Interval transcendentals

eqcert/src/interval.js:94 throws on non-integer exponents, and the library has no transcendental functions at all. Four separate things in this tree stop there. Written down because the convergence is invisible when each is written up on its own.

Status — read this before anything below

Advance 2026-08-01: sound interval exp, log, sin, cos, tanh ship in eqcert/src/transcendental.js (battery eqcert/tests/test-transcendental.js, in make check-eqcert). Consumers: softmax H/PR + continuous-β on attention geometry (exported); tanh/sin/cos unblocked for edge-of-chaos / aerospace / Path 5 arithmetic (instances still to build). Still open: fractional powers, erf, full aerospace report.

Four things, one wall

Aerospace — blocked, and the standing instruction is do not send a report until the Rust kernel exists, because an uncertified report under a certification brand is the one liability that cannot be repaired. Path 5, certified price formation — the same wall sits on the critical route to a December deadline. Path 6, the edge of chaos — needs exp for Gaussian quadrature and tanh/erf for the activation. Any MFG instance with a non-polynomial Hamiltonian or an exponential coupling — which is most of them outside this tree's current corpus.

Why this is a frontier page and not a ticket

Because the honest accounting changes the priority. Each of the four reads as “blocked on engineering” in isolation, which sounds like a reason to work on something else. Together they say the opposite: one bounded piece of arithmetic unlocks a commercial track with a clock, a research bet that has been waiting for an instance, and an entire class of MFG problems currently out of reach.

It is also the only item in the set whose scope is genuinely known. Rigorous interval exp, log, sin/cos, tanh and fractional powers with correct outward rounding are solved problems with published algorithms — MPFI, Arb, CAPD and Boost.Interval all ship them. This is implementation against a known specification, not research.

What must be true of it, and this is where the care goes

Outward rounding must be provable, not asserted. The five that do not bend include outward-rounded arithmetic where a bound is claimed and every check can go red. A transcendental that is merely accurate is worse than none: it makes every downstream certificate look sound while resting on an unproved bound — the exact species as the radiiPolynomial self-map/contraction gap, one layer lower and under everything.

the falsifier

Directed-rounding test vectors where the true value is known to lie outside a naively-rounded interval, plus mutation: an implementation with the rounding mode flipped must turn a passing certificate red. If a wrong rounding mode does not break a downstream certificate, the certificate was not using the bound.

The honest risk

Scope creep into a general-purpose interval library. The temptation is to build what Arb built. The defence is that this tree needs a named, closed list of functions, each with test vectors and a mutation proof, and nothing else — and that the list should be derived from the four consumers above rather than from completeness. A library nobody's certificate depends on is a tax charged in attention, which is the standard this tree already applies to gates.