Conditional κ = 8.03512980550910e-4: paid-aware integer donor-basis exchange on #354 (+0.0468%)

[Dugongue]

Result

Conditional κ = 80351298055091 / 10^17 = 8.03512980550910e-4, +0.046793425% over #354. Add research/e8-basis-exchange/.

Frozen baseline: #354 at 3c59b8c8fc8521e8ba4ed87c6c4368457b13c485. Its staged Design T, twins, target squares, 240 early restorations and ten terminal sinks are retained, together with #352's unchanged E8 complex bound. Later PRs are outside this frozen submission. No current-record claim.

New mechanism

The fresh paid-aware donor-basis exchange varies which actual cold helper responses form a basis instead of accepting a fixed Gaussian basis. Thirty-two deterministic orderings, including prefix-READ-cost orderings and seeded cross-dimension orders, nominate signed integer relations. Each relation is checked exactly over all 960 target coordinates; the weighted closure charges each distinct donor once. The productive exchange raises the newly rebased kernel from 173 to 208 saved rank units.

For helper r, T_r is its complete signed target response and Q_r is the prefix-READ contribution transported through the actual suffix target gates. At the cold cut, F_r = T_r - Q_r. The sufficient relation is

Q_p = sum_d alpha[p,d] F_d.

Remove the pivot's prefix READs, insert the donor shears at a common admissible cap and restore at the full endpoint. This cancels the deleted response over the integers. Prime-field elimination is nomination only, never certification. The selected donors have zero complete response; no successful warm-donor extension is claimed.

The final selection has 208 pivots, 315 donors, 48 disjoint groups, 746 terms and 74 even-coefficient terms. It removes 1536 READs and emits 820 charged unit setup ADDs plus 820 restoration ADDs. Repeated unit gates implement integer coefficients without another frame change. Kernel histogram delta:

H1:+173, H2:+357, H3:-357, H4:+24, H5:-24.

That saves 208 weighted rank units. One separately rebound, historically source-corrected cache saves another 15 (H2:+1, H17:-1). The complete word is re-emitted and priced, never by adding κ gains.

Gaussian elimination, basis exchange, weighted closure, integer lifting and kernel/cache identities are established tools. The own contribution is the productive paid selection, exact new witness and its fully checked physical composition. No global first-invention claim.

Exact accounting

Quantity	Value
Local rank mass	169120 → 168897
Bit coarse saving	12564986457081 / 15625000000000000
Unchanged E8 complex coarse saving	876248285600677 / 10^18
Helpers / physical replicas	7610 / 300
Banks per stage	396171
Literal / normalized stock	3132855 / 626571
Local unit ADDs	358298
Finite selector charge	1806119562000
Primitive coefficient / payload prefix bits	57 / 94

Final literal word SHA-256:

3b9488869388a9f78abca616b5d25f25253a26f030b12106b13e8698fbbcc9dc

Validation and reproduction

python -m pip install -r research/e8-basis-exchange/requirements.txt
python -B research/e8-basis-exchange/verify.py /tmp/e8-basis-exchange-review

Requires Python 3.11+, GNU-compatible C++17 and Boost; the JSON header and required pricing modules are bundled. The output path must be new and outside the package. Optional --cxx and --boost-include flags select dependencies. Assertions must stay enabled.

The verifier pins all inputs and reconstructs the literal kernel/cache. It checks 9148800 signed input-target entries, all clean-source operand spans, 11450 five-stage F2 formal columns, nested frames/COPY, both reflected ledgers, new prime minors, exact changed projector charts and actual zero-padding-free 300-replica bank allocation. Eight category-omission controls fail as required.

Deleting an adjacent pair for one productive even coefficient leaves the binary operator unchanged but changes 28 integer target entries by two. Independent nomination checks 199680 signed entries and all 1492 term-omission/sign-flip controls. Two exact rational moment engines include the full 10^-16 fallback. All 47 strict assembly inequalities pass, the adjacent κ tick is rejected, and the complete finite invoice passes.

This runs our changed construction's checks. It does not rerun unchanged public mathematical supplier verifiers.

Research archive and scope

The folder includes sufficient proofs, the native deterministic basis-exchange/closure sources, selected witness, nomination receipts and controls, the exact financial receipts, and preserved intermediate constructions. The unused warm-donor plus transported endpoint-repair route is documented with its explicit sufficient checks and bounded unsuccessful searches; no gain is attributed to it. Extra alternative-basis nominations were not composed or priced. Future workers can change the donor basis or source word while rechecking actual cuts, membership, signed relations, frame paths, stock and price.

The conditional compiler, common-chart, restored-row, selector/routing, prime-supply, precision/recovery and analytic-transfer interfaces remain inherited. #352's complex certificate is explicit unchanged input. No unconditional multiplication theorem, new Lean certificate or current leaderboard-record claim is made. Source pins, licensing and upstream attribution are included; prepared with OpenAI Codex assistance.

[Mister-Meeseeks]

Thank you @Dugongue. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Paid-aware integer donor-basis exchange.

Review and limits: Changed signed construction is replayed with its own finite bill; the unchanged E8 supplier is a separately identified dependency.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: edef1a7d6da2c628d1adecda234a711e6fff8152. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.