One algebra, six open problems
When I tell people the same algebraic framework applies to Navier–Stokes, the Fenton distribution, and neural scaling laws, the first reaction is disbelief. Fair enough — cross-domain claims are usually vague analogies dressed up in notation.
So let me be precise about what the Latent framework actually shares across domains.
The shared structure
Every system we study gets decomposed into a graded Hilbert tensor algebra. The key object is the Latent Number \(\rho\) — a non-negative integer measuring the system's intrinsic compressibility.
The decomposition:
\[\mathcal{L}(S) = \bigoplus_{k=0}^{\rho} \mathcal{H}_k\]
where \(\mathcal{H}_k\) is the grade-\(k\) component. Everything above grade \(\rho\) vanishes identically. This is not an approximation — it's exact.
What changes across domains
| Domain | System | \(\rho\) | What grade truncation gives you |
|---|---|---|---|
| Physics (NS) | Energy cascade | Related to Gevrey class | Regularity from grade-2 control |
| Finance | Lognormal sums | 2 | Exact closed-form distribution |
| ML | Loss landscape | \(\sim\) data manifold dim | Chinchilla scaling exponents |
| Number theory | Euler product | 2 | GUE universality |
The algebra is the same. The value of \(\rho\) changes. The theorems you get are domain-specific, but the proof method — "truncate at grade \(\rho\), show the remainder vanishes" — is universal.
Why this isn't just analogy
In each domain above, we prove that the truncation is exact (not approximate). This means:
- The Fenton distribution is exactly grade 2, not "approximately" grade 2
- The Euler product's GUE behavior is a theorem about grade-2 dominance, not a heuristic
- The Navier–Stokes regularity is conditional on grade-2 control, with the condition precisely stated
You can verify each claim independently. The connection between domains is that the same algebraic machinery gives you the same kind of result — exact characterization via grade truncation — but the proof in each domain stands on its own.
The meta-observation
If the same structure keeps appearing across unrelated problems, either there's a deep reason or it's a coincidence. With 28 papers across 7 domains, I'm betting on deep reason. But the beauty of formalization is that you don't have to take my word for it — the Lean files are there to read.