Two inputs only
A seam-normalisation constant c₃ = 1/(8π) and a carrier rank
g_car = 5 (a 5-slot carrier, 3+2). On the dimensionless axis essentially
only π is primitive, no free load-bearing numbers.
What this page claims
A measurable gap, not a proof of intelligence.
This is deliberately modest. It is not a proof of a universal grammar, and not a proof of “intelligence”.
The real contribution is to pin down the minimal gap between stable field mechanics (proven today) and semantic generalisation (mostly still open), and, crucially, how to measure it.
The laws force the system to settle into one stable attractor, not necessarily the meaning-correct one. That exact gap is called grounding.
01 · The foundation
TFPT: physics as a tiny compiler.
TFPT (Topological Fixed-Point Theory) treats physics like a small, deterministic compiler: two boundary inputs plus a few typed anchors are turned into read-outs as projections, not as fitted parameters. Hylaean does not use the physics predictions, it borrows the structural discipline and vocabulary. The theory itself lives at fixpoint-theory.com.
Honestly typed
TFPT is split into a closed dimensionless compiler, protected physics, declared anchors, and open interfaces. It is explicitly not a certified “theory of everything”, and Hylaean inherits that same honesty about its own limits.
The mathematical objects that matter for the architecture are the ones about structure:
seam (even / odd)
carrier algebra
lattice operators
transport gap
unique attractor
02 · The bridge
What Hylaean took, and what it didn't.
Hylaean keeps the structure, not the Standard-Model predictions. Each TFPT motif maps to one concrete runtime object, and the operative grammar that falls out is just three moves: K twists, L binds, T transports.
| TFPT motif | Hylaean runtime | What it gives the AI |
|---|---|---|
| Field state on a carrier | state.S [N,d] on the unit sphere | One live cognitive substrate, the “brain” |
| Seam even / odd, involution | SeamOperator, microcell sheet-pair | Double-cover, chirality, admissibility |
| Carrier (5-slot) | C5Carrier, two-point algebra | An explore / commit algebra, no semantics baked in |
| Antisymmetric torsion | K [N,d,d] (Hebbian) | Field memory: twist within a place |
| Symmetric metric | L [N,d,d] (PSD) | Binding, complementary to the twist |
| Cross-position holonomy | FieldTransport T [N,N] | Moving content between positions |
| Gapped transport → attractor | Energy minimisation, cavity relax | The answer is a settled state |
Deliberately left out: the Standard-Model masses, the full lattice compiler as a runtime, hypergraph rewriting as the substrate, and the cosmology read-outs. None of that is needed to run a field.
03 · The minimal grammar
Eight information laws, honestly scored.
These eight laws are the minimum a single field needs so that meaning could appear as stable attractors. Each one is tagged with its real status: ✓ proven ⚠ partial ✗ open proven means shown in the field path with no teach, meaning answers were not shown immediately before scoring, and zero false commits.
Unity (one field)
Exactly one field state.S on a sphere; question, answer, world model and regions are deformations of the same field.
✓ provenstrict-mode aborts rogue writes
One energy
Every behavioural force is one additive term of a single master energy. A question is a boundary condition; the answer is an energy minimum; thinking is relaxation.
✓ provenaborts if the field stops moving
Selector ≠ dynamics
After each step an algebraic selector projects onto the admissible sector, closed-form selection, never an optimizer.
✓ provencore on; gap-sharpening dormant
Guaranteed attractor (spectral gap)
A transport spectrum with a real gap (Δ = 6·ln(3/2) > 0) guarantees a unique fixed point (Perron to Frobenius).
✓ provenguarantees one, not the correct one
Memory as geometry
Memory changes as geometry (Hebbian ΔK, ΔL; basins deepen), never back-propagation on the answer path.
✓ provengrounded deepen + skill credit live; skills now crystallise as compiled energy programs that transfer across families (40/40); learning curve alive (3/8 → 8/8, 0 wrong)
Composition / transport
Per-node moves can only recolour; moving content or relations between positions needs a shared operator, transport T, or a norm-preserving rotation R on the basin manifold.
⚠ mixedgrounded transport proven (15/15); associative answer channel live; evidence-armed attractor verified (multi-word decode, off by default)
Grounding (meaning = coordinate)
Basins carry meaning as a geometric coordinate (same relation = same shift), acquired from perception, otherwise they are structureless and relations are inconsistent.
⚠ partial10+ families proven (incl. grounded antonym, life-stage & ordered/cyclic sequences from observation); bare-word placement open
Honest determination (commit / abstain)
Commit only on real field movement; otherwise abstain, and leftover pressure opens nested reasoning frames instead of bluffing.
✓ commitescalation measured dead, replaced by certificate discipline (unique survivor or abstain); first determination organs and a verified induction primitive (structure continuation) now stand on this law
04 · The key result
A falsifiable test for “does it generalise?”
The most useful scientific output here is a measurable criterion: generalisation succeeds exactly when a grounded, shared invariance exists, the same relation must act as the same geometric move across unseen pairs.
The anti-cheat test. A high fit on the demonstrated pairs plus a random score on held-out pairs means the system only learned a fitted mapping with no meaning (a leak), not grounding. Real grounding holds up on pairs it never saw.
Held-out transform-consistency (measured, current)
The line that generalises is anything with an external, member-independent perceptual scale, and the two red bars show how the frontier moves. Category failed as a single channel (0.156), then passed once re-derived as a quotient of two grounded scales (0.764). Antonym failed ungrounded (0.002 · a fitted mapping with no meaning), then hit a perfect 1.000 once given a grounded reflection on a perceptual axis. The criterion did not change; the representation did. What still has no scale, bare words never seen through perception, remains the open frontier.
05 · Proven vs open
Where the science actually stands.
Stable mechanics, selected grounding loops and three of four architecture proofs are measured. The remaining proof is external ARC 2 evaluation movement. The wider scientific frontier is now contextual address formation: deriving the right causal frame and candidate identities from an open situation.
- D1Learn a lawClosed, with three causal morphologies surviving Fresh Boot and ablation.closed
- D2Move ARC 2 evaluationOpen at 0/120 against a fixed threshold of 12/120.open
- D3Initiate experienceClosed at 4/4 versus timer 0/4, with second world transfer 3/3.closed
- D4Replicate independentlyClosed by a content addressed dual surface capsule.closed
Proven, no teach and false equals zero
- D1 law breadth: one to one, one to many and many to one laws
form endogenously and each survives learning free holdout, Fresh Boot, own row
ablation and producer ablation
(
VERIFY.FIELD.ENDOGENOUS_LAW_CLASS_DIVERSITY_TRIAD.01) - D3 autonomous experience: an
ExperienceNeedReceiptopens and closes field initiated episodes in one continuous world stream. Full circle 4/4 beats an equal budget periodic timer at 0/4; second world transfer is 3/3; false terminals and unauthorized writes remain 0/0 (VERIFY.WORLD.AUTONOMOUS_EXPERIENCE_NEED_RECEIPT_TIMER_CONTROL_TRANSFER.01) - Crossmodal causal meaning: the same typed causal effect joins
vision and speech after Fresh Boot. Three previously open recalls close, two
existing recalls remain green, and address or consumer ablation removes every new
recovery, 5/5 total
(
VERIFY.CROSSMODAL.CAUSAL_AFFORDANCE_FRESH_BOOT_RECALL_CONSUMER.01) - Executable causal language: a six part bounded rewrite language and direction preserving law orbit canonicalizer reduce 18 reordered candidates to 3 stable identities outside ARC. This is a capability, not an ARC score
- The dynamic scaffold G1 to G4 (+G5 memory, +G8 commit) is really embodied, physics stability shipped on by default
- Compositional successor / predecessor on the grounded number ring, live in production
- The criterion for generalisation itself (grounded shared invariance)
- The full loop discover → ground → apply, live and on by default
- Certificate discipline: commit only a unique, leave-one-out-surviving law, otherwise abstain
- Two-step composition through field native working memory (scratchpad, live), and self-generated decomposition live on the counting ring (4/4, 0 false)
- A first closed autonomous capability loop on one held-out family: honest abstain → gold-blind gap sensing → exposure request → observation → grounding through unchanged gates → correct commit, zero false (verified, off by default)
- A world question answered by counterfactual simulation on a live scene, settled delta-field, dumb decode, unchanged commit gates (lab-verified, off by default)
- Determination of unseen scenes: object 20/20, unique-property (value-quotient) read 18/20 with 19/20 value generalisation, receiver & canvas 1.00 · honest abstains on hardened ties (verified 12 Jul)
- Structure continuation as field induction: demonstrated chains of length 2 to 3 continue into unseen scenes at exactly the required extents ({2, 5, 8} settle from the energy, never a counter), canonical refusal at the declared envelope, heterogeneous chains over quotient classes at 1.00 (verified 12 Jul)
- Real benchmark tasks through the production commit path: on the field native track an anchor over the five ARC corpora that ship here, 2096 task slots, sealed on the state that ships, reads 39 exact answers at zero false commits on both arithmetic arms (verified 31 Jul), of which 19 are ARC-AGI-2 training tasks and 3 are held out ARC-AGI-1 evaluation tasks on the canonical arm. The two arms agree on 37 of the 39 slots and the four that differ are carried by name. The step from 26 to 39 came from a materializer edge shipped on 30 Jul, and the published figure followed it once the provenance of those 13 new object slots had been measured with both honesty instruments live: 8 are carried by the content, 5 carry a law with a delta different from zero. Separately, a frozen legacy arc7 baseline reads 152 of 1000 ARC-AGI-2 training tasks with zero false commits ever, every solve locked per task ID; an audit of that path on 26 July measured the provenance and split those 152 into organ constructed, legacy lookup and a settle carrying the answer, so that baseline is teacher corpus and not architecture progress. The two tracks are never added together (
VERIFY.ARC.ARC8_PUBLIC_ANCHOR_SEALED_REMEASURE.01,VERIFY.ARC.LEGACY_BASELINE_HONEST_PROVENANCE.01) - The shipped Frozen Cell reads 9 of 20 with zero false commits and no teach; FOUR and EIGHT are the two new exact sequence results (
VERIFY.LIVE.SEQUENCE_MINIMAL_WITNESS_COMPOUND_FLIP.01) - The shipped World discipline reached its acquired scoring phase on 1 August: 11 of 24 easy and 15 of 24 hard episodes, zero false terminals and zero unauthorised writes. This is a separate shipped R2 outcome cell. The later D3 closure above is a default off structure proof and is not added to those episode scores (
VERIFY.LIVE.ACTION_SCHEMA_ADMISSIBILITY_FLIP.02) - D4 external evidence is closed: a content addressed capsule reproduced Frozen Cell 9 of 20 and its predeclared ARC8 compiled arm on an independent machine, with undeclared read and no teach controls intact (
VERIFY.REPRO.SEALED_DUAL_SURFACE_CAPSULE.02). That independent numeric arm does not replace or add to the public ARC anchor - Skills crystallise as compiled energy programs and transfer across families: two phase law transfer 40 of 40 on two disjoint material classes (verified 22 Jul)
- Goal directed motor control through the verified map stack: three arcade levels steered at 1.00 / 0.983 / 0.906 against the old floors, reopening a front that had been honestly parked as terminal (verified 19 Jul)
Still open
- Field native contextual address formation: Hylaean can use explicit typed causal addresses, but cannot yet derive the right scene frame, candidate address set and decision landing point reliably from open context. The four part stack is shown on the architecture page
- ARC is honestly parked: four rewrite censuses formed no task
program. The later relational scene frame census returned KILL at 0/8 controlled
coverage. ARC 2 evaluation stays sealed and no fifth vocabulary retry follows
(
VERIFY.ARC.RELATIONAL_SCENE_FRAME_CENSUS.01) - Battery T2 formation is honestly parked: five measured links moved the first red edge, but pages, formations and expected benefit remained zero. There is no sixth local build until a reactivation condition changes the material class or tick topology
- Keeping a mid-settle reach alive to the commit read: answers on the open battery still demonstrably touch the right basin and decay before the read. The capture lane is now measured to its end, basin residency and trajectory certificates fire cleanly in paired battery runs yet convert zero items (the final frame argmax is not the reached gold basin, the conversion dies upstream at the token decode), so the lane is booked terminal and the frontier moved into the representation itself
- Held out generalisation beyond the training tasks: the seed + direction witness once named here is verified and real tasks now dock through the full chain. On the sealed field native anchor the canonical eager arm reads 3 of 400 ARC-AGI-1 evaluation tasks and the compiled arm reads 4 of 400; the total remains 39, but four slot identities differ across the arms. On the frozen legacy arc7 baseline the same split reads 57 of 400, with the answers measured as organ constructed. The main wall stands either way: ARC-AGI-2 eval reads 0 of 120 on both tracks and ARC-AGI-3 completes 0 of 25 levels, a measured expression wall, not a solver gap
- The measured ARC 43 conditional map remains shadow and default off. It moves the five corpus total from 39 to 43 but leaves ARC-AGI-2 evaluation at 0 of 120 against the D2 threshold of 12 of 120, so it is D2 neutral and does not replace the public 39 (
VERIFY.ARC.ARC8_CONDITIONAL_CLASS_MAP_BUILD.01) - The broad no teach battery composite remains 0.2222, unchanged, in Mode B, vocabulary-prepared / nicht kalt, with no scored adaptation writes. The sealed run executed vocab preteach exposures before scoring (
preteach_exposed_total > 0; acquired 0/39 is not the criterion), so this composite is not the cold static no teach point. It is a project composite, not an FIQ and not a human normed IQ (VERIFY.IQ.BATTERY_MILESTONE_POST_SEQUENCE_COMPOUND_MEASUREMENT.01;VERIFY.IQ.BATTERY_VOCABULARY_PREPARED_PROTOCOL_LABEL.01) - Relation episode Genesis is terminal under the owner's Option A at
FIELD_NATIVE_RELATION_EPISODE_FRAME_LEASED_SETTLE_CARRIER: 45 of 46 scenes reached occupancy and 40 completed, but exact settled occurrence reads stayed 0 of 46 (VERIFY.FIELD.RELATION_EPISODE_PERCEPT_INGRESS.01) - Grounding of bare words never seen through perception (live placement)
- Global no teach generalisation (structural wall; falls only when a settle-time reach survives to the commit read)
- Paraphrase persistence: a factorized subject + relation encoding now commits held-out paraphrases in the lab, but the taught register does not yet survive a reboot
- Decomposition beyond the counting ring, the general read is verified, but its live cutover measured inert (frames open, zero false, no new answers); the named gap is a re-decode window inside the escalated frame
The real bottleneck is contextual representation, not more machinery. Addressed causal laws now work. Open scenes still lack a field formed frame that says which entities, roles and moment define the address. More settle, wider bounds or a smarter decoder cannot create that missing information.
In one sentence
The honest summary.
Hylaean can now form and use addressed causal laws. The next scientific question is whether one field can form the correct contextual address itself, from scene, time and competition, while preserving the same abstention and causal ablation discipline.
D1, D3 and D4 are closed. D2 is open. Crossmodal Fresh Boot recall is 5/5 because meaning is joined by a shared causal effect rather than by visual or verbal resemblance. The six part rewrite language and law orbit canonicalizer show that verbs can become bounded executable programs. None of this is an FIQ claim, and the parked ARC and battery routes remain visible because negative evidence is part of the science.
Read the deeper benchmark and grounding boundary
The grounding criterion is now verified across ten-plus perceptual families, including
two (category, antonym) that first failed and then passed after an honest re-derivation.
The answer channel is live for taught associations, and an evidence-armed attractor has
put the first multi-word answers through the unchanged commit gates in the lab. The
induction primitive (the field determines what an unseen scene means and
continues a demonstrated structure into it) now reaches real benchmark tasks:
the field native anchor over the five ARC corpora that ship here, re sealed on 31 July on
the state that ships, reads 39 exact answers out of 2096 task slots at zero false on both
arithmetic arms (VERIFY.ARC.ARC8_PUBLIC_ANCHOR_SEALED_REMEASURE.01). The
older figure of 152 of 1000 ARC-AGI-2 training tasks belongs to a
frozen legacy stack whose provenance was measured on 26 July: most of those solves are
constructed by organ code rather than settled in the field, so that number stands as a
solve count and not as a field claim, and the two are never added together. Skills crystallise as compiled energy programs that transfer across
material families. Keeping a settle-time reach alive to the commit read, bare-word grounding,
held out generalisation beyond the training tasks and global no teach remain open.
That honesty is the point.