# PP296 twin calibration: scoped interval certificates

30 September 2026 · Davenport–Heilbronn twin only · Riemann hypothesis open

This directory combines the PP296 as-received material, a portable same-library
Arb replay, a separate standard-library kernel check, and a new **independent
MPFR Euler–Maclaurin zero evaluator**. The latter uses directed 256-bit MPFR
arithmetic, an explicit Hurwitz remainder, exact rational contour vertices,
and segment-wide enclosures. Its corrected source and complete receipts are
under [independent_mpfr/](independent_mpfr/README.md). The zero evaluator does
not call Arb, python-flint, `acb.zeta`, or another Hurwitz-zeta implementation.

## Established scope

- **Uniform negative witness and adjacent positive rung:** exactly seven simple zeros lie in the seven
  stated 10⁻⁹ coordinate squares in 80 < Im ρ < 90, and
  the enclosing rectangle has winding seven. The independent MPFR replay
  proves these counts with seven Rouché inequalities and 319 zero-free contour
  segments. The independent standard-library arithmetic checker bounds the
  seven folded terms by G₁₆₅₈₉(7140.06626) < −1.796027803 × 10⁻⁶.
  PP296's analytic outside-zero contribution is less than
  5.211 × 10⁻¹⁴, giving the full-function bound
  G₁₆₅₈₉(7140.06626) < −1.796027751 × 10⁻⁶. The positive rung
  normalization therefore gives W₁₆₅₈₉(7140.06626) < 0. A separate
  [exact-dyadic interval replay](negative_interval/README.md) proves, for
  **every** real x in the closed interval [7139.83, 7140.28],
  G₁₆₅₈₉(x) < −2.57845315144859809 × 10⁻⁷ and
  G₁₆₅₈₈(x) > 7.62752058031253466 × 10⁻⁶. The positive normalization
  transfers both signs to W. These are interval-local statements, not a
  first-failure proof.
- **Positive prefix:** the independent evaluator isolates 179 disjoint simple
  upper zeros below height 260. Six lie below 20 by a symmetric base count;
  twelve direct strip windings contribute 173. Exact reflection symmetry
  forces the 169 uniquely isolated roots in line-centered boxes onto
  \(\operatorname{Re}\rho=1/2\); ten lie in five mirrored off-line pairs.
  Every low-box angle satisfies \(|\arg(\rho(1-\rho))|<0.0072\), with maximum
  recorded ratio bound 0.007199839572. The analytic zero strip and tail
  angle bound 5850/844993 < 0.0072 cover all higher zeros. Since
  2 · 218 · 0.0072 = 3.1392 < π, PP296's folded-kernel sector
  argument gives Wₖ(x) > 0 for every x > 0 and integer 1 ≤ k ≤ 218.
- **Extension through rung 227:** a sharper [zero-free-strip
  certificate](strip_227/STRIP_227_NOTE.md) bounds the high-zero angle so
  its sector contribution stays positive through k=227. The lowest
  off-line pair alone exceeds the uniform low-zero sector threshold, but
  its contribution is positive for 3600 ≤ x ≤ 15000. On each outer x
  range, a neighboring on-line term dominates the pair's two terms.
  The [exact-rational replay](rung_227/PROOF_227.md) checks the complete
  179-zero census, source hashes, all three x ranges, and strict margins.
  It proves Wₖ(x)>0 for every x>0 and integer 1≤k≤227. The earlier
  [sharp-angle note](sharp_angle/SHARP_ANGLE_NOTE.md) identifies 218 as the
  limit of the original uniform-sector proof, not a failed rung.

These statements concern this twin only. The first failing rung is **not**
identified: no global positivity proof covers every rung 228 through 16,588.
The current coarse high-tail sector argument stopping at 227 does not imply
that 228 fails. The
Riemann hypothesis remains open.

## Independent replay and QA

Run from `independent_mpfr/` with Python 3.14:

```text
python replay.py --scope negative
python replay.py --scope all
python summarize_receipts.py
python trace_audit/verify_mpfr_receipts.py
python trace_audit/verify_census_receipts.py
python trace_audit/test_mpfr_rounding_source.py
```

`replay.py` installs the pinned `gmpy2==2.3.1` into a local `deps/` directory
when needed. The `all` replay recalculates the seven-window certificate,
179 Rouché squares, symmetric base path, and twelve 20-unit strip contours.
The sibling `independent_zero_eval/mpfr_em.py` is a byte-identical
compatibility copy for the frozen symmetric-base producer's original path;
it is not an additional implementation. `replay.py` passes its pinned
dependency path to that child process.
The full run is substantial. Its [summary](independent_mpfr/REPLAY_SUMMARY.json)
checks current source hashes, all exact integer windings, contour continuity,
strict local inequalities, box disjointness, and coverage. Each strip receipt
contains all accepted segment endpoints, zero-free image rectangles, and
argument increments. The separate standard-library census audit rechecks
the printed decimal endpoints as exact rationals, the contour topology,
the 179 local inequalities, the base phase, and the analytic strip/angle
comparison; it does not reevaluate Hurwitz zeta. The [precision QA note](independent_mpfr/MPFR_PRECISION_QA_2026-09-30.md)
records a 53-bit leak detected during development, its fix, mandatory
self-tests, and the exclusion of every pre-fix MPFR receipt from this package.
The [exact reflection derivation](DH_FUNCTIONAL_EQUATION_NOTE_2026-09-30.md)
and [symmetric-base proof](independent_mpfr/BASE_SHORTCUT_PROOF.md) state the
analytic premises used by the new count.

The earlier [Arb replay receipt](REPLAY_RECEIPT.json) is preserved as a
historical same-library pass; its “independent evaluator pending” status was
superseded by this addendum. The original PP296 ZIP is retained verbatim as
`PP296_evidence_as_received.zip`. Its extra `2e-9` kernel field is not
generated by the shipped producer script, so the site uses the reproducible
`1e-9`/standard-library bound instead. The original prose says “13 strips”;
the actual count is twelve 20-unit strips plus one symmetric base region.
The original `0.0071997` angle target is below the lowest quartet's angle;
`0.0072` is the sufficient bound.