Research: two-block native-query encoding theorem and paid boundary obstructions

[aliromman]

PR #840 gives a centered DATA shear with three ordinary children, but leaves its conversion to the ordinary BIT supplier boundary open. This contribution proves that one fused block cannot realize any positive number of whole ordinary-interchange rows under an arbitrary binary linear encoding and its inverse, with restored dirty helpers. It then proves that exactly two such blocks suffice algebraically for three genuine rows over every odd prime-power modulus m>=11. The latter result uses explicit complementary square-zero factors; it is not a search claim.

Review files: Proof and scope · Evidence · Source pins and credit.

The proof counts the complete six-field native orbits, including short prime-power orbits and every spectator. It also gives an orbit-quotient charge for added adapters and treats different entry/exit frames separately. A literal two-block construction for one genuine row plus two restored dirty rows is compiled and checked on all 2,187 independent VALUE columns, then lifted to two spectator fibers and checked on all 4,374 columns. Its six full-volume children fail the recurrence even with free encoders. That failure applies only to this one-genuine-row construction.

For three genuine rows, the six children each have volume V/3 and width N/3, so the child-only coefficient is two. The full cost still includes affine streams, XOR scans, encoder generation/storage, every bit transfer, routing, head returns and child restoration. The supplied scan-per-bit encoder emission has a proved superlinear volume cost. A cheap uniform encoder, arbitrary-width layout and recursive hosting remain open. This is an enabling algebraic proof with scoped obstructions and an exact finite boundary construction; it is not an admitted supplier, a new kappa, or a formal proof of the full multiplication theorem.

Base: abdc6a7002abdbcd9e21bfc5e1a795784deacfdf. PR #840 is pinned at da2cd0dd2229e08b61b174a8835a2c2bb99d2230; PR #521, PR #677 and PR #990 pins, copied sources and notices are in research/native-query-boundary/SOURCES.json. Main was rechecked at ba82555c64e1879e4b7decce28186f208fcb8262, which selects PR #1042. The selected BIT exponent is unchanged and the complex side still binds. Neither the selected record nor upstream/ is changed.

Validation:

python3 -I -O -B research/native-query-boundary/verify.py
python3 -I -O -B research/native-query-boundary/check_package.py

The first command reports PASS_NATIVE_QUERY_BOUNDARY full_addresses=665444 prime_power_cases=13 dirty_columns=2187 escape_children=6 no_new_kappa. The package check passes relocation and rejects changed source bytes, changed source identity, a false child volume and Boolean/integer substitution. The mathematical replay rejects missing children, XORs, decoder transfers, helper corruption and a lost spectator fiber. Nine large-domain direct-sum templates are checked; no full m=11 encoder matrix is allocated. The base selected-record gate and staged diff check also pass. The full repository make verify and selected cold replay were not run for this isolated package; the validation note states that limit.

The local claim records the three candidate approaches and the reason for each pivot. No pinned coordination issue exists at the final check. Recent open-PR metadata through PR #1125 and the predecessor heads were refreshed; no matching native-boundary result appeared in the inspected scope. Predecessor credit is retained for Karl Appel (@applecrusher), @commdata2338, Wentao Li (@VictorLiwentao), Sagar Thakrar (@thakrarsagar), the upstream framework and selected suppliers. Contributed by Ali Romman (@aliromman) with substantial OpenAI Codex assistance in research, mathematics, implementation, verification and writing. No independent human review is claimed.

[Mister-Meeseeks]

100-PR batch review at e6a8003673d55929be6925b22f9ee645d03c33c3.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Native VALUE-rank/orbit obstruction for a single encoded fused block and two-block abstract encoding construction for three rows; efficient tape encoders are not supplied.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.