Checkpoint: oct08-extensions-1. Base: f4ef0715bbe2f4890b3dcea034757ef0c823053a. Owner: Codex root. Isolated checkout, no publication or integration authorized. Parallel writers use task-local route directories; only the owner imports results here.
Exact targets: (A) an explicit prime in every open seventh-power interval under the zeta 7/8 hypothesis, improving PR276's ninth-power statement through an audited eighth-power step by adding explicit zero-density information; (B) a parameterized finite-network lemma and its exact-DFT complexity consequence, with upstream transport lemmas identified; (C) a useful genus-aware partial-form rejection algorithm with measured gains and completeness limits; (D) a useful deterministic certificate algorithm beyond the existing Miller consequence, or a recorded overlap/obstruction.
Original requested breadth is preserved. A fresh-source survey selected B as an additional route. A bounded search or a model response never establishes novelty. Written proof, source verification, finite arithmetic and formal replay have distinct statuses. Each successful candidate needs a fresh audit.
Pinned prior branches: prime powers 1aff81ebbdb5b749c638a6450016e167f9b3f61e; nonresidue cc33a9a208609ac584a4f1c470d1808153dc8213; class number 3f136908426167ff97888a6ae8b89087223509e6. The class-number input does not supply a committed full list or an audit of every streamed discriminant. Cremona-Sutherland already supplies the GRH classification through h=1000, including even h.
Use one-core short computations. No Lean build, paid Modal, global account change, push, merge, publication, or writes to concurrent checkouts.
id: oct08_extensions
question: Which precise useful extension of the current arithmetic or exact-transform results survives independent source and mathematical checks?
frontier: Ninth-power intervals and nonresidue exponent eight under the seven-eighths hypothesis; class-number tables through 1500; fixed finite Fourier network
proposed_attack: Zero-density splitting, parameterized finite networks, genus-fiber counting and generator-based certificates in disjoint routes
dead_routes:
- Reoptimizing the same full-zero-density weight cannot remove its recorded logarithmic eighth-power loss
- A larger class-number cutoff alone does not satisfy the structural target
- Classical Miller consequences cannot be relabeled as new algorithms
required_oracles:
- Exact integer and rational checks with planted faults
- Arb enclosures and a separately derived numerical cross-check
- Primary-source theorem statements with exact hypotheses
kill_conditions:
- A source hypothesis fails or an implemented inequality differs from its stated contract
- A purported complete list omits a valid discriminant
- A complexity improvement hides work in an oracle or ignores the stated arithmetic regime
agents_may:
- Derive mathematical arguments and scoped obstructions
- Run bounded exact checks and independent audits
- Preserve reproducible artifacts in the isolated hunt
agents_may_not:
- Promote their own candidate as externally reviewed
- Claim universal truth from a finite experiment
- Publish or alter shared research work