We design what Artificial Superintelligence requires.
We build and test the architecture.

The mathematics of ASI is algebraic, not statistical

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 (v21.16).

MIA: Any trained model, upgraded to GRAIL

Complete formal definitions, migration proofs, and section references are in the technical brief.

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 coordinationCoordination on a Ramanujan graph (a sparse network with the best possible connectivity for its wiring): averaging across 2,184 units took 77 rounds, against 3,417 on a wrap-around grid of comparable size and the same wiringComplete · 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 geometryHyperbolic symmetry group checked three ways: element counts at lengths 1–5 match the predicted growth exactly, the growth rate matches the group’s fractal dimension to within 10⁻⁴, and symbolic simplification matched exact matrix arithmetic on 100,000 of 100,000 random pairsComplete · research scale
• GRAIL — invariant geometrySelberg trace formula on the modular surface, a classical identity linking its resonant frequencies to the lengths of its closed loops, checked numerically to within 3 × 10⁻¹⁵Complete · research scale
• GRAIL — invariant geometryRetrofit of existing models (MIA Tiers 0–2)Complete · research scale
• Integration and self-improvementSeed architecture assembled: the three components and five supporting modules (symbolic, spectral, arithmetic, language, self-improvement), linked by 36 cross-checks, each shown to reject a deliberately broken inputComplete · research scale
• Integration and self-improvementFault-injection test of the 36 cross-checks: across all 7,806 combinations of up to three faults, a build-up to all 36, and every on/off pattern of any four (942,480), exactly the faulted checks fired and no healthy one didComplete · 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

Seven discriminating demonstrations

Each targets an architectural difference, not a benchmark gap.

A complete graph has no centre: every face touches the other eight, so the face drawn in the middle shows a role, not a position the structure forces. Two placements are defensible.

How the nine faces work together

The nine faces are run the way an intelligence community runs its agencies. Two rules pull in opposite directions, and both are kept.

• ◆Compartmentation (avoids entanglement). Each face writes its own report on each candidate from its own data, without seeing the others’ reports. The reports stay independent sources, and no face’s conclusion gets repeated back through another face.
• ◆Fusion (uses every connection). The reports are combined using the methods of the intelligence community:
• ◆Source grading: each face carries a track record of how often its call agreed with the gate’s verdict. Its weight comes from that record, not from an assumption.
• ◆All-source score: the reliability-weighted sum of the faces’ reports, with abstentions carrying no weight.
• ◆Corroboration: a candidate gets a complete trial only when at least two positively graded faces support it.
• ◆Competing hypotheses (Heuer): candidates are also ranked by the weighted evidence against them, since disconfirming evidence is what separates hypotheses.
• ◆Collection management: trial slots left over go to the candidates the faces disagree on most, so the system learns where it is least sure.
• ◆Counter-intelligence: the arithmetic face can veto a candidate whose certificates fail. The trace store can discredit the spectral reports when the spectral record is inconsistent.
• ◆After-action review: the gate’s verdicts update every face’s grade. The fused score’s predictive power is compared with each single face’s, which tests whether fusing helps.

In the two phases these rules govern both directions of the map: which program, read out of a trained model, enters the language, and which program from the language is compiled into a new model.

K9 faces when the model learns nuclear physics

K9 is the network of Logarchéon’s nine coupled components. When the tokens are nucleons, levels or reactions instead of words, the faces take a physical meaning. Complete analysis →

In a model of nuclear data, GRAIL’s geometry defines a loss landscape, and its Laplace eigenfunctions are that landscape’s normal modes. When the group is arithmetic, Hecke operators commute with the Laplacian and add labels to those modes; the labels belong to the model’s geometry, not to nuclei. The labelled modes give a kernel basis in which a fit could be made mode by mode. Whether this helps training is untested; on small models, spectral jumps gave a head start but no lower final loss. Each stage →

Not a copy of the brain

The brain is one solution to intelligence, shaped by evolution under the limits of biology: slow signals, unreliable parts, a fixed energy budget, and no means, within biology, of copying itself exactly. A computation is not bound to those limits. The rules a program follows are chosen; mathematics is the one law it cannot break, and physics sets only the price in energy-momentum and spacetime. Every quantity in physics, its constants included, can be expressed in three dimensions: Time (T), Length (L) and Mass (M). The only exceptions are the dimensionless constants, which are pure numbers/ratios, and these ratios may also set limits on a physical ASI design. Within those bounds a machine can use what no nervous system can: exact arithmetic, exact symmetry, exact copies, proofs checked by machine. The brain still teaches. Once part of a nervous system is described in mathematics, the description reaches beyond its source: a quantity first recognised as a vector belongs to the family of tensors of every rank, and the mathematics admits all of them at once. In a computer, where the rules are chosen, each of those ranks can be given a meaning in intelligence, either from patterns that appear in data or by a designer’s assignment that is logically and mathematically consistent. The vector is one example; in Logarchéon’s development, one of its approaches treats every quantity of intelligence written in mathematics in this way. A model of the brain is a way into that larger space, not its boundary. The architecture is drawn from that larger space. Its question is not how the brain computes, but how far mathematics allows intelligence to be built.

Where the current build stands

This is a landing page. It states results and criteria only; how each result was obtained, and why the architecture takes its form, are withheld here. Proofs and implementations are rechecked against the stated claims by separate verification passes, run by AI models and agents from different companies, to detect drift.

Logarchéon’s current build (CEAS–Ψ–GRAIL) placed among 60 AI developments of 2026, each scored on twelve measured quantities in [0, 1], with no judgement scores. Ranking is by Pareto layer and by the mean of the twelve. Status as of 12 October 2026. Complete study →

Attention cost at scale (computed)

Replacing dot-product attention scores with GRAIL’s hyperbolic scores changes total training compute by a factor of 1.003–1.14 at GPT-1’s size (117 million parameters), 1.001–1.04 at GPT-3’s (175 billion), and 1.002–1.11 for a dense model of about 400 billion parameters (128 layers, width 16,384, context 8,192). The range depends on how many group words each score sums, from one to five, and on the cost of the distance function. These figures are computed from layer shapes and operation counts, not measured on models of those sizes.

Lorentz-model heads (one hyperbolic distance per head, in ℍ^d_k). Training-compute multiplier against dot-product attention; W is the number of group words summed per score, and c the cost of the distance function per query–key pair, in floating-point-operation equivalents.

Multi-channel ℍ² heads (several two-dimensional distances per head), a diagnostic mode run only during troubleshooting; the multiplier applies only while it runs:

On compute cost, the architecture can work at large scale: with Lorentz-model heads, the hyperbolic scores add between 0.1 and 14 percent to training compute at the three sizes computed, and the addition does not grow with model size. The multi-channel diagnostic mode, which costs up to 2.3 times as much while it runs, is used only for troubleshooting.

Frontier-scale models on a single node, with the seed’s safeguards intact (computed)

Running and testing a model of GPT-3 scale or larger usually takes a rack of GPUs. Compressing the model’s stored numbers brings this down to a few GPUs. Compression alone is now widely available; Logarchéon’s contribution is to compress without giving up what the seed architecture adds:

• Exact twins survive compression. The function-identical twin copies used for provenance and protected deployment stay exact after compression. This is proved, not estimated. Ordinary compression breaks it.
• Every compression step is checked and reversible. A step is kept only if the model’s measured accuracy and its causal answers are unchanged, and any step can be undone exactly.
• CEAS-trained models are predicted to be cheaper to store. Holding attention at its critical operating point leaves no idle parameters for compression to remove, and the theory predicts 1.4–2.5 times lower storage cost in the attention’s query and key weights than standard training. This prediction is being tested.

Each GPU is a current data-centre GPU with 80 GB of memory. “GPT-1/2/3 scale” refers only to the published parameter counts of those models, used as reference sizes; no row describes a particular company’s model. Figures are computed, not measured at these sizes, and exclude the memory for long contexts. For the largest models, the GPU count is set mainly by how much information a heavily trained model holds, which no method can compress away.

Next milestones. The seed architecture has been built and tested at research scale; compression is what takes it to large models. Each step is tested in order: models of up to 1.5 billion parameters first, then open models of 14 billion, 70 billion and about 235 billion parameters, the last on a single GPU, with CEAS temperature control, the twin checks and the causal checks applied to each. Results will be posted here as they are measured.

Applied to an open 32-billion-parameter model (measured)

The same controls were applied to an open-weight language model of 32.8 billion parameters. Each change was accepted only when a statistical bound passed on held-out text. Perplexity measures prediction error; lower is better.

Independent verification · status

This is a landing page. It states results and criteria only; how each result was obtained, and why the architecture takes its form, are withheld here. Proofs and implementations are rechecked against the stated claims by separate verification passes, run by AI models and agents from different companies, to detect drift.

1 · Measured quantities

This is a landing page. It states results and criteria only; how each result was obtained, and why the architecture takes its form, are withheld here. Proofs and implementations are rechecked against the stated claims by separate verification passes, run by AI models and agents from different companies, to detect drift.

Twelve quantities, each in [0, 1], computed only from measured numbers; an unmeasured quantity counts as 0.

“·” unmeasured (counts as 0); “—” not applicable. † Level 4 applies if U is read against the loop’s own earlier miss, which rises with wear, instead of against the frozen controller. Discovery values: OpenAI math model, 300 of 719 results with Lean formalisations (openai/math, 7 October 2026); AlphaEvolve, improvements on about 20% of more than 50 open problems (DeepMind, May 2025); Aristotle, Erdős problem #728 with a Lean proof, jointly with GPT-5.2 Pro (arXiv 2601.07421); AxiomProver, Fel’s conjecture with a public Lean proof (Axiom, February 2026).

2 · Gates and the discovery track

This is a landing page. It states results and criteria only; how each result was obtained, and why the architecture takes its form, are withheld here. Proofs and implementations are rechecked against the stated claims by separate verification passes, run by AI models and agents from different companies, to detect drift.

Levels 1–6 of the certification ladder are unchanged. Three gate qualifiers attach to any level, and a discovery track applies to any system.

Current build: Level 5 (4†) · gate A in progress · gates B and C not started · D0.

3 · The index line

This is a landing page. It states results and criteria only; how each result was obtained, and why the architecture takes its form, are withheld here. Proofs and implementations are rechecked against the stated claims by separate verification passes, run by AI models and agents from different companies, to detect drift.

M12 · mean of twelve measured quantities
0         20        40        60        80       100
|---------|---------|---------|---------|---------|
    ^8.3 34 systems
     ^9.8 H-JEPA
     ^10.2 C-JEPA
     ^10.4 Physical Intelligence π0.5/π*0.6
     ^10.8 Self-improving EFMs
      ^12.5 DeepSeek V4-Pro, Huxley-Gödel Machine
       ^13.9 Equivariant sci-ML
       ^14.7 AlphaEvolve
        ^16.2 HCP-DCNet
        ^16.7 10 systems
           ^22.2 ByteDance Seed 2.0 / Prover
            ^23.5 Harmonic Aristotle
            ^23.6 AlphaProof, Hyperagents
            ^24.9 Axiom Math
              ^28.9 OpenAI math model
                           ^53.4 Logarchéon current build

M12ᴾ · provenance-weighted
0         20        40        60        80       100
|---------|---------|---------|---------|---------|
   ^5.6 Equivariant sci-ML
    ^8.3 41 systems
      ^12.5 DeepSeek V4-Pro, Huxley-Gödel Machine
      ^12.7 AlphaEvolve
        ^16.7 8 systems
         ^18.1 Hyperagents
           ^22.2 ByteDance Seed 2.0 / Prover
            ^23.5 Harmonic Aristotle
            ^23.6 AlphaProof
            ^24.9 Axiom Math
              ^28.9 OpenAI math model
                 ^33.6 Logarchéon current build

4 · The leaders

This is a landing page. It states results and criteria only; how each result was obtained, and why the architecture takes its form, are withheld here. Proofs and implementations are rechecked against the stated claims by separate verification passes, run by AI models and agents from different companies, to detect drift.

Sorted by M12; tied scores share a rank. #ᴾ is the rank on the provenance-weighted reading. Pareto layer 1 holds the 4 developments that no other development matches or exceeds on all twelve quantities while exceeding on at least one. Seed is the third reading: M12 scaled by the pillar core, nonzero only where coordination, causation and invariance are all measured.

5 · Where the main developers stand

This is a landing page. It states results and criteria only; how each result was obtained, and why the architecture takes its form, are withheld here. Proofs and implementations are rechecked against the stated claims by separate verification passes, run by AI models and agents from different companies, to detect drift.

6 · All 60 developments

This is a landing page. It states results and criteria only; how each result was obtained, and why the architecture takes its form, are withheld here. Proofs and implementations are rechecked against the stated claims by separate verification passes, run by AI models and agents from different companies, to detect drift.

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.

Nuclear physics, fusion and gravitation

Logarchéon’s AI designs draw on statistical physics, hyperbolic geometry and causal inference, the same mathematics used in nuclear physics, fusion and gravitation. Some of the links are exact identities; others are analogies that still require testing. Complete analysis →

Where the designs map onto nuclear physics

New physics: theoretical results

Fusion

Magnetic-confinement fusion gives two close links: Greene’s residue, exact for the classification of periodic field lines, and the poloidal-angle twin, exact before the truncation of the series.

Inertial-confinement fusion overlaps with stockpile stewardship, so work stays on magnetic confinement and open fusion-energy data.

New physics: theoretical results

What the dimensional analysis gives

Placed in nuclear relations, the parameters of the designs acquire dimensions, as h acquired the dimension of angular momentum through E = hν. The dimension names the known quantity to compare a parameter with; it does not make the parameter a new constant. The table gives the new results; Propositions 14 and 15 come from DL ≠ LD. Definitions and proofs →

New physics: theoretical results

Where the designs enter nuclear science

Nuclear science divides into fourteen branches: eight of basic science and six of application. The table is an index of where the designs and their quantities enter each branch. Details →

Energy: fission power, inertial fusion energy and the stars

In fission, the capture and fission cross sections of fuel nuclei in the unresolved-resonance region come from average resonance parameters (Hauser–Feshbach; Gilbert–Cameron), and delayed-neutron yields and decay heat from β-decay strength; the designs enter through spacing-ratio flags on evaluated levels, Ψ's abstention in model fits, and fission-path searches that use the inertia tensor as metric. Barrier penetration needs the WKB or instanton action, which the twin step does not compute. In plasmas and stars the rate weight exp(−βE − √(E_G/E)) is the standard product of the thermal factor and the Gamow factor: E_G = 1.182 MeV for D–T, with the Gamow peak at 30.9 keV for kT = 10 keV; E_G = 493 keV for p + p, with the peak at 6.0 keV for kT = 1.34 keV. These are background numbers, not results of the designs. In neutron stars the mean squared rapidity of nucleons is κ = ⟨ζ²⟩ = 0.046 in symmetric matter at saturation density and 0.073 in neutron matter, and 0.094 and 0.147 at three times saturation.

Programmes for each branch (as of 20 October 2026)

The direct entry for a small business is the SBIR/STTR route; other programmes take a university or national-laboratory partner. Details →

Other fields that share the mathematics

The mathematics of Logarchéon is not specific to nuclear physics. Five fields share several of its tools at once; eight more share one or two. The closest match is protein folding: there the critical point of CEAS corresponds to the glass temperature of the energy landscape, and the gap score is T_g/T_phys, glass over physical temperature. Proposition and details →