Theorem Catalog
Every result is labeled for what it actually is. Proved facts are theorems. Everything else is clearly marked.
THEOREMPROPOSITIONCONJECTUREOBSERVATIONDEFINITION
T01 Infinite Zeros Barrier
THEOREM sin(x) has no finite real EML tree. Every finite real EML tree is real-analytic with finitely many zeros; sin(x) has infinitely many (π·ℤ).
T02 EML Universality
THEOREM eml(x,y) = exp(x) − ln(y) generates every elementary function as a finite binary tree. (Odrzywołek, arXiv:2603.21852, 2026.)
T03 Euler Gateway
THEOREM ceml(ix, 1) = exp(ix) = cos(x) + i·sin(x). A single depth-1 EML node with complex input gives Euler's formula.
T04 Log Recovery
THEOREM ln(x) = 1 − ceml(0, x) for all x > 0. Direct computation: exp(0)=1, so ceml(0,x) = 1 − ln(x).
T05 Phantom Attractor is a Precision Artifact
THEOREM The apparent attractor at ~6.2675 seen in EML gradient descent at double precision vanishes at 15+ decimal places. Not a true fixed point: ∇L ≠ 0.
T06 Tropical Self-EML
THEOREM In the tropical semiring (⊕=max, ⊗=+), teml(a,a) = max(a,−a) = |a|.
T07 BEST Routing — sin/cos via 1 node over ℂ
THEOREM Over ℂ, sin(x) = Im(ceml(ix,1)) and cos(x) = Re(ceml(ix,1)). Both from a single EML node with complex input. Follows from T03.
P01 EDL Not Complete over Addition
PROPOSITION The EDL operator family cannot represent addition using single-variable inputs and EDL grammar. Exhaustive enumeration to depth 5.
P02 EXL Node Counts
PROPOSITION Under EXL grammar: ln(x) achieves 1-node representation; pow(x,n) achieves 3-node representation.
P03 exp(x) is EML Depth-1
PROPOSITION exp(x) = ceml(x,1) is a single-node EML tree. EML-0 ⊊ EML-1.
P04 x² is EML Depth-2
PROPOSITION x² = exp(2·ln(x)) has depth 2. All power functions x^r are EML depth-2.
P05 sin(x) is EML-3 over ℂ
PROPOSITION sin(x) requires at least one complex exp/ln composition — depth 3 over ℂ under the full grammar.
P06 Depth-5 Im ≤ 0 Boundedness
PROPOSITION All EML₁ elements at depth ≤ 5 satisfy Im(z) ≤ 0. Computational verification (41,409 values).
P07 Depth-6 Phase Transition
PROPOSITION Depth-6 EML₁ elements can have positive imaginary part. First Im > 0 values appear at depth 6.
C01 i-Constructibility (T_i)
CONJECTURE i ∈ EML₁ (i is constructible from terminal {1} under strict principal-branch grammar). Status: OPEN. Depth-6 best: Im = 0.99999524.
C02 EML₁ Density
CONJECTURE EML₁ is dense in ℂ: for every z ∈ ℂ and ε > 0, there exists w ∈ EML₁ with |w − z| < ε.
C03 i as Accumulation Point
CONJECTURE i ∉ EML₁ yet i is an accumulation point of EML₁. Equivalently: δ(d) → 0 as d → ∞, but δ(d) > 0 for all finite d.
C04 EML Weierstrass Density
CONJECTURE For every continuous f: [a,b] → ℂ and ε > 0, there exists a depth-k EML tree approximating f to within ε. (Constructive Weierstrass for EML.)
T01–T07: Theorems (proved) · P01–P07: Propositions · C01–C04: Conjectures ·
Count: 7 theorems, 7 propositions, 4 conjectures ·
Paper: arXiv:2603.21852