K=1 Geometric Deep Learning and Chronogeometrodynamics

This repository currently contains the LaTeX manuscript source for:

K=1 Chronogeometrodynamics: Lorentzian Signature as a Geometric Precondition for the Schrodinger Representation

It also serves as the theoretical anchor for a proposed K=1 geometric deep learning program for language models: a pseudo-Riemannian framework in which Transformer hidden states are treated as evolving near Lorentzian metric structures rather than as purely Euclidean statistical features.

The central theorem-level claim is conditional: given a physically selected non-degenerate local metric block G, the Ornstein-Uhlenbeck/Fokker-Planck construction supports a Schrodinger-type wavefunction representation exactly when the local metric block has Lorentzian signature, equivalently det G < 0.

Repository contents

k=1 quantum.TEX                                  Main LaTeX manuscript
experiments/k1_throttle_v43_negative_replication.py
                                                 Standalone V4.3 DistilGPT-2
                                                 negative-Lorentz replication
experiments/k1_throttle_v44_gpt2_screen.py       V4.4 GPT-2 three-seed screen
experiments/k1_v45_gpt2_failure_diagnostic.py    V4.5 read-only diagnosis
experiments/k1_v46_gpt2_mask_holdout.py          V4.6 mask/full-Lorentz holdout
experiments/k1_v47_gpt2_pooled_audit.py          V4.7 pooled GPT-2 audit
results/audit_v43.json                           Preregistered ten-seed audit
results/audit_v47_summary.json                   Compact V4.7 evidence summary
docs/V43_RESULT_BOUNDARIES.md                    Interpretation boundary
docs/V47_EVIDENCE_AND_CLAIM_BOUNDARY.md          V4.7 claim boundary
docs/K1_LLM_EVIDENCE.md                          Research status through V4.7

The TeX file references several PDF figures:

• fig1_signature_gate.pdf
• fig2_oneway_barrier.pdf
• fig3_activetime_clock.pdf
• fig4_mu2_diagnostic.pdf

These figure files are not currently included in the repository, so a direct PDF build will require adding those files or temporarily removing/commenting the corresponding \includegraphics lines.

Main ideas

• A K=1 stability condition gives a critical damping parameter d_c > 0 if and only if det G < 0.
• In the local OU reduction near the K=1 surface, the Fokker-Planck equation can be mapped by a similarity transform to an imaginary-time Schrodinger-type structure.
• Since the OU construction requires real positive damping, the Schrodinger-type representation is supported only on the Lorentzian side.
• Treating Delta = det G as an effective dynamical variable gives an idealized one-way signature-boundary protection mechanism through Lorentzian-side noise scaling D_L(Delta) proportional to sqrt(|Delta|).
• The active-time and measurement-channel sections are presented as downstream operational closures, not as evidence for physical metric signature switching.

Geometric deep learning interpretation

The language-model interpretation takes the manuscript's signature gate as an engineering hypothesis:

• Hidden representations should be regularized or parameterized so their effective local metric remains Lorentzian (det G < 0).
• Token output may be studied as a boundary event near a critical determinant surface, rather than only as unconstrained softmax sampling. This repository does not establish physical wave-function collapse in language models.
• Optimization can be viewed as flow along a pseudo-Riemannian structure, replacing flat Euclidean updates with geometry-aware dynamics.

In this reading, the model update has the schematic form:

h_{l+1} = h_l + P_up (z_{l+1} - z_l)
z_{l+1} = z_l + dt (J_G - d_c I) grad V

where:

• G is the local effective metric of hidden states.
• d_c is the critical damping parameter, real and positive only on the Lorentzian side.
• J_G = alpha G^-1 J is the symplectic generator.
• det G < 0 marks the Lorentzian regime in which the wavefunction representation is supported.
• det G > 0 marks the Euclidean regime in which this representation is not supported by the K=1 OU-FP construction.

K1 LLM evidence through V4.7

A pre-mix scale-matched, full-layer negative-Lorentz adapter produced a small, reproducible cross-entropy improvement over matched residual, negative-Euclidean, and negative-random controls in DistilGPT-2 and in a secondary ten-seed pooled GPT-2 analysis. The GPT-2 pooled Test improvements were significant at paired 95% intervals and showed 9/10 wins against each control.

The effective direction was negative and is not OU attraction toward K=1. These experiments do not validate physical wave-function collapse or a Token-collapse transition. Absolute improvements were small and remain limited to the tested models, datasets, and adapter protocol.

Experiment history

Experiment	Status	Meaning
V4.3 DistilGPT-2 10 seeds	Passed	Negative-Lorentz engineering effect in this setting
V4.4 GPT-2 3-seed screen	Failed	Screening GO rule did not pass
V4.5 diagnosis	Exploratory	Suggested local seed/layer sensitivity
V4.6 fixed Mask	Failed	Fixed layer-mask hypothesis did not generalize
V4.6 full-Lorentz control	Positive held-out result	7/7 Test wins over residual
V4.7 GPT-2 pooled 10 seeds	Passed, secondary	Pooled common-control replication

V4.7 is a secondary pooled analysis of two consecutive GPT-2 experiments, not a newly preregistered standalone ten-seed experiment. The V4.4 screen failure and V4.6 fixed Mask failure are part of the evidence record and should remain visible.

See docs/K1_LLM_EVIDENCE.md and docs/V47_EVIDENCE_AND_CLAIM_BOUNDARY.md for the full evidence boundary.

Reproducibility entry points

The scripts are standalone Colab/GPU experiments:

python experiments/k1_throttle_v43_negative_replication.py
python experiments/k1_throttle_v44_gpt2_screen.py
python experiments/k1_v45_gpt2_failure_diagnostic.py
python experiments/k1_v46_gpt2_mask_holdout.py
python experiments/k1_v47_gpt2_pooled_audit.py

V4.3 compares frozen DistilGPT-2 with parameter-matched adapter branches under a fixed negative-sign protocol:

• Model: distilgpt2
• ID dataset: Salesforce/wikitext, wikitext-2-raw-v1
• OOD dataset: fancyzhx/ag_news, test
• Block counts: train=500, val=160, test=500, ood=500
• Sequence length: 96
• Adapter rank: 2 * planes = 16
• Ten paired seeds: 10103, 10301, 10501, 10709, 10903, 11113, 11311, 11503, 11701, 11909
• Matched parameter budget across residual, negative-Euclidean, negative-random, and negative-Lorentz branches
• Pre-mix branch normalization by per-token RMS over planes/components
• No post-hoc sign selection

The preregistered audit is stored in results/audit_v43.json. The compact V4.7 pooled GPT-2 summary is stored in results/audit_v47_summary.json.

Selected results

• V4.3: PASS_NEGATIVE_LORENTZ_SPECIFIC = true.
• V4.3 Test Lorentz-negative minus Euclid-negative: mean -0.0003550461, 95% CI [-0.0004488782, -0.0002612139], wins 10/10.
• V4.3 Test Lorentz-negative minus Random-negative: mean -0.0004974693, 95% CI [-0.0005776457, -0.0004172929], wins 10/10.
• V4.3 OOD Lorentz-negative minus residual: mean -0.0000541615, 95% CI [-0.0000815838, -0.0000267391], wins 9/10.
• V4.6 full-Lorentz held-out control: Test Lorentz minus residual mean -0.000601889, paired 95% CI [-0.000800628, -0.000403150], wins 7/7.
• V4.7 GPT-2 pooled Test Lorentz minus residual: mean -0.000583545, paired 95% CI [-0.000814308, -0.000352782], wins 9/10.
• V4.7 GPT-2 pooled Test Lorentz minus Euclid: mean -0.000545595, paired 95% CI [-0.000762316, -0.000328873], wins 9/10.
• V4.7 GPT-2 pooled Test Lorentz minus Random: mean -0.000613526, paired 95% CI [-0.000836661, -0.000390391], wins 9/10.

Suggested dependencies:

pip install torch transformers datasets numpy pandas matplotlib seaborn

Building the manuscript

Install a LaTeX distribution such as TeX Live or MacTeX, then run:

pdflatex "k=1 quantum.TEX"
pdflatex "k=1 quantum.TEX"

Run pdflatex twice so cross-references are resolved.

If the referenced figure PDFs are absent, LaTeX will stop at the first missing figure. To build a text-only draft, either add placeholder PDFs with the expected names or comment out the four \includegraphics commands in the TeX source.

Claim status

The manuscript separates its claims into three layers:

1. Theorem-level result: the conditional signature gate for the Schrodinger-type representation within the stated K=1 OU-FP construction.
2. Effective-dynamics consequences: one-way boundary protection and related equilibrium identities inside the effective model.
3. Operational closures: active-time reconstruction and the mu = 2 measurement-channel diagnostic.

The paper should be evaluated primarily on the first layer.

Citation

No formal citation metadata is included yet. If you use or discuss this work, please cite the manuscript title and author listed in the TeX source.

References

• Y.Y.N. Li, K=1 Chronogeometrodynamics: Lorentzian Signature as a Geometric Precondition for the Schrodinger Representation.
• S. Amari, "Natural gradient works efficiently in learning," Neural Computation, 1998.
• P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, 2008.
• J. Anandan and Y. Aharonov, "Geometry of quantum evolution," Physical Review Letters, 1990.

License

No license file is included yet. Add a LICENSE file before distributing or reusing this work under a formal open-source license.