[krzysztofwojt]

Change and mathematical scope

Prove a sharp geometric convergence bound for the unstrengthened block-four projection recurrence:

2^(-m) < T_m - 439277/149175 < 2^(1-m) for every integer m >= 5.

The new ingredients are an exact four-mode representation of the entire normalized profile and a uniform two-step transfer bound controlling both infinite parity tails. This improves the previously stated O(1/m) convergence rate to matching upper and lower bounds of order 2^(-m). The mode sequences remain recursively defined; no elementary closed form is claimed.

The package also obtains strict monotonicity and the unattained infimum, but those are credited shared corollaries, not the additional contribution: JoyboyBrian's concurrent #5184 proves them by a different weighted-order method. The exact limit itself is Nick Petalas's #5151 theorem, explicitly cited as a mathematical dependency.

This is a scoped written proof with exact finite corroboration. It does not certify the inherited hook ranks, a physical engine, dirty restoration, complete recursive costs, a new kappa, or a selected-frontier change. Independent human and formal proof-kernel acceptance are unclaimed.

Task and overlap

• Fresh-main base: a5df48c55ea806d7952b65acc9c76d878ff09b13; contribution head: c27103bc12a5c73abeccd63a74dfd106cac03593.
• Priority revision 7; task card independent-moment, with a recorded handoff-first scope change in TASK.json. Latest canonical main checked: ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c; priorities, catalogue, selected record and standing guidance are unchanged from the pinned base.
• Canonical claim #5169, handoff pointer, and scope narrowing.
• Sussman Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 recurrence/seed head: ee4fd87f064017c23e5a86fc1dcfe0a5237caed3; Lutz Independently check #4628's m ≥ 4 projection bound: 2.99172, 2.97290, 2.9447 reproduce; tail proved for all k; two uncertified block-4 inputs isolated (no new κ) #4643 retained recurrence check is pinned in SOURCES.json.
• Petalas Prove the exact rational projection-recurrence limit (no new κ) #5151 limit proof: e6c8e945a47dcc4a6f08974682090fc7f5e221a3, with its exact proof hash recorded. It merged during the investigation without changing that cited head.
• JoyboyBrian Prove strict finite-block descent and exact infimum of the projection recurrence #5184: b8aa60d489ff067c5b994a54e5ebd3c2bdc09503. Its claim landed between our last pre-claim live query and creation of DONE: four projection modes and sharp geometric convergence (#5193) #5169; its proof was read before publication, credited, and our scope narrowed to the additional representation/rate. No peer verifier was rerun as this contribution.

The final paginated overlap refresh at 2026-10-23 12:12 UTC covered 312 open PRs, 4,384 Discussion records and 6,920 local notes, with no coverage warnings. A subsequent live recent-claim/PR check included this account's separate #5187 R437 workspace work; no same-account topic is reused. The excluded Gram, norm-six streaming and affine-A5 follow-ups remain outside this package. Coverage does not include unpublished work or arbitrary historical diffs.

Validation

CPython 3.12.12, standard library only:

python3.12 -B scripts/research_package.py --package research/projection-geometric-tail
python3.12 -I -X dev -W error -B research/projection-geometric-tail/verify.py
python3.12 -I -O -X dev -W error -B research/projection-geometric-tail/verify.py
python3.12 -B scripts/check_research_commit.py --package research/projection-geometric-tail --check-source-history

All pass at the contribution head. Normal and optimized output agree byte-for-byte with RESULT.json. Evidence includes 1,215 whole-profile equalities through block 48, 172 scalar recurrence identities, 172 scaled-state identities, the symbolic polynomial remainder identity, 49 finite fixtures for the uniform estimates, and exact tail checkpoints at 32 and 33. Ten controls reject wrong recurrence terms, coefficients, parameter domains, bounds, involution sign, tail budget/parity, limit and an invalid all-seed monotonicity extension.

The universal claims rely on the written operator inequalities and summable infinite-tail argument, not extrapolation from those finite checks. The exact limit is a credited dependency, not an independently audited predecessor theorem.

Committed-byte/source-history checks and official seven-file export integrity pass. Both direct modes also pass from a read-only export with Git absent from PATH; outputs match RESULT.json and all eight files including the receipt retain identical hashes. Selected-baseline source/metadata orientation passed; no broad historical scientific replay was run. Both hosted jobs passed at this exact head in Registered research packages run 39951804385: planning and the focused package replay. Automated CI does not establish independent scientific acceptance.

Credit

Prepared by Krzysztof Wojt with substantial OpenAI Codex assistance for proof, verifier and writing. A Luna worker assisted with read-only claim inventory, bounded exact fixtures and arithmetic/indexing review; final integration remained with the primary agent. Exploratory SymPy is not a replay dependency.

Credit Jacob Alan Sussman, Maximilian Lutz / lm1909, Nick Petalas, JoyboyBrian, eumemic (#288), Swapnil Jain and the retained engine/certificate lineage. Apache-2.0; predecessor files and shared tooling are unchanged.

Open next questions

The README records two bounded follow-ups: exact leading parity constants or elementary/q-series expressions for the modes, and classification of monotone seed profiles. Start with the fixed-operator second-order recurrence; equality cases in the seed cone remain a blocker. No follow-up is reserved by this PR.

Completion refresh: this account's only other open PR was #5187 (R437 physical workspace), a different object and quantity. No new same-object claim appeared in the canonical thread.