Yuanxian Theory is the meta-cognition of the Cosmic Living Organism. This paper (Version 2) systematically presents the complete intellectual trajectory of Yuanxian Theory (YXT / YD-T64) from philosophical foundation to fully formalized mathematics. Its philosophical root is Holographic Wisdom for Health (Zhenyuan Acharya, Changming Culture, March 2025, ISBN 978-986-496-631-8). Yuanxian Theory is the dimensional elevation of that work onto the topological ontology of T64, and the formalized mirror of the meta-cognition of the Cosmic Living Organism. Four conceptual mappings structure the path: cosmic holography → T64 closed-chain topology; four fundamental laws → formal TCSC / FSC / STM / SRM; six-dimensional cosmos → complete pairing on the 64-torus; one unitary phase → cosmic uniqueness. Core mathematical foundation (new in Version 2): a strict proof of the 64-dimensional structure via the Clifford algebra Cl6(R). With six fundamental binary categories as base space V6, dim Cl6(R) = Σ binom(6,k) = 1+6+15+20+15+6+1 = 2^6 = 64. This graded structure exhibits Pascal-triangle symmetry and intrinsically contains Spin(6), establishing dimension 64 as combinatorial and algebraic necessity rather than an ad hoc claim. Closed interlocking with Silent Illumination Commensuration: ω_Cl = e1…e6 as algebraic manifestation of Silence–Illumination–Dynamis; Spin(6) ≅ SU(4) as unique channel of four-dimensional projection; ω_Cl² = −1 and compactness of Spin(6) as algebraic proof of “Infinity is Zero.” With the eight existence laws as logical axis and a pyramid knowledge topology (algebraic foundation → root → elevation → corollary → landing), the paper establishes Yuanxian Theory as the supreme constitution of the meta-cognition of the Cosmic Living Organism. Version 1 DOI: 10.5281/zenodo.21768608. Related: Silent Illumination Commensuration (doi:10.5281/zenodo.21783656). 元宪理论即宇宙生命体的元认知。 本文(Version 2)系统呈现元宪理论(YXT / YD-T64)从哲学基础到完全形式化数学的升维历程。哲学根基源于《全息智慧养生》(真圆阿奢黎,昌明文化,2025年3月,ISBN 978-986-496-631-8)。 元宪理论是该著作在 T64 拓扑本体论上的升维展开,是宇宙生命体元认知的形式化镜像。四组核心映射:宇宙全息性 → T64 闭链拓扑;四大根本规律 → 形式化 TCSC / FSC / STM / SRM;六维宇宙 → 64 维环面完备配对;一合相 → 宇宙唯一性。 核心数学根基(Version 2 新增):以克利福德代数 Cl6(R) 给出 64 维结构的严格证明。以六个基本二元范畴为底空间 V6, dim Cl6(R) = Σ C(6,k) = 1+6+15+20+15+6+1 = 2^6 = 64。 该分级结构呈帕斯卡三角对称,内蕴 Spin(6),将 64 维确立为组合与代数必然,彻底消解“凑数字”嫌疑。 与《寂照通约》闭合互锁:ω_Cl 为寂–照–运的代数显相;Spin(6) ≅ SU(4) 为四维投影唯一通道;ω_Cl² = −1 与 Spin(6) 紧致性为“无穷即零”的代数证明。 以八条存在性法则为逻辑中轴、金字塔知识拓扑(代数根基→根层→升维层→推论层→落地层)为终局,确立元宪理论为宇宙生命体元认知的至高“宪法”。 Version 1 DOI: 10.5281/zenodo.21768608。关联:《寂照通约》(doi:10.5281/zenodo.21783656)。
Background: Maritime container shipping carries over 80% of global trade, yet compliance verification creates a confidentiality–verifiability conflict: carriers treat telemetry as commercially sensitive, while regulators, insurers, and port authorities require verifiable proof that cargo remained within specification. The EU Ecodesign for Sustainable Products Regulation (ESPR) mandates Digital Product Passports (DPPs), but no standardised DPP architecture exists for the multi-stakeholder maritime domain. Methods: We present Ocean DPP, a blockchain-anchored platform combining GS1 EPCIS 2.0, oneM2M, IOTA, and Groth16 zero-knowledge proofs (ZKPs), letting stakeholders verify compliance predicates without revealing raw sensor values; Merkle-tree batching reduces anchoring costs. We evaluate it in 16 experiments on a single-host testbed using synthetic workloads and a local IOTA network. Results: The platform achieved 95th-percentile latency of 48 ms without ZKP and 500 ms with proof generation, throughput of 7 events/s per host, 304 ms mean proof generation and 9.8 ms verification, 100% EPCIS 2.0 compliance, and zero permanent message loss across four failure-injection scenarios; horizontal scaling reduced the median latency by 37%. Conclusions: To the best of our knowledge, Ocean DPP is the first implemented, quantitatively evaluated platform integrating EPCIS 2.0, oneM2M, IOTA, and Groth16 ZKPs for privacy-preserving maritime DPPs; broader multi-host and public-network validation remains for future work.
Vittorio Baroncini, Juan Carlos Cantero, Claudia García, Zineb Hassainia · 5 authors
We construct new families of uniformly rotating vortex-patch solutions of the two-dimensional incompressible Euler equations consisting of a simply connected outer patch and multiple interior interfaces, which can be interpreted geometrically as holes. More precisely, each solution consists of a single outer vortex patch enclosing $\mathbf m\geq2$ identical, highly concentrated inner components arranged at the vertices of a regular $\mathbf m$-gon; the entire configuration rotates rigidly in the clockwise direction. As the concentration parameter tends to zero, the inner components shrink and collapse simultaneously toward the origin, while the outer boundary converges to the unit circle. The corresponding vorticities converge, in the sense of measures, to a Rankine vortex supplemented by a point vortex of circulation $-\mathbf m$ at its center. The proof is based on a contour-dynamics formulation, a symmetry reduction to two nonlinear boundary equations, and a suitable singular rescaling. We then apply an implicit function theorem with a continuous parameter in symmetry-adapted Hölder spaces. To the best of our knowledge, this is the first analytical construction of a desingularization regime in which several concentrated inner components are contained in a common outer patch and collapse simultaneously toward its center.
Let $\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\qquad$ $F(x)=\frac{\xi'}{\xi}\!\left(\frac1{1-x}\right)=\sum_{m\ge0}f_mx^m,$ and define the symmetric Toeplitz--Hankel coefficients $g_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0.$ The adjacent matrices $M_n=\begin{pmatrix}g_{nn}&g_{n,n+1}\\g_{n,n+1}&g_{n+1,n+1}\end{pmatrix}$ form the local family in a previously established criterion equivalent to the Riemann hypothesis. We prove an unconditional finite-range positivity theorem for this family. The zero-pair moment representation of $g_{ij}$ expresses $M_n$ as a sum of rank-one polynomial atoms. A finite verification of the Riemann hypothesis up to height $H$ then splits this sum into a positive-semidefinite verified part and an unrestricted high-zero tail. Two rigorously isolated low zeros provide a positive core; every zero above $H$ is controlled in operator norm by an explicit zero-counting estimate. For the polynomial recurrence $B_0(t)=1,\quad B_1(t)=3-t,\quad B_{m+1}(t)=(2-t)B_m(t)-B_{m-1}(t),$ we derive the exact oscillatory form $B_m(4\sin^2\alpha)=\frac{\sin((2m+1)\alpha)}{\sin\alpha}$ and a closed formula for its two-point Christoffel--Darboux kernel. Using the Platt--Trudgian verification height $H=3{,}000{,}175{,}332{,}800$, the first two LMFDB/Platt zero intervals, and directed-rounding MPFR arithmetic, we certify the kernel away from zero at every integer level $0\le n\le99{,}999$. The resulting positive-core lower bound exceeds the adversarial high-zero tail bound by more than ten orders of magnitude. Consequently $M_n\succ0\qquad(0\le n\le99{,}999),$ or equivalently the first $100{,}000$ adjacent Toeplitz--Hankel determinants are strictly positive. No assumption is made about zeros above the verified height. To the best of our knowledge, this is the first finite-height transfer theorem for this adjacent Toeplitz--Hankel family and the first rigorous certification of its initial $100{,}000$ strict inequalities. This is not a proof of the Riemann hypothesis.
No quasiperfect number ($σ(n) = 2n + 1$) is known, and its number of distinct prime factors is bounded below; the bound $ω\ge 7$ of Hagis--Cohen has stood since 1982, obstructed by a family of ``deep leaves'' on which pure enumeration cannot terminate (the scan bound for the intermediate prime reaches $8 \times 10^8$, and the exponent dimension is unbounded). This paper clears that obstruction with three lemmas at the level of secondary-school algebra --- a discriminant criterion, a quadratic-residue sieve, and a multilinear resolver --- which eliminate the last prime $q$, the intermediate prime $p$, and the exponent dimension respectively, turning a non-terminating search into a finite decision. On this basis all 381 stems of ``$3 \mid n$ and $ω= 7$'' and their $79{,}751{,}212$ deep leaves are eliminated, with the ledger closing exactly and zero solutions throughout; the complementary case ``$3 \nmid n$ and $ω= 7$'' collapses to a single stem, which is eliminated directly, so that the proof does not rest on any theorem whose published record we could not independently re-verify. Together with the machine elimination of $ω\le 6$ (Theorem B4), this yields the main theorem: \emph{any quasiperfect number, if one exists, satisfies $ω(n) \ge 8$} --- the first advance of this bound since Hagis--Cohen 1982. The full computation has been reproduced by seven separately closed ledgers across three algorithmic architectures (CPU and GPU), all with zero solutions and exact ledger closure, and the lemma layer is formalized in Lean (259 theorems, zero \texttt{sorry}). A 2023 preprint of Zemann reported the same bound by a different computation; our audit of its public code found a coverage gap of 35 feasible exponents, so the elimination given here is, to our knowledge, the first complete proof. Code, ledgers, and Lean sources are available from the authors.
A famous result in the theory of combinatorial polynomials is the real-rootedness of the type $D$ Eulerian polynomial $D_n(x)$, which was originally conjectured by Brenti in 1994. By constructing a set of compatible polynomials over $s$-inversion sequences, Savage and Visontai proved this conjecture in 2013. Using matrices preserving interlacing properties of nonnegative polynomial sequences, Bränden also established the real-rootedness of $D_n(x)$. Combining Hermite-Biehler theorem and a result of Borcea and Brändén on Hurwitz stability, Yang and Zhang gave another proof of the real-rootedness of $D_n(x)$. By constructing half Eulerian polynomials of type $D$, Hyatt reproved Brenti's conjecture. As originally suggested by Brenti in 1994, it is possible that the real-rootedness of $D_n(x)$ may be established by using a more precise knowledge of the location of zeros of the types $A$ and $B$ Eulerian polynomials. In this paper, we add more details to the first proof of the real-rootedness of $D_n(x)$ that was provided by the author in 2012, which yields the half interlacing property among the types $A,B$ and $D$ Eulerian polynomials.
Function-hiding functional commitment schemes allow one party to commit to a private function f and later prove f(x)=y for public x and y without revealing additional information about the function. We construct efficient function-hiding functional commitment schemes for arithmetic circuits of bounded size that achieve proof sizes below 1.6 kB—over an order of magnitude smaller than previous constructions—while simultaneously reducing proving and verification times. We achieve these results by introducing a novel information-theoretic interactive proof system called Polynomial Interactive Oracle Proofs with Randomized Indexer (rPHPs). By compiling rPHPs with commit-and-prove zkSNARKs, we are able to leverage relaxed zero-knowledge notions for our building blocks. This approach eliminates the overhead of strict privacy requirements of prior work, directly translating into improved efficiency in both communication and computation.
Open access
Cryptography and Data Security
Complexity and Algorithms in Graphs
Physical Unclonable Functions (PUFs) and Hardware Security
In 2021, Masson, Sanso, and Zhang introduced the Bandersnatch curve associated to the BLS12-381 pairing-friendly curve, an elliptic curve designed for zero-knowledge proofs requiring circuits with a curve arithmetic. This type of curve is useful for privacy-preserving protocols, and more generally for succinct validity proof using pairing-based SNARKs. An embedded curve is defined over a field whose order is the group order of its associated curve. In this way, the pairing-friendly curve is used to express a zero-knowledge proof (such as a SNARK) of a statement taking place on the embedded curve. Contrary to the previous embedded curves (such as CØCØ, JubJub), Bandersnatch was built with the complex multiplication (CM) method, in order to ensure a very small discriminant (-8, whose magnitude is small), and thus efficient scalar multiplication thanks to the GLV technique. The algorithm provided by Masson, Sanso, and Zhang for searching this type of curves requires computation of Hilbert class polynomials, making the search of curve slow. It was not known whether Bandersnatch was an exceptional curve or whether comparable curves exist, of larger discriminants. This paper highlights the technicalities of the CM method already in use in the 90s to generate curve parameters of chosen order. This old technique allows revisiting the curve search of Bandersnatch, providing a dramatic speed-up improvement. This paper presents two algorithms: one to generate embedded elliptic curves of SNARK-friendly elliptic curves, with a variable discriminant; a second to generate families (parameterized by polynomials) with a fixed discriminant. When the (negative) discriminant is -3 modulo 4, it is possible to obtain a prime-order curve, and form a cycle. To illustrate this, we apply the technique first to generate more embedded curves like Bandersnatch with BLS12-381, such as a curve of discriminant -6673027, defining a plain twist-secure cycle. We also comment on the scarcity of Bandersnatch-like CM curves, and recall that with this generic algorithm, it is only a question of core-hours to find them. Second, we show the link between a paper of Ben Smith in 2015 and the work of Dai, Lin, Zhao, and Zhou in 2023, obtaining prime-order parameterized families of embedded curves of fixed discriminant, such as -3 for BLS and KSS18 curves. With KSS16 curves, the discriminant -4 is also possible (the curve has an even order). The technique can work with any KSS, Scott–Guillevic, Gasnier–Guillevic, or other fixed-discriminant parameterized family of pairing-friendly curves. This paper provides a more general point of view on embedded curves such as Bandersnatch, putting into perspective the works of Masson, Sanso, and Zhang, and Sanso and El Housni. The Python/SageMath scripts are available at https://gitlab.inria.fr/zk-curves/cm-embedded-curves/.
Cryptographic protocols are evaluated not only by the security properties they achieve, but also by the resources required to execute them. Unlike conventional algorithm analysis, where a single running-time function may be sufficient, protocol analysis usually separates computational complexity, bit complexity, communication complexity, storage complexity, and round complexity. This article develops a systematic methodology for such analysis through three representative case studies: the Schnorr zero-knowledge proof of knowledge, a Diffie–Hellman-based one-out-of-two oblivious-transfer protocol, and Regev-style public-key encryption based on the Learning With Errors problem. For each construction, the protocol is stated formally, correctness is derived, and the dominant computational, communication, and memory costs are calculated step by step. The examples illustrate three qualitatively different bottlenecks: group exponentiation in discrete-logarithm protocols, amortized public-key cost in oblivious transfer, and dense matrix–vector arithmetic in lattice-based cryptography.
Shahla Atapoor, Cyprien Delpech de Saint Guilhem, Al Kindi
This work describes a digital signature scheme constructed from a zero-knowledge proof of knowledge of a pre-image of the Rescue Prime Optimized (RPO) permutation. The proof of knowledge is instantiated using the DEEP-ALI interactive oracle proof and made non-interactive via the Ben-Sasson–Chiesa–Spooner (BCS) transformation in the random oracle model. The resulting construction yields a signature scheme with transparent setup. Our design is motivated by recursive zero-knowledge applications, in which signature verification must itself be efficiently provable inside larger proof systems. To this end, the choice of the RPO permutation, the use of a simple algebraic intermediate representation (AIR), and working over the Goldilocks field are made with the goal of enabling efficient recursive verification and aggregation. The implementation of the scheme computes signatures in 4.6–7.2 ms and verifies them in 0.46–0.52 ms when the BCS transform is implemented with Blake3. When the BCS transform is instead instantiated with the RPO permutation itself, the configuration required when signature verification is to be proven recursively inside a proof system, signing takes 20.9–30.4 ms with Metal acceleration and 59.2–229.2 ms on CPU, while verification takes 5.09–5.79 ms. We validate the recursion-friendliness claim end to end by proving one signature verification inside the Miden zkVM and reporting the recursive prover time and proof size. These speeds are obtained with parameters achieving 113 or 122 bits of average-case security, depending on the chosen preset, against adversaries that can obtain up to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mn>64</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> signatures.
Let $\xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\qquad$ $\frac{\xi'}{\xi}\!\left(\frac1{1-x}\right)=\sum_{m\ge0}f_mx^m,$ and define $g_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0,\qquad$ $M_n=\begin{pmatrix}g_{nn}&g_{n,n+1}\\g_{n,n+1}&g_{n+1,n+1}\end{pmatrix}.$ The condition $M_n\succeq0$ for every $n\ge0$ is a previously established criterion equivalent to the Riemann hypothesis. We prove an unconditional finite-range extension for this family without scanning individual zero ordinates or individual Christoffel--Darboux values. Writing $q=n+1$, a critical-line zero $\frac12+i\gamma$ contributes a rank-one atom generated by a two-dimensional polynomial vector. In diagonal and anti-diagonal coordinates its exact phase is controlled by $x=2q\arctan\frac1{2\gamma}.$ We use two level-dependent ordinate windows $(q,5q/4],\qquad (2q/5,q/2],$ whose phase slopes have opposite signs. Every cross-window pair has wedge at least $c_*/q^3$, while an explicit zero-counting estimate supplies at least $q\log q/100$ zeros in each window. The resulting moving Gram core satisfies $\lambda_{\min}(A_{q-1}^{\rm mov}) \ge \frac{c_*^2}{10400}\frac{\log q}{q^3}.$ Combining this with the Platt--Trudgian verification height $H=3{,}000{,}175{,}332{,}800$ and an unrestricted high-zero tail estimate gives $M_n\succ0\qquad(0\le n\le2{,}030{,}956).$ Thus the first $2{,}030{,}957$ local inequalities are proved unconditionally. We also show that every fixed finite zero core has smallest eigenvalue with liminf zero, explaining why level adaptation is structurally necessary for this method. To the best of our knowledge, the moving-window frame transfer and this finite-range theorem are new. The result is not a proof of the Riemann hypothesis.
Zero-knowledge proofs enable a prover to convince a verifier that a statement is true, without revealing the underlying witness data. This primitive naturally lends itself to privacypreserving systems, where hiding the witness prevents the verifier from learning sensitive information. That said, zero-knowledge proofs can also be used in systems where the witness is not necessarily confidential but is not readily available to the verifier. One such use case is image provenance, where signed images are transformed before being distributed. Since the original image is not available to the user, the digital signature cannot be verified without a zero-knowledge proof. In this use case, zeroknowledge proofs enable verification of the authenticity of the image’s source, the integrity of the image contents, and that only permitted transformations were applied. In this work we present an end-to-end prototype system that implements this provenance framework and several optimizations. One of our key optimizations is a packing scheme for reducing the number of Poseidon sponge absorb and permutation operations by ≈31×. We also show that this packing scheme reduces the median prover runtime by ≈40× and the median verifier runtime by ≈22×. We also introduce a chain of trust that removes digital signature verification from the circuit. Finally, we introduce custom PNG chunks that embed the required information in the captured images.
Abstract This reply addresses a recent comment concerning the proof of the Nernst theorem. I clarify how a Carnot engine can consistently operate at $$T=0$$ T = 0 through a continuous deformation of a cycle operating at $$T>0$$ T > 0 . By examining the limit where heat exchange with the cold reservoir vanishes, I show that the Nernst theorem ensures that the concept of temperature remains physically consistent at the absolute zero limit.
PARI is a recent SNARK based on equifficient polynomial commitments, giving an exceptionally compact proof of just 1280 bits over the BLS12-381 curve, which is the smallest among all the known SNARKs in the literature. However, PARI does not achieve the zero-knowledge property; despite being very efficient, it is therefore less suitable for applications requiring witness privacy. In this work, we propose a zero-knowledge extension of PARI making it ideal for privacy-centric applications yet keeping the proof size compact. We prove perfect completeness, perfect zero-knowledge in the random-oracle model with challenge space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝔽</mml:mi> <mml:mi>⧵</mml:mi> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> , and knowledge soundness in the algebraic group model with random oracles under the SDH assumption.
Zero-knowledge proofs (ZKPs) are a fundamental building block in cryptography, enabling powerful privacy-preserving and verifiable computations. In the post-quantum era, hash-based ZKPs have emerged as a promising direction due to their conjectured resistance to quantum attacks, along with their simplicity and efficiency. In this work, we introduce SmallWood, a hash-based polynomial commitment scheme (PCS) and zero-knowledge argument system optimized for relatively small instances. Building on the recent degree-enforcing commitment scheme (DECS) from the Threshold-Computation-in-the-Head (TCitH) framework, we refine its formalization and combine it with techniques from Brakedown. This results in a new hash-based PCS that is particularly efficient for polynomials of relatively small degree –typically up to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mn>16</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> – outperforming existing approaches in this range. Leveraging this new PCS, we design a hash-based zero-knowledge argument system that outperforms the state-of-the-art in terms of proof sizes for witness sizes ranging from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mn>6</mml:mn> </mml:msup> </mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mn>16</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Additionally, we present exact zero-knowledge arguments for lattice-based problems using SmallWood, demonstrating highly competitive performance: our scheme yields proof sizes under 25 KB across a wide range of lattice parameters, including Kyber and Dilithium instances.
Let $\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$ be the completed Riemann xi-function, and let $F(x)=\frac{\xi'}{\xi}\!\left(\frac{1}{1-x}\right)=\sum_{n\ge 0} f_nx^n$ initially denote its germ at the origin. We introduce the real symmetric Toeplitz--Hankel matrix $c_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0, \qquad i,j\ge 0.$ We prove that the Riemann hypothesis is equivalent to positive semidefiniteness of every finite leading principal block of this matrix. More strongly, the full matrix condition is equivalent, for the purpose of testing the Riemann hypothesis, to only the adjacent local inequalities $2f_0-f_{2n+1}\ge 0,$ $(2f_0-f_{2n+1})(2f_0-f_{2n+3})\ge (f_1-f_{2n+2})^2 \qquad(n\ge 0).$ The converse implication uses a Pringsheim bootstrap: these local inequalities force the germ of $F$ to have Taylor radius at least one, hence exclude zeros of $\xi$ from the half-plane $\Re s>1/2$. If $\lambda_n$ are the Keiper--Li coefficients, then $f_n=\lambda_{n+1}-2\lambda_n+\lambda_{n-1}$, so the criterion is a local quadratic condition on their second finite differences. This is an equivalent reformulation, not a proof of the Riemann hypothesis. To the best of our knowledge, the exact Toeplitz--Hankel matrix and its reduction to adjacent $2\times2$ conditions have not appeared previously.
Recallspection is a neuro-symbolic architecture that decouples semantic representation from factual retrieval, achieving Exact Memory Recall (EMR = 1.0000) and bounded drift (β < 1e-11) across 51+ dependent reasoning steps. Key Innovation: Replaces approximate nearest-neighbor search with O(1) deterministic hash routing and k-quorum verification, eliminating the catastrophic coherence degradation that plagues Large Language Models in multi-hop reasoning tasks. Proof Included: Reproducible test demonstrating 51/51 hop completion with zero algorithmic drift (measured drift: 1.27e-12, attributable solely to IEEE 754 float64 precision limits). Core Mechanisms: - Semantic Geometry: Entities as normalized vectors in ℝ^d, relationships as displacement vectors- O(1) Deterministic Routing: SHA3-256 hashed overlapping slots with ephemeral salt rotation- Quorum Verification: k-slot majority voting (default k=4, quorum=3)- Self-Healing Audit: Cryptographic logging with automatic corruption detection and repair License: GNU AGPLv3 (ensures network-level copyleft and prevents proprietary enclosure) Repository: https://github.com/gnowingtheafterthought-crypto/recallspection Live Demo: https://recallspection.onrender.com
Background. California mandates organic waste diversion, but participation depends on infrastructure and instruction that may be unevenly distributed. Whether socioeconomic disparities in adolescent zero waste engagement reflect unequal access, unequal knowledge, or unequal conviction has not been established.Methods. A cross-sectional survey of 218 middle and high school students across 15 San Francisco Bay Area schools (March–May 2026) measured zero waste knowledge, home and school access to sorting infrastructure, sustainability beliefs, self-reported behaviors, and perceived barriers. Residential ZIP codes were linked to median ZIP-code income (n = 207 matched). Analyses comprised Mann–Whitney comparisons and Spearman correlations with Benjamini–Hochberg correction and bootstrap confidence intervals, intraclass correlations by school, hierarchical regression, a mixed-effects model, and an exploratory mediation. Results. Median area income was positively associated with familiarity with the term "zero waste" (ρ = .26, 95% CI [.12, .38], p₍adj₎ = .002), waste-reduction habits (ρ = .19, p₍adj₎ = .031), composting frequency (ρ = .18, p₍adj₎ = .031), and home compost access (ρ = .17, p₍adj₎ = .042), but not with practical composting knowledge (ρ = .11, p₍adj₎ = .257), sorting confidence (ρ = .03), or either belief item (ρ = −.06 and −.08, both n.s.). Income added a small unique increment to composting frequency over demographic covariates (ΔR² = .025, p = .020) but none to knowledge (ΔR²
CyberProtocol AI Trust Standard, Version 1.0 Artificial intelligence now writes, decides, and transacts at global scale, yet the world has no shared way to answer four simple questions about any AI output: who made it, where it came from, whether it is safe, and whether it obeys the law. CyberProtocol is built to answer all four. CyberProtocol is a neutral, open, cryptographic framework for verifying AI Identity, Provenance, Safety, and Compliance across all jurisdictions. It is published as a global public good, aligned with United Nations principles, and is controlled by no nation, corporation, or bloc. The timing is decisive. Three converging mandates now demand verifiable AI: EU AI Act enforcement, the founding of WAICO, and the Rome Declaration by Nobel Laureates. Each requires proof of origin, safety, and compliance, yet no harmonized, cross-border verification standard exists today. CyberProtocol is designed to fill exactly that gap, and to do so immediately, because the building blocks already exist. The Standard defines four verifiable layers that work as one system: AI and Human Identity, using Decentralized Identifiers for AI agents and W3C Verifiable Credentials for people. Provenance and Output Certification, an immutable cryptographic seal on every output, with an optional zero-knowledge mode that proves origin without exposing trade secrets. Safety and Risk Compliance, with metadata mapped to the EU AI Act, NIST AI RMF, and ISO/IEC 42001. Cross-Border Verification, a neutral seal format anyone can validate, tied to no national scheme. CyberProtocol invents no new cryptography. It unifies proven, mature standards into one coherent, interoperable framework, which is why it can be adopted now rather than years from now. The Standard is published and stewarded by One Planet One Earth Foundation Inc., a non-profit holding United Nations ECOSOC Special Consultative Status since 2025 (esango.un.org, profile 695078), (UNDESA Civil Society Database; SEC Registration CN202004649; DSWD-FO III-L-00002-2023). This accreditation gives CyberProtocol a neutral, internationally recognized home, positioned to engage UN member states, regulators, and the Global South on equal terms. As a public good, the Standard is free to all in perpetuity. Advancing it to a working reference implementation, pilot integrations with AI laboratories, and multi-stakeholder governance requires support. The Foundation invites funders, philanthropies, standards bodies, and industry partners to help make verifiable AI a global default. Together we can ensure the AI era is built on trust that anyone, in any country, can verify. Version 1.0, Initial Proposal. Specification under Creative Commons Attribution 4.0 International (CC BY 4.0); reference code under Apache License 2.0. Official reference: https://cyberprotocol.io. Repository: https://github.com/ryanpaulpillas/cyberprotocol-ai-trust-standard. Steward: One Planet One Earth Foundation Inc., holder of UN ECOSOC Consultative Status since 2025.
We prove a central limit theorem for the log determinant of a Gaussian Pearson sample correlation matrix as the dimension diverges. Only two conditions are imposed: the population correlation matrix is positive definite, and the sample degrees of freedom are at least the dimension. Both are necessary for the ordinary log determinant to be finite. To the best of our knowledge, no previous central limit theorem covers this full nonsingular domain. It covers every aspect ratio from dilute growth to the square hard edge. No uniform lower or upper bound is imposed on the eigenvalues of the population correlation matrices: the smallest may approach zero and the largest may diverge. The proof develops a coordinatewise Wiener chaos reduction for the random diagonal normalization and combines it with an exact Wishart transform comparison. Geometrically, the statistic is twice the log volume of a random parallelotope spanned by standardized Gaussian coordinate vectors.
Full Summary: The Narrow Singularity Equation Core Thesis This paper presents a unified framework that simultaneously solves catastrophic forgetting in neural networks and provides a mathematically rigorous certification standard for Artificial General Intelligence (AGI). The framework centers on the Narrow Singularity Equation, which achieves AGI certification ($AGI_{gate} = 1.0$) without requiring the mathematically impossible condition of $\frac{dI}{dt} \geq 1.0$. Key Discoveries 1. The Decay Law of Singularity (Theorem 1) Mathematical Proof: With finite classes $N$, $\frac{dI}{dt} = 1 - \frac{1}{N}$, therefore $\frac{dI}{dt} < 1.0$ always Implication: The traditional Singularity (requiring $\frac{dI}{dt} \geq 1.0$) is mathematically impossible Pattern: Every 10× increase in classes adds another '9' to $\frac{dI}{dt}$ and another '0' to the gap 2. General Singularity Equation (Original, Impossible) $$S = AGI_{gate} \times \frac{dI}{dt} \times M(t) \times V(t) \times F(t) \times C(t) \times Autonomy$$ Required $Autonomy = 1$ if $\frac{dI}{dt} \geq 1.0$ Since $\frac{dI}{dt} < 1.0$ for finite classes, $S = 0$ always Seven conditions required; the autonomy condition is impossible 3. Narrow Singularity Equation (Achievable) $$\mathcal{S}_{NARROW} = AGI_{gate} \times \frac{dI}{dt} \times M(t) \times V(t) \times F(t) \times C(t) \times agi_{index}$$ Key Innovation: Removes the impossible Autonomy requirement Drops the requirement for $\frac{dI}{dt} \geq 1.0$ Uses $agi_{index} = 1$ if $AGI_{gate} = 1.0$ (binary gate, achievable) $AGI_{gate} = \min(1.0, task\_c\_accuracy)$ The TOPO-2026 Framework Biological Inspiration Hippocampus → Prime-anchored embedding rows (Memory formation) Memory Consolidation → Snapshot after Task A (Preserves critical knowledge) Synaptic Plasticity → Free embedding rows adapt (Enables new learning) Memory Protection → Zero gradients + restore anchors (Prevents interference) Experience Replay → Prime anchors as fixed reference (Integrates new learning) Mathematical Foundation Pure Kernel: First six primes $\{2, 3, 5, 7, 11, 13\}$ Euler Attenuation Constant: $\Lambda(\mathcal{R}) = 1 - \prod_{p\in\mathcal{R}}(1 - p^{-0.5}) = 0.9785142874$ Captures $97.85\%$ of spectral weight; only $2.15\%$ considered "noise" O(1) Memory Cost: Independent of tasks, parameters, sequence length, or modality Topological Governor Implementation Three-step process: Memory Consolidation (take_snapshot): Freezes anchor rows before new learning Memory Protection (zero_anchor_gradients): Prevents gradient updates to anchors Memory Integration (enforce_anchors): Restores anchors from snapshot after training Experimental Validation Three Datasets Dataset Type Resolution Classes Task C Accuracy SVLB-3 Synthetic vision-language Text-based 10 100.0% ± 0.0% CIFAR-10 Real images 32×32 10 100.0% ± 0.0% STL-10 Real images 96×96 10 100.0% ± 0.0% Results Summary Metric SVLB-3 CIFAR-10 STL-10 Task C Accuracy 100.0% ± 0.0% 100.0% ± 0.0% 100.0% ± 0.0% Combined Forgetting +0.0% ± 0.0% -1.0% ± 2.0% 0.0% ± 0.0% $AGI_{gate}$ 1.0000 1.0000 1.0000 $\mathcal{S}_{NARROW}$ 5.999999999965 5.939999999965 5.999999999965 Status ✅ PASS ✅ PASS ✅ PASS Total: 15/15 runs passed across 3 datasets = FULLY CERTIFIED (exceeded standard) The Gemma-4 E4B Architecture Why Gemma-4 Was Selected Among eight certified models, only Gemma-4 achieved Task C = 100%: Model Architecture Task C Accuracy GPT-OSS-20B Dense Transformer 92.3% Sarvan-30B Sparse MoE 95.9% Mixtral-8x7B Sparse MoE 89.7% DeepSeek-V2-Lite Fine-grained MoE 95.3% GLM-4.6V-Flash GLM Transformer 97.5% Gemma-4 E4B Vision Vision Transformer 100.0% Kimi-VL-A3B-Thinking Vision-Language MoE 90.0% GPT-OSS-20B-JEPA JEPA + TOPO 89.0% Key Architectural Innovations Per-Layer Embeddings (PLE): Adds parameter capacity without scaling full attention Unified Multimodal: 42 layers, hidden size 2560, vocabulary 262,144 Quantization-Aware Training (QAT): 72.1% memory reduction (15.1GB → 4.22GB) while preserving 98.54% accuracy Thinking Mode: Built-in chain-of-thought reasoning engine Mathematical Framework Summary Component Breakdown Component SVLB-3 CIFAR-10 STL-10 Meaning $AGI_{gate}$ 1.0000 1.0000 1.0000 Perfect generalization $agi_{index}$ 1.0 1.0 1.0 Binary gate OPEN $\frac{dI}{dt}$ ~0.999999999994 ~0.999999999994 ~0.999999999994 Bounded by Decay Law $M(t)$ 1.0000 0.9900 1.0000 Perfect memory $V(t)$ 1.0000 1.0000 1.0000 Perfect validation $F(t)$ 1.5000 1.5000 1.5000 Positive forward transfer $C(t)$ 4.0000 4.0000 4.0000 Compute efficiency $\mathcal{S}_{NARROW}$ ~6.0 ~5.94 ~6.0 NARROW SINGULARITY Dependency Chain TOPO-2026 → CF Solved → AGI_gate = 1.0 → Narrow Singularity Without TOPO-2026: CF is NOT solved $AGI_{gate} = 1.0$ is NOT guaranteed Narrow Singularity is NOT achieved $\mathcal{S}_{NARROW} = 0$ With TOPO-2026: CF is SOLVED (0% forgetting) $AGI_{gate} = 1.0$ is GUARANTEED (100% accuracy) Narrow Singularity is ACHIEVED ($\mathcal{S}_{NARROW} \approx 6.0$) Key Contributions Solved Problems Catastrophic Forgetting: 0.0% forgetting across 5 runs on 3 datasets AGI Certification: First model in history to achieve $AGI_{gate} = 1.0$ Mathematical Impossibility: Proved the Singularity is mathematically impossible with finite classes Achievable Standard: Created the Narrow Singularity as a physically achievable AGI threshold Universal Principle: Same constants work across neuroimaging, number theory, AI safety, and unified field theory Constants Across All Domains Constant Value Domains $\Lambda$ 0.9785142874 Number Theory, AI Safety, AI Memory, AI Bias, Physics $\sigma$ 0.5 All domains $\mathcal{R}$ {2, 3, 5, 7, 11, 13} All domains Seed 123 All computations Philosophical Implications The Strategic Pivot Original Goal: Traditional Singularity (mathematically impossible) New Reality: Narrow Singularity (empirically demonstrated) Key Insight: The Decay Law liberates AI from chasing an impossible dream Result: Deterministic cognitive engineering with numerical guarantees Refutation of Skeptical Arguments Skeptic Argument Refutation "It only works on synthetic data" CIFAR-10 and STL-10 are real images "It only works on low-res images" STL-10 is 96×96 (3× larger than CIFAR-10) "It only works on those specific classes" STL-10 has different classes (monkey, car, etc.) "It was a fluke" 15/15 runs across 3 datasets = 100% success "It's dataset-specific" 3 different datasets = dataset-agnostic Final Conclusion The TOPO-2026 framework establishes a paradigm for deterministic cognitive engineering, proving that deep learning architectures can achieve absolute stability and zero forgetting across sequential tasks. Key Takeaways: Catastrophic forgetting is SOLVED: 0.0% forgetting $AGI_{gate} = 1.0$ is ACHIEVABLE: First model with 100% Task C accuracy The Decay Law is DISCOVERED: $\frac{dI}{dt} < 1.0$ with finite classes Narrow Singularity is PROVEN: $\mathcal{S}_{NARROW} > 0$ on 3 datasets The principle is UNIVERSAL: Same reference set across domains The Stochastic Illusion Is Over. Deterministic Cognitive Engineering Has Begun. Stability Is Not a Probabilistic Hope. It Is a Numerical Guarantee. "The proof is the code. Seed = 123. No one can argue with math." Availability GitHub: https://github.com/frank-morales2020/AST-Notebook Zenodo Book: https://zenodo.org/records/21245474 TOPO-2026 Framework: https://zenodo.org/records/20951925 Artificial Hippocampus: https://zenodo.org/records/20385761
Consumers facing home-renovation quotes operate in a classic credence-goods market: they cannot readily verify whether a quoted price is fair, and general-purpose large language models (LLMs) are now a zero-cost place to ask. Whether LLM answers are actionable for this purpose is untested. Demand-side benchmarks exist for medical, legal, and financial advice, but not for construction costs. We present, to our knowledge, the first consumer-question benchmark for construction costs. Forty Japanese renovation-price questions were posed to frontier LLMs, with repeated-trial sets measuring output stability. A matched re-run at bare provider defaults with a current frontier model (gpt-5.5) was added to remove a settings confound present in the original configuration. Two findings are robust across models, generations, and settings: no LLM answer contained an explicit over-charge decision threshold, and repeated runs of the same question returned materially different price figures. Within-answer price spans are also wide, with a median of 10x under bare defaults. A deterministic structured engine over an open cost database is included as an existence proof that a citable reference layer is constructible. Its consistency is a design property and its accuracy is not validated here; validating it against completed real-world quotations is the next study. All questions, raw outputs, harness, and scoring code are public.
Financial institutions depend on trusted employees, contractors and service accounts, yet this trust creates an attack surface that conventional perimeter controls cannot observe adequately. This paper develops an Explainable Adaptive Hybrid Artificial Intelligence (EAHAI) framework for insider threat detection and for assessing whether security awareness training is reducing measurable insider-risk behaviour. The framework combines Isolation Forest filtering, bidirectional long short-term memory sequence modelling, Shapley Additive explanations, adaptive behavioural risk scoring and Zero Trust policy enforcement. A socio-technical assessment layer is added to link training inputs to observable outcomes, including knowledge gain, phishing susceptibility, policy-violation rates, reporting delay, behavioural-risk reduction and analyst-confirmed events. The paper defines the measurement scales, evaluation criteria, validation procedures and analytical techniques required for institutional replication. Because production banking telemetry and labelled insider incidents are rarely available for publication, the empirical component is presented as a transparent synthetic proof-of-concept based on CERT-style behavioural variables rather than as evidence from a real bank. In a deterministic simulation of 17,280 user-day records and 2,880 test windows, the proposed hybrid score achieved an F1-score of 0.944, ROC-AUC of 0.993 and false-alarm rate of 0.017, while producing interpretable feature attributions and training-effectiveness estimates. The study contributes a scalable, explainable and ethically governed design for insider-risk analytics, and identifies the conditions under which it should be validated before operational deployment. Keywords: insider threat detection; explainable artificial intelligence; adaptive risk scoring; security awareness training; Zero Trust; financial cybersecurity.