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 depth map of elementary functions. Depth 0 (arithmetic), depth 1 (exp, softplus), depth 2 (log, negation), depth 3 (oscillatory via complex bypass), depth ∞ (non-constructible over ℝ). →
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 — each a minimal EML tree. →
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 F16 operator that can express every elementary function. Eight of the sixteen are complete: EML, EDL, EXL, EAL, EPL, LEAd, ELAd, ELSb. →
Completeness theorem (T26–T28): An F16 operator is complete iff it contains exp(+x) with no domain restriction on self-composition. One structural rule explains all sixteen cases. →
Cosh preserves ELC: cosh/sinh/tanh are arithmetic combinations of exp(±x); the hyperbolic functions stay inside ELC(ℝ). sin/cos do not. Lean-verified in HyperbolicPreservation.lean. →

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. 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 at k = 1 (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). Complete; one-node representation of exp(x) + 1 via EAL(x, e). →
EDL: edl(x, y) = exp(x) / ln(y). Complete; 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. →
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). Approximately complete — generates exp(−x) but not exp(+x) as a finite real tree. →
EPL / ELMl: Power primitive. ELMl(k, x) = exp(k · ln x) = xᵏ is a 1-node F16 operator for x > 0 (UpperBounds.lean). →
EXL: exl(x, y) = exp(x) · ln(y). Complete; source of the period-3 Sharkovskii regime in the EML family. →

F

F16: The sixteen binary operators built from exp(±x), ln(±y), and the four combiners {+, −, ·, /}. Monogate's base operator family. →
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 (T01): No finite real EML tree equals sin over all of ℝ. Any finite EML tree is real-analytic with isolated zeros; sin has zeros at every integer multiple of π. →
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). 1-node F16 operator. →
Lean 4: Proof assistant developed by de Moura. Monogate's 14 proof files compile under Lean 4 + Mathlib with zero sorries in 13 files and 2 sorries in one partial file (InfiniteZerosBarrier.lean Part D). →
LEdiv: lediv(x, y) = ln(exp(x) / y) = x − ln(y). 1-node F16 operator. Key routing for subtraction. →
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 F16 operator. 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 are minimal EML trees. →
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 optimal routing table mapping common arithmetic operations to the minimal number of F16 nodes. v5.3 positive-domain total is 14n across 10 ops vs 73n naive — 80.8 % savings. →

T

T01: The Infinite Zeros Barrier theorem. Partially Lean-verified (analyticity lemmas all at 0 sorries; depth-k zero-count bound waits on o-minimal Mathlib). →
T02: EML universality — every elementary function equals a finite EML tree over algebraic constants with the Euler Gateway. →
T03: Euler Gateway: ceml(ix, 1) = exp(ix), so sin / cos are 1-complex-node under complex EML. →
T15: Weierstrass universality for EML — any continuous function on a compact interval is uniformly approximable by finite EML trees. →
Trust score: A quantitative summary of how well a project's claims are supported. Monogate's v3 card declares 0.97 (11 Lean-verified files, 4 reproducible benchmarks, 315-equation catalog with zero prediction error). →

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. →