geDIG $\mathcal{F} = \Delta EPC - \lambda(\Delta H + \gamma \Delta \beta_1)$

A structural fitness score for knowledge graphs —
can one equation decide when to restructure? Three independent structures: $\Delta EPC \approx$ "metric (distance)", $\Delta H \approx$ "measure (probability)", $\Delta \beta_1 \approx$ "topology (loops)" $\mathcal{F} < 0$ means information gain exceeds structural cost — the system should commit the change. Maze: 60% → 98% goal-reach (15×15). Transformer: F tracks model quality across 8 models.

🧠 Try Live Demo (HF Spaces) Why Three Terms? View Paper (Draft)

$\Delta EPC$ (Metric) $\Delta H$ (Measure) $\Delta \beta_1$ (Topology) Helmholtz F = E − TS

Visual Intuition

Why does F need three terms? Same edit cost. Completely different outcomes.

Case A — Insight

→

EPC1

$\Delta\beta_1$+1

$\Delta H$+0.4

F −0.4

F < 0 → Commit

Case B — Routine

→

EPC1

$\Delta\beta_1$0

$\Delta H$+0.3

F +0.7

F > 0 → Reject

Case C — Collapse

→

EPC1

$\Delta\beta_1$−1

$\Delta H$−0.2

F +2.2

F > 0 → Reject

All three cases have EPC = 1, yet $\Delta\beta_1$ diverges to +1, 0, −1. KL divergence sees only $\Delta H$ and cannot distinguish Case A from B. F sees all three.

(λ = 1, γ = 1)

Full interactive version (EN) → (JA) →

Interactive Graph Demo — see F in real-time as a graph grows

Visualizing "The Pulse"

Real-time simulation of the gauge $\mathcal{F}$ as the graph grows.
Click anywhere in the graph to inject a new query node and see if it triggers an Insight (DG) or Ambiguity (AG).
AG (Ambiguity Gauge): 0-hop error (High Cost).
DG (Discovery Gauge): Multi-hop shortcut (Insight Found).

Observation Guide:

Click near a cluster: $\Delta EPC$ is low, but $\Delta IG$ is also low. $\mathcal{F}$ stays high (Reject).
Click between clusters: If a shortcut is found, $\Delta SP$ spikes (negative cost), driving $\mathcal{F} < \theta$ (Insight!).

IDLE

Click to Add Node

Base Node

Query

1-hop (Ego)

2-hop+ (Insight)

Gauge Telemetry

Structural Cost ($\Delta EPC$) --

wiring cost (normalized)

Information Gain ($\Delta IG$) --

Entropy ($\Delta H$) --

Shortcuts ($\Delta SP$) --

Total Gauge ($\mathcal{F}$) --

Threshold: < 0.30 IDLE

Manual Override

Use Cases

Where can a single thermodynamic gauge make a difference?

Multi-Hop QA (RAG)

F acts as an accept/reject gate for retrieved information. AG detects novelty; DG validates structural integration. Tested on HotPotQA with real LLM inference.

Result (GPT-4o-mini, dev-500)

EM 38.0%

F1 53.7%

Try Live Demo (HF Spaces) → Experiment details →

Status: Archived experiment

Autonomous Graph Navigation

A partial-observation maze agent builds a persistent knowledge graph. F decides when to explore (AG) vs. exploit (DG). Wake-Sleep-Wake cycle with three-layer search.

Result (15×15 maze, 12 seeds)

Goal-reach 98%

Baseline ~60%

Experiment code →

Status: Active & reproducible

Transformer F-Trajectory

Layer-by-layer $\mathcal{F}$ decomposition across hidden states. Tests whether F tracks model quality as a structural signature.

Result (8 models)

GPT series shows monotonic improvement in $\Delta R^2_{\text{struct}}$ with model scale.

Experiment code →

Status: Preliminary

The Mechanism

Two gates govern intelligent processing. One equation drives both.

AG/DG Gating

Attention Gate

"Is this surprising?" — 0-hop novelty detection. When prediction diverges from input, AG fires and triggers processing.

Computational analogy: noradrenaline

Decision Gate

"Does $\mathcal{F}$ decrease?" — Multi-hop structural validation. If restructuring reduces F, the system commits the change.

Computational analogy: dopamine

*The neurotransmitter correspondence is a computational analogy, not a physiological claim.

Theoretical Backbone

Helmholtz Correspondence

The canonical reading $\mathcal{F} = \Delta EPC - \lambda(\Delta H + \gamma \Delta \beta_1)$ is cost − gain (economic). A second reading $(\Delta EPC - \Delta \beta_1) - \Delta H$ exposes a Helmholtz-like structure:

$U$ (internal energy) ↔ $\Delta EPC - \Delta \beta_1$

$T$ (temperature) ↔ $\lambda$ (information temperature)

$S$ (entropy) ↔ $\Delta H$ (Shannon)

Same equation, different bracketing. One equation, multiple readings, one canonical.

geDIG exposes an operational correspondence to the Helmholtz free energy $F = U - TS$, as well as to FEP (minimizing surprise) and MDL (maximizing compression). These are secondary interpretations — the paper body uses the economic reading (cost − gain) as canonical.

*Open problem: whether structure ≡ probability admits a rigorous mathematical equivalence is left to theorists (information geometry, MDL, MaxEnt). This work commits only to engineering validation of the scalar $\mathcal{F}$.

Experiments & Results

Empirical tests of F across different domains.

Experiment	Scale	Key Result	Status
Maze 15×15	12 seeds, 250 steps	98% goal-reach (baseline ~60%)	Reproducible
Maze 25×25	Active	Graph-persistent DG + 10D vector extension	In progress
OCR (vector-based-cnn-ocr)	Font→handwriting	18K params / 73.53% acc. — DG channel is 5× more info-efficient than AG	External repo
HotPotQA	dev-500, GPT-4o-mini	EM=38.0%, F1=53.7%	Archived
Transformer F-decomp	8 models	GPT series: $\Delta R^2_{\text{struct}}$ improves with scale	Preliminary

Reproduce (Maze 15×15)

Requires .venv with networkx, numpy, etc.

.venv/bin/python3 experiments/maze/run_experiment_query.py \
--maze-size 15 --max-steps 250 --seeds 12 \
--search-mode threelayer --vector-mode extended

See experiments/maze/README.md for full CLI reference.

Resources

Papers, code, and open questions.

Paper

Research Paper

"geDIG: Gauge what Knowledge Graph needs"

v6.0 Draft

JA Version (v6) EN Version (v6) JA Overview Slide (Landing PDF)

Code

Source Code

miyauchikazuyoshi/InsightSpike-AI

Research framework for the thermodynamics of intelligence

View Repository Browse Experiments

Quick Start

Install: pip install -e .

Test: make test

Open Research Questions

We welcome collaboration on these specific questions:

F-trajectory reproduction:
Does F show structured layer-wise behavior on untested models?
β₁ vs SP as structural term:
When does topological Δβ₁ outperform metric ΔSP? TDA expertise welcome.
Scaling to 70B+ models:
We hypothesize (λ, γ) converge at sufficient scale. GPU resources needed.

How to engage: open an Issue, PR, or DM on X: @kazuyoshim5436.

Citation (BibTeX)

@software{miyauchi2026gedig,
  title={geDIG: graph edit Distance and Information Gain},
  author={Miyauchi, Kazuyoshi},
  year={2026},
  publisher={Zenodo},
  doi={10.5281/zenodo.19454110},
  url={https://doi.org/10.5281/zenodo.19454110}
}

DOI: 10.5281/zenodo.19454110