Bryan Ehrlich
I'm interested in how a physical system learns about itself and uses what it learns. This is an independent side project, with a few mathematical detours. I do it for fun, and I'm trying to make the arguments precise enough to check.
The current work includes a classification of mathematical update rules and a draft about the limits of one self-modeling framework. Neither establishes that self-modeling selects quantum mechanics. The connection to conscious experience is still an open question.
Current work
A. Sequential Products on Finite-Dimensional Euclidean Jordan Algebras: A Complete Classification v2 preprint
Classifies the sequential products allowed by seven specified axioms on finite-dimensional Euclidean Jordan algebras. On simple algebras of rank at least three, the real, quaternionic and exceptional cases have only the Lüders product; the complex case has exactly a one-parameter family. Rank-two spin factors have a separate classification, with more freedom in the complex qubit and in higher dimensions.
This is a mathematical result within the assumed Jordan setting. Part of the classification is also proved in Lean 4.
B. Limits of Self-Modeling in Quantum Reconstruction: Effect-Valuedness, Reciprocity, and Local Tomography as Non-Derived Inputs working draft under revision
Examines what one explicit self-modeling framework implies and which additional assumptions a quantum reconstruction needs. The draft treats effect-valuedness, reciprocity and different meanings of local tomography separately. Its conditional route to complex quantum mechanics still requires extra hypotheses.
Paper A in Lean
The formal proofs cover the real and complex matrix classifications on all effects, including the unique twist parameter for complex matrices of size at least three, and the separate complex-qubit classification. The quaternionic and Albert files check the central-imaginary argument for strictly positive first effects diagonal in a fixed standard frame, with an explicit Archimedean-order hypothesis.
The classification of spin factors of vector dimension at least four, the dynamical-correspondence theorem and the classification on abstract simple algebras remain unformalized. The two linked snapshots passed separate audits of 166 and 213 modules on October 3, 2026. The audited declarations depend only on Lean's three core axioms. Those audits cover the named module sets, not every file in either repository.
Earlier work
Older drafts are kept here for the record, including the ones I withdrew or retracted. Their original titles are preserved.
7. The Standard Model from Self-Modeling: Gauge Structure from the Observer-Universe Interface exploratory
An older attempt to connect exceptional Jordan algebras with Standard Model structure. Kept as an exploratory draft, with its assumptions and gaps.
5. Quantum Mechanics from Self-Modeling: Deriving Complex C*-Algebraic Structure from a Single Operational Premise withdrawn
Withdrawn. The claimed derivation of complex quantum mechanics did not hold. The original remains available as a record of that attempt.
6. The Self-Modeling Basin Is Exceptional Supergravity retracted
Retracted. Matching an algebraic construction to a supergravity model did not derive gravity's dynamics. The original remains available here.
1. Experiential Measure on the Structure Space of Self-Modeling Systems exploratory
An early proposal for a density functional on self-modeling structures and a conjectured connection to experience.
2. Exponential Suppression of Transient-Basin Contributions in Trajectory-Weighted Markov Chain Measures
An earlier argument about transient and metastable contributions to a trajectory-weighted Markov-chain measure.
2b. Theorem A: Lemma Assembly
Companion lemma statements, references and dependencies for the Markov-chain argument.
3. Lipschitz Stability of the Experiential Density Functional
An earlier stability calculation for the proposed density functional under perturbations of the transition kernel.