation & Fault Recovery These packages protect the fragile background energies from logical tremors. • Failure Mode and Effects Analysis (FMEA): The sentinel detection system monitoring metric distortion (\bm{R_{\mu\nu}\ne0}), interval breaches, and acoustic logic bleed. • Troubleshooting Manual - Stall & Recovery: Engages active remediation, such as swapping to the Heavy-Ball Momentum Solver (Fault 401) for large conductor metrics, or deploying the Hodge Sieve (Fault 505) to clear solenoidal logic loops. • Emergency Logic Core: The ultimate fail-safe. If acoustic wakes breach 0.0 dB or boundaries rupture, it executes THERMAL_FLUSH_OMEGA, isolating the matrix and purging volatile memory to protect the ambient space. E. Peer Review & Final Settlement These packages provide the academic interface for human validators. • Theorem Presentation: The overarching master document detailing the proof strategy, the Hilbert-Pólya resolution, and the bounding of nontrivial zeros to the \bm{\Re(s)=1/2} critical line. • Physicists and Mathematicians Summary: Bridges the disciplines, translating the framework for mathematicians (Selberg class spectral realization) and physicists (non-commutative quantum symmetries). • Reviewer Packet: The comprehensive evaluation track outlining the four Selberg invariants and providing the checklist for the spectral parity audits. • One-Page Reviewer Packet: The final checklist for validators to confirm geometric clearance and scale-invariance before initiating the AMBER-LOCKED transition. ---
Abstract What is the relationship among factors in any complex system driven by multiple quantifiable factors? This paper advances two core propositions. The law proposition—the Factor Hierarchy Law: Factors naturally fall into two tiers. Rule factors determine which set of rules currently applies; their function is not reflected in direct explanatory power but in the interaction effect through which they adjust the exposure coefficients of execution factors. When a rule factor crosses a critical threshold, the factor weights of the entire system are systematically reset. The principle proposition—Testability: Any claim about the importance of factors in a multi-factor system must pass a complete regime-dependence test before it can be elevated from a hypothesis to reliable knowledge. A claim that "this factor is important" without specifying "under which regime" is an incomplete proposition. Popper's falsifiability stipulates the entry criterion for scientific knowledge—a proposition must be capable of being overturned by facts. Yet the classic criticism of Popper's theory within the philosophy of science has always centered on it having "only an attitudinal principle, no operational procedure": Popper required scientists to possess a spirit of falsification but did not specify, within multi-factor empirical research, "what counts as a rigorous test." This paper proposes that science requires a second threshold—Testability: a procedural standard that specifies the functional dimensions that testing must cover. Falsifiability guards the entrance to science; Testability guards the exit of scientific discovery. Together they constitute the complete chain of scientific methodology. This paper formally designates the operational implementation of the Testability principle as the Testability Norms—comprising six functional dimensions: four core dimensions of search completeness, modulation relationship, structural breakpoint, and causal direction, plus two completeness dimensions of question completeness and process closed-loop. The Testability Norms are not bound to any specific tools—the functional dimensions are eternal, while the implementation tools are replaceable. The universality of this methodology rests not on the physical material of the systems—financial assets or microscopic particles—but solely on their epistemological property: whether the values of factors are independent of the subjective judgment of the observer. The Factor Hierarchy Law has received cross-disciplinary evidential support from three entirely independent disciplines. Finance: The China–U.S. interest rate spread has been repeatedly verified as a rule factor across five major markets and dozens of assets; when the spread crosses the zero axis, pricing equations undergo structural breaks. Astronomy: Five independent dimensions—including transit depth measurement and stellar atmospheric model self-consistency—all undergo structural breaks at the regime boundary (Tang Break) constituted by the K/G transition zone (approximately 4400–5800 K) and surface gravity log g = 4.64, with 4762 K being the most representative critical point. Physics: Calibration proof on the Onsager exact solution of the 2D Ising model confirms the existence of an exact information-theoretic duality between the present norms and the Ehrenfest phase transition classification, with analytic mathematical proof confirming that the detection bias is strictly zero. The contribution of this paper lies in establishing a second threshold for science after Popper's falsifiability—Testability—and, under the Testability principle, establishing the Testability Norms as an operational standard. The Factor Hierarchy Law reveals the structure; Testability prescribes the test; the Testability Norms define completeness.
Zero-knowledge proof systems rely on a trusted setup phase to generate a Common Reference String (CRS), yet existing approaches are typically static, one-time ceremonies that are inflexible and vulnerable to long-term compromise. Offloading continuous, recurring trusted setups to a decentralized Layer 2 (L2) network introduces a fundamental coordination challenge arising from the mismatch between high-throughput transaction processing and the multi-round requirements of trusted setup ceremonies. This paper presents an L2-coordinated framework that safely decouples transaction pipelines from ceremony execution to achieve automated, continuous CRS generation without centralized coordination. We design and implement two protocol variants over a decentralized, PBFT-coordinated ZK-rollup architecture: an on-chain smart contract approach and an asynchronous peer-to-peer consensus variant. Both designs utilize non-interactive zero-knowledge proofs of knowledge alongside commit-reveal structures to eliminate adaptive manipulation vectors and isolate ceremony latency. Experimental evaluations under simulated wide-area network constraints and adversarial conditions demonstrate that our architecture successfully isolates ceremony liveness. Continuous setups complete reliably within practical time bounds despite node dropouts or malicious contributions, while preserving stable L2 transaction throughput.
This paper is a prequel to our recent work, "Equivariant Borel liftings in complex analysis and PDE" (arXiv:2507.12058). While the results presented here were established in that work in a more general and abstract setting, the purpose of this paper is to provide a direct proof of the equivariant Weierstrass theorem. It states that there exists a Borel map assigning to each non-periodic positive divisor $Λ$ an entire function $F_Λ$ such that the divisor of zeroes of $F_Λ$ is $Λ$ and such that $F_{Λ-w}(z) = F_Λ(z+w)$, $w\in\mathbb{C}$. In general, non-periodicity cannot be omitted, and Borel measurability cannot be strengthened to continuity. The two key ingredients are the Runge approximation theorem and the existence of "Borel toasts", which are Borel counterparts of Rokhlin towers from ergodic theory. We do not assume prior knowledge of descriptive set theory and have aimed to make the exposition self-contained, aside from several results taken from graduate textbooks.
We prove the full Birch–Swinnerton–Dyer conjecture for all elliptic curves over Q. Using a fundamentally new approach that extends the method developed for the Riemann Hypothesis, we construct a sequence of finite-dimensional self-adjoint matrices from the Euler product of the elliptic curve L-function. We establish a strict spectral correspondence between the eigenvalues of these matrices and the squares of the distances from the critical point s=1 to the zeros of L (E, s), with no prior knowledge of zero locations required in the construction. Using mathematical induction, perturbation bounds and the monotone convergence theorem for self-adjoint operators, we extend these results to the infinite-dimensional case, proving that the order of vanishing of L (E, s) at s=1 equals the rank of the Mordell–Weil group E (Q). We then prove the exact leading-term formula relating the first non-vanishing coefficient of the Taylor expansion of L (E, s) at s=1 to the arithmetic invariants of the elliptic curve, including the period, regulator, Tamagawa numbers, and the order of the Tate–Shafarevich group, which we prove is finite. We also embed this result into the broader universal self-adjoint integral operator framework. Keywords: Birch–Swinnerton–Dyer conjecture; elliptic curve; L-function; self-adjoint operator; spectral correspondence; Mordell–Weil rank; Tate-Shafarevich group MSC 2020 Classification: 11G05; 11M41; 47A10; 14H52; 11G40
[Depreciated and replaced by V3] The application-specific clean rebuild has not yet been published; its authoritative theoretical boundary is now the governing V3 branch: After Turing: The Fold Machine - An Exact, Parameter-Free and Machine-Closed Derivation of Classical Computational Science from Smithian Fold Theory; From Fold to Consciousness: An Exact, Zero-Parameter and Machine-Closed Foundational Reconstruction of Consciousness and Cognitive Science from 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. Full paper v1.1 — supersedes the pre-paper (From One Axiom to Master-Level Chess — and the Law Inside Neural Networks). Built from scratch by one woman, working alone, in under twenty-four accumulated hours: where a score falls short it marks an implementation gap at measurement time, never a limit of the mathematics — the gains between releases are the finding. v1.4 adds the fold eye (vision as exact integer Walsh spectra, self-certified by integer Parseval per image, recognition of seen images with no image model in the loop) and the graduation score (blind head-to-head vs the teacher, tallied per question-territory; the teacher retires as wins cross the majority lock) -- and documents the 2026 convergence: DeepSeek Engram arrives at deterministically-addressed exact memory from the gradient side, and two independent results place the optimal curriculum at p = 1/2, the fold lock. v1.6: the full omnimodal engine (the voice via Kokoro, the fold ear -- sound as Parseval-certified integer Walsh spectra, video composed from frames + sound), speaker-transparent reasoning threads, and 32/32 end-to-end empirical verification of the entire architecture including persistence across process death. v1.7: removal-proof omnimodality, measured -- every supporting model is a teacher with an exit: a sound taught once by the synthesis teacher is re-spoken from the engine's own exact counted record in 0.00s with no model; a sound heard once is recognized natively with no transcriber; 34/34 end-to-end verification. v1.9: zero-model perceptual learning (the human observer -- a novel image learned and re-recognized at share 1.00 with no model in the loop); agentic self-knowledge (the observer reads the engine's own source, measured); the hourly progress instrument with a committed pre-boot birth line; one-tap y/n closure. v2.0 (flight-ready): the full modern-agent toolkit (live web search/fetch, paginated reading, in-file grep -- every call held as a training trace), the 43-domain everything-curriculum under the fold-only law, SOTA 1-1 benching on the public MMLU test split with the newborn baseline committed, generation closure (the Learning Law reaches generate() itself), and 36/36 end-to-end verification. v2.1: the ReAct law (reason-act-observe enforced in-turn; narrated intent without an act is detected and forced), reasoning trained on the observer's NATIVE thinking tokens (STaR-gated) with both minds' full thinking streamed to the user, and document intake (a sent file is reading -- inboxed, counted, persistent). Three connected results and the architecture they force. First, a pre-registered, self-certifying spectral instrument shows trained neural-network weights carry placement-law in the dyadic (Walsh) basis: 18/18 unanimous on validated released models; the law concentrated in transformer expansion projections and token embeddings across three unrelated architectures (up to 230x chance in GPT-2), attention at chance; strictly training-caused (He-initialised controls at 1.0x); surviving 4-bit deployment quantization. A recipe map from 124M to one trillion parameters shows the law tracks training recipe, not scale or architecture — strongest carrier DeepSeek-R1-671B at 43–47x — and loud-recipe weights transform under the fold's transformation group exactly as solved game-theoretic value fields do. Second, the "learned similarity space" is a counted object: word kinship as exact co-occurrence shares reproduces semantic family structure (quark → lepton, neutrino, proton) with zero parameters and zero gradients. Third, UnisonAI: a complete language architecture in which every LLM mechanism — memory, attention, similarity, learning, prediction, generation — is replaced by a machine-verified law of the Smithian Fold Theory, zero trained parameters end to end. On identical held-out text the fold-native engine outperformed its trained transformer twin (cross-entropy 1.289 vs 1.888) after reading the corpus once (26 seconds) against 48,000 gradient readings (21 minutes per seed). Deployed as a live, continuously-learning agent whose teaching loop also runs autonomously: a teacher model asks, judges, and closes the learning law itself, and the engine self-plays against its own held lessons. Negative results reported in full with their scopes. Companion to The Smithian Fold Theory of Everything (DOI: 10.5281/zenodo.21182469; 307 suites, 1,844 forced checks, 0 failures). Engine and records: github.com/MettaMazza/UnisonAI and github.com/MettaMazza/Smithian-Fold-Theory-Of-Everything.
Jin Ah Seo, Kun Hwa Lee, Vijayan Sugumaran, Jo Yeon Park · 5 authors
We build and evaluate a concrete Zero-Knowledge Machine Learning (ZKML)-based pipeline for epidemic diagnosis and show that it can enforce computational integrity without exposing raw medical data in a Web3 setting. In response to security challenges posed by centralized data handling in medical AI applications, particularly during public health crises such as COVID-19, ZKML offers a privacy-preserving alternative by combining machine learning and Zero-Knowledge Proofs (ZKP). We experimentally applied ZKML to a CNN (Convolutional Neural Networks)-based COVID-19 diagnostic model, achieving 87% accuracy and 0.35 loss. All proof generation and verification processes were executed entirely off-chain, with the verified outputs represented as committed public_vals recorded on-chain via smart contracts. To ensure authenticity, the system enforces dual ECDSA signature verification from both the model provider and the data provider. This mechanism prevents unauthorized submissions and confirms the validity of the result before it is stored on-chain. The system was tested under both normal and adversarial conditions, demonstrating robust and reliable operation. By enabling decentralized trust and self-sovereign control over data, this architecture aligns well with Web3 principles. The results indicate that ZKML can support the development of privacy-preserving and verifiable AI systems.
Open access
Adversarial Robustness in Machine Learning
Privacy-Preserving Technologies in Data
Artificial Intelligence in Healthcare and Education
This preprint formalizes the Cross-Agent Governance Alignment (CAGA) problem: the challenge of verifying mutual governance compatibility between autonomous AI agents operating under distinct organizational policy regimes, without disclosing proprietary governance structures. As AI agents increasingly coordinate across institutional boundaries in regulated industries (healthcare, finance, cross-border data exchange, supply chains), existing governance models prove insufficient. Current frameworks assume either a single organizational authority or full policy transparency between participants. Neither assumption holds in multi-stakeholder settings where governance constraints encode confidential risk tolerances, regulatory interpretations, and competitive strategy. This paper: Defines governance domains and cross-domain interactions in formal terms, adopting the triadic verdict space (ALLOW, DENY, ABSTAIN) of the execution-time authorization framework Introduces the governance alignment predicate Φ(Dᵢ, Dⱼ, τ) Formalizes the CAGA problem under an honest-but-curious threat model Identifies required solution properties spanning correctness, privacy, determinism, evidentiary sufficiency, and composable security, including the requirement that alignment protocols produce tamper-evident authorization artifacts sufficient for independent third-party replay, consistent with the Replay requirement of the Five Tests Standard (5TS) Demonstrates that CAGA is irreducible to existing paradigms, including agent communication protocols, federated learning, secure multi-party computation, single-organization governance architectures, and blockchain-based transparency systems We argue that CAGA constitutes a zero-knowledge coordination problem at the intersection of AI governance, cryptographic protocol design, and multi-agent systems. The paper deliberately stops at problem formalization and does not disclose protocol constructions or implementation mechanisms. By precisely defining the problem space and evaluation criteria, this work establishes the foundation for rigorous solution development and provides a formal framework against which candidate governance-alignment protocols can be assessed. Version 1.1 (July 2026) retitles the paper to make explicit that CAGA is formalized as a zero-knowledge coordination problem for cross-organizational AI governance; aligns terminology with the Five Tests Standard (5TS) v1.2.0 and the FERZ authorization-artifact vocabulary; adopts the triadic verdict space in the governance domain formalization; and adds a companion reference to Execution-Time Authorization for AI Agents (v2.1), which develops the formal architecture of the single-domain authorization boundary. The problem formalization, threat model, and irreducibility argument are unchanged from the February 2026 release (v1.0). Keywords: AI governance, multi-agent systems, zero-knowledge proofs, cross-organizational coordination, governance alignment, deterministic governance, authorization boundaries, authorization artifacts, Five Tests Standard
In this project, I provide a complete, human-readable description for every one of Mathlib4's 9,150 modules — the mathematics library of the Lean 4 proof assistant — stating what each module contains, who uses it, and, wherever the names alone would leave it ambiguous, how it differs from its similarly named neighbors. Coverage is total rather than representative: every directory and every file, described against one fixed, fully specified reference snapshot, released as an independent, open-source resource for the Lean and Mathlib community — not an official product of either. Every entry in this glossary, without exception, is checked against the actual Mathlib4 source at the reference snapshot (Lean 4.29.1, Mathlib4 commit 1ad783f9bf, 2026-05-09): of 9,150 entries, 9,107 carry Complete status and 43 carry Benchmark Theorem status; zero are Pending, and zero are Needs Review. Ismail's Glossary covers the full Mathlib4 hierarchy — 1,129 directories and 8,021 files across six depth levels, spanning all 32 of Mathlib's top-level mathematical domains, from algebra and analysis to category theory and measure theory. Each entry carries six structured fields (path, name, type, parent path, depth, description), so the same data serves a human reader and a retrieval pipeline equally well.The Glossary JSON. The complete dataset, all 9,150 entries, in machine-readable form for any AI platform or retrieval pipeline.The RAG JSON. A flat, embedding-ready export with each entry pre-merged into a single field, for retrieval-augmented-generation systems that want a drop-in data source.The Claude Skill. A self-contained bundle that installs the glossary as an active, queryable reference inside Claude, so Mathlib navigation answers are grounded in current data rather than a language model's frozen training-time memory.The Master Spreadsheet. The live, community-editable source of truth, with a static snapshot published alongside it for anyone who needs a fixed, citable copy.The Interactive Website. A searchable glossary tree plus a dedicated visual Atlas of all 32 top-level domains, built for orientation rather than lookup, alongside a Lean 4 syntax reference and a getting-started guide. To this project's knowledge, no existing Mathlib tool — declaration search engine, in-editor tactic, or auto-generated documentation — provides complete, structural, plain-language coverage of the library at this depth; each presupposes that the user already knows, at least approximately, what they are looking for. All data is provided in full transparency and community contribution is actively encouraged: the complete glossary, every deliverable described above, and the moderated contribution workflow are at github.com/M-Ismail-ZA/IsmailsGlossary. For any feedback, corrections, or collaboration, please contact me via the email address listed on the paper.
Open access
2 source records
Mathematics, Computing, and Information Processing
CapIX is a decentralized physical infrastructure network (DePIN) designed to aggregate volatile, untrusted consumer-grade hardware and public container fleets into a single high-performance compute fabric. This technical report presents a practical dual-runtime execution model that addresses the core multi-tenant security challenges on untrusted permissionless hosts. On hardware-virtualization-capable bare-metal nodes, CapIX deploys lightweight micro-VMs isolated by hardware Trusted Execution Environments (TEEs). On standard unprivileged edge containers, it utilizes strict process-level sandboxing, syscall filtering allow-lists, and direct GPU passthrough to achieve native CUDA/ROCm execution performance. The system coordinates fleet routing using a transparent, multi-factor weighted geometric scoring engine resistant to economic manipulation. To achieve scalable, lightweight state verification without the prohibitive overhead of full zero-knowledge proofs on every job, the architecture introduces an interactive bisection fraud game that isolates deterministic instruction disputes to a single step for targeted zkVM proof generation, relying on probabilistic canary spot-checking for non-deterministic workloads.
Open access
2 source records
Security and Verification in Computing
Physical Unclonable Functions (PUFs) and Hardware Security
[Depreciated and replaced by V3] The former monolithic corpus is replaced by the complete V3 branch-paper series: There Is No Nothing: A Premise-Free Operational Foundation and an Open Verification Platform for Smithian Fold Theory; From Nothing to Fold: A Premise-Free, Parameter-Free and Machine-Closed Foundation for Smithian Fold Theory; From Fold to Mathematics: An Exact, Parameter-Free and Machine-Closed Derivation of Mathematical Foundations from Smithian Fold Theory; From Distinction to Information: An Exact, Parameter-Free and Machine-Closed Derivation of Information Science from Smithian Fold Theory; After Turing: The Fold Machine - An Exact, Parameter-Free and Machine-Closed Derivation of Classical Computational Science from Smithian Fold Theory; The Quantum Fold Machine - An Exact, Parameter-Free and Machine-Closed Derivation of Reversible and Quantum Computation from Smithian Fold Theory; From Fold to Physics: An Exact, Parameter-Free and Machine-Closed Reconstruction of Physical Science from Smithian Fold Theory; From Fold to Chemistry: An Exact, Parameter-Free and Machine-Closed Reconstruction of Chemical Science from Smithian Fold Theory; From Fold to Materials: An Exact, Parameter-Free and Machine-Closed Reconstruction of Materials Science from Smithian Fold Theory; From Fold to Life: An Exact, Zero-Parameter and Machine-Closed Foundational Reconstruction of Biology and Life Sciences from Smithian Fold Theory; From Fold to Medicine: An Exact, Zero-Parameter and Machine-Closed Foundational Reconstruction of Medicine and Health Sciences from Smithian Fold Theory; From Fold to Consciousness: An Exact, Zero-Parameter and Machine-Closed Foundational Reconstruction of Consciousness and Cognitive Science from Smithian Fold Theory; From One World to Earth: An Exact, Zero-Parameter and Machine-Closed Foundational Reconstruction of Earth and Environmental Sciences from Smithian Fold Theory; From One Sky to Cosmos: An Exact, Zero-Parameter and Machine-Closed Foundational Reconstruction of Astronomy and Cosmology from Smithian Fold Theory; From One Relation to Society: An Exact, Zero-Parameter and Machine-Closed Foundational Reconstruction of Social and Collective Sciences from Smithian Fold Theory; From One Law to a Working World: An Exact, Zero-Parameter and Machine-Closed Foundation for Engineering Translation from 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. The complete theory of everything with ZERO free parameters and zero axioms — every fundamental constant derived, counted, and machine-verified from one object (the One) and one operation (the fold), in exact arithmetic, reproducible in one command. Headline results: The fine-structure constant, EXACT: 1/α = 503846395469/3676744786 = 137.035999177180855… (0.009σ from CODATA 2022; the next digits …177181 and the resolution of the 5.4σ Rb/Cs discrepancy called in advance) Koide relation Q = 2/3 exactly (5 digits); quark ratios to 0.005–0.09%; mp/me to 0.0094%; (MPlanck/mp)² = 2¹²⁷; Higgs mH = v/2, λ = 1/8 All eight nuclear magic numbers exactly, zero fitted couplings → the island of stability at element 126 (SMITHIUM, Z=126, N=184, A=310), the periodic table ending at element 137, the chemistry of element 126 written before its synthesis Neutrinos: Σmν = 0.0583 eV (normal ordering, pre-registered); the neutrino is Majorana — neutrinoless double-beta decay MUST occur Cosmology: dark/baryon = 279/52 = 5.3653 (measured 5.3643); the Hubble tension = 3305/3048 = 1.0843175 (measured 1.0843230); w = −1 exactly; the cosmological-constant problem dissolved; dark matter identified (a many-body neutral baryon of two new forces — no WIMP recoil, ever) Two new forces predicted (couplings 4/5 and 6/7, 24+48 carriers, twelve new fermions with fully derived spectra) with exact grand unification at unison Kolmogorov 2/3 & 5/3 = the strong coupling; Gutenberg–Richter b = 1 = Zipf; the Parker Solar Probe 400 keV proton cutoff computed with zero local inputs (399.714 keV) Mathematics: Collatz as a forced contraction; Goldbach as fold antipodes; the Riemann critical line as the unique self-antipodal axis; Artin's density counted Pre-registered, falsifiable: a decision-dated prediction ledger where every row names its kill condition — plus a zero-parameter chess engine as a computing demonstration Reproduce everything: clone the repository, run make -C verify prove — 306 proof suites, 1,832 forced checks, requiring only a C compiler. An engine that HALTS on any fitted value enforces zero-parameters mechanically. How this began. This theory was not an attempt at a theory of everything. Its author set out to answer a prior question — what mathematics actually is — and, finding that zero, the negatives, and the continuum are inherited conveniences rather than derived truths, rebuilt mathematics from the only thing no observer can be without: the observation itself (the One), observing itself and the world (the fold), assuming nothing else, because the empirical world contains no no-grape and no negative-grape. The physics in this paper was not constructed on top of that foundation; it fell out of it. What follows is the record of how far "no assumptions" reaches — which turned out to be all the way. On the numbers this theory declines to use. Before the physics, the arithmetic — because this theory works entirely in the exact positive rationals of the interval (0, 1], with no zero, no negative quantities, no irrational and no imaginary values anywhere on the derivation side, and a reader trained on the modern continuum should know that this is a return, not a heresy. Humanity counted, surveyed, built, traded, and predicted eclipses for thousands of years with no zero-as-number: Babylonian and Egyptian mathematics had none, and Greek mathematics — the mathematics of Euclid and Archimedes, sufficient for a geometry that stood alone for two millennia — rejected the void as a number outright. Zero as an arithmetic object enters with Brahmagupta around 628 CE, travels through the Islamic mathematicians into Fibonacci's *Liber Abaci* in 1202, and is then resisted in Europe for centuries: Florence banned the new numerals in 1299, and as late as the sixteenth century Cardano was calling negative roots "fictitious." Zero and the negatives only settle into full European legitimacy across the 1600s–1800s. The irrationals are later still as numbers: the Pythagoreans met the incommensurability of √2 as a crisis and confined such magnitudes to geometry, where they stayed — ratios of lengths, not numbers — until Dedekind and Cantor constructed the real line in 1872. The continuum physics inherited is a nineteenth-century formalism, adopted for convenience of analysis, not a primordial necessity discovered in nature. No measurement has ever produced an irrational number; no detector has ever clicked zero times in a way that distinguishes absence from the absence of a detector. This theory takes that history at face value: the domain is what humanity always actually computed with — exact ratios of a whole — and the results below are the demonstration that this domain is not merely philosophically tidier but sufficient for the constants of nature, which the continuum, with its ~26 fitted dials, never derived at all. The foundation. The theory has zero free parameters and, in the precise sense proven in its own engine, zero axioms: its single premise is itself a machine-checked theorem. Given only that there is not nothing, the One (the whole, 1), its domain (0, 1], and the fold — the doubling map x → x + x with whole Ones cast out — are forced: the engine enumerates every parameter-free self-map of bounded size, runs them, and proves the fold is the unique generator (non-injective, recurrent, and — strengthened beyond any proxy — covering the entire residue class of its orbits, verified by an independent census). The two structural generators of everything that follows, binary 2 and colour 3, are not inputs: they are read off the fold's own period spectrum as its two smallest orbit periods. Every downstream quantity is then forced, counted, derived, and verified from these; the engine halts — by construction, provably, with exit code 1 — the moment any fitted, chosen, or untraceable value is introduced. Measured values appear only on the comparison side, never inside a derivation. Every number below is exact rational arithmetic; the entire corpus recomputes from the One in one command on a bare C compiler. The fine-structure constant, exact. The covering ladder of the fine-structure constant is counted and it terminates: each self-similar order promotes exactly one of the covering cube's three directions from the down-depth 5 to the up-depth 7 (5³ → 5²·7 → 5·7² → 7³ — one successor per rung, no choice anywhere, four rungs = colour + 1, then no successor exists). Read to the end of its own structure, 1/α = 503846395469/3676744786 = 137.035999177180855… — exactly. Against CODATA 2022 (137.035999177(21)): 0.009σ. The leading order 34259/250 = 2⁷ + 3²(251/250) is itself six parts per billion from measurement; the orders collapse by more than a thousandfold per rung; and the terminal value stands as, to our knowledge, the first exact closed value of α ever stated. Two pre-registered calls follow (Part XVI): the digits at 2×10⁻¹¹ are …177181, and the live 5.4σ Rb/Cs photon-recoil discrepancy resolves at 137.0359991772 — the fold's value lies 2.6σ below the rubidium result and 4.9σ above the caesium one. The assembly survives no substitution: nine alternative shapes, five second-order refinements, seven sub-promotions, four mis-built covering volumes, and every generator mutation are each machine-rejected. Mass. The charged-lepton cubic is the m = 3 case of one emergence
Modern supply chain management systems increasingly rely on distributed architectures to ensure transparency, integrity, and trust between participants. Blockchain technology provides a promising foundation for such systems; however, traditional consensus mechanisms introduce high computational overhead, energy inefficiency, and privacy risks. These limitations are particularly critical for small and medium-sized enterprises (SMEs) with constrained computational resources, that they are using to expand on their traditional informational systems and not to integrate distributed technologies into the work process, as setup process for blockchain tools is more complex than centralized approach. This paper proposes a private, dockerized blockchain architecture for supply chain management that combines the Proof of Friendship (PoF) consensus mechanism with Zero-Knowledge Proofs (ZKP). By integrating a private, dockerized framework with the Proof of Friendship consensus and Zero-Knowledge Proofs, this architecture enables resource-constrained enterprises to achieve a high-performance decentralized network that simultaneously ensures sub-second transaction validation through social trust metrics, robust protection of competitive business intelligence via cryptographic privacy, and seamless cross-platform deployment through containerization, ultimately overcoming the traditional trade-offs between system transparency, operational cost, and data confidentiality in global trade. PoF extends Proof of Stake by incorporating social trust indicators, including transaction success rate and geographic diversity of validators, enabling resource-efficient and decentralized consensus. ZKP mechanisms are integrated through an off-chain prover module, allowing transaction correctness to be verified without revealing sensitive business data. The proposed approach enhances cybersecurity, data confidentiality, and system scalability while reducing computational costs. Simulation results demonstrate improved resistance to Sybil attacks, reduced validator centralization, and acceptable transaction latency for corporate blockchain deployments.
Modern supply chain management systems increasingly rely on distributed architectures to ensure transparency, integrity, and trust between participants. Blockchain technology provides a promising foundation for such systems; however, traditional consensus mechanisms introduce high computational overhead, energy inefficiency, and privacy risks. These limitations are particularly critical for small and medium-sized enterprises (SMEs) with constrained computational resources, that they are using to expand on their traditional informational systems and not to integrate distributed technologies into the work process, as setup process for blockchain tools is more complex than centralized approach. This paper proposes a private, dockerized blockchain architecture for supply chain management that combines the Proof of Friendship (PoF) consensus mechanism with Zero-Knowledge Proofs (ZKP). By integrating a private, dockerized framework with the Proof of Friendship consensus and Zero-Knowledge Proofs, this architecture enables resource-constrained enterprises to achieve a high-performance decentralized network that simultaneously ensures sub-second transaction validation through social trust metrics, robust protection of competitive business intelligence via cryptographic privacy, and seamless cross-platform deployment through containerization, ultimately overcoming the traditional trade-offs between system transparency, operational cost, and data confidentiality in global trade. PoF extends Proof of Stake by incorporating social trust indicators, including transaction success rate and geographic diversity of validators, enabling resource-efficient and decentralized consensus. ZKP mechanisms are integrated through an off-chain prover module, allowing transaction correctness to be verified without revealing sensitive business data. The proposed approach enhances cybersecurity, data confidentiality, and system scalability while reducing computational costs. Simulation results demonstrate improved resistance to Sybil attacks, reduced validator centralization, and acceptable transaction latency for corporate blockchain deployments.
Motion prediction –forecasting the future trajectories of surrounding vehicles, pedestrians and cyclists is a safety-critical component of the autonomous-driving pipeline that must run in real time on embedded hardware. State-of-the-art predictors, however, are trained on compute clusters and are too large to run on a single consumer GPU, placing both ends of the contemporary pipeline out of reach for an individual researcher. This thesis asks how small a competitive trajectory predictor can be made before its accuracy degrades, and whether the lost accuracy can be recovered through knowledge distillation without enlarging the model or worsening the calibration a downstream planner depends on. The study uses HiVT, a transformer-based Laplace-mixture predictor that is small enough to be both trained and run on a single GPU, evaluated on the Argoverse 1 benchmark. The accuracy–capacity trade-off is first characterised by sweeping the embedding width (128, 64, 32, 16) and locating the point at which a from-scratch student falls measurably below the teacher. The mode-permutation problem is then identified: because HiVT trains its mixture modes with a winner-takes-all loss, the mode slots of two independently trained models do not correspond, so any distillation term that aligns modes by index supervises the student with self-contradictory targets. To resolve this, a permutation-invariant mixture negative-log-likelihood objective is derived that treats the teacher’s modes as an order-free set of soft targets and supports unequal mode counts, with a proof of invariance. Experiments show that a mean-target variant of this objective recovers roughly 84% of the HiVT 32→ HiVT-64 capacity gap (−9.2% minFDE over a matched non-distilled baseline) at zero added inference cost, but degrades full-distribution calibration (mixture NLL +41%, calibration error 5×) by discarding the teacher’s predictive variance. A distribution-matching objective that also distils the teacher’s per-mode scales removes this penalty entirely, leaving the student better calibrated than both the non distilled baseline and the teacher while retaining the full geometric gain. The benefit grows as the student shrinks: at width 16 (55× smaller than the teacher) distribution-matching distillation improves minFDE by −22.7%—roughly 2.5× the width-32 gain—recovering ∼81% of the width-16→width-32 gap, with calibration improving rather than degrading. Distillation thus buys close to a full size-class of accuracy for free, and most where capacity is scarcest. A final efficiency analysis quantifies the deployment frontier: parameter and memory savings are fixed and unconditional (15× at width 32, 55× at width 16), whereas the single-scene latency speed-up is far sublinear and batch-dependent (on CPU ∼3× online, rising to ∼5.5× under modest batching), locating the compression benefit primarily in memory footprint. Overall, the answer to how small a competitive HiVT can be made is encouraging: with a permutation-invariant, calibration-preserving distillation loss, a 55×-smaller student reaches roughly the accuracy of an un-distilled model nearly four times its size at no calibration cost.
This paper establishes an exact information-theoretic duality between econometric regression topology and statistical mechanics. We demonstrate that an autonomous, data-driven Quadruple Test, grounded in the Factor Hierarchy Law, can blindly detect, precisely quantify, and correctly classify thermodynamic phase boundaries in perfect mathematical equivalence with the Ehrenfest paradigm, without any prior knowledge of Free Energy functions. The validation platform is the two-dimensional Ising model, one of the few systems in statistical physics possessing a mathematically rigorous exact solution (Onsager, 1944; Yang, 1952). Verification proceeds in two logical stages: Stage A (pristine algebraic validation) on Onsager's exact solution, and Stage B (stochastic robustness testing) on finite-lattice Monte Carlo simulations. We openly declare that because the data derives from the known Onsager-Yang formula, the contribution is not an independent empirical discovery of new physics, but rather the rigorous proof of an exact informational duality between two independent frameworks. In the language of metrology, this is not an endogeneity flaw but a mandatory calibration requirement—before a telescope is deployed to observe unknown deep space, it must first be calibrated against a known, invariant light source in a controlled laboratory. Positive Controls and Asymptotic Convergence (9 items): Exhaustive search blindly locks onto the critical temperature at Tc = 2.260 (deviation 0.009 at a coarse step size of 0.01). A grid-refinement study demonstrates monotonic convergence: the deviation shrinks to zero within 6-decimal precision at a step size of 0.001, and further converges to ~10⁻⁸ under a Golden Section Search—the absolute limit of 64-bit double-precision machine arithmetic. An analytical proof formally demonstrates that the Chow F-statistic achieves a unique global maximum exactly at T₀ = Tc; therefore, in the analytical limit, the localization error is strictly zero. The Chow test at Tc yields F = 118,074 against a null control of F = 3.12 (a 266-fold difference), with permutation test p = 0.000. The interaction term is overwhelmingly significant (p = 0.000, ΔR² = 0.991). A symmetry-breaking regime switch at the external field boundary h = 0 is detected with Chow F = 989.41 (p = 0.000). Interaction R² peaks sharply at Tc (deviation 0.03). Multi-response-function validation (specific heat C, nearest-neighbor spin correlation) and anisotropic validation (three Jx/Jy ratios) all lock onto their respective theoretical Tc values with deviations under 0.007. A synthetic double-break dataset is tested with both breaks successfully detected. Negative Controls (4 items): A 3,000-temperature-point exhaustive scan over 4 response variables finds no false positive of comparable magnitude to the true peak (maximum artifact F = 481 vs. Tc peak F = 78,754,162; a signal-to-noise ratio of 164,000:1). Monte Carlo simulations (L = 16, 32, 64, 128; 8 observables including the Binder cumulant U₄ and multi-body correlation functions) successfully detect the Tc break in all sizes; all cross-size candidate peaks are excluded by the criterion of F-value decay with increasing lattice size. Curvature artifact tests confirm that Chow F for a smooth sigmoidal curve does not diverge with sample size, maintaining a stable ~11-fold gap from the true break. Robustness (4 items): Under 10% Gaussian noise, Chow F remains at 22.3. F-values grow strictly monotonically with sample size (100 → 1,000: 11,436 → 118,074), confirming genuine physical signal characteristics. F-values grow overall with lattice size L (L = 16 → 128: 170.9 → 289.2, Kendall τ = 0.33), confirming qualitative consistency with Fisher Finite-Size Scaling theory. Detection accuracy remains invariant under anisotropic conditions. Physical Scaling (3 items): Chow F(h = 0) establishes a strictly monotonic mapping with the order parameter M_sp—F-values decay monotonically from 691 million at T → 0 to 55 at T → Tc, spanning 7 orders of magnitude and tracing the full lifecycle of the order parameter. This decay curve precisely mirrors the physical vanishing process of latent heat. The F-statistic's ~38-fold amplification effect is proven to originate from the quadratic structure of the F-statistic based on the sum of squared residuals (M²)—the theoretical lower bound β_F / β_M ≥ 2 is empirically confirmed (ratio 1.97 ≈ 2), with the actual 38-fold amplification representing the composite contribution of the quadratic structure and residual difference structure. This algebraic guarantee proves that the amplifier property of F is an intrinsic feature of its mathematical structure, not a sampling accident. Interaction R² peaks at Tc at 0.9992 (deviation 0.03). Core Theoretical Contributions: Contribution 1: Informational duality between the Factor Hierarchy Law and the Ehrenfest classification. This paper rigorously proves two distinct regime-switching topologies with fundamentally different statistical signatures—"Rule-Reset" (interaction-dominated, p = 0.000, ΔR² = 0.991) and "Direction-Reversal" (intercept-jump-dominated, interaction p = 0.978). Rule-Reset maps precisely onto Ehrenfest's second-order phase transition, and Direction-Reversal maps precisely onto Ehrenfest's first-order phase transition. This correspondence is not an empirical coincidence, but a functional duality—a bijective informational mapping exists between the calculus operations on the thermodynamic potential (∂G/∂h, ∂²G/∂T²) and the statistical operations of regression geometry (Δ Intercept, Δ Interaction Slope). The Factor Hierarchy Law independently arrives at all conclusions of the Ehrenfest classification purely through regression analysis of observational data, without any knowledge of the Free Energy function. Contribution 2: Chow F-statistic as an informational proxy for the order parameter and an early-warning signal. This paper discovers and proves that Chow F(h = 0) is a statistical proxy variable for the thermodynamic order parameter M_sp—their relationship is not a linear mapping, but a nonlinear high-gain amplification guaranteed by the quadratic structure (M²) of the F-statistic. The 38-fold amplification effect has been confirmed through algebraic root analysis. This enables Chow F to serve as a more sensitive early-warning signal than the order parameter itself in complex systems where the order parameter is difficult to measure directly. The complete decay curve of F(h = 0), which monotonically attenuates to zero at Tc with rising temperature, provides a definitive statistical proxy for the vanishing of latent heat. Contribution 3: Interaction R² as a precise proxy for second-order transition intensity. Interaction effect incremental R² peaks at Tc at 0.9992, with a deviation of only 0.03. This provides a precise quantitative metric for the "Rule-Reset" switching topology within the Factor Hierarchy Law. Methodological Contribution: This paper completes a "Severe Test" (sensu Deborah Mayo) of the Quadruple Test, establishing both the sensitivity (all positive controls passed) and specificity (all negative controls passed) of the methodology. A total of 22 independent verification checkpoints—spanning five dimensions (9 positive controls, 4 negative controls, 4 robustness checks, 2 statistical rigor checks, and 3 physical scaling checks)—are all passed. The analytical proof further confirms that the localization error of the method is strictly zero in the analytical limit. Cross-Disciplinary Integration: Together with the interest-rate-spread regime switch discovered by Tang (2026a–2026f) across five major financial markets (institutional systems), the Tang Break (a five-dimensional stellar regime boundary at 4762 K) discovered by Tang (2026h, 2026i, 2026j) across five independent astronomical dimensions (physical observation systems), and the informational duality proven in this paper on a first-principles physics model, the Factor Hierarchy Law has now received evidential support from three completely independent disciplines. This paper provides the physics cornerstone for the Law—proving that the hierarchical structure of Rule Factors and Execution Factors, and the critical behavior of regime switches, are not accidental products of data noise, but an informational dual of thermodynamic symmetry-breaking structures, a universal principle by which complex systems self-organize. Much like the historical realization that information-theoretic entropy reflects thermodynamic states, this paper demonstrates that regression variance partitioning serves as a direct informational proxy for physical symmetry structures.