# Initial algorithm feasibility calculations

These calculations illustrate source parameters; they are not observed software benchmarks. Loose worst-case bounds can overestimate practical costs. Favorable asymptotic exponents can also hide construction, memory, precision and startup costs.

## Subset sum exponent saving

Comparing only `2^(0.5n)` with `2^(0.49n)` gives an ideal ratio of `2^(0.01n)`: about 1.32 at 40 items, 2 at 100 items and 1,024 at 1,000 items. Both searches remain exponential. At 1,000 items the improved expression is still `2^490`; the ratio alone cannot establish commercial tractability. Different constant factors and word-operation costs are excluded from this illustration. [Subset Sum manuscript](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Subset-Sum-in-Time-2-power-0-49n-October-4-2026/subset-sum.pdf).

## Matroid allocation constant

The promised reward fraction `2^-310` is approximately `4.794 × 10^-94`. A positive absolute constant resolves a theoretical question while giving no useful minimum revenue assurance at that scale. Practical rules could perform better, but that requires implementation and evaluation beyond this bound. [Matroid scope statement](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/111.md).

## Integer and Fourier logarithmic savings

The integer-multiplication exponent saving `2^-182` is approximately `1.631 × 10^-55`. The Fourier saving is `10^-13`, in an exact-complex arithmetic model. Comparing the Fourier complexity expressions at `n = 2^30`, while excluding hidden constants, gives a fractional ratio improvement of only about `3.03 × 10^-13` when natural logarithms are used. This illustrates the tiny exponent change, not a performance prediction. A different fixed logarithm convention and constants further prevent treating it as a benchmark. [Integer multiplication manuscript](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Integer-multiplication-below-n-log-n-September-23-2026/paper.pdf), [Fourier scope](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/130.md).

## Graph switching and scheduling bounds

The graph-switch mixing upper bound `2n^8` is `2 × 10^16` steps at 100 vertices. This is a worst-case guarantee, not an estimate of the chain's actual convergence on a particular graph. It motivates measuring an empirical regime while retaining honest certification limits. [Graph-switch scope](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/131.md).

The scheduling expression `(L+2)^150020` has base-ten logarithm about 161,899 even at illustrative `L = 10`, before the unspecified multiplicative constant. The theorem supplies no practical performance claim. This cannot be used to prove that all structural scheduling implementations are slow; it shows that this published upper bound gives no reasonable production budget. [Three-machine manuscript](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/paper.pdf).

## Initial implementation priorities

The research infrastructure candidates can provide value without converting the most difficult theorem constructions into practical algorithms. Conditional algorithm products should first demonstrate a realistic crossover, an input regime matching their assumptions, and a benefit in the buyer's actual workflow. No quoted prospective pricing or idealized savings establish profitability.