https://monogate.org Monogate Research: one operator for all elementary functions. Blog posts on SuperBEST routing, ELC characterisation, Lean-verified theorems, and the EML framework. en-us Fri, 01 May 2026 20:47:58 GMT https://monogate.org/blog/one-operator https://monogate.org/blog/one-operator Mon, 27 Apr 2026 00:00:00 GMT The NAND gate of continuous math. A single binary operation eml(x, y) = exp(x) − ln(y) generates every elementary function — and the structural fingerprint of an expression turns out to predict where it came from. https://monogate.org/blog/hear-the-math https://monogate.org/blog/hear-the-math Mon, 27 Apr 2026 00:00:00 GMT The best-selling synthesizer in history runs on a Bessel function. The Gibbs phenomenon's 9% overshoot is a theorem you can hear. Three interactive demos at 1op.io let you turn structural complexity into sound. https://monogate.org/blog/dynamics-counter https://monogate.org/blog/dynamics-counter Mon, 27 Apr 2026 00:00:00 GMT Hand a damped-oscillator equation to a computer and it can tell you, without knowing any physics, that there's one oscillation and one decay inside it. Across 193 expressions and 12 domains, this counter holds at ρ = +0.885. https://monogate.org/blog/built-with-claude https://monogate.org/blog/built-with-claude Mon, 27 Apr 2026 00:00:00 GMT 578 expressions, 50 Lean theorems, 5 PyPI packages, an npm port, a HuggingFace dataset, three websites, four interactive demos. Two weeks. One human. Here's what actually worked, what failed, and what the audit system caught before it reached the public. https://monogate.org/blog/which-way-does-the-transform-go https://monogate.org/blog/which-way-does-the-transform-go Thu, 23 Apr 2026 00:00:00 GMT Classical integral transforms partition into three ELC-direction classes. The direction is determined by the kernel. https://monogate.org/blog/what-we-got-wrong https://monogate.org/blog/what-we-got-wrong Thu, 23 Apr 2026 00:00:00 GMT Four things we retracted, corrected, or demoted during the 2026-04 foundation audit. What survived is stronger for it. https://monogate.org/blog/two-boundaries https://monogate.org/blog/two-boundaries Thu, 23 Apr 2026 00:00:00 GMT The elementary logarithmic closure is bounded by two structurally independent obstructions. Classical analysis guards one edge; classical algebra guards the other. https://monogate.org/blog/planck-elc-native https://monogate.org/blog/planck-elc-native Thu, 23 Apr 2026 00:00:00 GMT Six canonical electromagnetic formulas costed in F16 nodes. Planck's radiation law sits entirely inside the exp-log closure — unlike wave equations, which must cross to complex EML for cos. A double-angle identity inflates cost. https://monogate.org/blog/oscillation-boundary https://monogate.org/blog/oscillation-boundary Thu, 23 Apr 2026 00:00:00 GMT Across 315 tested equations, a clean dichotomy: oscillatory functions sit outside ELC with one exception — a non-elementary token. https://monogate.org/blog/olympiad-meets-eml https://monogate.org/blog/olympiad-meets-eml Thu, 23 Apr 2026 00:00:00 GMT Classical functional equations characterise exp and ln, and their solutions turn out to be minimal EML trees — often cheaper than the equations that define them. https://monogate.org/blog/fma-staircase https://monogate.org/blog/fma-staircase Thu, 23 Apr 2026 00:00:00 GMT We measured the node-cost decay across seven basis states on 222 elementary-function equations. One primitive dominates: fused-multiply-add. https://monogate.org/blog/conjugacy-explained https://monogate.org/blog/conjugacy-explained Thu, 23 Apr 2026 00:00:00 GMT The EAL self-map and the EXL self-map have completely different fixed points, yet both have derivative exactly 4.3164206… at those points. The answer is a one-line topological conjugacy via exp. https://monogate.org/blog/chaos-multiplicative-operators https://monogate.org/blog/chaos-multiplicative-operators Thu, 23 Apr 2026 00:00:00 GMT A 600-point parameter sweep across all 16 F16 operators shows that 12 of them collapse to period-2 dynamics, while the four multiplicative operators (EXL, DEXL, EXN, DEXN) exhibit long cycles, chaos, and a period-3 Sharkovskii signature. https://monogate.org/blog/relu-softplus-exact-error https://monogate.org/blog/relu-softplus-exact-error Wed, 22 Apr 2026 00:00:00 GMT How much accuracy you lose by approximating ReLU with the smooth softplus activation — to three decimals, this is a clean closed-form constant. https://monogate.org/blog/lambert-log-branch-attractors https://monogate.org/blog/lambert-log-branch-attractors Wed, 22 Apr 2026 00:00:00 GMT Iterate principal log on any seed in ℂ and you land at 0.318 + 1.337i. Use the k-th branch and you land somewhere else — at z_k* = −W_k(−1). Infinitely many complex attractors, one per integer, all provably repelling under exp. https://monogate.org/blog/hyperbolic-preserves-elc https://monogate.org/blog/hyperbolic-preserves-elc Wed, 22 Apr 2026 00:00:00 GMT sinh, cosh, and tanh map ELC inputs to ELC outputs. sin, cos, and tan don't. Machine-verified in Lean 4. With a 3-4-5 triple bonus. https://monogate.org/blog/exp-log-duality-at-fixed-points https://monogate.org/blog/exp-log-duality-at-fixed-points Wed, 22 Apr 2026 00:00:00 GMT Every repelling fixed point of exp is an attracting fixed point of log on its branch, with reciprocal multipliers. Machine-verified in Lean 4. https://monogate.org/blog/tan1-obstruction https://monogate.org/blog/tan1-obstruction Mon, 20 Apr 2026 00:00:00 GMT A single transcendence fact about tan(1) is the root cause behind three separate EML results: the multiplication lower bound, the depth-3 ceiling for standard functions, and the complex density behavior. https://monogate.org/blog/superbest-complete https://monogate.org/blog/superbest-complete Mon, 20 Apr 2026 00:00:00 GMT Every arithmetic operation now has a proved-optimal or exhaustively-bounded node count in the exp-ln operator family. Total: 21 nodes, 71.2% savings vs naive. https://monogate.org/blog/sixteen-operators https://monogate.org/blog/sixteen-operators Mon, 20 Apr 2026 00:00:00 GMT Every binary combination of exp(±x) with ln(y) via arithmetic — completeness classification of all 16 operators. https://monogate.org/blog/recip-one-node https://monogate.org/blog/recip-one-node Mon, 20 Apr 2026 00:00:00 GMT ELSb(0, x) = exp(0 − ln(x)) = 1/x. One node. SuperBEST v4: 18 nodes total, 75.3% savings. The reciprocal was never a division problem. https://monogate.org/blog/quantum-costs https://monogate.org/blog/quantum-costs Mon, 20 Apr 2026 00:00:00 GMT Partition functions, time evolution, density matrices, and quantum information geometry measured in matrix EML nodes. https://monogate.org/blog/geometry-costs https://monogate.org/blog/geometry-costs Mon, 20 Apr 2026 00:00:00 GMT 12 classical geometric primitives — hyperbolic distance, Lie group maps, curvature, conformal maps — expressed as EML operator trees. Total: 125n SuperBEST vs 345n naive, 64% savings. All exact. Updated for R16-C1 (recip = 1n). https://monogate.org/blog/eml-neural-networks https://monogate.org/blog/eml-neural-networks Mon, 20 Apr 2026 00:00:00 GMT # EML Meets Neural Networks https://monogate.org/blog/eml-negation https://monogate.org/blog/eml-negation Mon, 20 Apr 2026 00:00:00 GMT The neg gap is closed: exl(0, deml(x,1)) computes −x in exactly 2 nodes for all x ∈ ℝ, with no domain restriction. The SuperBEST table is complete. https://monogate.org/blog/depth-spectrum https://monogate.org/blog/depth-spectrum Mon, 20 Apr 2026 00:00:00 GMT Every function has a minimum node count. We now know the complete depth spectrum: 1, 2, 3, ∞ — and why depth-4 exists but contains no standard functions. Plus: multiplication drops to 2 nodes. https://monogate.org/blog/cost-theory-complete https://monogate.org/blog/cost-theory-complete Mon, 20 Apr 2026 00:00:00 GMT One formula predicts the SuperBEST node cost of any scientific equation. Proved, validated on 187 equations, and open-sourced. https://monogate.org/blog/cost-theory https://monogate.org/blog/cost-theory Mon, 20 Apr 2026 00:00:00 GMT Four structural classes, the cost decomposition theorem (T38), complexity classes O(1)/O(N)/O(N²), and the Linear Ceiling Conjecture (T39): a complete theory of how many EML nodes any standard scientific equation requires. https://monogate.org/blog/cost-of-everything https://monogate.org/blog/cost-of-everything Mon, 20 Apr 2026 00:00:00 GMT From Google's PageRank to your GPS to the NFL passer rating — every equation has a node count. Here are the ones that matter. https://monogate.org/blog/conditional-tan1 https://monogate.org/blog/conditional-tan1 Mon, 20 Apr 2026 00:00:00 GMT A thought experiment: if tan(1) could be built from EML trees, what would follow? The conditional chain connects to Schanuel's conjecture and would collapse the depth hierarchy. https://monogate.org/blog/completeness-characterization https://monogate.org/blog/completeness-characterization Mon, 20 Apr 2026 00:00:00 GMT 16 operators, one structural rule: exp(+x) with no domain restriction implies exactly complete. exp(-x) implies incomplete. -exp(x) implies approximately complete. The Exponential Position Theorem explains all 16 classifications at once. https://monogate.org/blog/chembio-costs https://monogate.org/blog/chembio-costs Mon, 20 Apr 2026 00:00:00 GMT How many operator nodes does it take to compute 40 standard equations from chemistry and biology? A systematic analysis using the SuperBEST v3 routing table. https://monogate.org/blog/calculus-costs https://monogate.org/blog/calculus-costs Mon, 20 Apr 2026 00:00:00 GMT Taylor series, integration, ODEs, autodiff, and Fourier transforms measured in EML nodes. https://monogate.org/blog/add-gen-2n https://monogate.org/blog/add-gen-2n Mon, 20 Apr 2026 00:00:00 GMT The only operation in SuperBEST costing more than 3 nodes was general-domain addition at 11n. It now costs 2 nodes. The table is complete. https://monogate.org/blog/157-equations https://monogate.org/blog/157-equations Mon, 20 Apr 2026 00:00:00 GMT A complete catalog of SuperBEST node counts for standard equations across 12+ domains — from 1-node trivialities to 2037-node error correction. Expanded from 157 (Monster Sprint) to 214 (COMP-ALL) to 295+ (domain-2 sessions: FIN, INFO, QM, THERMO, CHEM, BIO, ECON). https://monogate.org/blog/tight-zeros-bound https://monogate.org/blog/tight-zeros-bound Sun, 19 Apr 2026 00:00:00 GMT We proved that a depth-k EML tree has at most 2k+2 real zeros, and verified computationally that the true bound may be as low as 2 for all k ≥ 3. This strengthens the Infinite Zeros Barrier from qualitative to quantitative. https://monogate.org/blog/operator-zoo https://monogate.org/blog/operator-zoo Sun, 19 Apr 2026 00:00:00 GMT We applied the DEML incompleteness template to seven exp-ln operators. Six are incomplete. One is open. One surprise: a gate with the identity function built in. https://monogate.org/blog/mul-gap-closed https://monogate.org/blog/mul-gap-closed Sun, 19 Apr 2026 00:00:00 GMT The BEST router's mul entry drops to 3 nodes via exl(ln(x), exp(y)) = x·y. The lower bound is 3n, confirmed tight by exhaustive search. Gap fully closed. https://monogate.org/blog/fourier-beats-taylor https://monogate.org/blog/fourier-beats-taylor Sun, 19 Apr 2026 00:00:00 GMT sin(x) costs 101 nodes as a Taylor series in BEST routing. The same function is 1 complex EML node using Fourier. This 100x gap validates the lab's sound design and reveals a deep structural fact about the operator. https://monogate.org/blog/eml-sound https://monogate.org/blog/eml-sound Sun, 19 Apr 2026 00:00:00 GMT Each Fourier harmonic is one complex EML node. We measured timbre complexity for 5 instruments and found: Sine=1n, Clarinet=5n, Violin=12n. EXL is the most musically useful operator. https://monogate.org/blog/eml-no-fixed-points https://monogate.org/blog/eml-no-fixed-points Sun, 19 Apr 2026 00:00:00 GMT f(x) = exp(x) − ln(x) satisfies f(x) > x for all real x > 0. The gap is minimized at x ≈ 1.31 where f(x) − x ≥ 1.648. This is a theorem about the operator's self-interaction — and it separates EML from every other operator in the family. https://monogate.org/blog/eml-fractals https://monogate.org/blog/eml-fractals Sun, 19 Apr 2026 00:00:00 GMT Iterating exp(z)−k is Devaney's exponential family. We computed 8 operator fractal zoos, measured box-counting dimensions, and found DEML/EMN generate bounded strange attractors. https://monogate.org/blog/eml-closure-density https://monogate.org/blog/eml-closure-density Sun, 19 Apr 2026 00:00:00 GMT We enumerated EML constant trees to depth 7, tested 20 random complex targets, and tracked how close EML gets to π. Strong evidence for a density conjecture — but not yet a theorem. https://monogate.org/blog/completeness-trichotomy https://monogate.org/blog/completeness-trichotomy Sun, 19 Apr 2026 00:00:00 GMT Three completeness classes for exp-ln operators: exactly complete (EML), approximately complete (EMN), and incomplete (all others). Two new theorems prove EMN's exact limits and approximate power.