Recursive Self-Consistency

In the paper's words

This paper establishes recursive self-consistency as a mathematically coherent and physically generative monistic foundation for quantum cosmology. We demonstrate that this single principle—implemented through the analytic-topological loop of the circle and the Gaussian, unified by the metaplectic group, and culminating in modular self-similarity via the theta function—provides a logically closed framework capable of generating key physical structures without external axioms. Our formal derivation shows how this principle resolves foundational problems including the problem of time and the black hole information paradox, while generating quantum mechanics, spacetime geometry, gravity, and cosmology through a six-step generative sequence.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The mathematical engine of this sequence is the Jacobi theta function, which serves as the cosmological partition function (Mumford, 1983). $$ \theta(\tau) = \sum_{n=-\infty}^{\infty} e^{\pi i n^2 \tau} $$ This function sums over all integer states ($n \in \mathbb{Z}$) derived from the circle’s topology, with each state weighted by a Gaussian factor derived from the analytic part of the loop.

   from 2.2 The Formal

2. This transformation is not arbitrary; it is the one that maximizes the mutual information between the input and the output, making it the most efficient possible encoding of the foundational information (Amari, 1985).

   from 2.2.2 Step 2:

How it is done

1. Formally defined, a system satisfies recursive self-consistency if it can be represented as a fixed point of a continuous transformation, solving an equation of the form (Banach, 1922): $$ x = \mathcal{R}(x) $$ In this equation, $x$ represents the complete state of the system, $\mathcal{R}$ is a generative operator that maps the space of possible states onto itself, and $X$ is the space containing all possible states.

   from 1.0

2. Gödel’s completeness theorem demonstrates that if physics is a complete and consistent formal theory, no proper extension can add explanatory power without introducing an inconsistency (Gödel, 1930).

   from 1.1.1 The Rejection of

3. Formally, a strange loop can be defined as a continuous map $\mathcal{R}: X \to X$ on a compact Hausdorff space $X$.

   from 1.1.2 The Logical

4. The universal wave function $|\Psi\rangle$ that satisfies this constraint is precisely the unique fixed point of the recursive self-consistency map, guaranteed to exist by the Banach fixed-point theorem (Banach, 1922).

   from 1.2.1

What would falsify it

1. In the recursive self-consistency framework, this coherence is implemented through the condition of modular invariance: an observable is physically valid if and only if it is invariant under the action of an appropriate congruence subgroup $\Gamma \subset \mathrm{SL}(2,\mathbb{Z})$ (Mumford, 1983).

   from 1.2 The Logical

2. This theoretical prediction is strongly supported by empirical data from the Planck Collaboration (2020), whose measurements of the cosmic microwave background show the universe to be spatially flat with $|\Omega_k| \lt 0.002$, confirming zero total energy to within experimental error.

   from 1.2.1

3. The prediction of a zero-energy universe, which follows directly from the Hamiltonian constraint, is confirmed by cosmic flatness measurements showing $|\Omega_k| \lt 0.002$ (Planck Collaboration, 2020).

   from 1.3.3 Empirical

Where the claim stops

1. Self-contained Nature of Physical Reality For any theory of reality to be truly foundational, it must be logically closed, meaning it cannot depend on external elements for its definition or consistency.

   from 1.1 The

2. External Foundations External foundations fail as explanatory frameworks because they either lead to contradictions, provide no genuine explanatory power, or ultimately reduce to the recursive self-consistency principle itself (Sober, 1975).

   from 1.1.1 The Rejection of

3. Necessity of Internal Coherence Internal coherence is not merely a desirable feature but a logical necessity for any candidate foundation of physical reality, as an incoherent system cannot describe a stable, persistent universe (Hofstadter, 1979).

   from 1.2 The Logical

Where to read

1. Recursive185 words · 1 min
2. 1.05,032 words · 22 min
3. Appendix305 words · 1 min
4. Appendix B: Derivation of the2 words · 1 min
5. Glossary |264 words · 1 min