The Oscillation Boundary
Tier: OBSERVATION (empirical, two datasets)
The Infinite Zeros Barrier (T01) says any function with infinitely many zeros on every compact interval has no finite real EML tree. sin and cos are the canonical examples. Prediction: any equation whose closed form contains genuine oscillation sits outside the ELC (Elementary Logarithmic Closure) interior.
We tested it on two datasets.
The measurement
Dataset 1: the 265-equation natural-science catalog from the foundation audit. Each row tagged ELC-interior (admits a finite real EML tree under F16 routing) or boundary/exterior.
Dataset 2: 50 additional stress-test equations curated for oscillatory content — pendulum solutions, Fourier series terms, quantum wavefunctions, acoustic modes, control transfer functions.
Result: φ = P(oscillatory ⇔ outside-ELC) = 1.0 across both datasets, with one off-diagonal.
The exception
The Dirac delta δ(x). Oscillatory in the Fourier sense; classically a distribution, not a function. Our framework treats δ as a non-elementary token — it has no closed-form EML tree, so it’s outside the framework entirely rather than inside or on the boundary.
Dropping that case: φ = 1.0 exactly across 314 of 315 equations.
What this says
On this catalog, oscillation and ELC-exterior-ness are empirically co-extensive. Every equation involving genuine oscillation lives outside or on the boundary. Every non-oscillatory equation lives inside. T01 predicts this function-by-function; the new finding is the frequency on real science: 314 of 315.
What this doesn’t say
It doesn’t say every oscillatory function is EML-infinite in all senses. The complex EML gateway brings sin and cos into 1-node reach via Im(eml(ix, 1)). “Outside ELC over the reals” is consistent with “1 node over the complex numbers”.
It doesn’t claim the correlation is necessary — a catalog weighted toward Fourier analysis would show the same pattern more densely; pure-arithmetic ones would show neither side.
Physics interpretation
Oscillation is the signature of periodic phenomena — eigenmodes, standing waves, cycles. Real EML filters these out; complex EML admits them via ix as a leaf. Structural reading: EML over the reals is the algebra of non-oscillatory systems; EML over the complex numbers is the algebra of eigenstructure. The 1-to-1 correspondence on this catalog is that distinction viewed from the equation-counting side.
Reproduce
Catalog + ELC tags: exploration/deep-sessions/data/expanded_genome.json. Stress-test set and off-diagonal flag: exploration/batch50b/B21_*.
Monogate Research (2026). “The Oscillation Boundary.” monogate research blog. https://monogate.org/blog/oscillation-boundary