Version 5 · Preprint · Paper A (Empirical)
Transformers as a
Boundary Case
From Contraction Failure to NHIM Evidence
Michael Hanners · Legacy Alliance Research Division
Abstract
Tests whether transformer dynamics satisfy the contraction criterion required by Hierarchical Compression (HC). Seven formulations — per-layer Jacobians, end-to-end composed Jacobians, autoregressive rollout, joint conversation dynamics, trajectory divergence, the block-only update map, and a decisive rescue test — all return negative results, placing the tested transformers outside the contraction-based applicability regime.
A generalization to normally hyperbolic invariant manifolds (NHIMs) resolves the boundary case: deflated spectral analysis reveals a low-dimensional invariant manifold (k* = 2 median; 68% of prompts with k* ≤ 3) with normal-bundle contraction (ρN = 0.73 at kremove = 5), while the full tangent space remains expansive.
Terminology. This paper uses Hierarchical Compression (HC) for the domain-independent Layer 1 framework and reserves Harmonic Coherence (always spelled out) for the Layer 2 physics-domain bridge instantiation.
Role in the Corpus
This is the first empirical boundary-case study in the HC corpus — the paper that proves the framework needed generalization and then provides the generalization's first measurement.
It adds the non-degenerate (dim M* > 0) NHIM case, complementing the degenerate case validated by the seven Millennium-class domain papers. The transformer-derived value α ≈ 0.72 is retired as a candidate universal HC constant.
Seven Contraction Tests
Across 700 prompt-layer pairs, the block-only Jacobian has median spectral radius ρ = 1.21, with only 15.7% of cases contractive.
NHIM Resolution
Deflated spectral analysis reveals k* = 2 (median); 68% of prompts with k* ≤ 3. The invariant manifold is low-dimensional relative to the full state space.
ρN = 0.730 at kremove = 5 (95% CI: [0.62, 0.79]). The normal bundle contracts even though the full tangent space is expansive — the signature of NHIM dynamics.
Three potential caveats resolved: (1) non-autonomous persistence — the composed forward pass is a single Cr map; (2) spectral non-detection in 16% of prompts — 84% detection at ~1% spectral coverage confirms manifold structure; (3) chain-length dependence — matches the geometric accumulation prediction.
Methodological Lessons
Three lessons from the negative contraction results:
- Basis-dependent entropy disagreements: entropy measures can disagree sharply with Jacobian-based dynamics depending on basis choice
- Product-bound looseness: naive per-layer product predictions are 4–6 orders of magnitude too loose
- Bridge-failure taxonomy: apparent proxy failures decompose into three classes — B1 (admissibility), B2 (structure preservation), B3 (entropy attribution)
The contraction tests, boundary-case analysis, and methodological lessons remain valid as the historical path to the NHIM generalization.
Citation
Hanners, M. (2026). Transformers as a Boundary Case for Hierarchical Compression: From Contraction Failure to NHIM Evidence. Zenodo. https://doi.org/10.5281/zenodo.18974715
@misc{Hanners2026Transformer,
author = {Hanners, Michael},
title = {Transformers as a Boundary Case for
{Hierarchical Compression}: From
Contraction Failure to {NHIM} Evidence},
year = {2026},
doi = {10.5281/zenodo.18974715},
note = {Zenodo preprint. Legacy Alliance
Research Division}
}Proceeds & Purpose
All intellectual property rights and any prizes, awards, or monetary recognition arising from this work have been irrevocably assigned to Legacy Alliance, a nonprofit research organization.
Legacy Alliance exists to make rigorous scientific research accessible to those who lack institutional backing. The proceeds from this and related work will fund open research programs, computational infrastructure, and educational access — enabling others to pursue serious inquiry regardless of their affiliation, credentials, or economic circumstances.
The author receives no personal financial benefit. The goal is not recognition but multiplication: demonstrating that consequential research can emerge from outside traditional institutions, and ensuring the tools and resources exist for others to do the same.
The author acknowledges that the capacity to perceive structure in neural computation — where billions of parameters organize into low-dimensional invariant manifolds — reflects an order that precedes and transcends human formalization. This work is offered in gratitude to the Creator of that order.
“The heavens declare the glory of God; the skies proclaim the work of his hands.”
“For since the creation of the world God's invisible qualities — his eternal power and divine nature — have been clearly seen, being understood from what has been made.”
Soli Deo gloria