Knowledge as Dynamics: The Paper

Veso R&D has released Knowledge as Dynamics, a research paper on treating a neural network’s latent vector field as the knowledge itself: something that can be copied without data, merged across models, passed down generations of students, and priced before any transfer runs. The paper, code, and every run record are public.

A trained autoencoder moves every point of its latent space: iterating f = Enc(Dec(z)) displaces each point by V(z) = f(z) - z, and training carves attractors into that field. Prior work analyzed this field. This paper uses it as a transmission channel.

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Release facts

Item	Value
Title	Knowledge as Dynamics: Distilling, Composing, and Inheriting the Latent Vector Fields of Neural Networks
Authors	Elias Helou (Veso AI), Ivan Nemytchenko (independent researcher)
Version	v1.4, July 15, 2026
Repository	github.com/Veso-AI-Open-Source/knowledge-as-dynamics
License	MIT, paper PDF included
arXiv	Pending
Evidence	Every reported number traces to a JSON run record in runs/
Hardware	One Apple laptop, MLX, no torch; full suite under an hour

The object

• An autoencoder induces a dynamical system on its own latent space: z(t+1) = f(z(t)) with f = Enc(Dec(z)).
• The displacement field is V(z) = f(z) - z. Attractors and their basins summarize what the network learned.
• A fresh student trained only to match V, with no training data and no teacher weights, inherits the teacher’s basins of attraction.
• Standard distillation matches static snapshots (outputs, features, relations, Jacobians). This channel matches the dynamics.

Figure 1. The core effect. Left: the teacher’s latent field; black stars are the attractors training carved. Right: a student rebuilt from field queries alone, no data, no weights. Its attractors land on the teacher’s; 84% of test points reach the same basin in both networks (0.86 mean across seeds).

Figure 2. The field as a channel. Four operations, all data-free: transfer, merging, lineage, and a pre-transfer audit.

The five laws

Law	Statement	Key number
1. Sufficiency	The field alone transfers the teacher’s basins; static channels do not	Field 0.86 vs output 0.53, features 0.15, scratch 0.00 (ceiling 0.99)
2. Order	Transmission reliability changes discontinuously where the teacher’s field becomes fully convergent	Seed spread 0.09 to 0.75 below the transition; 0.690, 0.685, 0.685 at it
3. Resolution	Mean field error is the wrong observable; it anti-correlates with basin transfer	Lookup carrier: 7.6x worse NMSE, 3.5x better basin agreement
4. Composition	Two disjoint specialists union into one data-free student, but only in a designed frame	Joint 0.82 vs behavioral merging 0.13, grafted side 0.00
5. Heredity with selection	Knowledge survives five generations of re-distillation; the selection lives in the lossy carrier, not the channel	Lineage 0.91 to 0.83 over 5 generations, attractor sets stable

Figure 3. Four ways to copy a network. Field-matched copies reproduce the terrain. A network retrained from scratch on the same data builds unrelated terrain (agreement 0.00). Output distillation recovers part; feature matching almost none.

The alpha instrument

The paper’s main practical result is a teacher-only measurement that prices a transfer before it runs.

• Perturb probe points by radius epsilon; measure the fraction whose endpoint basin flips. Fit f(epsilon) proportional to epsilon^alpha.
• Alpha is the uncertainty exponent of the teacher’s basin boundaries; boundary dimension is D_b = d - alpha.
• The measurement takes minutes and needs no student.

Validation	Result
Newton fractals (attractors known analytically)	Estimator matches box-counted boundary dimension; pricing law basin = 1 - f(delta) predicts all 12 students to 0.01 to 0.03
Magnetic pendulum (physical roughness dial)	Alpha tracks the damping dial; ordering holds at both query budgets
MNIST autoencoder	Alpha = 0.45 predicts the 0.69 transfer ceiling (0.71 measured) where convergence fraction 0.97 predicts nothing
Across all systems	Alpha orders distillation quality at Spearman +1.0; reliability follows it too (spread vs alpha, rho = -1)

Figure 4. The pre-transfer audit. Scope note from the paper: read the full flip-rate curve at the copy’s achievable error scale, never the bare exponent; quantitative prediction holds in contracting maps.

One lineage result is worth stating on its own: down a five-generation chain, alpha climbs while fidelity falls. Copies of copies grow more definite and less faithful. Single copies can reproduce their teacher’s fractal boundary geometry exactly, because the geometry is generated by iterating the learned map, not stored in it.

External validation

• Within a day of v1.0, an independent researcher filed a from-scratch replication with a four-part re-diagnosis as a GitHub issue.
• Every claim was re-tested inside the original harness. Two confirmed decisively, one confirmed with a stronger result, one did not replicate (reported with a teacher-weights offer).
• Two laws were restated as a result. The paper says which, and the v1.0 diagnosis it overturned remains in the lab log.
• The issue’s author is a co-author from v1.2.

Issue claim	Outcome in our harness
2D losses belong to the SGD carrier, not the channel	Confirmed: a lossless lattice carrier hits union 0.99 with flat lineage
Mean NMSE anti-correlates with basin transfer	Confirmed: NMSE-selected models score 0.000 basin agreement every seed
The high-dimension wall is budgetary, not fundamental	Confirmed and extended: two-hop distillation reaches 0.31 to 0.50 in half the seeds from 61x fewer queries
GP carrier reaches 0.48 from 16k pairs	Not replicated (0.095 +/- 0.06); divergence is teacher-level, weights offered

Scope

• Quantitative alpha prediction is demonstrated in contracting maps; high-dimensional parametric field matching remains open (best MNIST parametric arm: 0.19 against a 0.71 ceiling).
• Lineage chains are single-seed; the paper discloses this and the best-of-5 ancestor selection.
• The laws table in the paper separates demonstrated from conjectured, per law.
• Method kinship: this is the same validate-the-mechanism approach as our JEPA reproduction, scaled up to a full paper.

Reproduce

git clone https://github.com/Veso-AI-Open-Source/knowledge-as-dynamics
uv sync
uv run python -m m5_fielddistill.validate   # Law 1, ~3.5 min
uv run python -m m5_fielddistill.gi         # Laws 4 and 5, ~3 min
uv run python -m m5_fielddistill.alpha      # the instrument, ~1 min

@misc{helou2026knowledgedynamics,
  title  = {Knowledge as Dynamics: Distilling, Composing, and Inheriting
            the Latent Vector Fields of Neural Networks},
  author = {Helou, Elias and Nemytchenko, Ivan},
  year   = {2026},
  url    = {https://github.com/Veso-AI-Open-Source/knowledge-as-dynamics}
}

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