The Derivable and the Selected

An Exactly Solvable Coupled-Map Lattice as a Case Study in the Limits of Deriving a Theory of Everything

A synthesis-and-perspective paper on a single question asked in two registers: how much of a physical world can be fixed by consistency and algebra alone, and where must that derivation stop and hand off to a choice made from outside the equations?

What this is (and isn't). This is a synthesis / perspective article, drafted in dialogue with Claude (Anthropic). It advances no experimental claim and does not claim an original solution to quantum gravity. Part I is a self-contained, numerically verified treatment of the diffusively coupled logistic map lattice; every analytic result there is already in the nonlinear-dynamics literature and is attributed in the "Relation to prior work" section (nothing in Part I is claimed as a new discovery). Part II reviews standard, cited results in quantum gravity, string theory, holography, and cosmology. The contribution of the paper is the framing: an explicit schema for the boundary between what consistency derives and what it can only select, together with a worked, provable instantiation of that schema.

The thesis in one picture

Any sufficiently rich deterministic theory passes through the same four stages:

Stage	Coupled logistic lattice (provable)	Theory-of-Everything frontier (perspective)
S1 Derivation	dispersion relation, exact Lyapunov sum rule + closed form, exact two-cycle, parameter-free fluctuation bound	effective field theory of gravity and its finite quantum correction; supersymmetry as the unique symmetry extension
S2 Obstruction	no Turing instability — double stochasticity provably forbids finite-wavelength amplification	Coleman–Mandula / Haag–Łopuszański–Sohnius / Weinberg–Witten
S3 Emergence	a ℤ₂ phase field with tanh domain walls; extensive KS entropy as field-like thermodynamics	spacetime emergent from entanglement (AdS/CFT, Ryu–Takayanagi)
S4 Selection	the equations fix every threshold but not the realized attractor, nor the invariant measure that sets the true control parameter	the string landscape (~10⁵⁰⁰ vacua); the ~20 constants are selected, not derived; anthropic Λ; self-reference ceilings

The lattice is not a model of quantum gravity. It is a proof of concept that the derivation–selection boundary is a sharp, sometimes provable, structural feature of deterministic systems — which lends concreteness (though not proof) to the conjecture that a completed Theory of Everything is less a master equation than a finite-information, self-referential structure that predicts a distribution of observers and contains our particular constants as a dial it cannot, from the inside, resolve.

Contents

File	Description
derivation_and_selection.tex	LaTeX source (arXiv-style, article class)
derivation_and_selection.pdf	Compiled paper (16 pages)
fig_phase.pdf	Phase diagram: λ_max(r, ε) with the analytic boundaries overlaid
fig_multi.pdf	Dispersion, synchronization windows, and Lyapunov spectra
Makefile	Build the PDF from source
LICENSE	CC BY 4.0
.gitignore	Ignores LaTeX build artifacts

Building from source

Requires a TeX distribution (TeX Live or MiKTeX) with amsmath, amssymb, amsthm, mathtools, mathrsfs, bm, graphicx, booktabs, caption, authblk, enumitem, microtype, xcolor, and hyperref.

make          # runs pdflatex twice (resolves the ToC and cross-references)
make clean    # removes .aux/.log/.toc/.out artifacts

Or manually:

pdflatex derivation_and_selection.tex
pdflatex derivation_and_selection.tex

Citation

No DOI yet. For a citable, archival version, deposit the PDF on Zenodo or OSF and update the entry below.

@misc{ideatrino2026derivation,
  author       = {Ideatrino},
  title        = {The Derivable and the Selected: An Exactly Solvable
                  Coupled-Map Lattice as a Case Study in the Limits of
                  Deriving a Theory of Everything},
  year         = {2026},
  howpublished = {\url{https://github.com/Ideatrino/derivation-and-selection}},
  note         = {Synthesis and perspective article; drafted in dialogue with
                  Claude (Anthropic)}
}

License

Released under Creative Commons Attribution 4.0 International (CC BY 4.0). You may share and adapt the material with attribution.