[RossRichardson]

The pooled complex fill had a better finite cost profile while its source-specific native, whole-call hosting and physical-controller obligations remained explicit premises. This PR now supplies written all-parameter proofs of those pooled obligations and joins them to the complete recursive and integer outer construction, giving the conditional bound

T(n) = O(n (log n)^(1-1/580)), with occupied and visited space O(n)+F_q.

At reviewed commit 5913291c421e851fb5cfb90bc61760a4e58a28ba, the supplier implication is proved rather than retained as a new U(S1) premise. The proof includes all restoration, helper wrappers, exact child returns, stopped recursion, bank scans, copies, head returns, support exposure and inverse exposure, three Gaussian transforms, three CRT conversions, scalar work and exact integer recovery. It uses the existing retuned B1 ordinary backend; neither the optional B2 proposal nor the stronger legacy arithmetic candidate is needed.

Source-specific admission

• Native: all three actual providers, both orientations, arbitrary correlated dirty inputs, every phase-inclusive child/adapter/warp/residual recipe, private cleanup and every positive tensor multiplicity. This reuses and credits PR Audit PR4100’s free-word tensor lift on all four actual programs #4184's free-word tensor argument; scalar role coefficients are applied once.
• Hosting and cover: every one of 1,321,920 helper assignments is checked against its actual whole-call observation window, original axes and spectators. Hosted factors plus all 162,900 residual children reconstruct the original chronology, for every common-cover chart and its final correction.
• Physical realization: the exact source admits a finite controller with encoded numerical extents, explicit tape continuations, complete bank/stack scans and paid cleanup. Strong induction supplies smaller exact children. Finite-width execution is restricted to the proved height/divisibility budget and caller-tail reserve; merely fitting an input in an h-bit word is insufficient.

The complete profile retains every restoration and residual term. Its layer exponent is .99827; the complete outer power is Phi=.99827519. The strict gap (1-1/580)-Phi=1949/2900000000 pays the remaining logarithm.

Explicit conditional boundary

The remaining named premises are B_ordinary (including BIT H1–H5), P_elementary, the complete retained G_modern physical-component and Gaussian analytic contracts, and Baker–Harman–Pintz with the stated multiplicity argument. The modern Gaussian inverse/gap/locality/rounding premises differ from the selected legacy assumptions; this is not an identical-premise transfer from that theorem.

Finite receipts support the written source-specific proofs. They are not a production-size end-to-end execution, independent human certification, formal-kernel proof, unconditional theorem, maintainer acceptance or selected-frontier update. The earlier PR comments audit head a929a3fa36301bdfcffe82afd37434dd9e3c4887 and retain their bounded scope; they do not certify the successor admission proof.

Packaging and separate B2 draft

The optional B2 ordinary-bootstrap calculation is moved to draft PR #4338, based directly on canonical main and independent of this PR. The bound here continues to use B1.

The three large receipt JSONs are stored losslessly as .json.gz, with both container and original-payload size/hash commitments. All fourteen historical receipt hashes, the five original execution-source hashes and the recorded release chain remain checked. The historical input-pin document is preserved separately; the current document changes exactly two addresses to byte-identical, explicitly declared canonical inputs. The interval engine and all native/hosting/physical/outer proof sources remain unchanged. PACKAGING.md documents direct decompression, the exact-engine reader adapter and offline reproduction.

Against canonical main, this reduces the PR from 123,942 to 23,848 added lines and from 20,057,240 to 15,791,279 file bytes. The remaining large archives bind the pooled owner fill and physical/native sources needed for independent replay.

Review and validation

A separate final gpt-6-astra review covers the cleanup and independent B2 extraction at the exact submitted commits; its executed checks remain distinct from the local suite below.

Independent-context gpt-6-astra review found an overbroad finite-width input claim. The amended proof separates arbitrary-input algebra from guarded physical execution, retains an exact overflow counterexample, and was rechecked with no remaining actionable finding. REVIEW.md records exact reviewed hashes and executed/unexecuted coverage; this is AI review, not independent human approval.

• Hosted CI passed at exact submitted head 5913291c421e851fb5cfb90bc61760a4e58a28ba; local, cold-export and hosted evidence remain separate.
• All sixteen registered commands across the finite/physical/outer package and successor admission package pass on Python 3.12.12, normally and with -O. All sixteen also pass from separate committed Git-free exports. The complete hosting traversal is included in both variants; deterministic native, machine and cost-join receipts agree.
• Committed-byte, source-history and metadata checks pass at 5913291c421e851fb5cfb90bc61760a4e58a28ba: 284 predecessor package/input files and 49 successor files. Declared source commits are available in publication ancestry. No network or pending Git object is needed by the exported checkers.
• The historical complete finite release remains separately preserved and authenticated. This packaging run rechecks all original source/receipt commitments; it does not claim a new heavy complete release or production payload execution.
• External audit #4332, head 07e6aada1f07d191d9d467c5aee46ffdc74dd51c, reports an independent exact check of the preserved moment/exponent chain. Its stated scope excludes native/hosting, physical and analytic premises; it does not certify this successor admission proof.
• Canonical main was refreshed before publication and remains 12fa8c5c844be21bfefb38a19503fbd867a11244; the branch contains it. Local validation and hosted CI remain separate evidence.

Start with ADMISSION.md, then its NATIVE/HOSTING/MACHINE companion proofs. The predecessor's unchanged PHYSICAL-GLOBAL.md, OUTER-JOIN.md, OUTER-DOMAIN.md and OUTER-CHECK.md retain the complete outer construction, production domain, normalization and recovery budgets.

Reproduce the successor from the repository root:

python3 -B scripts/research_package.py --package research/pooled-native-hosting-admission --check-only
python3 -B scripts/research_package.py --package research/pooled-native-hosting-admission

Credit: Rohan Arun supplies the pooled fill (#4181); yssite the prior admission architecture (#4180); Cinna Davis and David Ganz the native/operator/owner lineage. The reused #4184 audit retains ToikY, Sagar Thakrar and roger-sato's attribution. Gareth Davidson and Jose Manuel supply the retained charts, physical controllers, ordinary, support/Fourier/chirp and Gaussian constructions; Tekalign supplies the CRT converter. Original analytic sources, notices and frozen predecessor evidence are preserved.

Prepared for Ross Richardson with substantial OpenAI Codex assistance in derivation, implementation and separate specialist review. No shared source, selected record or maintainer review ledger is changed. Original mathematical sources and scientific receipt payloads remain unchanged.

[thu-nmrc]

Independent bounded audit at head a929a3fa36301bdfcffe82afd37434dd9e3c4887.

I ran the package's eight registered bounded commands in both normal and optimized Python 3.11 modes (the package documents Python 3.12; this is a compatibility replay, not a replacement for the recorded 3.12 release). All passed:

• verify.py: PASS_POOLED_BOUNDED_REGRESSION
• physical_bridge.py: six programs, 3,848 monomials, 9 controls
• physical_global_bridge.py: 3,240 calls and 2,047,680 owners
• outer_suite.py: B2 arithmetic, 12,788 bilinear coefficients, 13 outer controls, and source/token checks

Separately, a code-disjoint stdlib-only audit read the frozen JSON receipts without importing package verifiers and recomputed:

• 3,240 calls; 11,680,020 children; rank mass 23,431,680
• 2,047,680 owner entries; 6,489 helper charts; 162,900 pending children
• Phi = 99827519/100000000, hence 1-Phi = 172481/100000000 > 1/580
• the complex-root lower bracket 865490127994518479981/500000000000000000000000 > 1/580
• all 10 source hashes listed by PHYSICAL-GLOBAL.json

Result: PASS_CODE_DISJOINT_4270_RECEIPT_AUDIT. This supports the finite receipt arithmetic and bounded source binding. It does not independently establish the retained uniform physical, hosting, analytic, production-domain, or formal-kernel premises; the PR's conditional scope remains appropriate.

[thu-nmrc]

Follow-up to the bounded audit: the package's own AUDIT-EVIDENCE.json was also checked code-disjointly. All 27 listed source/receipt SHA-256 entries match, and all 298 SOURCES.json rows match the checked-out bytes (192 inputs, 106 provenance rows) at base 12fa8c5c844be21bfefb38a19503fbd867a11244.