Author: Alex Maybaum
Field: Theoretical physics / foundations
This repository develops the Observational Incompleteness (OI) framework: a programme for asking what laws are accessible to an observer embedded inside a finite, reversible system with only partial access to its state.
The framework is inspired by the same broad kind of self-reference constraint that appears in Gödel and Turing: an embedded system cannot, in general, obtain complete access to itself. OI studies the physical consequences of that limitation.
Scope note. Throughout the project, “derived” means derived relative to the named structural and empirical premises. The framework is best read as a compression of physics onto a small set of commitments, not as a derivation of known physics from the bare fact of observation alone. See the book §4.7 and
SM§8.3.
| If you want | Start with |
|---|---|
| The foundational finite-observer result | papers/Main.md |
| A short, self-contained phenomenology paper | papers/Juno.md |
| The current formal-verification programme | verification/README.md |
| The live list of proved, conditional, open and refuted obligations | verification/ROADMAP.md |
| The full book-length exposition | book/README.md |
| The physics papers | papers/ |
At finite observational horizon, the framework relates three descriptions:
finite stochastic law (S)
⇅ exact
finite reversible realization (D)
⇅ exact
fixed-basis unitary/Born representation (Q_fb)
The representation equivalence is universal at this finite-record level. The specifically OI content is the embedded-observer structure: coupling, hidden capacity, persistence and history readback constrain what a faithful realization must contain.
Main develops this finite-horizon correspondence together with the hidden-memory and recurrence results. Later operational/composite refinements, Bell questions and physics-layer bridges are tracked separately rather than folded into the base theorem; their current status lives in the verification roadmap.
The repository deliberately separates three levels of claim.
The physics core is developed in Main, SM, GR, Substratum, Structure, Methodology, and the focused Juno paper. Formal and computational certificates live under verification/.
Several later physics conclusions require explicitly named hypotheses or bridge statements. These are not silently promoted to consequences of OI. Important examples include:
-
H-link — identifies the exact six-link cubic representation with a physical gauge carrier; the single-copy clause gives the
$K=6$ reading. - H-cust — the custodial kinetic/condensate premise used by the stabilizer route.
- H-Bell — the open compatibility requirement that preparation-indexed graphs (i) supply the required ontic parameter dependence, (ii) preserve operational no-signaling, and (iii) preserve the metric/Ollivier--Ricci structure strongly enough for the continuum-curvature step.
- other manuscript-specific bridge hypotheses and obligations recorded in the papers and verification roadmap.
Applications outside the physics core are exploratory and carry no evidential weight for the core framework:
papers/Complexity.md— the structural chain toward evolution and complexity;papers/Computation.md— computation and complexity theory;papers/Medicine.md— medicine;papers/Bioinformatics.md— computational biology;- the consciousness discussion in the book.
Each extension should be judged on its own test-or-break conditions.
| Work | Role |
|---|---|
Main |
Finite embedded observers, hidden memory, the finite-horizon |
SM |
The |
GR |
The cosmological-horizon route to the gravitational and dark-sector programme, with assumptions and open calculations stated where they enter. |
Substratum |
Reconstruction and substratum gauge structure; the proved converse is scoped to the local propagating lattice/gauge residue under its stated hypotheses. Bell-inclusive existence is conditional on H-Bell; its uniqueness is open. |
Structure |
Observation/gauge hierarchies, universality classes of embedded observers, and comparison with other unification programmes. |
Methodology |
Developmental draft on the framework's foundations, methodology, and axiom structure. |
Juno |
Focused presentation of the parameter-free solar-mixing value and its retrodictive/forward-test status. |
Juno presents
The value was derived after JUNO's first measurement, so the existing agreement is a retrodiction, not a prediction made in advance. The paper treats JUNO's design-lifetime precision as the forward test. Full numerical and classification details are in the Juno paper and the relevant physics papers rather than being duplicated here.
The verification suite is the authoritative place for machine-checked status. It has three main layers:
verification/lean/— dependency-free Lean 4 kernels plus concrete numerical probes;verification/lean-mathlib/— the Mathlib-basedOIBridgetheorem programme;verification/coverage/— the manuscript-to-certificate coverage ledger.
CI checks the proof layers, probes, coverage and governed receipts. Exact theorem counts and current obligations change as the programme advances, so this README intentionally does not hard-code those counts.
The verification architecture and flagship formal results are described in verification/README.md, and the live obligation queue is verification/ROADMAP.md; both are linked under Start here. To run the zero-import checks, see verification/lean/VERIFYING.md; the narrower zero-import formalization roadmap is verification/lean/ROADMAP.md.
incompleteness/
├── papers/ Research papers and lattice-computation source
├── book/ Book-length exposition
├── verification/ Lean proofs, Mathlib bridge, probes, coverage and receipts
├── tools/ Build, audit and release-gate tooling
└── README.md
papers/Explainer.md— older overview, superseded by the book and retained for reference.papers/oi_lattice_code/— lattice Monte Carlo and related numerical source.
The core papers are tabulated above; the conjectural-extension papers are listed under 3. Conjectural extensions.
The verification programme changes faster than this README should. Rather than duplicating a potentially stale list of “open” claims here, the repository keeps the authoritative status of each obligation in verification/ROADMAP.md.
That roadmap tracks, among other areas, operational/composite refinements; the K1–K3 / K∞ / Kₙ pre-quantum kinematics programme; physical-carrier identification; Standard-Model bridge conditions; Bell-compatible completion; gravity/running obligations; and continuum emergence. Individual items may be proved, conditional, refuted or open; use the roadmap for the current verdict.
The Incompleteness of Observation: A Unified Framework from Quantum Mechanics to Computational Biology develops the framework across 20 chapters with expanded exposition, appendices, glossary and bibliography.
The reader's guide is book/README.md (linked under Start here); the compiled manuscript is book/The-Incompleteness-of-Observation-FULL.pdf.
The repository is archived on Zenodo under concept DOI 10.5281/zenodo.19060318, which resolves to the latest version. For a claim tied to a particular release, cite that release's version DOI.
Maybaum, A. (2026). The Observational Incompleteness Framework. Zenodo. https://doi.org/10.5281/zenodo.19060318
This repository contains both code and authored research material:
| Scope | License |
|---|---|
Source code — including papers/oi_lattice_code/, verification/, tools/, and other program source |
MIT, per LICENSE |
| Manuscripts — the papers and book | CC-BY-4.0 |
LICENSE is intentionally the repository's machine-detected MIT license file; the manuscript licence applies by scope to the authored research works.
Alex Maybaum — Independent Researcher
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