It from Fix: the Kernel Principle

Author: Fabien Polly (@infinition) Status: exploratory research, computational toy models Version: 0.3.0

This repository contains a speculative research note in theoretical physics, the toy models built to probe it, and, importantly, the experiments that failed. Nothing here claims to be a theory of quantum gravity. What you will find is a research process, documented honestly enough that someone else can pick it up, including its dead ends.

The idea in one paragraph

Three well-known statements in physics share the same algebraic shape. The Wheeler-DeWitt constraint says the quantum state of the universe is stationary in time. The renormalization-group fixed-point condition of asymptotic safety says fundamental couplings are stationary in scale. Structural conservation under an update map says a relational kernel is stationary under discrete evolution. The working hypothesis of this project is that these are three faces of one selection principle: the physically realized objects are the stationary ones, and observed dynamics is what the non-stationary degrees of freedom look like from inside. The paper calls this the Kernel Principle.

What this repository actually establishes

Being precise about this matters more than the idea itself.

Reproduced results. The fixed point (1/4, 1/64) of the Einstein-Hilbert truncation, the complex critical exponents (5 +/- i sqrt(167))/3, the dimensional window for the complex pair, and the f(R) truncation behavior are reproductions of published results by Litim and by Codello, Percacci and Rahmede. They are verification work, not new physics.

Elementary but exact. The chirality threshold of the projected 2D toy model, complex eigenvalues exactly when w^2 > (a-d)^2/4 + b^2, is textbook linear algebra. The related statistical fact that asymmetry raises the fraction of complex eigenvalues is a known subject in random matrix theory (real elliptic Ginibre ensembles). We keep these because they anchor the toy-model dictionary, not because they are discoveries.

A fit, not a prediction. A parameter search over chiral ring kernels matches the Litim spiral ratio to 3e-5. With enough free parameters this is expected. The repository says so explicitly.

Negative results, the most useful part. A series of controlled experiments tried to make gravitational attraction emerge from persistence and information exchange in relational graphs. Under equal-budget controls and accessibility-matched null models, every unbiased rule failed to produce attraction. Persistence-only reinforcement is repulsive (about -3.3%). Automatically survival-selected exchange is repulsive (about -1%). Only a hand-selected exchange halo attracts, which demonstrates a mechanism but not emergence. If you work on graph-based emergent gravity, these dead ends may save you time.

Earlier tests of the same idea

This note is one branch of a program running since 2024, and the principle it states has already been tested quantitatively in another sector. The strong form (measured Standard Model couplings as a global stationarity optimum of the renormalization-group flow) is falsified: the optimum requires alpha_s(M_Z) around 0.048 against the measured 0.1179. The weak form (stationarity as a Planck-scale boundary condition) is neither confirmed nor refuted against current functional renormalization-group computations. See docs/PROGRAM_LINEAGE.md for the numbers and raw data. Reading this repository without that context would overstate how untested the idea is.

Repository map

• paper/ the original v0.1.0 LaTeX note and PDF, kept as the historical record, and the current arXiv source in paper/v0.3.0/ (adds the earlier quantitative tests, the cross-program negative results, and a lineage appendix).
• docs/ theory evolution, experiment ledger, verdicts, limitations, roadmap. Start with docs/RESULTS_AND_VERDICTS.md if you only read one file.
• docs/PROGRAM_LINEAGE.md the earlier branches of this program (LVS, RAPC) with the key numbers the paper cites, and their raw result files under docs/lineage/.
• experiments/ all retained scripts, figures and JSON results, including failures.
• src/ compact reusable implementation of the common kernel operations.
• reproducibility/ environment notes and which experiments have exact scripts.

Quick start

python -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt
python experiments/kernel_principle_poc/kernel_principle_poc.py
python experiments/kernel_discriminant_test/kernel_discriminant_test.py

Some experiments are expensive parameter sweeps. Read docs/EXPERIMENT_LEDGER.md before rerunning them. Not every experiment kept its exact script; see reproducibility/STATUS.md for which results are byte-reproducible and which are documented reconstructions.

Where this could go

The roadmap (docs/ROADMAP.md) defines what a real positive result would require: a local update rule with no attraction term in the objective, universal response across probe types, survival under matched null models, and at least one quantitative prediction that was not fitted. The current experiments do not cross that bar, and the project treats that as a finding rather than a failure to hide.

Related programs worth reading before continuing this line of work: quantum graphity (Konopka, Markopoulou, Smolin), causal sets, Jacobson's thermodynamic derivation of the Einstein equation, Verlinde's entropic gravity, and Sornette's review of discrete scale invariance for the meaning of complex critical exponents.

Citing

See CITATION.cff. If you use anything from this archive, please state its toy-model status clearly. This is exploratory work and should be cited as such.

License

MIT for the code. The paper text and figures are CC BY 4.0.