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.

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.

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, Fourier scope.

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.

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.

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.