Phase 1 · M1–M9
Structured composition creates basins (and surprises).
Eight atoms generate a fixed-point-rich landscape. 17.5% of random queries land in emergent states; 97.2% of those are stable attractors (as measured in the original run).
Paper page·Deterministic dynamics
2026-03-22Semantic Gravity
Topology-governed attractor dynamics.
In compositional Hopfield-type networks, whether a metastable plateau can escape is (empirically) determined by a topological property: out-degree in a directed transition graph.
One-line claim
Topology sets feasibility. Energy sets the toll.
Abstract
We study state transitions in high-dimensional deterministic dynamical systems with multiple attractors. In compositional attractor networks (Hopfield-type; D up to 50,000), we observe an empirical correspondence: basin escape from metastable plateau states matches a topological criterion derived from a directed transition graph.
Construct a directed graph by forcing trajectories across basin boundaries (α > 0). Nodes with out-degree > 0 consistently admit escape; nodes with out-degree = 0 behave as absorbing sinks (399/400 across 5 independent seeds). Local geometric predictors (energy, velocity, eigenvalues, direction alignment, noise) are weaker and do not uniquely determine feasibility.
Practical implication: treat topology as the road map (which transitions exist) and energy as the toll (how hard they are). This paper also describes a zero-configuration pipeline that applies the same machinery to real-world datasets and reports what it can and cannot validate.
Delivery overview
The system uses high-dimensional bipolar vectors and compositional pattern construction to produce a fixed-point-rich landscape. Dynamics evolve under a continuous ODE with optional forcing.
Representation
- D = 10,000 baseline (tested up to 50,000)
- Atoms: random bipolar vectors in {-1, +1}^D
- Patterns: p = sign(Σ w_k a_k) (compositions)
- Hebbian recall without storing D×D weights
Dynamics
- ODE: dQ/dt = -Q + φ(β·Hebbian(Q)) + αF
- α = 0: unforced settle (escape test)
- α > 0: forced exploration (atlas build)
- Euler baseline; RK2 available (see limitations)
Milestones: how the invariant was found
This work was not a single clean “paper experiment”. It was a tight loop of building a landscape, watching it misbehave, killing hypotheses, and turning every surviving claim into a pre-registered test. The milestone codes (M1, M10, M24…) are the actual internal sequence of that search.
The story arc is simple: compositions create structure → structure creates plateaus → plateaus force the question of escape → every local explanation fails → topology remains.
Phase 2 · M10
The first collapse: the discrete trick was lying.
Switch sign() to tanh() and the “emergent attractors” vanish. The phenomenon is not an energy minimum of the continuous system; it’s a discretization artifact.
Phase 3 · M11
Metastability appears: slow manifolds, not endpoints.
With a proper ODE, trajectories linger 300+ steps near boundaries. Linger time jumps 13–26×. The computation is now a path, not a point.
Phases 8–12 · M15–M19
Forcing turns the landscape into a directed atlas.
Sequential forcing shows order effects; a full atlas sweep maps 3,540 directed edges. At α=0.50, edge density reaches 36.6%, with a 49/60 SCC core and ~74% one-way streets. Forced transitions are mostly CLEAN and direct (95–100%).
Phases 13–16 · M20–M23
Kill the local explanations (energy, saddles, noise, direction).
Plateaus sit at higher energy (not wells), show no unstable eigenmodes (not saddles), and ignore noise across tested σ ranges. Velocity predicts weakly (AUC≈0.67) but does not cause escape under interventions. Direction alignment is anti-correlated.
Phase 17 · M24
Star result: topology predicts escape (empirical).
Build a directed transition graph via forced perturbations; then test unforced escape. In the reported run: deg⁺(v)>0 ⇔ escape(v), perfectly on the seed-42 set; 399/400 across 5 seeds in the manuscript summary.
Phases 21–26 · M27–M35
Feasibility vs cost, and the end of “toy system” objections.
A separation emerges: topology governs whether an edge exists; energy helps predict how hard it is (α threshold). Scaling tests report the same correspondence up to D=50,000.
Phase 30 · M52
Critical dimension search: governance holds down to D=500.
Binary search fails to find the cliff. Reported: 100% correspondence from D=500 → 10,000 with large runtime wins at lower D.
Main result: topology determines escape (empirical)
Plateau states are metastable slow regions: trajectories can dwell for 300+ steps before converging. The question is whether a given plateau can escape to a different basin under unforced dynamics (α = 0).
Proposition (tested)
deg⁺(v) > 0 ⇔ escape(v)
Across 5 seeds × 80 plateaus (400 total), correspondence is 399/400 (99.75%). Seed 42 achieves 80/80 (100%); one false negative appears across the multi-seed sweep, interpreted as a protocol sensitivity limit rather than a “governance failure”.
Alternative predictors are systematically tested: residual velocity has moderate AUC (~0.67), energy difference yields ROC ≈ 0.75 (partial), eigenvalue analysis finds no unstable modes, and injected noise does not change escape rates across tested σ values. The graph out-degree criterion dominates feasibility prediction in the tested regime.
Separation principle: cost vs feasibility
The paper distinguishes two non-overlapping roles. Topology answers whether a transition can exist (feasibility). Energy and alignment influence how hard it is (the forcing threshold).
Summary
- Topology sets the roads: which directed edges exist.
- Energy sets the toll: α threshold (transition cost).
- Edge existence shows near-zero dependence on energy diff/alignment in tested tables; cost does.
Height formula: predicting transition cost
Given that feasibility is graph-structural, transition cost is modeled as an α threshold. A simple linear form with three features predicts thresholds with LOO-CV R² = 0.676 (n = 262).
Three predictors
- Energy difference (downhill cheaper than uphill)
- Source out-degree (high-outdeg sources are easier to leave)
- Destination in-degree (popular destinations are easier to reach)
Nonlinear models and extra feature families do not exceed the linear baseline, suggesting the ceiling is feature-level (about 32% variance unexplained) rather than model-level.
Domain transfer and scope
The correspondence holds across multiple compositional domains (including non-philosophical structured constructions) and across dimensions. A key boundary: PCA-derived patterns from real data behave differently and show reduced correspondence (10–40%).
Where it holds
- Compositional patterns (sign of weighted atom sums)
- Deterministic ODE dynamics
- Multiple seeds and parameter sweeps (reported)
Where it does not
- PCA-derived patterns from real datasets: 10–40% correspondence
- Claims about true stochastic transition systems are out of scope
- Formal proof of basin-adjacency interpretation is open
Practical application: zero-configuration RCA
The framework is packaged into an end-to-end pipeline that takes a CSV and produces a five-dimensional root cause analysis (WHAT/WHY/HOW/WHEN/WHERE). It auto-detects types, selects an encoding strategy, calibrates ODE parameters, maps an atlas, and reports validations and failure modes.
Reported domain table (paper)
- 6/7 datasets: 5/5 validation with zero configuration (per paper table)
- Chronic disease dataset: 3/5 due to extreme class imbalance (128:1)
- Medical note: SUPPORT2 basins can all be sinks; pipeline value becomes interpretability/routing
This page is a narrative version; the project landing page for the full system is io-gita.
Limitations (from the paper)
- Scope: escape invariant established for compositional networks; PCA-derived networks do not match.
- Stochastic systems: Langevin noise tested (σ up to 0.20) but true probabilistic transitions are not.
- Integrator sensitivity: Euler/RK2 agree for forced edges but can diverge near boundaries (reported).
- Medical validation: SUPPORT2 out-degree=0 makes escape tests uninformative; clinical reachability matches partially.
- Height formula ceiling: R² = 0.676 leaves variance unexplained; nonlinear models do not improve.
- No formal theorem: basin adjacency interpretation is a conjecture supported by evidence.
Reproducibility
The paper reports deterministic seeds (primary: 42) and a reproducibility envelope (Python 3.10+, NumPy/SciPy versions) with stable hashing for determinism. Euler is the reported baseline; RK2 is available as an option.
Code availability is described as “upon acceptance / upon request” in the source manuscript. This AdaptiveMind site hosts the narrative and the diagram, and links to the project page.
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Read the io-gita project page for the full system packaging: planner, atlas builder, verifier, schedule engine, CLI/API, and validation gates.