We design what Artificial Superintelligence requires.
We build and test the architecture.
Logarchéon's seed architecture combines three components: global coordination (CEAS), causal reasoning (Ψ), and invariant geometric representation (GRAIL). Each component is specified, and each has been built and tested at research scale.
Scaling grows the machine. CEAS coordinates it globally. The Ψ-operator supplies causal operator logic. GRAIL supplies invariant geometric cognition. Our thesis: scalable real-world Artificial Superintelligence needs all three. As a structural research programme, the same architecture is also being developed to seek correspondences with established physics across scales — from condensed-matter criticality to gravitation and cosmology — without adding postulates beyond what each domain already requires.
The Seed Architecture
A nonlocal collective coordination variable φ(t) that reduces effective computation-graph diameter to sub-diameter scale, resolving the coordination bottleneck that limits locally bounded architectures at scale. Grounded in finite-size criticality and thermodynamic phase analysis. Initializes via a single forward-pass measurement.
A family of maps Ψα : X × U × C → X with causal semantics that distinguish observed correlation P(Y|X) from intervened outcomes P(Y|do(X=x)), enabling certified reasoning about consequences of actions — not merely patterns in prior data. Supports constrained inverse reasoning and safe behavioural editing without retraining.
Computations generated from group-preserved invariants I(gq, gk) = I(q, k), replacing coordinate-dependent inner products so that generalisation is orbit-consistent across symmetry groups. Produces infinite families of distinct but functionally identical twin models. Root insight: general relativity study, 2009–2010.
Design principles
Each component answers a distinct engineering constraint. The technical brief sets out the principles and the tests behind them.
A global coordination channel reduces coordination rounds.
Two causal models can agree on all observations yet disagree on interventions.
Invariant kernels generalise across a symmetry orbit by construction.
The complete triad outperforms its reduced variants in a pre-registered ablation, complete at research scale.
The mathematics of ASI is algebraic, not statistical
Answers: given data, how well can a function be approximated?
Gradient descent, probably approximately correct (PAC) learning, Vapnik–Chervonenkis (VC) dimension, measure-theoretic probability, stochastic processes, random matrix theory — this tradition is largely answered at the engineering level. Transformers trained with Adam on cross-entropy loss work. The theory for why they generalise exists.
Statistical math remains foundational for training dynamics, generalisation bounds, information theory (entropy H(β) in CEAS), and Bayesian inference in the Ψ-causal layer. It is not replaced — it is de-centred.
Answers: what can a system structurally represent, regardless of data volume?
Category theory, algebraic topology, algebraic combinatorics, representation theory, number theory (automorphic forms, Langlands programme), algebraic geometry (moduli spaces), programming language theory (PLT; type theory). These fields describe what an architecture can represent, independent of data volume.
The GRAIL kernel is an automorphic (Γ-invariant) kernel. Coordination limits are graph-theoretic; causal identifiability is a question of logic. Each calls for an algebraic solution.
Each transition preserved the previous mathematics as infrastructure. Calculus did not stop mattering when group theory entered. Statistics does not stop mattering when algebraic structure enters.
There is a second meaning in this progression. Each new mathematical tool did not merely solve existing problems more efficiently — it made previously invisible problems visible for the first time. Differential geometry did not improve Newtonian calculations; it made gauge invariance a question that could be asked. Group theory did not speed up classical mechanics; it revealed that conservation laws and symmetries are the same thing. The tool creates the visibility.
This is the operating principle of Logarchéon: whenever possible, solve the ASI problem correctly first — using the right mathematics at the right level of the stack — and then apply it to whatever other domains it reaches for humanity. Not because ASI is more important than those domains, but because a correctly built ASI framework will reveal problems in those domains that cannot yet be formulated, the way calculus made celestial mechanics possible not by improving arithmetic but by doing three things at once: making a new class of problem statable for the first time, providing the mechanism to solve it, and encoding in its own structure the clues about where solutions are to be found. The tool asks, enables, and guides simultaneously.
Algebraic mathematics occupies the top of the stack. Statistical mathematics is the bottom. The bottom is infrastructure. The frontier is at the top.
Persistent homology tracks which features survive training. Coordination-round bounds depend on graph diameter.
The GRAIL automorphic kernel Kβ(q,k) = ∑γ∈Γ exp(−β d(q,γk)) is an automorphic (Γ-invariant) kernel. The Langlands programme connects symmetry groups to analytic objects; GRAIL takes its kernel from the automorphic side of that correspondence.
Category theory is not one more algebraic field — it is the language that connects all the others. Markov categories contain probability theory as a special case. The Curry–Howard–Lambek correspondence unifies type theory, logic, and category theory.
The space of all model architectures with a given property is a moduli space. The MIA migration tiers are a stratification of this space. Deforming a model via RSI is a path in this geometric space.
Programming language theory is not separate from the algebraic fields — it is their computational instantiation. Homotopy type theory: types are topological spaces.
Random matrix theory bridges both traditions. Statistical side: eigenvalue structure of weight matrices and Hessians. Algebraic side: Montgomery’s conjecture connects RMT to the Riemann zeta function. RMT is being enriched by both directions simultaneously.
“Statistical mathematics tells you how well a system approximates. Algebraic mathematics tells you what a system can and cannot represent. For building systems that approach ASI, the binding constraint is representational — what the architecture can structurally express — not approximation quality.”
Why this is not ANI — and why scaling alone does not get there
ANI (Artificial Narrow Intelligence) — systems restricted to bounded tasks or domains, including every current large-scale language model built on next-token prediction and parameter scaling: GPT-4 through GPT-6, Claude, Gemini, Grok, DeepSeek, Qwen, and their successors. AGI (Artificial General Intelligence) — broad, robust competence across most cognitive task families, at least at the level of a competent adult human. ASI (Artificial Superintelligence) — performance exceeding the best human across most major cognitive domains, with robust transfer to new task families not represented in training.
The distinction between these tiers is not about benchmark scores or parameter count. It is architectural and mathematical. The table below is derived directly from the formal certification criteria and scenario analysis in the lecture notes (v21.5).
Before operating systems, each program managed its own hardware directly — purpose-built for one task, not composable, not self-coordinating. Current frontier models are this: large standalone programs that excel within their training distribution but cannot rewrite their own subsystems or enforce causal validity across novel domains.
An operating system is not a bigger program — it is the coordinating layer that makes computation composable and self-managing. This architecture is the analogous layer for intelligence — with three specific mechanisms. GRAIL preserves every physical observable as a coordinate-independent tensor: the representation is invariant by construction, not by approximation. CEAS correlates signals across the entire model simultaneously — passing intelligence between computational units no matter how far apart they are, so distant but relevant connections are not cut off by architectural locality. The Ψ-operator acts on what those two layers produce: conducting causal analysis and certified inverse reasoning — because the operators transfer structurally, not by distributional similarity to training data. The architecture is designed so that causal, geometric, and coordination structure generalises across task families. The OS made computation compositional. This makes intelligence compositional across structurally related domains.
ANI solves tasks. An ASI seed improves the process that solves new task families.
A system can score at the highest human level on every standard benchmark and still be ANI if it succeeds only within its training distribution. The distinction is not performance — it is whether the system can intervene causally, generalise over symmetry orbits, and improve its own architecture inside a verified closed loop with a falsifiable audit record.
- ✗A large language model (LLM), however large, operating at observational-correlation-only causality
- ✗An agentic pipeline wrapping a correlation-based LLM in causal-sounding instructions — wrapping produces a "causal-shaped" system, not a causal one
- ✗Advanced computing hardware (quantum, neuromorphic) without goals, world models, or causal agency
- ✗A system with superhuman performance in narrow domains only — that is ANI by definition
- ✗A system with high benchmark scores but high scaffolding dependence — scaffolded performance is not autonomous intelligence
- ✗Any system whose claimed improvements cannot be reproduced from logs, checkpoints, and pre-registered benchmarks alone
A single mathematical origin
Note on independence. The ASI seed does not assume Anti-de Sitter geometry, does not require a conformal field theory dual, and makes no claim that depends on the physical truth of the AdS/CFT correspondence. The architecture draws from classical mathematics that predates AdS/CFT by decades: spectral theory and heat kernel methods, Seeley–DeWitt coefficients, free energy functionals from statistical mechanics, and operator algebra methods from functional analysis. Convergence via Brouwer's fixed-point theorem (existence) and Knuth's TAOCP Vol. 1 §1.1 + pigeonhole principle (halting). No string-theoretic scaffolding required.
All three components emerged from independent study of classical mathematics — spectral theory, differential geometry, statistical mechanics, and operator algebras — encountered in part through the AdS/CFT literature. The study predates transformers, learned word vectors, and modern AI tools. The root insight predates the attention mechanism by years.
predates transformers
notes v21.5
independently tested
4 independent)
Filed 2025
MIA: Any trained model, upgraded to GRAIL
Metric-Invariant Architecture (MIA) is the general class of which GRAIL is a strict specialisation. MIA replaces every scalar dot-product primitive with a group-preserved invariant F(dₘ(q, k)), where I(g·q, g·k) = I(q, k) for all isometries g. The critical consequence: a legacy model — including any Transformer trained on Euclidean dot products — can inherit twinhood and geometric properties at runtime without discarding what it learned.
Four tiers, not three. The path from a plain pre-trained model to complete GRAIL is a gradient: zero-step arithmetic replacement, lightweight β fine-tuning, LoRA + hyperbolic projection, and complete retraining from scratch. Each tier is independently verifiable. The author's research-scale CEAS–Ψ–GRAIL builds satisfy all four tiers by construction; the retrofit of existing models through Tiers 0–2 is complete at research scale.
GRAIL ⊂ MIA (strict inclusion)
GRAIL = MIA + orbit-jump + automorphic kernels + CEAS β-control
Merely storing tensors in geometric memory without replacing arithmetic primitives does not confer twinhood. Both formal inheritance conditions must hold — verified in the technical brief.
| Property | Tier 0 · MIA retrofit Arithmetic replacement only — zero gradient steps | Tier 1 · CEAS β β-thermostat fine-tuning — hundreds of steps | Tier 2 · LoRA + metric LoRA + ℍd projection — thousands of steps | Tier 3 · Complete retrain Train from scratch with triad priors |
|---|---|---|---|---|
| Twinhood Fg·θ(gx) = Fθ(x) |
✓ | ✓ | ✓ | ✓ |
| Entropy corridor H(β) ∈ [H★ − δ, H★ + δ] |
✗ | ✓ | ✓ | ✓ |
| Susceptibility sharpening χL ~ Lγ/ν (conjecture) |
✗ | ✓ | ✓ | ✓ |
| Orbit generalisation unseen g ∈ G, εtwin ≤ 10−6 |
✗ | ✗ | ✓partial → complete | ✓ |
| Automorphic kernels Kβ(q,k) = Σγ∈Γ e−β d(q,γk) |
✗ | ✗ | ✓with ℍd projection | ✓ |
Twinhood only. Replace dot products with F(dₘ(q,k)) and wrap with (ψ, π). Zero gradient steps. No other GRAIL properties are transferred.
Adds entropy corridor and susceptibility sharpening. β is an algebraic consequence of having adaptive temperature — no orbit generalisation required.
Adds orbit generalisation and automorphic kernels via LoRA adapters + ℍd projection on Q, K. A small adapter layer. Tractable on a single graphics processing unit (GPU).
All five properties at the theoretical optimum. The author's research-scale CEAS–Ψ–GRAIL builds are at this tier by construction.
Complete formal definitions, migration proofs, and section references are in the technical brief.
Encrypted-in-use deployment
The seed architecture includes a λ-secure runtime for deployments requiring encrypted-in-use computation. Buyer-held keys. Policy-gated interfaces. No canonical plaintext during execution. Operates on your hardware or within your cloud tenancy.
V1 — λ-native models
Geometry built into model architecture and training dynamics from the ground up. Maximal integration. Principled semantics. Tighter control of canonicalization. For long-lived sovereign AI assets where architectural integrity is non-negotiable.
V2 — Exported wrapper (NN/LLM)
Adoption-first path. Wraps existing models and runtimes without complete re-architecture. Fastest path to pilots. Reduces reusable plaintext exposure in in-use pipelines via constrained interfaces and protected representations.
V3 — VM/OS/runtime posture
Extends the same non-canonical in-use discipline to OS/VM/runtime boundaries for general-purpose compute — not solely AI. Covers cloud instances, on-premises deployments, hypervisor surfaces, and complete artifact lifecycle control.
Why homomorphic encryption (FHE), secure multi-party computation (MPC) and trusted execution environments (TEEs) fall short
FHE: 10³–10⁶× overhead, impractical for large neural pipelines. MPC: communication latency dominates at scale. TEEs: shift trust to vendor firmware, not zero-trust. All reintroduce plaintext through telemetry, caches, or debug steps.
High-assurance missions
The long-term home for Logarchéon is environments where ASI architecture and encrypted-in-use AI are mission-critical — not marketing.
National Security / Defense / Intelligence — U.S., NATO, and Coalition Partners
- U.S. Intelligence Community (IC) agencies and Department of Defense (DoD) components requiring encrypted-in-use AI at operational scale
- NATO member-state defense and intelligence agencies operating under shared threat environments
- Coalition and allied-nation partners requiring sovereign AI execution without data exposure to third-party infrastructure
- Defense and intelligence industrial base injecting hardened AI into mission-critical systems
- Systemic finance and critical infrastructure with real-world failure modes
Research grants & regulated enterprise
- U.S. DARPA / IARPA / ONR and allied-nation equivalents (DSTL, DRDC, DST Group, and NATO STO programmes) — research and development funding for ASI and formal-verification infrastructure
- Healthcare, pharma, aerospace requiring intellectual property (IP) protection and sovereign execution
- Cloud and hardware vendors licensing encrypted-in-use runtime infrastructure
Law, founders & civil organizations
- Law firms that cannot upload privileged material to public AI APIs
- Privacy-first founders treating their data as the strategic moat
- High-confidentiality civil, humanitarian, and intergovernmental organizations
The stack in plain language
The page is simple on purpose. Underneath, the work draws on original results in geometry, spectral theory, statistical physics, and causal inference.
Future implementations. The same design is intended to run on progressively more capable real-world substrates — classical hardware today; quantum and topological substrates as the research frontier advances.
Core research pillars
- CEAS: nonlocal entropy-temperature coordination; finite-size criticality; collective variable φ(t) that reduces coordination rounds.
- Ψ-operator: all four Pearl rungs in a single operator algebra; constrained inverse design; causal identification with explicit, checkable criteria.
- GRAIL: group-invariant metric replaces the Euclidean dot product; automorphic kernel construction; orbit-consistent by construction.
- Recursive self-improvement: gated closed-loop improvement; machine-checked invariants; no manual edits inside the measured cycle.
Language architecture — correctness by construction
Every language has a bug topology — a map of what mistakes are structurally impossible to write. For a self-modifying system running autonomous improvement loops, the key question is not which language is fastest but which class of bugs each language makes impossible. The language choices here were reached through rigorous PL-theoretic analysis, including several revisions.
#lang causal-dsl makes it impossible to conflate P(Y|X) with P(Y|do(X)) at compile time. contract-out enforces RSI invariants at module boundaries.#lang defines new languages at the reader level (before parsing), not just at macro expansion. #lang causal-dsl makes it impossible to conflate observation and intervention at the source level. Chez Scheme’s macros cannot do this. Racket CS runs on Chez Scheme anyway, so Racket’s call/cc performance is Chez Scheme’s.(eval new-state) is Racket’s normal operation; Haskell requires the GHC compiler application programming interface (API) or Template Haskell. The RSI loop has side effects by definition — it modifies the system. Haskell’s monadic purity adds overhead without solving the self-modification problem.call/cc in Racket cannot replicate without weeks of infrastructure: tabling (automatic query memoisation), CLP(FD/R) (constraint propagation over physical laws), and meta-interpreters that can generate entirely new causal reasoning languages in a few dozen lines. The Racket–Prolog boundary is not a seam within a computation — it is an interface between two paradigms. Across paradigms, the interface is the contribution. Racket generates functional/syntactic languages; Prolog generates logic/constraint languages; together they generate next-generation languages in both paradigms simultaneously.World model — (M, G, β)
The world model is not a separate design decision — it is already determined by the three components. M is the Riemannian manifold (GRAIL), G is the symmetry group acting on M (GRAIL), β is the inverse temperature controlling information density (CEAS). The Ψ-causal layer adds structural causal equations over M. The Ψ-causal engine is the world model: prediction is what results from marginalising out the causal structure; intervention, counterfactual, and inverse design are what result from using it completely.
Hierarchical structure (object → part → subpart) requires exponentially many Euclidean dimensions without distortion. The same structure admits an injective map into ℍd with constant distortion. LeCun’s JEPA and DreamerV3 both suffer this silently. GRAIL uses the right geometry.
Every other world model is fixed architecture trained once. In this framework the RSI loop, now running at research scale, can modify the world model’s causal graph structure, manifold geometry, symmetry group assumptions, and β-schedule. The world model improves its own theory of the world. No existing public approach does this.
LeCun is right that world models are necessary. Fei-Fei is right that 3D structure matters. DreamerV3 is right that planning in latent space is efficient. All are solving Rung 1 with increasingly good architectures. None have the causal structure for Rungs 2–4.
Where to read more
Technical reviewers, cryptographers, and ML researchers who want the mathematics, proofs, and working code:
- CEAS, GRAIL, Ψ-Operator — component pages
- CV — academic background and prior work
- Email for NDA-gated technical briefs and evaluation materials
Progress to Certification
Every commitment of the programme, marked as it stands. “Complete” means built and passed its pre-registered test; “research scale” means on research-scale tasks, before the production model.
- FoundationsLecture notes, formal verification layer and reproducible test harnessComplete
- FoundationsLanguage-level self-improvement prototypeComplete
- FoundationsGRAIL-native research buildComplete
- Ψ-operator — causal engineCausal identification with machine-checked derivationsComplete · research scale
- Ψ-operator — causal engineCounterfactuals and inverse design with memoryComplete · research scale
- Ψ-operator — causal engineSafe behavioural editing without retrainingComplete · research scale
- Ψ-operator — causal engineCausal reversal and action-order demonstrationsIn progress
- CEAS — global coordinationCoordination scaling under the locality boundComplete · research scale
- CEAS — global coordinationLearned global coordination at scaleIn progress
- GRAIL — invariant geometrySymmetry generalisation to unseen group elementsComplete · research scale
- GRAIL — invariant geometryAutomorphic kernel inside the architectureComplete · research scale
- GRAIL — invariant geometryRetrofit of existing models (MIA Tiers 0–2)Complete · research scale
- Integration and self-improvementTriadic integration and the complete ablationComplete · research scale
- Integration and self-improvementClosed self-improvement loop with a machine-checked gateComplete · research scale
- Integration and self-improvementWorld-model self-improvementComplete · research scale
- Integration and self-improvementAll 23 test batteries, including multi-cycle improvement and scaling exponentsComplete · research scale
- Integration and self-improvementSelf-improving control of a drifting physical systemComplete · research scale
- Integration and self-improvementPhysical-ASI seed evidence (Cert Level 5): all 23 test batteries, all four Pearl rungs, GRAIL orbit generalisation, scaling exponents, self-improving control, hidden benchmark committed in advanceComplete · research scale
- Integration and self-improvementPublic audit bundle for independent replicationIn progress
- ProductPublic deployment with free user accessIn progress
- ProductSelf-improvement applied to production codePlanned
- Long horizonASI claim: cluster-scale evidence across ten or more domainsIn progress · groundwork
- Long horizonPhysics correspondences; quantum and topological substratesResearch
#lang causal-dsl enforcement, memory, audit. Homoiconicity + call/cc for RSI reversion. Chez Scheme performance underneath.Racket + Lean 4
CompleteThe architecture’s own primitives, implemented from first principles
CompleteSWI-Prolog + Lean 4
Complete · research scaleDemo E
Complete · research scaleDemos A + B · public release with the product
In progressDemo C
Complete · research scaleDemo D
Complete · research scaleCert Level 3
Complete · research scaleCert Level 4 · Demo F
Complete · research scaleCausal structure, symmetry and temperature schedule
Complete · research scaleMulti-cycle improvement and scaling exponents
Complete · research scaleCert Level 5
Complete · research scaleEvery script, result and certificate
In progressFastAPI · Oracle Cloud ARM · Supabase
In progressCert Level 6
In progress · groundworkSeven discriminating demonstrations
Each targets an architectural difference, not a benchmark gap.
A dataset where X appears to prevent Y in observed data, but causes Y under intervention. Correlation-only predictors give the wrong direction, and the open-source language model tested never recognised the cases that cannot be identified. The Ψ-causal engine computes P(Y|do(X=x)) by SCM mutilation — correct whenever the effect is identifiable.
“Open drawer then grasp” ≠ “grasp then open drawer.” Order-insensitive models treat these as the same, and the open-source language model tested chose the same order for every pair. The Ψ-causal engine has [Ta, Tb] built in — the commutator is measurable and non-zero.
On N-bit global parity, CEAS reduces coordination rounds via the rank-one collective variable φ(t). Crossover with local attention visible at a measurable scale threshold on a reproducible chart.
Train on a set of group elements; test on unseen elements from the same group. GRAIL: zero error gap. Euclidean transformer: degrades. I(gq, gk) = I(q, k) holds by construction, not by approximation.
Given a target state, compute the action sequence that causes it. Learned inverse models pattern-match forward — they do not invert physical dynamics outside their training distribution. The Ψ-causal engine solves this by construction.
A defined minimum of closed-loop cycles with no human edits inside the loop. Lean 4 gate active throughout. Once the product launches, user queries are designed to feed the continuous fine-tuning cycle.
A controller that keeps its own physics current. As the machine it drives wears and drifts, the closed loop revises its causal model of the dynamics, admits only the terms that predict what happens next and drops the rest, and the Ψ-causal engine plans every action through the revised model. A model fitted once and frozen falls behind; the self-improving loop does not.
Where the current build stands
Logarchéon’s current build (CEAS–Ψ–GRAIL) placed among 51 AI developments of 2026, each scored from measured numbers on one [0, 100] line. Status as of 2 October 2026. Complete study →
1 · Measured ladder quantities
Eight quantities, each in [0, 1], computed only from measured numbers.
| System | C | Ψ | G | R | U | H | B | S | P | Strict I | Bounded I | Alt A | Level |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Logarchéon current build | 1 | 1 | 1 | 1 | 0.574 | 0.5 | 0 (·) | 1 | 1 | 75.9 | 75.9–88.4 | 75.9 | 5 (4†) |
| Hyperagents | 1 | · | · | 1 | 0.332 | 0.5 | · | · | 0 | 0 | 0–85.4 | 35.4 | 2 |
| GPT-6 Astra | 1 | · | · | 0 (≤0.5) | · | 1 | · | · | 0 | 0 | 0–87.5 | 25.0 | 2 |
| GPT-6.1 Sol, GPT-5.6 Sol, Opus 5, Opus 5.5, Opus 4.8, Gemini 3.8 Flash, Grok 4.6, Kimi K3, Qwen3.8 (27B), GLM-5.3, Seed, Sakana AI Scientist, Co-Scientist, AlphaProof, Aristotle | 1 | · | · | · | · | 1 | · | · | 0 | 0 | 0–100 | 25.0 | 2 |
| Darwin Gödel Machine | 1 | · | · | 1 | · | · | · | · | 0 | 0 | 0–100 | 25.0 | 2 |
| MACE / GNoME | 0.667 (≤1) | · | 1 | · | · | · | · | · | 0 | 0 | 0–100 | 20.8 | 2 (via G) |
| DeepSeek V4-Pro | 1 | · | · | · | · | 0.5 | · | · | 0 | 0 | 0–93.8 | 18.8 | 2 |
| Huxley-Gödel Machine | 1 | · | · | 0 | · | 0.5 | · | · | 0 | 0 | 0–93.8 | 18.8 | 2 |
| Self-improving embodied FMs | 1 | · | · | 0 (≤0.05) | 0.300 | · | · | · | 0 | 0 | 0–67.5 | 16.3 | 2 |
| π*0.6 | 1 | · | · | 0 | 0.25 (≤0.5) | · | · | · | 0 | 0 | 0–93.8 | 15.6 | 2 |
| AlphaEvolve | 1 | 0.238 | · | 0 | · | · | · | · | 0 | 0 | 0–99.6 | 15.5 | 2 |
| GPT-5.5 / 5.3 | 1 | 0.085 | · | · | · | · | · | · | 0 | 0 | 0–94.2 | 13.6 | 2 |
| Gemini 4 Argon, Muse Spark, V-JEPA 2-AC, Absolute Zero, Gemini Robotics 2, GR00T, AgiBot GO-2, AlphaFold 3 | 1 | · | · | · | · | · | · | · | 0 | 0 | 0–100 | 12.5 | 2 |
| Calibration: GPT-4 on CLadder | 1 | 0.117 | · | · | · | · | · | · | 0 | 0 | 0–85.1 | 14.0 | 2 |
| Calibration: Euclidean transformer inside the build | 1 | — | 0.300 | — | — | — | — | — | — | — | — | — | — |
“·” unmeasured (strict value 0; bounded value runs over [0, 1]); a narrower bounded range is given in parentheses. “—” not applicable. Bounded I runs from the strict value to the value with every unmeasured cell at 1. † Level 4 applies if U is read against the loop’s own earlier miss, which rises with wear, instead of against the frozen controller.
2 · The index line
STRICT INDEX (necessity-gated)
0 20 40 60 80 100
|---------|---------|---------|---------|---------|
X ← 33 systems at 0 (Astra, Opus 5.5, Gemini 4, Hyperagents, DGM, AlphaEvolve, robots, provers …)
L (Logarchéon build) 75.9
bounded: build [75.9 ===== 88.4] EFM [0 ====== 67.5] Hyperagents [0 ======= 85.4]
Astra [0 ======= 87.5] most others [0 ========== 100]
ALTERNATIVE A (arithmetic mean, no necessity gate)
0 20 40 60 80 100
|---------|---------|---------|---------|---------|
^12.5 Argon, Muse, V-JEPA, robots, AF3
^15.5-16.3 AlphaEvolve, π*0.6, EFM
^18.8-20.8 DeepSeek, HGM, MACE
^25.0 ARC-verified LLMs, DGM, provers, Co-Scientist, Sakana
^35.4 Hyperagents
^75.9 Logarchéon build
3 · The leaders on each reading
| # | Ungated reading | Score | # | Strict reading | Score |
|---|---|---|---|---|---|
| 1 | Logarchéon current build | 61.4 | 1 | Logarchéon current build | 61.4 |
| 2 | Hyperagents | 60.5 | 2= | AlphaEvolve | 52.3 |
| 3 | AlphaEvolve | 55.9 | 2= | Hyperagents | 52.3 |
| 4 | Darwin Gödel Machine | 55.8 | 4 | Darwin Gödel Machine | 50.0 |
| 5 | Equivariant scientific ML | 54.0 | 5= | Equivariant scientific ML | 49.2 |
| 6 | Self-improving robot models | 53.0 | 5= | V-JEPA 2-AC | 49.2 |
| 7 | AlphaProof | 52.7 | 5= | Self-improving robot models | 49.2 |
| 8 | V-JEPA 2-AC | 52.1 | 8= | AlphaProof, Absolute Zero, Huxley-Gödel Machine | 46.9 |
B1 novelty, B2 breadth per resource, B5 protocol discipline (each 0–10); Q the measured ladder index. With the conservative ladder index (64.2), the current build scores 58.7 and ranks second on the ungated reading.
4 · The 51-development mixed line
Show all 51 developments
| # | Development | Group | Rank (0–12) | Level | B1 | B2 | B5 | Q-ung. | Ungated | Q-str. | Strict |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | Logarchéon current build (CEAS–Ψ–GRAIL) | Logarchéon | 1 (aim 8→10) | L5 own rule | 4.5 | 5 | 8 | 75.9 | 61.4 | 75.9 | 61.4 |
| 2 | Hyperagents | Self-improvement loops | 4 | L1 | 8 | 6 | 6 | 35.4 | 60.5 | 0 | 52.3 |
| 3 | AlphaEvolve | Self-improvement loops | 4 | L1 | 8 | 6 | 6 | 15.5 | 55.9 | 0 | 52.3 |
| 4 | Darwin Gödel Machine | Self-improvement loops | 4 | L1 | 8 | 5 | 6 | 25 | 55.8 | 0 | 50.0 |
| 5 | Equivariant sci-ML | Science, proof & geometry | 1 | L2 | 7 | 5 | 7 | 20.8 | 54.0 | 0 | 49.2 |
| 6 | Self-improving EFMs | World models & robotics | 8 (partial) | L1 | 7 | 6 | 6 | 16.3 | 53.0 | 0 | 49.2 |
| 7 | AlphaProof | Science, proof & geometry | 1 | L1 | 7 | 4 | 7 | 25 | 52.7 | 0 | 46.9 |
| 8 | V-JEPA 2-AC | World models & robotics | 8 (partial) | L1 | 7 | 6 | 6 | 12.5 | 52.1 | 0 | 49.2 |
| 9 | Harmonic Aristotle | Science, proof & geometry | 1 | L1 | 6 | 4 | 8 | 25 | 51.9 | 0 | 46.2 |
| 10 | Huxley-Gödel Machine | Self-improvement loops | 4 | L1 | 7 | 5 | 6 | 18.8 | 51.3 | 0 | 46.9 |
| 11 | DeepSeek V4-Pro | Chinese models | 4–5 | L1 | 6 | 6 | 6 | 18.8 | 50.5 | 0 | 46.2 |
| 12 | Absolute Zero Reasoner | Self-improvement loops | 4 | L1 | 7 | 5 | 6 | 12.5 | 49.8 | 0 | 46.9 |
| 13 | Co-Scientist | Self-improvement loops | 3 | L1 | 6 | 5 | 6 | 25 | 49.6 | 0 | 43.8 |
| 14 | Sakana AI Scientist | Self-improvement loops | 3–4 | L1 | 6 | 5 | 5 | 25 | 47.3 | 0 | 41.5 |
| 15 | AlphaFold 3 | Science, proof & geometry | 1 | L1 | 6 | 5 | 6 | 12.5 | 46.7 | 0 | 43.8 |
| 16 | DoWhy / y0 / Ananke | Science, proof & geometry | 0 | L2 | 5 | 3 | 8 | 12.5 | 43.7 | 0 | 40.8 |
| 17 | UAE K2 Think / Falcon H1R | Rest of world | 3 | L1 | 4 | 5 | 7 | 12.5 | 42.9 | 0 | 40.0 |
| 18 | Physical Intelligence π0.5/π*0.6 | World models & robotics | 8 (partial) | L1 | 6 | 5 | 4 | 15.6 | 42.8 | 0 | 39.2 |
| 19 | Kimi K3 | Chinese models | 4–5 | L1 | 5 | 5 | 4 | 25 | 41.9 | 0 | 36.2 |
| 20 | Anthropic Opus 5/5.5, Mythos | US frontier models | 5 | L1 | 4 | 4 | 6 | 25 | 41.2 | 0 | 35.4 |
| 21 | Genie 3 | World models & robotics | n/a | n/a | 6 | 4 | 4 | 12.5 | 39.8 | 0 | 36.9 |
| 22 | GPT-6 Astra | US frontier models | 5 | L1 | 5 | 4 | 4 | 25 | 39.6 | 0 | 33.8 |
| 23 | Apertus 1.5 | Rest of world | 2–3 | L1 | 2 | 4 | 9 | 12.5 | 39.0 | 0 | 36.2 |
| 23 | Gemini Robotics 2 | World models & robotics | 8 (partial) | L1 | 5 | 5 | 4 | 12.5 | 39.0 | 0 | 36.2 |
| 23 | Intern-S2 | Chinese models | 4–5 | L1 | 5 | 5 | 4 | 12.5 | 39.0 | 0 | 36.2 |
| 23 | NVIDIA Cosmos / GR00T | World models & robotics | 8 (partial) | L1 | 5 | 5 | 4 | 12.5 | 39.0 | 0 | 36.2 |
| 27 | ByteDance Seed 2.0 / Prover | Chinese models | 4–5 | L1 | 4 | 5 | 4 | 25 | 38.8 | 0 | 33.1 |
| 28 | Korea sovereign FMs | Rest of world | 3 | L1 | 4 | 5 | 5 | 12.5 | 38.3 | 0 | 35.4 |
| 29 | AgiBot GO-2 | World models & robotics | 8 (partial) | L1 | 5 | 5 | 3 | 12.5 | 36.7 | 0 | 33.8 |
| 29 | Hyperbolic nets incl. HELM | Science, proof & geometry | 1–2 | L2 | 5 | 3 | 5 | 12.5 | 36.7 | 0 | 33.8 |
| 29 | OpenAI research intern | Self-improvement loops | 4 | L1 | 5 | 4 | 4 | 12.5 | 36.7 | 0 | 33.8 |
| 32 | Qwen3.8-Max | Chinese models | 4–5 | L1 | 4 | 5 | 3 | 25 | 36.5 | 0 | 30.8 |
| 33 | Wayve | World models & robotics | 1 | L1 | 5 | 4 | 3 | 12.5 | 34.4 | 0 | 31.5 |
| 34 | Gemini 3.5–3.8 / 4 Argon | US frontier models | 5 | L1 | 4 | 4 | 4 | 12.5 | 33.7 | 0 | 30.8 |
| 35 | GLM-5.3 | Chinese models | 4–5 | L1 | 3 | 5 | 3 | 25 | 33.5 | 0 | 27.7 |
| 36 | Isomorphic Labs | Science, proof & geometry | 1 | L1–L2 | 5 | 3 | 3 | 12.5 | 32.1 | 0 | 29.2 |
| 37 | Grok 4.6 | US frontier models | 4–5 | L1 | 4 | 4 | 2 | 25 | 31.9 | 0 | 26.2 |
| 38 | Meta Muse Spark | US frontier models | 5 | L1 | 4 | 4 | 3 | 12.5 | 31.3 | 0 | 28.5 |
| 38 | MiniMax M2.7/M3 | Chinese models | 4 | L1 | 4 | 4 | 3 | 12.5 | 31.3 | 0 | 28.5 |
| 40 | Mistral AI | Rest of world | 3–4 | L1 | 3 | 4 | 4 | 12.5 | 30.6 | 0 | 27.7 |
| 41 | Unitree | World models & robotics | 8 (partial) | L1 | 3.5 | 4 | 3 | 12.5 | 29.8 | 0 | 26.9 |
| 42 | ERNIE 5.1 | Chinese models | 4–5 | L1 | 4 | 4 | 2 | 12.5 | 29.0 | 0 | 26.2 |
| 42 | Figure / Skild | World models & robotics | 2–3 | L1 | 4 | 4 | 2 | 12.5 | 29.0 | 0 | 26.2 |
| 44 | Inkling | US frontier models | 3–4 | L1 | 3 | 4 | 3 | 12.5 | 28.3 | 0 | 25.4 |
| 44 | Microsoft MAI | US frontier models | 3–4 | L1 | 3 | 4 | 3 | 12.5 | 28.3 | 0 | 25.4 |
| 44 | Sarvam-105B | Rest of world | 2–3 | L1 | 3 | 4 | 3 | 12.5 | 28.3 | 0 | 25.4 |
| 44 | SEA-LION v4 | Rest of world | 2–3 | L1 | 3 | 4 | 3 | 12.5 | 28.3 | 0 | 25.4 |
| 44 | Tencent Hunyuan | Chinese models | 3–4 | L1 | 3 | 4 | 3 | 12.5 | 28.3 | 0 | 25.4 |
| 49 | Cohere + Aleph Alpha | Rest of world | 2–3 | L1 | 2 | 4 | 3 | 12.5 | 25.2 | 0 | 22.3 |
| 49 | Sber / Yandex | Rest of world | 2–3 | L1 | 2 | 4 | 3 | 12.5 | 25.2 | 0 | 22.3 |
| 51 | Amazon Nova 2 | US frontier models | 2–3 | L1 | 2 | 4 | 2 | 12.5 | 22.9 | 0 | 20.0 |
Sorted by the ungated score; tied scores share a rank. Developments absent from the quantified table receive the floor value Q-ungated = 12.5.
Generating a better language is formulating a better theory
RSI Level 2 changes the language itself — what can be written and what is rejected before execution.
Each proposed language change passes a gate: it may only make more errors impossible, must keep every known-good program valid, and must improve on the error suite.
The loop cannot regress, always terminates, and blocks over-broad changes.
A research build now runs this loop on a GRAIL-native model. The language grows by abstracting the operations that succeeded, and every extension passes the gate with a certificate checked in Lean 4.
On sixteen tasks the model had never seen, the grown language corrected the model clearly better than the language it started from, without a single gradient step and without loss on the patterns it already knew. The architecture’s own identities held to machine precision throughout, and every edit was an isometry checked by Racket and certified by Lean 4.
When weights and programs live on the same manifold
Weights and inputs share one geometry, so a program edit and a geometric edit can be treated alike. Seven capabilities follow from this; details are in the technical brief.
The GRAIL-native research build realises this. The architecture's own primitives are implemented from first principles, every parameter is a point of the geometry, and the identities the theory predicts — function-identical twin models, and training that commutes with the symmetry group — hold to machine precision. Edits to the model are themselves programs of isometries, checked before they run.
Retaining pre-trained knowledge
A protocol for importing pre-trained knowledge without breaking invariance is specified in the technical brief.
From theory to implementation
Theory-to-implementation mappings are provided in the technical brief.
William Huanshan Chuang
Mathematician and sole founder of Logarchéon Inc., a one-person C-Corporation structured as an IP-first research laboratory. The work sits at the intersection of geometry, control theory, statistical physics, and artificial intelligence.
The designation Artificial Superintelligence Architect reflects a design principle rather than a marketing claim: no intelligence system — computational or otherwise — can remain reliably aligned with the world it operates in without continuous coupling to causal measurement. The architecture is built for that coupling, not as an external constraint, but as its fundamental operating mode. On advanced hardware substrates, this coupling becomes more precise: a quantum-substrate CEAS with Ψ-mediated coordination would replace classically approximated nonlocal correlations with genuine quantum correlations, narrowing the boundary between the computational model and the world it represents. Advanced computing is not intelligence; this architecture on an advanced substrate may be. That distinction is the research frontier.
All three components of the ASI seed architecture were conceived during independent study of classical mathematics — spectral theory, differential geometry, and statistical mechanics — encountered in part through the AdS/CFT literature, before transformers, before learned word vectors, and before modern AI tools. The root insight predating all modern attention mechanisms dates to 2009–2010 study of general relativity.
AI tools — including proprietary trained agents and recursive agentic systems — were used to verify proofs and accelerate documentation under human direction. All core claims, mathematical structures, and inventive concepts are human-originated. All patent claims are human work.
Start a quiet conversation.
If you work in national security, defense, or intelligence — U.S., NATO, or coalition — research, or high-assurance compute — or if you want to evaluate the ASI seed architecture under NDA — the starting point is simple.
Request a technical brief
A 30–45 minute briefing on your mission and constraints, followed by a scoped proof-of-concept on your hardware or within your cloud tenancy. Claims are bounded by written scope and acceptance criteria. No unbounded promises.
founder@logarcheon.comU.S. Patent Portfolio · 9+ Applications (2025) · Principal: 64/067,703 · Some materials subject to U.S. export regulations