Measuring the Path to Superintelligence · 12 October 2026

Abstract

The central question

Are coordination, causal reasoning and invariant geometry necessary on the way to superintelligence, and, if they are required at least as functions, which 2026 AI developments have demonstrated them, measured rather than described?

The thesis advanced here

(i) Causal reasoning at Pearl’s second and third rungs is necessary as a function, by a proven theorem. (ii) Nonlocal coordination is necessary as a function only where communication rounds are counted, and ordinary global attention already supplies it. (iii) Exact invariance is a proven efficiency, not a proven necessity, and becomes necessary only where exactness on unseen group elements is part of the task. (iv) The joint necessity of the three is not proved. (v) When 60 developments are scored on twelve measured quantities, with no judgement scores, Logarchéon’s current build is one of four developments that no other development dominates, and ranks first on the mean of the twelve (53.4 against 28.9 for the next) and on its provenance-weighted reading (33.6 against 28.9). (vi) The lead rests on the three pillar quantities and the closed loop; the build trails on held-out verification, verification and discovery. Each clause is defeasible by a measurement named in the method.

How the thesis is tested

Each pillar’s necessity is checked against published theorems. Twelve quantities, each in [0, 1], are computed by stated formulas from measured numbers: coordination C, causal reasoning Ψ, invariance G, the closed self-improvement loop R, control improving across cycles U, held-out verification H, breadth B, scaling S, verification V, separation X, negative controls N and discovery D. Developments are ranked by Pareto layer on the twelve, then by their mean M12; a provenance-weighted reading M12ᴾ multiplies every self-measured quantity by H. Sensitivity readings bound every result.

Second thesis

The lead measured here is a lead in demonstrated evidence, not in capability. The build’s measurements come from its own synthetic tasks and are held by the developer, and most pillar cells of other systems are unmeasured. Running one battery on every runnable system, with a hidden set held by a third party, is the only route to a comparison that is measured throughout.

Present Position

What this section is for: the state of the field and of the ladder at the date of the study, before any scoring.

As of 12 October 2026 the most capable public systems are OpenAI’s GPT-6 Astra, Anthropic’s Claude Opus 5.5 and Google’s Gemini 4 Argon. On the ladder’s 0–12 Rank scale they sit at Rank 5 (Emerging AGI). On the Certification Levels they sit at Level 1 read literally and at Level 2 read functionally. Global attention gives every transformer the coordination function, while no frontier system has published a group-invariance measurement or a current causal-rung score, and every frontier self-improvement loop still has human gating.

Mathematical verification moved in the ten days to 12 October. On 7 October OpenAI released the record of an unreleased mathematics model (openai/math): 719 results grouped into 372 families, 300 of them with Lean formalisations and checker challenges. Axiom’s AxiomProver, Harmonic’s Aristotle, AlphaProof and ByteDance’s Seed-Prover also prove in Lean. These systems lead on verification and discovery; none has a measured causal, invariance or closed-loop component. On 5 October AMI released H-JEPA, a hierarchical world model, with code and weights.

The systems nearest the ladder’s middle rungs remain research-scale loops: Meta’s Hyperagents, Google DeepMind’s AlphaEvolve, the Darwin Gödel Machine and the Huxley-Gödel Machine. Each runs twenty or more cycles without manual edits, and none has separable causal, invariance and coordination components. Embodied systems (DeepMind’s self-improving embodied foundation models, Physical Intelligence, Gemini Robotics 2, NVIDIA GR00T) improve control, mostly between human-run releases.

Logarchéon’s current build meets Certification Level 5 under its own certification rule: all 23 test batteries pass at research scale, simulated control improves across self-improvement cycles on validation and on unused hidden seeds, and a hidden benchmark committed by hash in advance is consistent with validation. Since 12 October each candidate step’s Lean certificate is compiled before the step is written, and a step whose certificate fails is rejected; the certificates are not yet chain-linked from step to step. Two of the ten hard exclusion conditions are not cleared (transfer to an unseen task family; a falling self-improvement rate) and two are partly cleared (one task family per loop; no adversarial benchmark). Level 6 is at the groundwork stage.

The two scales of the ladder

Certification Level	Pass criterion (lecture notes)
1 · ANI	Passes one task or one domain.
2 · Architecture demonstration	At least one of CEAS, Ψ, GRAIL beats its baseline independently.
3 · Triadic integration	J₁₁₁ below all seven ablation variants.
4 · Recursive triadic improvement	All three modules improve over K ≥ 20 closed-loop cycles, with no manual edits inside the loop.
5 · Physical-ASI seed	Level 4, plus all three Pearl rungs and inverse design, invariant generalisation, nonlocal coordination, control improving across self-improvement cycles, and ablation superadditivity.
6 · ASI claim	Level 5, plus best-baseline-beating performance across at least ten domains, an improvement process that itself improves, scaling exponents that predict larger hardware, and consistent private and adversarial benchmarks.

Rank	Capability class	Rank	Capability class
0	Classical automation	7	Expert / Exceptional AGI
1	Narrow AI	8	Embodied AGI / Physical AI
2	Foundation / frontier AI	9	Artificial superintelligence
3	Agentic AI	10	Physical ASI
4	AI-amplified R&D	11	Recursive ASI
5	Emerging AGI	12	Complete-spectrum ASI
6	Competent AGI

Certification Levels measure evidence about one architecture; Ranks measure capability class. There is no Certification Level 7: a “7” exists only as Rank 7 (Expert AGI) or as RSI Level 7 (closed-loop successor design) in the notes’ separate table of self-improvement levels.

Take-awayThe frontier leads in capability class and the provers lead in verification; the ladder’s middle and upper rungs are populated only by research-scale systems, and Logarchéon’s current build is the one development that claims Level 5.

Summary of Findings

Causal reasoning is necessary as a function. The Causal Hierarchy Theorem shows that answering rung-2 and rung-3 questions requires information at that rung or above; an explicit causal engine is one way to supply it, not the only way.
Nonlocal coordination is necessary as a function only in a round-count model, and global attention or a global mean already supplies it. On the build’s own scaling tasks a plain global mean matched CEAS exactly.
Exact invariance is a proven efficiency (a |G|-fold effective sample gain, and excess risk equal to the non-invariant part of the predictor), necessary only where exactness on unseen group elements is required.
The joint necessity of the three pillars is not proved; the independence of the three obstructions is asserted, not shown.
Four developments are non-dominated on the twelve quantities: Logarchéon’s current build, the OpenAI mathematics model, AxiomProver and AlphaEvolve. Each leads on something the others lack: the build on the pillars and the loop, the provers on verification and discovery, AlphaEvolve on discovery with a measured inverse-design term.
On the mean of the twelve, the build ranks first of 60 at 53.4, ahead of the OpenAI mathematics model (28.9), AxiomProver (24.9), AlphaProof and Hyperagents (23.6), Aristotle (23.5) and Seed-Prover (22.2). The frontier general models tie at 16.7.
On the provenance-weighted reading the lead narrows to 33.6 against 28.9, because the build’s own measurements are held by the developer (H = 0.5) while the provers’ results are checked externally (H = 1).
The first place survives every sensitivity reading tested: with control read against the loop’s own miss and cautious bounds on the small causal samples together, the build scores 47.1 (M12) and 30.5 (M12ᴾ).
The build trails on H, V and D: no third-party hidden set, verification that gates each step but does not yet link the steps, and no resolved open problem (D0).
The build fails Level 6 on four counts: breadth B = 0, no prediction to larger hardware tested, an improver that grew once (+0.0041; sign-test p = 0.40), and no adversarial set.

Method

What this section is for: every rule of measurement and aggregation, fixed before any system is scored, with the exclusions stated as prominently as the rules.

Rule 1 · Pillars are required as functions

Following the ladder strictly means requiring the three functions, not the three named modules: any mechanism that provides a pillar’s function by a mathematically equivalent route counts. Each function is tested by a number measurable on any system (Part I).

Rule 2 · Twelve quantities, each in [0, 1], from measured numbers only

Quantity	Formula	Rule for edge cases
C · coordination	1 − q	q is an architectural fact or a measured round count. Global attention gives q = 0; local message passing in 3-D gives q = 1/3.
Ψ · causal	mean of four terms: â₂, â₃, abstention rate, inverse-design success; â = clip((a − c)/(1 − c), 0, 1)	Chance c = 0.5 for binary items (CLadder); c = 0 for numeric or open-ended items (the build’s demos, CausalReasoningBenchmark, inverse design). A pooled score with no rung split and an unverified definition (CausalDS, secondary source) is excluded.
G · invariance	min(1, acc_unseen / acc_seen) on held-out group elements, or 1 when exact invariance holds by construction	ARC-AGI is never used for G. Learned or augmented invariance with no measured error counts as unmeasured.
R · loop	min(1, K/20)·(1 − h)	K counts cycles that modify the system’s own code, weights or scaffold; h = 0 when the published procedure has no human step inside a cycle.
R2	fraction of cycles that improved	Used only in the Level 4 threshold (≥ 0.6).
U · control across cycles	g·min(1, cycles/2); g = (E₀ − E₁)/E₀, or (P₁ − P₀)/(1 − P₀) for success rates	A single improvement phase counts half; an unknown cycle count is treated as one phase.
H · held-out verification	1, 0.5 or 0	1: scored by a third party on a set the developer could not see, with no third-party finding of overstatement. 0.5: a developer-held hidden set (hash-committed included), a developer-chosen held-out domain, a third-party run on a public set, or a third-party finding of overstatement. 0: none reported.
B · breadth	min(1, n/10)	n counts domains with independent measurement at or above the best human.
S · scaling	1 if a fitted exponent’s confidence interval excludes 0 against a baseline, else 0	Log-linear fits without an exponent and interval score 0.
V · verification	mean of three terms: the share of headline claims with a machine-checked proof; 1 if an external checker confirms the published statements with the standard axioms only; 1 if certificates gate each step before it is written and are chain-linked	A gate without chain linkage counts 0.5 in the third term. A proof checked only after the run does not gate.
X · separation	min(1, log₁₀ r / 2)	r is the pre-registered ratio over the strongest baseline at the largest measured scale, on a quantity indexed by problem size. r = 10 gives 0.5; r = 100 gives 1. A ratio of indices never counts.
N · negative controls	share of six theorem-backed controls passed	A tampered certificate is rejected; an injected wrong step is refused in the loop; a 2n-cycle against two n-cycles is at chance without labels; an equivariant sampler is refused above the factor-of-IID threshold; the next Liouville value is at chance; the engine abstains on forced-flow reachability past the certified horizon.
D · discovery	min(1, log₁₀(1 + n) / 2)	n counts recognised open problems resolved with a publicly checkable proof or construction. One problem gives 0.151; ten give 0.521; 99 or more give 1.

Rule 3 · Level thresholds

Level	Threshold rule (pillar equivalence allowed)
L1 · ANI	Any measured score on one task.
L2 · Architecture demonstration	C ≥ 0.9, or Ψ ≥ 0.5, or G ≥ 0.9.
L3 · Triadic integration	All three pillars meet their L2 thresholds, and a 2³ ablation shows J₁₁₁ below all seven variants.
L4 · Recursive improvement	L3, R = 1 (K ≥ 20, h = 0), R2 ≥ 0.6, and every module improved inside the loop (battery R3).
L5 · Physical-ASI seed	L4, every Ψ term ≥ 0.8, G ≥ 0.9 with ε_twin ≤ 10⁻⁶, C = 1, U > 0, H ≥ 0.5, and ablation superadditivity.
L6 · ASI claim	L5, B = 1, S = 1 with a confirmed prediction on larger hardware, a significant measured rise of the improver (R_t+K > R_t), and H = 1 including an adversarial set.

Rule 4 · Three readings and one ranking

M12 = 100 · mean(C, Ψ, G, R, U, H, B, S, V, X, N, D)

M12ᴾ = (100/12) · [C + H + V + D + H · (Ψ + G + R + U + B + S + X + N)]

Seed = M12 · (C · Ψ · G)^1/3

M12 weights the twelve equally. M12ᴾ counts a self-measured quantity completely only when its measurement was held out from the developer (H = 1), and at half when the developer held the hidden set. Seed scales M12 by the geometric mean of the three pillar quantities, so it is nonzero only where coordination, causation and invariance are all measured. The ranking does not depend on the weights: a development is placed below another only when the other matches or exceeds it on all twelve quantities and exceeds it on at least one (Pareto layers); M12 orders developments inside a layer.

Rule 5 · Gates and the discovery track

Qualifier	Criterion
A · Verified	Each step certified in Lean before it is written, certificates chain-linked, and a wrong step injected in every run refused.
B · Measured	Positions from shared public test sets, head to head, and one battery on a hidden set held by a third party (H = 1).
C · Separated	A pre-registered asymptotic separation of at least 10× over the strongest baseline on one axis; C⁺ at least 100×.
D1–D4 · Discovery	D1 reproduces known results under verification; D2 at least one recognised open problem resolved with a checkable proof or construction; D3 at least 10; D4 at least 100.

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

Exclusions

No judgement scores. Novelty of mechanism, breadth per unit of resource and protocol discipline are not scored; every number on this page is computed from a measured quantity.
No credit for peer review, outside evaluation, replication or adoption except through H: every development is judged on its own evidence.
Unmeasured counts as zero. A development is not credited with a quantity its record does not report.
Instruments and components with no measurable system are left out: METR, the CLadder-type causal benchmarks, the UK AI Security Institute and the Muon optimiser.
Enablers with no mechanism to score are left out: NVIDIA compute, TSMC, Huawei Ascend, national compute programmes, and announced laboratories with no public system.
Developments absent from the quantified table are transformer-based with only C = 1 measured, and receive M12 = 8.3.
Release results are counted as reported. The Lean files of other developments were not compiled for this study.
Hidden-benchmark seed values are not reproduced.

Take-awayEvery number on this page follows from the rules above; changing a rule changes the result, and the sensitivity readings in Parts II and IV show by how much.

I · Are the Pillars Necessary?

What this section is for: to settle, from published theorems, which pillar is required, in what sense, and by what measurable criterion.

The lecture notes (v21.8) prove three narrow theorems: a round separation (local message passing needs at least D rounds, where D is the diameter, while one all-reduce suffices; physical time is Ω(N^1/d) for either model), a two-model non-identifiability example with a worst-case error floor of 1/4, and a finite-group Reynolds risk identity. The broader triadic necessity theorem builds the three capabilities into its definition of scalable superintelligence and asserts that “these three obstructions are independent” without a separating construction; “Architecture from Theory” is labelled a conjecture.

Pillar	Verdict	Theorem behind it	Criterion measurable on any system
CEAS	Necessary as a function, proven only in a unit-cost-broadcast (round-count) model; not necessary in physical time	Round separation (notes); LOCAL t-hop lower bound (Ghaffari); GNN depth×width bounds (Loukas)	Rounds exponent q in T(N) ∝ N^q on globally sensitive aggregation: local q = 1/d, functional CEAS q ≈ 0. Any all-reduce qualifies (global attention, a global mean, a bus, a tree).
Ψ · rungs 2–3	Necessary as a function, proven	Causal Hierarchy Theorem (Bareinboim, Correa, Ibeling & Icard): rung collapse occurs only on a measure-zero set of models; meagre-set version (Ibeling & Icard); robust agents imply an approximate causal model (Richens & Everitt)	Rung-2 and rung-3 accuracy above chance; correct abstention on non-identifiable queries; the 1/4 error floor on observationally equivalent pairs.
Ψ · inverse design	Unproven as a separate necessity; reduces to rung 2 plus planning	None found	Success rate in reaching a target under constraints, against a forward-only baseline.
GRAIL	Efficient, not strictly necessary; a symmetry prior is necessary only if exact invariance on unseen orbit elements is required	Excess risk equals ‖f^⊥‖² (Elesedy & Zaidi); \|G\|-fold sample gain (Tahmasebi & Jegelka); canonicalisation is equivalent (Kaba et al.)	Retention on held-out group elements; ε_inv = E‖F(gx) − F(x)‖²; any map that factors through X/G qualifies.
Triad jointly	Unproven	Independence asserted, not shown	Main effects of a 2³ ablation.

Three concessions that bound the verdicts

The notes state that standard transformer self-attention already realises an aggregate primitive, so the round theorem does not by itself distinguish CEAS from transformers.
On the build’s own R4 tasks a global mean matched CEAS exactly, and the entropy-corridor controller failed its CEAS hypothesis test (mean −0.002), while the β-controller passed (+0.092).
GRAIL’s necessity argument is a no-free-lunch argument that does not exclude canonicalisation or learned invariance, which grows with scale in vision models (Gruver et al.).

The Causal Hierarchy Theorem does not require an explicit causal-model engine: causal statements in training text, or experiments run through tools, can carry rung-2 information. Following the ladder strictly therefore means requiring the three functions, and treating GRAIL’s function as required only because the ladder writes exact invariance into its definition.

Take-awayOf the three pillars, only the causal function is proven necessary without a cost-model caveat; a proof-weighted index gates on causal reasoning and coordination, and measured causal accuracy is the pillar frontier systems have yet to report.

II · The Twelve Measured Quantities

What this section is for: the twelve quantities computed for each leading development, the three readings, and how far each result moves under alternative readings.

“·” means unmeasured and counts as 0; “—” not applicable. The table lists every development at or above M12 = 16.7; all 60 are in Part III. † Level 4 applies if U is read against the loop’s own earlier miss, which rises with wear, instead of against the frozen controller.

Development	C	Ψ	G	R	U	H	B	S	V	X	N	D	M12	M12ᴾ	Seed	Layer	Level
Logarchéon current build (CEAS–Ψ–GRAIL)	1	1	1	1	0.574	0.5	·	1	0.167	·	0.167	·	53.4	33.6	53.4	1	5 (4†)
OpenAI math model (unreleased)	1	·	·	·	·	1	·	·	0.472	·	·	1	28.9	28.9	0	1	L1
Axiom Math (AxiomProver)	1	·	·	·	·	1	·	·	0.833	·	·	0.151	24.9	24.9	0	1	L1
AlphaProof	1	·	·	·	·	1	·	·	0.833	·	·	·	23.6	23.6	0	2	L1
Hyperagents	1	·	·	1	0.332	0.5	·	·	·	·	·	·	23.6	18.1	0	2	L1
Harmonic Aristotle	1	·	·	·	·	1	·	·	0.667	·	·	0.151	23.5	23.5	0	2	L1
ByteDance Seed 2.0 / Prover	1	·	·	·	·	1	·	·	0.667	·	·	·	22.2	22.2	0	3	L1
Equivariant world model (Wang)	1	·	1	·	·	·	·	·	·	·	·	·	16.7	8.3	0	2	L2
Darwin Gödel Machine	1	·	·	1	·	·	·	·	·	·	·	·	16.7	8.3	0	3	L1
Anthropic Opus 5/5.5, Mythos	1	·	·	·	·	1	·	·	·	·	·	·	16.7	16.7	0	4	L1
Co-Scientist	1	·	·	·	·	1	·	·	·	·	·	·	16.7	16.7	0	4	L1
GLM-5.3	1	·	·	·	·	1	·	·	·	·	·	·	16.7	16.7	0	4	L1
GPT-6 Astra	1	·	·	·	·	1	·	·	·	·	·	·	16.7	16.7	0	4	L1
Grok 4.6	1	·	·	·	·	1	·	·	·	·	·	·	16.7	16.7	0	4	L1
Kimi K3	1	·	·	·	·	1	·	·	·	·	·	·	16.7	16.7	0	4	L1
Qwen3.8-Max	1	·	·	·	·	1	·	·	·	·	·	·	16.7	16.7	0	4	L1
Sakana AI Scientist	1	·	·	·	·	1	·	·	·	·	·	·	16.7	16.7	0	4	L1
Calibration: GPT-4 on CLadder	1	0.117	·	·	·	·	·	·	·	·	·	·	9.3	8.3	0	—	—
Calibration: Euclidean transformer inside the build	1	—	0.300	—	—	—	—	—	—	—	—	—	—	—	—	—	—

How the non-trivial cells are computed

Logarchéon current build. Ψ = mean(24/24, 5/5, 26/26, 100/100) = 1 with c = 0. G = 1.000/1.000 on SL₂(ℝ), with ε_twin ≈ 10⁻¹³. R = min(1, 40/20)·1 = 1, and R2 = 0.80 (31–33 of 40 cycles). U = mean over six seeds of (1 − loop/frozen) = 0.574, multiplied by min(1, 24/2) = 1. S = 1 from the token-axis difference interval [0.692, 0.936]. V = 0.167: each candidate step’s certificate is compiled before the step is written, but the certificates are not chain-linked, so the third term counts 0.5; the other two terms are 0. N = 0.167: one of the six controls passes. X = 0: no pre-registered separation has been measured. D = 0.
OpenAI mathematics model. V = mean(300/719, 1, 0) = 0.472: 300 of 719 results carry Lean formalisations, the release ships checker challenges against the standard axioms, and no step gate is reported. D = 1: the release reports resolutions of more than 99 recognised open problems. H = 1 from the external checker.
AxiomProver. V = 0.833; D = log₁₀2 / 2 = 0.151 for Fel’s conjecture with a public Lean proof.
Aristotle. V = 0.667; D = 0.151 for Erdős problem #728 with a Lean proof, jointly with GPT-5.2 Pro.
AlphaEvolve. The target was matched or beaten on about 75% + 20% of 50+ problems, so inverse-design success = 0.95 and Ψ = 0.95/4 = 0.238; improvements on about 20% of more than 50 open problems give n = 10 and D = 0.521.
Hyperagents. U = (0.372 − 0.060)/0.94 = 0.332 over 100 iterations of robotics reward design. H = 0.5 because the held-out domain was chosen by the developer.
Self-improving embodied FMs. g takes three values: (0.875 − 0.63)/0.37 = 0.662 on the real robot, (0.75 − 0.45)/0.55 = 0.545 in simulation, and 0.595 on BananaTable. Their mean is 0.601; one phase halves it to U = 0.300.
π*0.6. The primary abstract reports that the method roughly halves the task failure rate, so g = 0.5; the cycle count is unknown, so one phase halves it to U = 0.25.
GPT-4 on CLadder. Ψ = [(0.628 − 0.5)/0.5 + (0.606 − 0.5)/0.5]/4 = 0.117.
H = 1 assignments come from ARC Prize semi-private scores, live IMO contests, blind peer review, external proof checkers and an independent experiment. DeepSeek receives 0.5 because NIST CAISI found it performed worse than its own report suggested.

Sensitivities

Reading	Build M12	Build M12ᴾ	Build Seed	Rank
As measured	53.4	33.6	53.4	1st on both
U read against the loop’s own miss (U = 0)	48.6	31.2	48.6	1st on both
Wilson 95% lower bounds on the small Ψ samples (Ψ = 0.815)	51.9	32.9	48.4	1st on both
Both of the above	47.1	30.5	44.0	1st on both
A third-party hidden set confirms the build’s results (H = 1)	57.6	57.6	57.6	1st on both

The index line

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 Huxley-Gödel Machine, DeepSeek V4-Pro
       ^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 Huxley-Gödel Machine, DeepSeek V4-Pro
      ^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

Take-awayThe build’s lead is made of the pillar quantities and the loop; the provers’ scores are made of verification and discovery; the two kinds of evidence do not yet overlap in any one development.

III · The 60-Entity Line

What this section is for: every development on one [0, 100] line, under both readings, with its Pareto layer.

M12 is the mean of the twelve quantities; M12ᴾ multiplies every self-measured quantity by H. Layer 1 holds the 4 developments that no other development matches or exceeds on all twelve quantities while exceeding on at least one.

Reading

Indicator Table

LEVEL = certification level (functional) · C Ψ G R U H B S V X N D = the twelve quantities · · = unmeasured

#	Development	Group	Level	C	Ψ	G	R	U	H	B	S	V	X	N	D	M12	M12ᴾ	Seed	Layer

Take-awayOn both readings Logarchéon’s current build leads; the frontier general models tie at 16.7 on M12, behind the provers and the research-scale self-improvement loops.

IV · Firmness of the Lead

What this section is for: how much of the first place survives a change of reading.

Reading	Logarchéon build	Nearest rival	Margin	What the margin rests on
M12, as measured	53.4 (1st)	OpenAI mathematics model 28.9	+24.5	The pillar quantities C, Ψ, G and the loop R, S, which no other development reports together.
M12, conservative (U = 0, Wilson bounds on Ψ)	47.1 (1st)	OpenAI mathematics model 28.9	+18.2	The same quantities, with control and the small causal samples read cautiously.
M12ᴾ, as measured	33.6 (1st)	OpenAI mathematics model 28.9	+4.7	Halved credit for the build’s developer-held measurements; complete credit for the prover’s externally checked results.
M12ᴾ, conservative	30.5 (1st)	OpenAI mathematics model 28.9	+1.6	The narrowest margin on this page.
Pareto layer	Layer 1	OpenAI mathematics model, AxiomProver, AlphaEvolve	shared	No weights: no development matches or exceeds the build on all twelve quantities, and the build does not dominate the other three.

Take-awayThe first place on M12 is firm; on the provenance-weighted reading it rests on a margin of 1.6 to 4.7 points, and a third-party hidden set would move it either way.

V · What the Numbers Cannot Say

What this section is for: the exclusions of the evidence, stated as prominently as its findings.

Equal numbers do not mean equal capability. The build’s values of 1.0 are exact results on its own synthetic tasks: 24 causal questions, the SL₂(ℝ) and PSL₂(ℤ) groups, a two-dimensional point mass, 16 causal blocks and at most 40 cycles on one workstation, measured against correlation-only baselines and SmolLM2-135M. Frontier numbers come from public benchmarks of a different order of difficulty; ARC-AGI-3 measures novel-task generalisation, not group invariance.
Most pillar cells of other developments are unmeasured, and an unmeasured quantity counts as 0. M12 records demonstrated evidence; it does not bound what those systems could show if measured.
The ladder was written around one architecture. Its Level 5 checks (ε_twin ≤ 10⁻⁶, a hash-committed hidden set, an R2 rate) are quantities the build was designed to report, the coordination term gives every transformer the same score as CEAS, and V, X and N are defined by the build’s own plan.
Equal weighting is a choice. M12 and M12ᴾ weight the twelve quantities equally; the Pareto layers do not depend on weights, and they place four developments together in the first layer.
Verification of the build is internal. H = 0.5: the hidden set is held by the developer. Proofs and implementations are rechecked against the stated claims by separate verification passes, run by AI models and agents from different companies; no outside party has rerun the build.
Other developments’ values are read from their public records. Their Lean files were not compiled for this study, and the OpenAI mathematics model itself is unreleased.
Two hard exclusion conditions of the examination are not cleared (transfer to an unseen task family; a falling self-improvement rate), and two are partly cleared (one task family per loop; no adversarial benchmark). The Level 5 verdict applies the certification report’s rule and is stated together with these conditions.
Level 6 is not met. Breadth B = 0; no prediction to larger hardware has been tested; the improver grew once (+0.0041, 9 of 16 tasks won, one-sided sign-test p = 0.40); no adversarial set exists.

Take-awayThe line measures demonstrated evidence at research scale; it does not measure capability, and it is not an outside verification.

VI · Plan of the Study

What this section is for: the measurements, in order, that can change this page.

Gate A (verified). Link the certificates from step to step, so that each step’s certificate starts at the state where the previous one ended, and inject a wrong step in every run; then rerun and pass a separate verification pass. Moves V and N.
Gate B (measured). A fresh hidden set held by a named third party, with one battery run on their hardware; head-to-head runs on shared public sets. Moves H, and with it every term of M12ᴾ.
The same battery on other systems. Open-weight models with ARC Prize scores (Qwen3.8-27B, DeepSeek V4, GLM-5.3, Kimi K3) and frontier APIs where access exists: C from rounds against N, Ψ from the 24-question suite and the per-rung splits, G from seen/unseen retention on SL₂(ℝ) and PSL₂(ℤ), R and U inside the same 40-cycle gated loop. Turns unmeasured cells into measured ones.
Gate C (separated). Three pre-registered axes on which a 10× to 100× ratio is possible: orbit sample efficiency to radius 7, error under confounding to 10⁶ samples, and coordination rounds to N = 2¹⁶. Moves X.
Breadth and discovery. Two domains measured against the best-human baseline under a published protocol (B), and open problems attempted under verification (D).
Release. Publish the audit bundle (scripts, logs, Lean files and used seeds) so that every number above can be checked from the files alone.

Take-awayThe next measurement that can change this page is a hidden set held by a third party.

Sources

Bareinboim, Correa, Ibeling & Icard, “On Pearl’s Hierarchy and the Foundations of Causal Inference” (causalai.net/r60.pdf); Ibeling & Icard (arXiv 2107.08558); Richens & Everitt (arXiv 2402.10877); Zečević et al. (arXiv 2308.13067).
Ghaffari, LOCAL model notes (MIT); Loukas (arXiv 1907.03199).
Elesedy & Zaidi (arXiv 2102.10333); Tahmasebi & Jegelka (arXiv 2303.14269); Kaba et al. (arXiv 2211.06489); Gruver et al. (arXiv 2210.02984).
CLadder (arXiv 2312.04350); CausalReasoningBenchmark (arXiv 2602.20571); ARC Prize results (arcprize.org).
Hyperagents (arXiv 2603.19461); self-improving embodied foundation models (project page); π*0.6 (arXiv 2511.14759); AlphaEvolve (Google DeepMind).
OpenAI research-automation report (Help Net Security, 7 September 2026); NIST CAISI on DeepSeek (The Decoder).
OpenAI, openai/math (GitHub, 7 October 2026): overview, contents, Lean formalisation list and checker challenges; Axiom, Fel’s conjecture with a public Lean proof (February 2026); Aristotle and GPT-5.2 Pro on Erdős problem #728 (arXiv 2601.07421); AlphaEvolve open problems (Google DeepMind, May 2025).
H-JEPA (arXiv 2610.06805); Wang, exact equivariance in world models (arXiv 2606.03003).
Logarchéon, Lecture Notes: From Physical AI to ASI, v21.8 (October 2026), including Research Build I–XII; certification battery report and pass-by-pass results record of the build.

Measuring the Pathto SuperintelligenceEvaluation of 12 October 2026