Two Boundaries of ELC
Tier: OBSERVATION (classical theorems; new observation is the joint interpretation)
We recognise two structurally independent obstructions that eject a function from the Elementary Logarithmic Closure. They come from completely different branches of mathematics.
Boundary 1: T01 — the Infinite Zeros Barrier (analytic)
A real function with infinitely many zeros on every compact interval has no finite real EML tree. sin, cos, tan sit outside real ELC because any finite composition of exp, log, and arithmetic over ℝ produces a real-analytic function with only finitely many zeros on any compact set. The barrier is analytic — it’s about the function as an analytic object on the real line.
Oscillation is the dominant witness: on our 315-equation catalog, φ = P(oscillatory ⇔ outside-ELC) = 1.0 with one off-diagonal (the Dirac delta — not a function).
Boundary 2: Abel-Ruffini — the Solvability Barrier (algebraic)
Eigenvalues of a general symbolic matrix are roots of its characteristic polynomial:
- n ≤ 4: closed-form radical expressions (quadratic, Cardano cubic, Ferrari quartic). Inside ELC.
- n ≥ 5: no radical formula exists. Abel (1824) and Galois (1832) proved this by showing the symmetric group S₅ is not solvable. General quintic eigenvalues lie outside ELC.
The barrier is algebraic — the obstruction is the structure of a finite group, not analytic behaviour. Entirely independent of oscillation: a generic 5×5 matrix need not have any sin or cos anywhere, yet its eigenvalues are still outside ELC.
Why both must coexist
Neither subsumes the other. A generic matrix without trig terms still has n ≥ 5 eigenvalues outside ELC — T01 doesn’t catch these. Conversely, sin(x) lies outside ELC by T01 regardless of any polynomial structure. The two obstructions fence the elementary closure from different sides.
Algebraic statement:
ELC ⊆ (non-oscillatory closure) ∩ (radically-solvable closure)
Two classical theorems, two centuries apart, constraining the same complexity class from independent directions.
What this opens up
If two independent obstructions exist, how many more are there? Candidates we haven’t placed in the picture:
- Lindemann-Weierstrass (transcendence of π, e) — transcendence obstruction
- Risch elementarity (elementary antiderivatives) — integration-closure obstruction
- Picard-Vessiot (ODEs solvable in elementary functions) — differential-algebraic obstruction
Each is a theorem about closure under a specific operation; each could in principle eject functions from ELC that survive T01 and Abel-Ruffini. Whether they do, and whether they collapse into each other, is an open structural question we’re now pursuing.
Reproduce
Abel-Ruffini eigenvalue costs for n = 2..5 and the classical solvability argument: exploration/symbolic-math/determinants_and_matrices.py (DET-4). Oscillation-boundary evidence for T01: exploration/batch50b/B21_*.
Monogate Research (2026). “Two Boundaries of ELC.” monogate research blog. https://monogate.org/blog/two-boundaries