# Research algorithm feasibility and semantic workbench Edition 9 October 2026. Current decision: **Prototype now**. Prices and product uses remain hypotheses. No independent source-proof verification or buyer validation has been completed. ## Research finding Existing dossiers document extreme finite constants behind headline complexity improvements. New reviews add delayed k-server activation at 2^T_P-1 and the exact logspace derandomization compiler's explicit resource-bound construction. A language-class equality is not automatically a usable compiler or speed improvement. 41 declared definitions across three computational interfaces have now received selected source-level semantic comparison. ## Problem and buyer Research commercialization teams need to distinguish a novel theorem, a computable construction, a practical implementation and a useful customer workflow. A formal result can match its intended model while still leaving a costly operational bridge. ## What the finding could enable The workbench can produce source-pinned implementation gates with exact quantifiers, numerical parameter derivations, semantic definition records and finite controls. Use it as part of the shared evidence/model-assurance product to support an investment decision. ## Technical and commercial limits No intended-model mismatch was detected in the selected 41 definition comparisons; this is not trusted kernel or whole-proof verification. LogspaceEquality selects equality of binary language classes, not the paper compiler or numerical resource bounds. The compiler assumes supplied all-input/all-coin polynomial-time and logarithmic-workspace promises; it does not discover them. It preserves decisions under an error gap, not a random output distribution. ## Minimal architecture Pinned source and declaration references -> manual semantic observations -> source-vs-application contract -> exact parameter worksheet -> bounded reference/benchmark evidence -> reviewed build/defer decision. Thirteen curated documentary contracts exist. A complete manifest does not authenticate the supplied evidence or prove applicability. ## Existing alternatives and differentiation Internal technical review, expert consulting and existing benchmark tools are substitutes. Value requires a material missed assumption, changed engineering decision or reduced preparation time. Lexical equality of displayed definitions is only a useful control, not an automated semantic proof. ## Monetization hypothesis Hypothesis: AUD 3,000-12,000 per assessment and a subscription only after reuse is demonstrated. An illustrative AUD 8,000 project at 40 assumed hours and AUD 180/hour leaves AUD 800 before overhead; 45 hours exceed the fee. Overlapping evidence, semantic and parameter modules do not represent separate paying accounts. ## Validation experiment The compiler uses P=1000(d_code+2)^4(a+b+2), rho=100(P+2), S=1+(t+1)(2rho+1)(g+2), K=4S, c=2S+1, H=2^((u+100)^2)*(q_D+1). Even the conservative minimal-parameter envelope gives c>=2,592,016,043 and H>=2^10202 for the declared upper bounds. These are not runtime lower bounds. No compiler/library transition list was executed. Have a qualified reviewer compare ten operational claims selected from thirteen contracts and measure material gaps, false alarms and preparation time. ## Conditions to reject or defer Defer compiler speedup claims until a concrete implementation and useful resource measurements exist. Reject a subscription if the workflow remains bespoke or reports cannot improve an engineering decision. Never turn a loose upper bound into a claim of algorithmic impossibility. ## Next concrete action Review the remaining seven semantic interfaces/24 declared entries, enrich exact source locations for older contracts, and prepare the ten-claim expert comparison. Keep finite reference validation separate from the unrestricted source algorithms. ## Source evidence Repository sources are pinned to revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. Selected scope notes and manuscript statements have been reviewed to the extent described above; these links do not represent successful kernel checks. ### Family 109 Subject: Integer multiplication below $`n\log n`$. - [Integer multiplication below n log n](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Integer-multiplication-below-n-log-n-September-23-2026/paper.pdf) ### Family 111 Subject: One-sample matroid prophet inequalities against an almighty adversary. - [One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/One-Sample-Suffices-for-Matroid-Prophet-Inequalities-against-an-Almighty-Adversary-September-23-2026/final.pdf) - [Formal scope notes](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/111.md) ### Family 124 Subject: Polynomial-time scheduling on three identical machines. - [A Polynomial-Time Algorithm for Three-Machine Unit-Job Scheduling](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/paper.pdf) - [Formal scope notes](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/124.md) ### Family 130 Subject: Exact Fourier transforms below $`n\log n`$. - [An explicit power saving for the exact discrete Fourier transform](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-explicit-power-saving-for-the-exact-discrete-Fourier-transform-September-25-2026/main.pdf) - [Finite tensor savings and exact Fourier circuits](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026/main.pdf) - [Formal scope notes](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/130.md) ### Family 139 Subject: Subpolynomial query complexity for log-concave sampling. - [Subpolynomial query complexity for well-conditioned log-concave sampling](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026/article.pdf) - [Formal scope notes](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/139.md) ### Family 107 Subject: Matrix multiplication with exponent at most 9/4. - [An Upper Bound of 9/4 for the Matrix Multiplication Exponent](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/paper.pdf) - [Complex Matrix Multiplication Below 2.258 and Rectangular Bounds](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026.pdf) - [Staggered extraction for exact matrix multiplication over every field](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026.pdf) - [Formal scope notes](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/107.md) ## New local evidence [Computational feasibility](../../FEASIBILITY-2026-10-09-v5.md), [semantic comparison](../../plans/definition-semantics-review-2026-10-09-v2.md), [eighteen utilities](../../PROTOTYPES-2026-10-09-v6.md). All sources remain pinned to the cited revision. Additional pinned sources: [logspace scope](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/103.md), [k-server scope](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/110.md), [transport scope](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/374.md).