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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.
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 purpose. The breadboard limit-cycled and looked nothing like the simulation; the envelope held anyway, because safety rides on the saturation, not on good control. Here is the receipt, the one place a skeptic would push, and exactly what we do not claim.
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 inside.
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.
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.
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.
A bounded research ledger for where EML helps, where protected standard math wins, and which claims remain blocked.
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.
Forge now has a saturation-deshelf packet: finite clamp-shelf collapse, pre-clamp pressure replay, boundary-structure recovery, and a MachLib clamp-invariant obligation.
The Monogate boundary-event rescue suite now has four packet-backed lanes and a unified Forge manifest.
Forge now has a guard-clamp overflow rescue packet: raw overflow-wall failure, bounded guarded evaluation, guard-rescue transition, and MachLib output-safety obligation.
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.
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.
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.
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.
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.
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.
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.
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.
A practical guide to the proof-carrying rescue suite manifest: what the packets mean, what they prove, and what they deliberately do not claim.
A compact status table for Monogate's boundary rescue operators: Forge evidence, MachLib bridge status, and publication state.
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.
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.
Node-count and wall-clock benchmarks across 8 elementary functions. BEST dispatch vs naive EML.
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.
From 957 claimed theorems to 18 honest ones. What the audit process revealed and why it matters.
deml(x,y) = exp(−x) − ln(y) cannot construct exp(+x) or neg(x). Structural proof + exhaustive search over 861,952 trees.
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.