Owner-authorized RH research, performed in an isolated worktree from 8941a80. This is one bounded attempt, not a change of the objective to a simple-zero proportion. The original checkout and earlier research records are read-only. No publication, hosted compute, paid jobs or Lean build.
Compare the arithmetic multi-prime orbit construction with the exact floor-level Mertens representation. Preserve the joint target |S_smooth + Sigma1 + Sigma2| = O_epsilon(N^(1/2+epsilon)) from ../prime_pair_error/FACTORIZATION_NEXT_STEP.md. Work on Sigma2 alone cannot silently discharge the other terms. Keep all orbit boundaries.
Owner-directed continuation: keep the bound controlled
The follow-up request asks for a constructive mechanism that prevents bound growth, rather than another report of a failed majorant. Compare the signed joint sum with a block residual-energy construction in the established Baez-Duarte criterion. Its target remains RH, not a simple-zero proportion. May add energy_*.py, energy_*.json, and ENERGY*.md in this directory. Earlier evidence stays unchanged; the old fixed-P kill condition closes only that majorant, not this continuation. Reuse existing Gram machinery read-only; do not run its large ladder. Test a predeclared Mobius shell witness at N=4,8,16,32,64 with ball arithmetic and a small independent comparison. A globally nonincreasing error is not convergence to zero: retain a named quantitative decrease inequality as unproved until proved. No full-suite rerun, heavy compute, paid job, public push or new campaign. The completed mathematical challenge may add a residual-matched fallback and a trace-based moment bound, on exactly the same five shells. Preserve the originally declared shell and compare, rather than retrofitting its reported values or claiming the adaptive direction is always better.
id: mobius_orbits
question: Can complete multi-prime Mobius orbits supply cancellation beyond the p=2 majorant in the RH-linked Sigma2 sum?
frontier: the p=2 absolute-value majorant has leading order N log^3 N divided by 192; the signed joint D_N bound remains open
proposed_attack: retain the exact complete-orbit floor kernel and every incomplete orbit, then compare with an exact floor-level Mertens expansion
dead_routes:
- treating the p=2 constant gain as an improvement of the N exponent
- dropping incomplete orbit boundary terms
- promoting finite arithmetic checks to an all-N RH estimate
required_oracles:
- integer sieve cross-checked against independent trial division
- exact integer and rational enumeration of direct and regrouped sums
- separately written mathematical challenge of the fixed-P asymptotic
kill_conditions:
- an exact direct sum disagrees with its orbit or floor-level reconstruction
- missing-boundary and sign lesions are not detected
- a fixed-P gain changes only a constant rather than the growth rate needed for RH
agents_may:
- derive and challenge candidate identities in this directory
- run small serial exact arithmetic checks
- record an unreviewed ordinary argument with every dependency explicit
agents_may_not:
- claim RH from a finite check
- claim novelty or promote their own claim
- alter the core or earlier hunts
- publish or launch heavy computation