Monogate Research: one operator for all elementary functions. Blog posts on SuperBEST routing, ELC characterisation, Lean-verified theorems, and the EML framework.
A chain-order hierarchy theorem and an effective validity threshold — both flagged 'hard' by outside review months ago. We stopped repeating the label and went looking for the actual obstruction in each. One real bridge theorem came out of it. Neither question closed. Both are now precisely located instead of vaguely deferred.
sin was the specific target. It turns out sin was never the point — no finite EML tree can equal ANY nonconstant, continuous, periodic function, full stop. We built genuine Extreme Value Theorem machinery to get there, then found the proof didn't need it: periodicity alone does the work an infimum was supposed to. Honest scope inside.
MachLib runs on axioms. The last post showed Mathlib's ℝ models each one, by hand. This post makes that an always-on invariant: enumerate the axioms from the kernel, decide 'witnessed' by typechecking an interpretation — never by name — and diff both directions so it fails loud. The teeth were real: the audit rejected an axiom that was actually false, an open-interval Rolle a name-matching check…
MachLib's real numbers are an axiomatized interface, kept Mathlib-free for build speed. There's now a machine-checked witness that those axioms are consistent: Mathlib's ℝ models every one of them, each #print axioms bottoming out in Lean's three. The analytic finite-zeros theorem, once postulated, is now proved. Honest scope inside.
We lifted an earlier rolle-only, machine-checked finiteness proof from one hardcoded tower of iterated exponentials to arbitrary exponential-type Pfaffian chains at every depth — with one honest hypothesis we do not round off: positivity.
The finite zero-count bound for iterated exponentials now reaches e^(e^(e^x)), unconditionally and with the Khovanskii-citation axiom removed. Proven from Rolle's theorem alone. Honest scope inside.
A saturating guard keeps a plant's state inside a safe envelope for all time, for any controller, under any bounded disturbance. We proved it in Lean (sorryAx-free), turned the proof into a number — and then made that number a machine-checked theorem too — and measured it holding on a real FPGA, on a noisy breadboard, and on a genuinely nonlinear plant, while we injected an actuator fault on…
One verified math source compiles to software, RTL, and a GPU shader. We stopped trusting the model and ran each artifact for real — C on gcc, RTL in Verilator, a shader on an NVIDIA GPU, and the datapath on an Arty A7 FPGA — checking every output against a Lean-proved error bound. Each one held. Building it found six real bugs. Here is the receipt, and here is exactly where it doesn't hold.
The Infinite-Zeros Barrier — the line between functions you can write as a finite EML tree and ones you can't — turns out to be the compact (rotational) factor of a differential Galois group. We connect the two, turn 'is this function representable?' into a computation from a differential equation, validate the special-function registry against it, and machine-check the core in Lean. Honest scope…
A 63-function corpus through 6 software backends, then 17 multi-function modules, then a Lean proof-emit survey. Four real Forge bugs and two eFrog bugs surfaced and fixed upstream with regression coverage. Hardware-target survey blocked on Pro license. Honest scope inside.
MachLib now has a second, fully constructive proof of the SingleExp Khovanskii zero-count bound, built on a polynomial canonicalizer instead of the ExpPolyBridge embedding. Same theorem, different machinery, same axiom footprint. Honest scope inside.
We shipped a constructive Khovanskii framework on MachLib, then built the CI dashboard the framework deserved. The dashboard caught us over-counting on its first run.
MachLib now ships a finite zero-count bound for polynomial-in-(x, eˣ), proven modulo an axiomatized analytic base. A Forge-emitted Butler-Volmer kernel obligation closes on top of it. Honest scope inside.
A bounded research ledger for where EML helps, where protected standard math wins, and which claims remain blocked.
High-dimensional volume collapse explains why EML tree search hits corners, log-domain cliffs, overflow walls, and phantom-attractor behavior. The Monogate stack now has Forge traces, IR evidence, and MachLib theorem targets for it.
Forge now has a precision-escape packet for a finite phantom-attractor trace: low-precision stalling, higher-precision sensitivity, escape to an interior event, and a MachLib precision obligation.
Forge now has a guard-clamp overflow rescue packet: raw overflow-wall failure, bounded guarded evaluation, guard-rescue transition, and MachLib output-safety obligation.
The Monogate boundary-event rescue suite now has four packet-backed lanes and a unified Forge manifest.
A compact status table for Monogate's boundary rescue operators: Forge evidence, MachLib bridge status, and publication state.
A practical guide to the proof-carrying rescue suite manifest: what the packets mean, what they prove, and what they deliberately do not claim.
Forge now has a saturation-deshelf packet: finite clamp-shelf collapse, pre-clamp pressure replay, boundary-structure recovery, and a MachLib clamp-invariant obligation.
A narrow Forge trace now demonstrates the Monogate stack's first end-to-end boundary rescue shape: raw domain-wall failure, log-domain lift, rescue packet, and MachLib positive-coordinate obligation.
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.
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.
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.
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.
Classical integral transforms partition into three ELC-direction classes. The direction is determined by the kernel.
Four things we retracted, corrected, or demoted during the 2026-04 foundation audit. What survived is stronger for it.
The elementary logarithmic closure is bounded by two structurally independent obstructions. Classical analysis guards one edge; classical algebra guards the other.
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.
Across 315 tested equations, a clean dichotomy: oscillatory functions sit outside ELC with one exception — a non-elementary token.
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.
We measured the node-cost decay across seven basis states on 222 elementary-function equations. One primitive dominates: fused-multiply-add.
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.
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.
How much accuracy you lose by approximating ReLU with the smooth softplus activation — to three decimals, this is a clean closed-form constant.
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.
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.
Every repelling fixed point of exp is an attracting fixed point of log on its branch, with reciprocal multipliers. Machine-verified in Lean 4.
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.
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.
Every binary combination of exp(±x) with ln(y) via arithmetic — completeness classification of all 16 operators.
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.
Partition functions, time evolution, density matrices, and quantum information geometry measured in matrix EML nodes.
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).
# EML Meets Neural Networks
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.
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.
One formula predicts the SuperBEST node cost of any scientific equation. Proved, validated on 187 equations, and open-sourced.
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.