Glossary

53 terms. Every one links to the page that explains it in depth.

A

Adam: Kingma+Ba 2014 optimizer. In EML cost accounting it costs 31 nodes per parameter per step (post-NN-13 re-audit, down from 37 when the bias-correction scalars 1 − βᵗ are shared across parameters). →
AM-GM: Arithmetic mean ≥ geometric mean. In EML nodes the arithmetic mean is 4n and the geometric mean is 3n — so the more expensive tree bounds the cheaper one above. →
Atlas: The /atlas map of elementary functions by atlas depth, which counts nested exp/ln applications and treats arithmetic as free: 0 (arithmetic), 1 (exp, softplus), 2 (log, negation), 3 (oscillatory via complex bypass), ∞ (non-constructible over ℝ). Atlas depth is not EML tree depth, under which ln x needs exactly 3 and x + 1 exactly 4 on (0, ∞). →
Attractor: A fixed point or cycle that nearby orbits converge to under iteration. Lambert fixed points z_k* of exp are log-attracting. →

B

Box-counting dimension: A fractal-dimension estimator that counts boxes of side s required to cover a set, fit log-log. Used in the S3 morph to measure in-set boundary roughness. →

C

CapCard: A JSON file that declares a project's verifiable capabilities, costs, proofs, and test coverage. Monogate's card is served at /capability_card.json and /.well-known/capcard.json. →
CapCard v3: Current schema version. Adds eml_metrics, neural_metrics, agent_usage, and agent_queries so agents can filter tools by computable numbers. →
Cauchy equation: One of three functional equations — additive f(x+y)=f(x)+f(y), multiplicative f(x+y)=f(x)f(y), logarithmic f(xy)=f(x)+f(y). Their continuous solutions are cx, eᶜˣ, and c·ln x. With c as a leaf, F16 trees for them take 3, 2 and 2 nodes: LEdiv(0, F13(c, DEML(x, 1))) for all x, F13(c, EML(x, 1)) for all x, and LEdiv(0, F13(−c, x)) for x > 0 (checked numerically); whether those trees are the cheapest is not shown. →
Cobweb diagram: A visualisation for iteration of a real map: y = x line plus f(x), ladder drawn to show orbit. Used in the Conjugacy Viewer. →
Complete operator: An operator that can express every elementary function. EML is one (the published universality result). Among the census operators, seven more are conjectured complete over ℂ: EDL, EXL, EAL, EPL, LEAd, ELAd, ELSb; over ℝ, EAL is not, since every real EAL tree is nondecreasing. →
Completeness conjecture (T26–T28): The conjecture that an operator of the census list is complete iff it contains exp(+x) with no domain restriction on self-composition. One structural rule would explain all sixteen cases, but none of its directions has a proof, its LEX domain argument is refuted, and over ℝ it fails for EAL. →
Cosh preserves ELC: cosh/sinh/tanh are arithmetic combinations of exp(±x), so the hyperbolic functions stay inside ELC(ℝ); sin/cos do not. The three identities are Lean-verified in HyperbolicPreservation.lean; the ELC-closure step is not itself a Lean statement. →

D

d(d) = 3: A placeholder in early drafts now retired. Do not use. →
DEML: deml(x, y) = exp(−x) − ln(y). The negated-exponential variant of EML, F3 of F16. One-node representation of exp(−x) (deml(x, 1) = exp(−x)). →
Depth hierarchy: EML-0 ⊊ EML-1 ⊊ … the class of real values / functions constructible in at most k EML node applications. Strict at k = 0, 1; conjecturally strict for all k. Lean-verified: EML-0 ⊊ EML-1 (exp_not_constant, EMLDepth.lean). →
Domain coloring: A plot of a complex function where hue = argument and brightness = modulus. Used in the Zen Garden renderer. →

E

EAL: eal(x, y) = exp(x) + ln(y), a census operator (/framework's F14, also written EAL there, is exp(x + ln y)). Conjectured complete over ℂ; not complete over ℝ, where every EAL tree is nondecreasing. One-node representation of exp(x) + 1 via EAL(x, e). →
EDL: edl(x, y) = exp(x) / ln(y), a census operator, not in F16. Conjectured complete over ℂ; multiplicative counterpart of EML. →
ELAd: elad(a, b) = exp(a + ln b) = eᵃ · b. Hybrid-operator shortcut; 1-node SuperBEST construction for multiplication after one ln setup. F14 of F16 (written EAL / ELAd on /framework). →
ELC(ℝ): Elementary Log–Constructive class over the reals. The class of real values (or functions) expressible as a finite real EML tree over algebraic constants. Non-oscillatory elementary constants live here. →
EMN: emn(x, y) = ln(y) − exp(x) = −eml(x, y), a census operator; F16's F2, EMLn, is exp(x) − ln(−y) instead. Conjectured approximately complete (T24). No EMN tree for exp(−x) is written down, and that no finite real EMN tree gives exp(+x) is also a conjecture. →
EPL / ELMl: Power primitive. ELMl(k, x) = exp(k · ln x) = xᵏ is one F16 node for x > 0: F13 on /framework, written EXL / EPL there (UpperBounds.lean). →
EXL: exl(x, y) = exp(x) · ln(y), a census operator. It is not in F16: /framework's F13 is also written EXL but is exp(x · ln y). exl(0, x) = ln x in one node, which /superbest's table counted until 2026-09-13 (its 14n total); inside F16, ln x needs 2 nodes, since no single F16 node equals it and LEdiv(0, F13(−1, x)) does, and the table now counts it that way. Conjectured complete over ℂ; source of the period-3 Sharkovskii regime in the EML family. →

F

F16: The sixteen binary operators listed on /framework as F1–F16, the list the Lean lower bounds in AddLowerBound.lean and MulLowerBound.lean quantify over: EML and its sign and branch variants (F1–F8), LEdiv (F9), LEdivn (F10), LEAd (F11), LEAdn (F12), exp(x·ln y) = yˣ (F13), exp(x + ln y) = y·eˣ (F14), (−y)·eˣ (F15) and exp(ln x + ln y) = x·y (F16). On this site F16 means this list. Two other lists of sixteen appear, and the pages using them say so: the census of Sixteen_Operator_Census.tex (EML, EMN, EAL, EXL, EDL, EPL and their exp(−x) versions, with LEAd, ELAd, ELSb and LEX), which the completeness pages classify, and the taxonomy lock's F16 orbit, which /superbest's high-impact tables count and which leaves LEAd out. →
Feigenbaum δ: The universal ratio ≈ 4.6692 of successive period-doubling bifurcation intervals. EXL's period-3 cascade gives area ratios 4.60 / 4.44 / 4.45 — within 3.7 % of δ. →
Fractal Studio: Interactive explorer for the EML-family Mandelbrot sets with visual / audio / sequencer / orbit / morph modes. →

I

Infinite Zeros Barrier (T01a, T01b): The zero-counting route to: no finite real EML tree equals sin over all of ℝ. Isolated zeros alone rule nothing out, since sin is real-analytic with isolated zeros too; the route needs a bound on how many zeros a tree can have, which is open (T14). Lean covers sin's infinitely many zeros (T01a) and trees of depth 0 and 1 (T01b). The conclusion itself is proved in Lean by periodicity (MachLib, sin_not_in_eml_any_depth_unconditional). →
Isomorphism family: A set of equations from different scientific domains sharing the same EML tree. Beer-Lambert / radioactive decay / compound interest form a 5-node isomorphism family. →

L

Lambert W: Inverse of z·eᶻ. Fixed points of exp on the slit plane are z_k* = −W_k(−1). Multiplier duality (EMLDuality.lean) says deriv(exp)·deriv(log) = 1 at each fixed point. →
LEAd: lead(x, y) = ln(exp(x) + y). Softplus primitive; LEAd(x, 1) = softplus(x). F11 of F16, one node. The taxonomy lock's F16 orbit, which /superbest's high-impact tables count, leaves it out; with it, log-sum-exp of two terms is 2 nodes, LEAd(x, EML(y, 1)). →
Lean 4: Proof assistant developed by de Moura. Monogate's Lean library (monogate-lean) builds under Lean 4 + Mathlib; two files carry documented sorries (InfiniteZerosBarrier.lean Part D, ChainOrderAdditivity.lean). Every Lean claim on this site is re-checked before each deploy. →
LEdiv: lediv(x, y) = ln(exp(x) / y) = x − ln(y). F9 of F16, one node. Key routing for addition and subtraction: lediv(x, deml(y, 1)) = x + y and lediv(x, eml(y, 1)) = x − y. →
Li-Yorke: Period 3 implies chaos (1975). EXL's c-plane has a ~40 % period-3 region (NN / deep-session S10) — Li-Yorke chaos coexists with visibly stable 3-cycles. →

M

Mandelbrot set: The set of c-values for which the iteration zₙ₊₁ = op(zₙ, c) stays bounded starting from z₀ = 0. Monogate has eight F16-operator Mandelbrot sets side by side. →

N

Node: One application of an operator: an F16 operator unless a page names another list. The unit of computational cost in this framework. →

O

Olympiad functional equation: Classical problem class where f is characterised by an identity over (x, y). Their continuous solutions have small F16 trees; whether those are the cheapest is not shown. →
Operator Morph: Cinematic animation (1−t)·op₁ + t·op₂ that interpolates between two F16 Mandelbrot sets. Reveals non-monotonic area dips near t ≈ 0.4. →

P

Period-3 island: Connected region of the c-plane where the iterate has period 3. EXL has two large mirror-symmetric islands centred at c = −1.392 ± 1.993j (deep-session S11). →
Periodic Table of Equations: Visual organisation of the 315-equation catalog by node count (row) and domain (column). The equation-genome K=5 clustering shows domain purity ~random — equations cluster by math, not by field. →
PGC: Positive Growth Criterion. Internal audit rule used during the F16 census. →
Power mean: (aᵖ + bᵖ)^{1/p}. Cost hierarchy: geometric 3n < arithmetic 4n < general p 5n < harmonic 8n. →

R

RMSNorm: LLaMA-style root-mean-square normalisation. 4097n at d = 512 in SuperBEST — 58 % cheaper than LayerNorm's 9728n. →

S

Sharkovskii ordering: Period 3 ⊢ every other period. A period-3 orbit in an interval map forces orbits of every positive integer period (1964). →
Softplus: f(x) = ln(1 + eˣ). The cheapest EML-native smooth activation — 1 node via LEAd(x, 1). Its derivative is sigmoid, which is 5n. →
Sorry: A placeholder in a Lean 4 proof that stands in for an unfinished step. Each sorry is one admitted fact. →
SuperBEST: The routing table mapping common arithmetic operations to their cheapest known constructions, counted in F16 nodes, the sixteen operators on /framework. Only sub, add, mul and div have lower bounds in Lean, over F16. The v5.3 positive-domain total is 15n across 10 ops vs 73n naive — 79.5 % savings; counting ln x = EXL(0, x), a census operator, as one node gives 14n / 80.8 %. Over all reals, the basket of 6 ops (exp, neg, add, sub, mul, div) takes 18n vs 54n naive — 66.7 %. →

T

T01: EML universality: every elementary function is a finite EML tree (Odrzywołek, arXiv:2603.21852). A local Lean formalization covers its definitional form. →
T01a: sin has infinitely many zeros: Part A of the Infinite Zeros Barrier. Partially Lean-verified (analyticity lemmas all at 0 sorries; depth-k zero-count bound waits on o-minimal Mathlib). →
T03: Euler Gateway: ceml(ix, 1) = exp(ix), so sin / cos are 1-complex-node under complex EML. Part of T_EULER_LEAN. →
T15: Zero counts of small EML trees (an observation): among all 862,118 real EML trees over {1, x} with at most 8 internal nodes, the most sign changes found on [−2, 2] was 3. No zero bound has a proof. Not the Weierstrass density claim, whose argument has a gap. →
Trust score: A quantitative summary of how well a project's claims are supported. Monogate's capability card no longer declares one: the manual score was removed on 2026-04-27 because nothing reproduced it. →

W

Well-known URI: RFC 8615 convention — discoverable metadata at /.well-known/<name>. Monogate's CapCard is mirrored at /.well-known/capcard.json. →

Z

Zen Garden: Living complex-plane EML interpreter with audio-reactive mode and domain-coloring overlays. →