GAn isomorphism is not an algorithm

Non-Newtonian arithmetics are built by transport: x (+) y = alpha^-1(alpha(x) + alpha(y)). When alpha is a bijection onto all of R, as the cube chart is, the result is the same field as ordinary arithmetic, with the same theorems. The exp chart is not of that kind: exp maps onto the positive reals only, so log(e^x + e^y) is positive-real addition carried back to R, and it has no finite additive identity (x (+) z = x would need e^z = 0). Either way the transport is exact, which makes it tempting to say the change of arithmetic is free. As mathematics it is; as computation it is not.

Compute x (+) x through the exp chart in float64 and the answer overflows at x = 709.0896, although the exact result is a perfectly ordinary number. Below about -745 it underflows instead.

generator alpha (x (+) y = alpha^-1(alpha(x) + alpha(y)))

reformulation

exact (native x + x is always here)wallwallsubnormal-700-35003507001e-161e-121e-81e-41x (computing x (+) x)relative error vs exact
Red dots: the chart computation. Red ticks at the top: results lost (overflow to Infinity or underflow to 0) although the exact value is representable. Positive-real addition carried to R (no finite additive identity, so not a field on R); the chart arithmetic is exact as mathematics but loses range and precision that native arithmetic keeps.
exact value
709.8931472
chart computation
Infinity (rel. err Infinity)
reformulated
709.8931472 (rel. err 0)
top wall, closed form ln MAX - ln 2
709.08957
top wall, measured here by bisection
709.08957
bottom wall, measured here
-745.13322
lost of 400 sweep points
18 chart, 0 reformulated

The precise statement

In type-2 computability with the Cauchy representation, for alpha a bijection of R: if alpha and alpha^-1 are computable, the transported operations are computable, the transported constants are computable reals, and decidable sets stay decidable. Parameter-free first-order truths carry over by the isomorphism alone.

Complexity needs more: polynomial-time alpha and alpha^-1 with a polynomial modulus of continuity on the domain in use. exp on an unbounded domain lacks that, which in float64 shows up as the walls above.

Where it breaks

The algebra does not imply computability. alpha(x) = x + c with c a non-computable real is smooth and isometric, yet 0 (+) 0 = c, so the transported addition cannot be computed on ordinary names; a symbolic representation that stores alpha(x) still works.

The walls are a property of the algorithm, not of the arithmetic: m + log1p(exp(-|x - y|)) gives 709.7931471806 at x = 709.1 where the naive chart gives Infinity. In the subnormal band precision decays first: at x = -740 the naive relative error is 3.49e-6.

Prior art and sources

Classical results it reduces to

  • Pour-El and Richards 1989
  • Weihrauch 2000
  • Ko 1991 (complexity of real functions)
  • Tarski (decidability of real-closed fields)

NNC anchors

  • Simple Fractal Calculus From Fractal Arithmetic (aerts_czachor_kuna_2018_simple_fractal_calculus)
  • Non-Newtonian Calculus (grossman_1972_nnc)

OpenAI families named (context only)

  • Hilbert's tenth problem over Q (family 004)
  • Integer multiplication below n log n (family 109)

Model-written and unreviewed. No result on this page depends on one.

Write-up and checks: docs/research/nnc-openai-crossanalysis-2026-10/X3/FINDINGS.md. Every number on this page is computed in your browser by lib/crossovers/, tested against independent closed forms in lib/crossovers.test.ts.