← Hylaean The science behind
Two foundations sit under the whole system: TFPT — the physics theory it borrows its structure from — and a minimal grammar of eight information laws that say what a field must obey so that language, thinking and creativity can appear as stable attractors. This page lays both out — and scores each one honestly.
What this page claims
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 (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.
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.
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:
02 — The bridge
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
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 and zero false commits.
Exactly one field state.S on a sphere; question, answer, world model and regions are deformations of the same field.
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.
After each step an algebraic selector projects onto the admissible sector — closed-form selection, never an optimizer.
A transport spectrum with a real gap (Δ = 6·ln(3/2) > 0) guarantees a unique fixed point (Perron–Frobenius).
Memory changes as geometry (Hebbian ΔK, ΔL; basins deepen) — never back-propagation on the answer path.
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.
Basins carry meaning as a geometric coordinate (same relation = same shift), acquired from perception — otherwise they are structureless and relations are inconsistent.
Commit only on real field movement; otherwise abstain, and leftover pressure opens nested reasoning frames instead of bluffing.
04 — The key result
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.
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
The real bottleneck is representation, not more machinery. This week proved it twice more: holding the question steady makes answers worse (resolving the question is the transport), and escalation/re-relaxation lifts nothing — while a representation change (grounded reflection) took antonym from 0.002 to a perfect 1.000. First the vocabulary must be grounded; depth comes after.
In one sentence
Under energy, projection, a gap, memory, transport and an honest commit, a single field reliably produces stable attractors — but general intelligence only follows necessarily once meaning is learned as a shared, acquirable invariance of the field's geometry and an answer channel puts that meaning where the commit gate can see it.
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 missing answer channel has a verified first building block. Bare-word grounding and global no-teach remain open. That honesty is the point.