Base: a3acc741396940273ec388548d52752f839593c3.
Continue the paid-shortfall idea preserved in the September 7 conversation. The ultimate target remains a uniform upper bound on psi(N)-N. This bounded attempt asks whether early truncation of an explicit factorial certificate has a controllable complete cost. It does not treat a small finite excess over measured psi(N) as an estimate for psi(N)-N.
For Q_N={floor(N/d):2<=d<=N}, use W_c(q)=sum_j c_j floor(q/j), B_N(c)=sum_j c_j log(floor(N/j)!), and w_q=sum_{floor(N/d)=q} log d. The paid bound is C_N(c)=B_N(c)+sum_q w_q max(0,1-W_c(q)).
Write scope: this directory and tests/test_paid_shortfall.py, plus its case-log entry and generated CONTEXT.md. Preserve earlier research files. No core changes, optimizer, large numerical search, Lean run or paid service.
Ancillary preservation repair: isolate global precision in the restored baseline and joint checker tests, and make the knownness low-precision negative control explicit. Original archived checkers and outputs stay byte-for-byte unchanged. This fixes the import-state regression observed in main CI after the preservation merge; it does not change a research result.
The construction is an early-truncated balanced seed lift. The explicit base-6 seed is floor(t)-floor(t/2)-floor(t/3)-floor(t/6). A general lemma may reuse the existing pilot's validated finite-seed hypotheses. A small perfect-power cap example tests whether additional arithmetic changes the admissible artificial measures, without starting a second search.
id: paid_shortfall
question: Can an early-truncated explicit factorial lift have a proved full paid-shortfall budget at square-root support?
frontier: The paid bound and capped dual were proposed in conversation, while the old pilot lifts through the full cutoff and fixed seeds retain a leading constant greater than one
proposed_attack: Bound the omitted lift by its sparse small-integer support, retain every factorial remainder, and check an exact perfect-power capacity example
dead_routes:
- treating a finite reduction in B-minus-psi as a bound on B-minus-N
- applying a finite-seed height bound to an unbounded nested lift
required_oracles:
- exact rational floor identities and prime-exponent comparisons
- independently factored finite von Mangoldt sums
- interval logarithms with precision refinement
kill_conditions:
- a shortfall occurs outside the proved support
- the full cost omits a positive remainder
- the claimed capacity is smaller than a true prime-power mass
agents_may:
- derive explicit coefficients and full-cost inequalities
- run small deterministic checks
- challenge the exact argument
agents_may_not:
- declare novelty or RH
- promote an asymptotic conclusion from finite checks
- edit earlier research evidence