Honest Gaps

What is proved, what is not, and what is next.

The point of this page is to be direct. A lot of science communication overstates what is known. This theory has genuine results and genuine gaps. Both are listed here without hedging.

What is genuinely proved

The following results are exact, derived from first principles, and verified computationally:

α = log(5/2) exactly. The mutual information decay exponent is determined by the Weingarten integral η2 = 2/5. This is a closed-form theorem, not a fit.

Hyperbolic geometry. The Gromov boundary of the infinite binary tree is the boundary of H². This follows from the definition of the tree.

Ryu-Takayanagi. The holographic entropy formula holds in the annealed sense. Exact moments confirm this analytically.

Ultrametricity at 98.3%. Confirmed on 1 million triples across multiple independent trees.

Supercritical cascade. η4/η2² = 65/56 > 1 establishes the model is in the supercritical regime of multiplicative cascades — the source of the divergent fluctuation law.

What is not yet proved

de Sitter space. Our universe has a positive cosmological constant. It is de Sitter, not anti-de Sitter. The model works cleanly in AdS. Extending to de Sitter is open — and shared with the entire holographic program.
Einstein’s equations. The linearized Einstein equations emerge from the entanglement structure, following Faulkner and Van Raamsdonk’s program. The normalization constant is off by a factor of ~16.7. The structural argument works; the calibration does not yet close.
The exact fluctuation exponent. The analytic lower bound on β is (1/2)·log(65/56) = 0.0745. The exact value requires computing Var[log c(V)] in closed form — a log-moment that lies outside the polynomial Weingarten machinery. The Monte Carlo estimate is β∞ = 0.101 ± 0.001.
Dynamics. The current model is static. A full quantum gravity theory needs the tree to evolve in time. This is the next generation of the problem.

What is next

The immediate computational target is η12 — the twelfth-order Weingarten moment. This requires summing over 479 million permutation pairs and is currently running on GPU hardware. Once in hand, we will have the first six free cumulants of the amplitude distribution, which may reveal a closed-form recursion for the full moment sequence.

The deeper goal is to derive the full Einstein equation from the genealogical structure, not just its linearization. Until that is done, this theory is a strong candidate for the mechanism behind holographic geometry — not yet a proof.

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