Research: dirty echo, verifier repair, and priced bit-sharing boundaries

[androolloyd]

PR628's binary dirty commutator has quadratic factor-call growth and an executable phase-alias admission defect. This contribution adds a subquadratic multiway echo, a one-line alias guard, an exact full-transform/bit integration experiment, and bit-sharing cost gates that include physical owner restoration. The integration works functionally, but its complete recursive bill rejects this direct adapter as a route to larger κ. This is a construction refinement, executable repair and scoped negative research result.

Local construction and verifier repair

The multiway echo composes noncommuting factor shears while restoring arbitrary borrowed data. Fixed fanout 8 reduces the leaf-call exponent from 2 to approximately 1.602452 with O(log n) borrowed arrays. At 512 factors the exact count falls from 262,144 to 21,952, using 21 arrays instead of 9. The flat variant takes 4n−4 calls and n−1 arrays. The complete inherited error recurrence is included; this is a classical local primitive, not an admitted multiplication supplier.

At pinned PR628, phases is x passes admission even though phases must remain read-only. For x=phases=[1,1], outer b=1/4 and eight fractional bits, execution returns x=[1,15/16]. The submitted patch rejects source/target/scratch phase aliases before mutation. Replay applies the actual patch to the pinned Git source. More precision does not repair this exact counterexample.

Full-transform and bit integration

research/dirty-echo-integration/ implements:

• An all-role complex adapter and Boolean address-chunk interchange, including outputs on every role borrowed during execution.
• Complete normalized synthetic Fourier transforms over C[y]/(y^r+1), stored bit-reversed frequency order, signed monomial twiddles, opposite transforms and correctly normalized convolution.
• Independent exact references, instruction/interface corruption controls, source pins and deterministic evidence.

Three shears per pair complete the all-role adapter. Even granting one recursive child per additive leaf and making routing, phases, storage, preparation and precision free, its moment is 3 N(n)/(2 n^(1-b)). Since N(n) >= 4n-4, its moment at saving zero is at least 6-6/n >= 3. This rejects this adapter's bill for every nonnegative saving; it is not a lower bound for other integrations or suppliers.

Two exact counterexamples prevent silently substituting different operators: replacing the triangular Newton factor by its cyclic counterpart fails at radix 4, with a rank-r−1 boundary correction; the radix-3 cyclic kernel is singular on a length-5 cycle over GF(4). The latter rules out the unrestricted finite-field analogue, not restricted lengths or Boolean address permutations.

At κ=0.01, both retained supplier savings must exceed 1/99. With unchanged widths and volumes, uniform reductions of all paid terms would need to exceed approximately 2.70% for pinned #589 and 2.47% for pinned #659. These are exact-moment diagnostic targets, not feasible fractional deletions or replacement suppliers. No κ gain is attributed to the echo integration.

Bit sharing priced before construction

research/bit-role-sharing-budget/ extracts the actual PR589 local paid profile and all 661 Boolean columns, retains the complete frozen hosting remainder and fallback, and compares role-sharing budgets at the necessary bit saving 1/99 for κ=0.01.

• Equal-call pooling saves dispatches but not volume-weighted recursive work. A fixed-address-permutation rank argument also excludes fewer queried role volumes under arbitrary invertible XOR row charts and restored private dirt. Different address-dependent charts and native primitives are outside this model.
• Even ideal rank-8 consolidation of every local rank unit fails the target. Making all retained-total copies free still fails. The corresponding bit-only outer ceilings are below 0.002644610326897 and 0.007315906873875; neither is an achieved supplier.
• With the original rank mass and boundary remainder, cap 55 fails and cap 56 is the first passing relaxation. A hypothetical integer rank-57 invoice allows 431 extra rank-1 repair calls globally and needs over 98.48% of local rank mass fused. The eight-stage interpretation applies only to one port's active spaces; cross-port sharing has a larger span.
• A different arithmetic target is ideal rank-8 consolidation plus 100 net paid rank units removed per invocation after hosting: 99 fails, 100 passes. This is not achieved by simply deleting helpers or XORs.

No replacement supplier was built: the simple sharing classes fail even under favorable complete-cost bounds, and the passing high-rank/mass-reduction targets still lack an operator and hosting plan. No new κ is claimed. The independent positive-atanh engine reproduces the current bit root and agrees with the inherited Mercator engine on 133 complete invoices. Normal/optimized evidence is byte-identical; 108 small coded-query programs, 384 positive-control basis columns, source/scatter/expected-evidence mutations and a relocated minimal replay are checked.

Standalone command: sh research/bit-role-sharing-budget/replay.sh /tmp/bit-role-sharing-budget. Expected final line: PASS_BIT_ROLE_SHARING_REPLAY. The combined command below now includes this screen. Final overlap review through #716 distinguishes #705/#708/#714/#715; no new pinned coordination issue was found.

Concrete bit-sharing candidate and physical hosting gate

research/bit-sharing-hosting-gate/ follows the 100-net-unit arithmetic target with an exact scalar candidate and a source-bound reconstruction of all 126,432 frozen helper reservations. Neither candidate is admitted as a supplier.

• The nine-stage source-bank masks have one exact dependence: the last source is the XOR of the preceding eight. Nine running accumulators per replica can therefore share the source part of the retained decoder totals at the scalar level. The actual snapshots have rank-nine dirty contamination. The proposed owned-storage plan adds 288 persistent rows and nine temporary rows; source extraction, copy, clear and XOR costs are listed. ADDRESS-chart alignment is still unproved. Even gifting it, the maximum COPY credit is only 7 rank units per invocation, with target moment 1.026605906228685.
• A hypothetical free producer eliminating 13 helper paths removes 104 local rank units, but the explicit idle owners must complete all 104 vacated directions: zero net credit. The new complete profile is priced after coalescing those directions at the owners' original boundaries. Every one of the 439 single-helper eliminations has the same mass conservation. These favorable screens do not assert a replacement producer exists.
• The previous 100-unit screen granted rank-8 consolidation everywhere. Keeping the 222 rank-7 data climbs makes 100 and 101 fail; 102 first passes that narrower relaxation. More fundamentally, under the current data/cover boundary identity, even zero COPY cost saves only 63 per invocation. Reaching 100 requires a changed boundary term.
• A scope control changes the hosting remainder: free elimination of all helper work plus perfect data consolidation still fails with COPY retained, but passes if COPY is also granted free. Thus 100 is not a universal necessary condition for other suppliers; the complete moment remains decisive.

The package verifies all 888 source-bank columns, retained source/dirty ranks and reflected copy forms, reconstructs chronological windows and disjoint owners independently of the parent allocator, and compares 449 complete invoices with two exact rational engines. Normal, optimized and relocated minimal replays agree; six ownership/admission controls plus source/receipt corruption controls are rejected. No new supplier, physical cache realization, all-size transfer or κ is claimed. The construction goal remains unmet.

Standalone: sh research/bit-sharing-hosting-gate/replay.sh /tmp/bit-sharing-hosting-gate. The combined replay includes it. The final overlap refresh through #762 distinguishes the fixed P111 COPY-floor proofs #757/#761 and R420 ghost-return audit #760; these are credited context, not imported dependencies.

Paired boundary target and exact owner plan

research/bit-paired-boundary-gate/ changes the boundary contract instead of withdrawing helper paths from fixed owners. For orthogonal P and Q, incoming source/target frames P/Q and outgoing Q-perp/P-perp give the exact local operator with diagonal D_(I-P-Q) and off-diagonal D_I. The existing nine-stage bank schedule composes it to the full all-role exchange. Rational charts at all 111 ports, the complete permutation algebra and all 18 omitted-bridge controls are checked.

An explicit realization by wrapping the old word is priced, including two rank-two endpoint children per port, 439 owned zero-initialized helper rows, every new idle boundary and ordinary padding to preserve m=64 and W=28416. Its moment exceeds 1.7015, so it is rejected before supplier construction.

A different, unrealized local word with rank(Q)=2 would have rank-five data climbs. The package gives a concrete chronological plan for 296 whole width-eight helpers: 85,248 exclusive global reservations and all owner remainders. Charging every local rank unit as its own rank-one child gives moment 0.9947928245644905 at c=1/99+10^-9 and conditional net saving 197 1/3 units per invocation. Adding 31 rank-one repair children per invocation still passes at 0.9999126541717095, with net saving 166 1/3; 32 fails the rate test. These are conditional construction targets, not achieved improvements.

The actual PR589 scalar order cannot realize the new endpoints by monotone re-framing: its least frame closure forces all 111 sources to full rank eight during cleanup. A new local source/read/restoration topology is required. The Gaussian appendix separately shows that the reported 80-rational-rank COPY floor, under the stated B_K slot model, bounds every stage count below c=0.0078; it does not re-prove that external floor or exclude other delivery models.

Standalone: sh research/bit-paired-boundary-gate/replay.sh /tmp/paired-boundary-new-output. Normal and optimized evidence agree; relocated minimal replay, source/receipt corruption controls, owner timing/collision controls, two exact price engines and selected-record integrity pass. The combined driver includes this package; its unchanged older stages were not repeated in this increment. The full goal remains unachieved: no paired local supplier, finite controller admission or new kappa is claimed. Final overlap metadata was checked through #802, including #791/#801 and the updated #717.

Reproduce and validation

Full contribution, after fetching the exact dependencies listed in the README:

sh research/cyclic-dirty-commutator-audit/replay.sh /tmp/dirty-echo-sprint-replay

Expected: PASS_DIRTY_ECHO_ALIAS_REPAIR_AND_FINE_GRID. This includes the integration experiment and both bit-sharing cost gates. The integration alone requires no network or historical Git objects:

sh research/dirty-echo-integration/replay.sh /tmp/dirty-echo-integration

Expected: PASS_INTEGRATION_EXPERIMENT_WITH_REJECTED_SUPPLIER_BILL.

Completed checks include 300 alias-repair differential cases and 19 early rejections; 72 ideal echo cases, 744 rounded comparisons, 120 dense inverse/error checks and 62 aliases; 168 complex adapter basis cases, 11,616 symbolic Boolean rows, six complete transform/convolution configurations, 40 transform basis columns, 59 signed rotations, 12 integration corruption controls and six wrong-normalization controls. Functional evidence agrees under isolated optimized Python. Relocated minimal replay passes; altered source pins and expected evidence fail. Pricing refuses optimized execution because the inherited rational engine uses assertions.

The combined replay freshly reconstructs the pinned complex supplier and checks two rational moment engines and all 47 assembly inequalities. Selected-record integrity and staged whitespace checks pass. The inherited full bit circuit, root-wide suite, remote CI and proof-assistant verification are not claimed.

Pins, arithmetic appendix and attribution

Main base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Integration continuation base: 0e35f56cab3cae1eb0a99447135569d687c5695a. PR628: 0768f97d06c623b8211e651543c66135796b62bf; PR659: 4982f0ca66ffe0960b5a6451deb5b66b1039326b; PR635: a6895a69f8ad98ab45777b03e9e6f48888c39f6f. Exact dependency files and primary Newton source are pinned in the packages.

The separate arithmetic appendix refines only the final grid on unchanged PR659 to κ = 0.001656193765416678434092, a gain of 4.34092×10^-19. This is not a new frontier; newer pending proposals already report stronger values. The selected certificate and existing supplier profiles remain unchanged.

Assignment seed 67; claims and results are recorded locally. Queue refreshed through #693. #663 independently audits the original binary dispatcher. Dillon Amburgey's #689 at 0dedef12f05f9a06faf9d4e1c8b1462c0e3bff3a refines balanced ternary placement in our unchanged original generator; the new all-role lower bound allows any group ordering. Greer Guthrie's #679 at dce137e540acd2b689281afe222cd32c0ec1f1ae separately prices selector batching and joined volume. #677 concerns a different native ADDRESS query model. These are credited context, not consumed dependencies.

Credit Chafik Boukhalfa / chafreaky (#628), Ali Romman (#621), Jeremy Kothe (#659/#635), icekylinx and the owner/construction lineage, DreamingOfClouds/eumemic (#589), OpenAI's integer-multiplication and exact-DFT sources, and their retained classical/Bennett/catalytic-space notices. No literature-priority claim is made. Prepared for @androolloyd with substantial OpenAI Codex assistance, under Apache-2.0.

The integration is an indexed-read exact-arithmetic reference, not a fixed-tape implementation. Physical hosting, all-size compilation, O(p) precision guards, analytic transfer and inherited bit H1–H5 obligations remain open or conditional. Finite certificates do not formally verify the multiplication theorem. upstream/ and the selected frontier are unchanged.

[Mister-Meeseeks]

The 20-PR pilot triaged 0b4f514c60d9bc7786415d7cbc24754f24d6dc3b and is leaving this PR open for focused follow-up.

Remaining maintainer review: Split review into alias-guard regression, dirty-echo construction, and four priced-boundary packages. PR689 independently audits only its vendored ternary subset.

This is a bounded-review disposition, not a rejection or a request to repeat unrelated full-suite checks. Pilot record. Your contribution and original attribution remain in the queue ledger.