Bracketing Cobb–Douglas with rational arithmetic

This tree cannot evaluate K^α. The idea was to sandwich it between the two members of its own family that are rational functions. The sandwich holds for output and fails for the derivatives, and only the derivatives matter.

Status — read this before anything below

Nothing on this page is claimed, certified, enclosed or proved, and no literature gate has run. What it does contain is a measured refutation with a re-runnable falsifier and a passing green control. A prospectus that reads like a result is the defect; so is a kill stated without a witness.

Verdict 2026-07-31 — REFUTED BY COUNTEREXAMPLE, before the page shipped

The construction was proposed, the objection to it was written down, and then the objection was tested rather than left as a requisite. It took about thirty seconds and it killed the mechanism. Falsifier of record: research/_frontier/bracket-falsifier.py, which must print REFUTED and carries a green control.

What was proposed

eqcert/src/interval.js exports only add sub mul div neg sqr abs pow, pow refuses non-integer exponents at line 94, and there is no sqrt. So the models this tree can certify today are rational functions of the state and nothing else — and K^α is not one.

CES is Y = (αK^ρ + (1−α)L^ρ)^1/ρ, and the outer exponent is an integer at exactly two values: ρ = 1 (weighted arithmetic mean) and ρ = −1 (weighted harmonic mean). Both are rational. Cobb–Douglas is the limit ρ → 0, the weighted geometric mean, sitting between them by AM–GM–HM. The idea: certify the two neighbours and bracket the blocked one with nothing but + − × ÷.

Why it fails, with the witness

The power-mean inequality orders OUTPUT. The equilibrium depends on MARGINAL PRODUCTS. Factor prices are r = ∂Y/∂K and w = ∂Y/∂L, and a pointwise ordering of functions does not order their derivatives. The free half of the idea is the half that does not matter.

the counterexample — closed form, no finite differencing

At K = 1, L = 10, α = 0.7:

ρ	form	∂Y/∂K
−1	harmonic — `αK⁻²(α/K+(1−α)/L)⁻²`	1.313567
0	Cobb–Douglas — `αK^α−1L^1−α`	1.396684
+1	arithmetic — `α`	0.700000

The bracket its own family endpoints provide is [0.700000, 1.313567]. The value to be bracketed is 1.396684. It is outside. The map ρ ↦ ∂Y/∂K is not monotone here — it has an interior maximum near ρ = 0, which is exactly the point that needed to be enclosed.

Green control, same point: output does bracket — 1.369863 ≤ 1.995262 ≤ 3.700000. AM–GM–HM holds, as it must. The control passing is what makes the refutation a measurement rather than a broken script.

A sweep of eight (K, L, α) triples found 6 of 8 monotone in ρ and 2 not. Monotonicity is not a rare failure at the edge of the parameter space; it is simply false, and one witness is enough.

Is there a salvage? Yes, and it is smaller and duller

The specific mechanism is dead: there are only two rational members of the CES family, so there is no third point to bracket with. But the general move survives and is completely standard — bound x^α directly by rational functions with a certified remainder (best rational or Padé approximation with an outward-rounded error term), which needs no economic family at all.

That is worth saying plainly: the salvage is just a lightweight interval-transcendentals. It does not avoid the blocker, it implements a narrow slice of it. The attraction of the bracket was that it looked like a way around the missing library, and that is precisely the part that turned out to be false.

What the surrounding pairing still is

Independently of the bracket, MFG plus a production function is continuous-time heterogeneous-agent macroeconomics — Achdou, Han, Lasry, Lions and Moll, Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach (Review of Economic Studies, 2022), and beneath it Aiyagari, Huggett, Krusell–Smith. Lions is a co-author, so N is zero and there is no version of “new mathematics”. That verdict is occupancy, not structural, so under SCORING.json the pairing itself is not killed by this page — only the bracket is.

what remains attractive about the pairing, unverified

The coupling runs through the scalar aggregate K = ∫a dμ rather than a local density, so the equilibrium is a one-dimensional fixed point — small, and structurally like edge-of-chaos. Constant returns give Euler exhaustion (Y = K∂_KY + L∂_LY) as an exact identity, the same species of free invariant as MWD's Kirchhoff and MGG's pathwise relation. And these models drive real policy analysis, while everyone solves them by finite differences with no enclosure. None of that is gated.

and the obstructions the bracket was never going to fix

The borrowing constraint makes the household HJB a free-boundary variational inequality; the wealth state space is unbounded, so the ℓ¹_ν tail machinery does not transfer; the price coupling is not Lasry–Lions monotone, so that toolkit does not carry over; and Inada (∂Y/∂K → ∞ as K → 0) makes any interval bound on r degrade without bound near zero capital — a textbook representation refusal, and a clean test case for the discriminator refusal-taxonomy needs.

One thing that did not get tested, and would have bitten

The harmonic endpoint is built from negative integer exponents, and the header of interval.js records that pow had no test coverage and was wrong — returning [1,1] for every negative exponent, claiming to enclose 0.5 for pow([2,2],−1) and not doing so. Found 2026-07-28 with zero call sites: “a loaded trap, not a live wound.” Had the bracket survived, this path would have been the first live consumer of that function. Recorded because the next idea that reaches for a negative exponent inherits the same obligation.

Scored

N / A

Not reached. The mechanism does not work, so there is nothing to score. Under SCORING.json this is a STRUCTURAL kill of the bracket — it says the thing cannot work, not that someone got there first — and structural kills survive a rescoring.

Citation state: the macro references are from memory and none verified at source. The interval.js operation surface, the pow defect and the counterexample above WERE measured directly on 2026-07-31 and are reproducible by python3 research/_frontier/bracket-falsifier.py.