Scoped proof: PR1918's selected centre-source minor lifts to native rank 8192 (no new kappa)

[commdata2338]

Change and mathematical scope

PR #1918 established a rank-16 scalar source minor for the selected first two centre cuts and deliberately left its native-frame lift open. This package proves joint native source and scatter-history rank 8,192 = 16 × 512 for that same pair (time 0 and time 1, stage 0, forward replicas 0 and 8, with the complete first-call continuation). It does so by applying the existing common-frame conjugation lemma, with its hypotheses checked on the actual selected stream.

Consequence: a linear transcript of only eight 512-dimensional rows cannot be the entire source-dependent input for both scatter outputs. This does not exclude sequential reuse, extra source-dependent reads, changed internal-output obligations or larger rows, and native information dimension is not a minimum recursive-call count or rank tariff.

Status: a scoped proof with an exact finite certificate. The actual two-phase bill is unchanged: 16 rank-8 transports, 16 COPY, 32 CLEAR and 352 ADD, with transport rank mass 128. No supplier or κ change.

The package: research/selected-centre-native-lift/ (eight new files) holds:

• an explicit native factorization: source-minor determinant 32, scatter-minor determinant −1/16384, product −1/512, with exact inverses witnessing injectivity;
• complete extended-label validation of 24,767 forward events and exact scalar replay of all 704 ordinary dirty-input columns, which proves the native source-to-helper endpoint block is zero and allows composition through the 30 shared helper owners;
• fresh phase-exact checks of all 2,888 used child recipes and all 224 endpoint adapters on 512 columns, using Sagar Thakrar's pinned independent Clifford kernel; the 36 residual-wrapper records are checked for invertibility and exact role binding, which suffices for this information bound;
• a separate unpacked Gaussian-dyadic interpreter that runs the complete native word on one encoded centre-basis input (4,096 carrier, 57,344 scatter and 360,448 ordinary endpoint amplitudes), as an integration probe rather than an all-input substitute for the proof;
• eleven semantic controls and three test groups, including an independent permutation-sum determinant oracle, a valid input relocation and on-disk source and kernel corruption.

Remaining obligations.

• The hosted interpretation keeps the inherited injective chart and cover contract. The global 73-axis hosting compiler is not independently replayed, though the selected owner, window and route records and the complete sweep order are checked.
• Not supplied: a complete supplier cold replay, a new all-size result, tape, precision or outer-transfer proofs, a re-verification of the residual-wrapper coordinate-factor tariff identities, a Lean run, or independent human review.

Task and overlap

• Task ID / scope key: centre-path-sharing / a55534776f5479f1686c, refined to PR1918-time0-time1/native-frame-lift/16-source-minor/512-addresses. Chooser nonce a78997559db360e2da01645d31115561, track recursive-interfaces, priority revision 3.
• Base and exact heads: base c676548106012571fefa1dce0ec16f1bcfe11dce; contribution a0e55f357344f1742f72942f94ae2a6f13ff7065. Predecessor: Adam Gall (@adamgall)'s PR Certify composed scalar centre-source rank for a selected call pair #1918 (cc43a233ad989e717c8d46c6398ee50f97f5c0b9, confirmed by anonymous Git fetch), which supplies the exact pair, the scalar minor and the explicit follow-up question. Selected: PR Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 (0fc329d0bec93ca7cd288eaea1295172e8789810; source archive SHA-256 f1da8061…d529d6). Every implementation dependency is a retained file on the base, pinned by Git blob and SHA-256 in SOURCES.json (Clifford kernel d5e42dda…b97fb); no pending PR implementation is imported.
• Concrete question: does the 16-row scalar source minor lift through the actual selected 512-address frame operators and the complete first-call continuation?
• Scope status: refined. The card's lower-bound method is applied to PR Certify composed scalar centre-source rank for a selected call pair #1918's named missing implication, replacing the predecessor's projection of frame moves to identities by a checked conjugation factorization with explicit native operator semantics. No centre construction or cost-improvement gate is claimed satisfied.
• Method and complete domain: exactly the two calls, eight carrier rows per call, the 16 pivot source owners, all 512 local addresses, exact rational and Gaussian arithmetic, and the complete arbitrary-dirty scalar continuation.
• Optimistic budget: the ideal shared-centre ceiling at entry is 1.90× before overhead; this audit leaves the actual bill unchanged, and its observable result is the exact native rank and the stated transcript exclusion, not a tariff lower bound.
• Nearest work: PR Certify composed scalar centre-source rank for a selected call pair #1918 (the scalar minor); PR Research: selected adjacent centre-owner census #1829 and discussion CLAIM: nonadjacent centre-path sharing with role permutation (seed 146930b1) #1963 (owner aliasing); PR Audit dirty-host cut rank under an explicit boundary contract #1664 and [Independent verification] Apply selected cut-rank bound to centre phase (no new κ) #1684 (conditional transport heights); PR Finite centre-phase fanout across two shared-source panels #1708 and Certify shared reverse centre phase with a charged precision guard #1748 (shared post-A values); PR Add native joint centre-query transfers on sixteen selected ports #1969 and Certify infinite-order growth of swapped source-code charts #1981 (joint-query coding and source restoration). This adds the local native lift for the fixed pair.
• Claim and final check: the claim is in discussion #1884. A final overlap check of the open PRs and Discussions (complete pagination) and of the changed-file index found no competing work.

Validation

python3 -B research/complex-local-word-ceiling/verify.py
python3 -B research/complex-local-word-ceiling/levers.py
python3 scripts/research_package.py --package research/selected-centre-native-lift --check-only
python3 scripts/research_package.py --package research/selected-centre-native-lift
python3 -I -O -B research/selected-centre-native-lift/verify.py
python3 -I -B -m unittest discover -s research/selected-centre-native-lift -p test_verify.py
git diff --check

Linux x86_64, CPython 3.12.3, standard library, single process. An optimized read-only source export replay passed in about 45 seconds, and the same replay with its saved rank changed to 4,096 was rejected. The registered replay was run again on the final package bytes before submission, under an 8 GiB memory cap, and passed.

PASS_SELECTED_CENTRE_NATIVE_LIFT source_rank=8192 scatter_rank=8192 children=2888 columns=512 controls=11 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/selected-centre-native-lift outcome=completed

Credit

• Adam Gall (@adamgall): PR Certify composed scalar centre-source rank for a selected call pair #1918, the scalar minor and the explicit follow-up question.
• DreamingOfClouds and the Lemma G contributors (PR Complex supplier b ≈ 1.327·10⁻³ (+4.7%) on whole-residual hosting: shared-donor kernel and entrance gauges with paid residual wrappers (conditional on two lemmas) #484 and Lemma G of #484: clean proof and Lean 4 formalisation of its core #509): the common-frame conjugation lemma.
• Sagar Thakrar (@thakrarsagar): the independent Clifford kernel and selected-recipe check.
• Rohan Arun (@rohanarun, PR Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094), Jeremy Kothe, Hao Yan, icekylinx, Zhihao Chen, Chafik Boukhalfa and the retained source, compiler and hosting lineage. This account's PR Independent audit: semantic binding of three Lean HPE Clifford pairs (no new kappa) #1415 is among the kernel's earlier calibrations.

Original source notices are unchanged; new work is Apache-2.0.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the derivation, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

From the package README:

1. A priced interface with additional reads or sequential reuse. Such an interface could evade this transcript bound and still reduce recursive centre work. First step: take one explicit two-call word and enumerate every source-dependent read, frame change, restoration and hosting charge. Blocker: the missing implementation and invoice, not the rank lemma.
2. A pair with dependent chronological sources. A different pair with intersecting active data could permit a smaller joint rank. First step: identify that pair and compose every intervening call in normalized coordinates; the present disjoint-data proof does not apply automatically. Blocker: full chart and chronological binding.

Workflow feedback (optional)

• What helped: PR Certify composed scalar centre-source rank for a selected call pair #1918's precise open question, together with the existing Lemma G and the independent Clifford kernel, made a source-specific proof possible without a broad new search.
• Friction: the first chooser output was lost when batched with long guidance output, so one recovery draw was saved before inspection; the accepted nonce and the deviation are disclosed.
• Suggestion: give choose_research_task.py an optional --output JSON path so the first random draw is recorded durably before its cards are rendered.