Grade: measured (float), throughout. No violation found, no support claimed. Every Li row in this file is graded measured, including the Arb-ball rows of phases 2 to 4 (AUDIT.md). Numbers quoted from the saved artifacts in this directory are pinned by tests/test_rh_resolve.py; numbers that were only printed by a session say so where they appear. Artifacts: probe_phase1.py, results_phase1.json and the files named below.
Current audit status (2026-10-04)
The n <= 58 enclosure claim below is withdrawn pending a sound complex modulus bound, outward-rounded remainder arithmetic, angle coverage, and a reproducible rerun. See AUDIT.md (formerly THEOREM.md) for exact defects and bounded checks. The saved rows still report positive lower endpoints for 1..58, but these are numerical evidence, not an established enclosure. Phases 2 to 4 below were regraded to measured on 2026-10-10 to match the audit; the wording claimed at the time is in git history and in AUDIT.md's historical section. Raw artifacts are unchanged.
What was measured
Route A, Li coefficients (Cauchy, unconditional): lambda_1..20 all positive at dps=30, min margin lambda_1 = 0.023095708966121033. Dps 20 vs 30 agree to 3.33e-21 max. Cauchy minus zeros (n<=8, 1000 zeros) is +2.4e-10..+1.5e-08, consistent with the tabulated truncation residual in zeta/li.py. Asymptotic at n=20 predicts 8.3507 vs measured 8.7693 (5 percent, inside the documented oscillating remainder, not a constant offset).
Route B, Weil explicit formula: Gaussian a=1 gives abs_diff 1.63e-21 (zero_side 0.0, arithmetic side 1.63e-21; the session's tail bound of 1e-21 is not saved); Fejer b=1 gives abs_diff 1.33e-04. Both are consistency checks of the two sides, not positivity results.
Route C, heat flow: Phi evenness defect 4.8e-32..3.5e-31 at u = 0.5, 1.0, 2.0; H_0(z) = (1/8) Xi(z/2) confirmed with max residual 3.77e-32. Identity machinery healthy, no RH signal.
Controls: rival battery imports and runs (constant-True claim is shared with all rivals, distinguishes False, as it should); Jensen scan d<=4, n<=2 all hyperbolic with min gaps 0.20..0.42 (session output, not saved; no violation, never support, docs/08). Precision response positive (3.3e-21 movement on dps change with cross-method convergence pattern).
Route chosen for phase 2
Li Cauchy positivity with ball-arithmetic numerics. Reason: it is the only lane whose sign carries information (zeros-route is structurally nonnegative), whose disk is unconditionally zero-free (RADIUS_MAX in zeta/li.py), and whose enclosure reduces to bounding one Cauchy integral, which the Arb backend in zeta/rigor.py can carry. Weil needs a new test-function construction first; heat needs an upper bound on Lambda, which no finite computation supplies.
Phase 2 obligation: enclose lambda_1..N on both Arb and mpmath.iv backends, cross-check, and state the exact N and widths (the mpmath.iv run was never done). Enclosed positivity to N would be a statement about 1..N only, not about RH. The uniform tail (all n) remains open and is named as the gap.
Phase 2: positive lower endpoints for lambda_1..3 (2026-10-03; measured)
Status: measured. On 2026-10-03 this phase was recorded as proved finite positivity; that grade is withdrawn, because the panel scheme's branch and angle-coverage obligations are unreviewed (AUDIT.md). Script enclose_li.py, panel quadrature in Arb balls, R = 1/2, K = 32768, prec 128 and 192.
Saved windows (real part; imag parts all contain 0):
n prec 128 prec 192 float check 1 [0.018865, 0.027334] [0.018519, 0.027681] 0.02309570896612103381 (closed form) 2 [0.075364, 0.109330] [0.074007, 0.110687] 0.09234573522804667039 (Cauchy float) 3 [0.155621, 0.259678] (not run) 0.20763892055432480379 (Cauchy float)
Every saved lower endpoint exceeds 0. The closed-form value for lambda_1 ([0.02309570896612103381 +/- 3.6e-38]) lies inside both quadrature windows: two independent routes agree, which is a measured agreement, not an enclosure.
Branch check, same run: min Re(xi) lower over all 32768 panels is 0.4951 > 0 (saved in branch_R0.5_K32768.json) and max |arg| upper is 0.0329 < 1 (session output, not saved). With angle coverage unproved (AUDIT.md item 3) this does not establish the branch on the whole circle. Zero-free disk uses only |rho| >= gamma1 > 14: every zero image satisfies |z| = |1-1/rho| >= 1-1/14 > 0.92 > 1/2.
Explicit dependencies: Arb ball soundness (python-flint 0.9.0); first-zero bound gamma1 > 14 (pinned in this tree against mpmath and PARI oracles); the Li generating identity (derived in zeta/li.py, defect-checked against the literal definition). Precision control: prec 128 vs 192 windows overlap; widths are panel-width dominated, as expected.
Scope, plainly: measured positive lower endpoints for n = 1, 2, 3. It says nothing about RH, which needs all n. The uniform tail is open.
Phase 3: extension to lambda_1..10 (2026-10-03; measured)
Radius tradeoff: error scales like |f-prime|_R/(K R^n), so R = 0.7 beats R = 0.5 above n = 5 (session comparison; no R = 0.5 rows above n = 5 are saved) and R = 0.3 loses badly (R^{-n} dominates; the saved R = 0.3 rows for n = 6..10 all have negative lower endpoints). At R = 0.7, K = 131072, prec 128:
n saved window float check 6 [0.776440, 0.878693] 0.8275660122823793 7 [1.038326, 1.210599] 1.1244601175709595 8 [1.323520, 1.607997] 1.4657556771470606 9 [1.619875, 2.081962] 1.8509160483825342 10 [1.908723, 2.649964] 2.2793393631931577
Combined with phase 2 (R = 0.5, K = 131072: n = 1..5 widths 0.002..0.18, all lower endpoints > 0), the saved lower endpoints are positive for n = 1..10 (measured, AUDIT.md). Branch check at R = 0.7 in-run: min Re(xi) lower 0.4869 > 0, max |arg| upper 0.1192 (session output at K = 131072, not saved; the saved check at K0 = 32768 in branch_R0.7_K32768.json gives 0.4854). Artifacts: enclosure_li_R0.7_K131072_p128.json and enclosure_li_K131072_p128.json (R = 0.5 rows for n = 1..5).
Ceiling note: width grows like n R^{-n}/K. Extrapolating the saved n = 10 row at R = 0.7 by that law, deciding n = 20 would need about three times K = 131072 (an estimate, not run). Finite extension has diminishing returns and no finite N implies RH. The tail needs new mathematics, not larger K.
Phase 4: midpoint scheme, positive lower endpoints to n = 58 (2026-10-03; enclosure grade withdrawn 2026-10-04)
Status: measured. First stated to n = 71, then withdrawn in part: re-derivation showed the midpoint remainder missed the oscillation terms. With them the bound G2 = F2 + 2n F1 + n^2 M exceeds the old M2c from n = 4 at R = 0.7 and from n = 5 at R = 0.8. Invalid rows were moved unmodified to superseded/ with CORRECTION.md, and the corrected bound re-ran. Corrected state: saved lower endpoints are positive for n = 1..58 (1..16 at R = 0.7/K = 16384, 17..58 at R = 0.8/K = 65536), with n = 59 mapped as the boundary ([-0.06, 111.57]). Mechanical check: coverage 1..58 complete, all lower endpoints > 0, all windows contain the float Cauchy values. The audit of 2026-10-04 (AUDIT.md) then withdrew the enclosure grade of these corrected rows: the complex modulus bound, outward rounding of the remainder arithmetic, and angle coverage are unproved. The panel-covering scheme (n <= 10) does not use the derivative remainder and agrees with these rows, but it is measured too.
Original phase-4 text (n = 71 claim) is preserved in git history and superseded/CORRECTION.md, not edited away.
Second scheme (enclose_li_mid.py): midpoint rule with near-point balls plus the remainder G2 (2pi)^3/(24 K^2), G2 from Cauchy's estimate on a larger circle whose M comes from coarse interval panels. For lambda_1 at K = 4096, R = 0.5 its saved window is about 9000x narrower than the panel-covering one.
Range with positive saved lower endpoints: every n = 1..58. Contours: R = 0.7 with rho = 0.85 (n <= 16, K = 16384) and R = 0.8 with rho = 0.88 (n = 17..58, K = 65536, K0 = 16384). Branch checks on the contours gave Re(xi) lower 0.29..0.49 (partly session output; see AUDIT.md, L2). Sample windows: n = 16: [5.70, 5.73]; n = 40: [30.19, 30.77]; n = 58: [11.72, 97.04]. Boundary mapped: n = 59 does not decide at this K ([-0.06, 111.57]). Coverage 1..58 re-verified mechanically, all lower endpoints > 0.
Scope: measured, about n <= 58 only. It is not RH and implies nothing uniform. The tail remains open and needs new mathematics. The nominal width grows like n^3 R^{-n}/K^2; the saved width grows by a factor of about 1.31 from n = 58 to n = 59 at fixed K.
Rival lane closed (2026-10-03)
Weil zero-side comparison, Gaussian h with a = 0.5, 1, 2: 87 DH on-line ordinates to height 150 (Z_dh sign hunt) plus the pinned off-line pair, vs 400 zeta ordinates. Both sides positive at all three widths (artifact rival_weil.json). Expected, and POWERLESS BY CONSTRUCTION: positive h sums positive over any zero set, so this comparison cannot distinguish zeta from DH. The distinguishing content lives in the arithmetic side (prime coefficients, where the Euler product enters), which no zero-side scan reads. This matches the outband hunt's ceiling: Weil positivity is shared structure, not an RH detector. Lane closed by argument plus this null, not by building DH prime machinery for a foregone null. The battery (zeta.epstein.battery) remains the live control for future candidate claims.
Disproof search (2026-10-03, measured, both lanes empty)
Li lane: positivity scan n = 1..100 (Cauchy, dps 25), zero violations, min margin lambda_1 = 0.023095708966121033, lambda_100 = 118.603775376791. Artifact disproof_li100.json. Jensen lane: hyperbolicity scan d <= 16, n <= 25 (416 rows, dps 30), zero non-hyperbolic instances, min root gap 0.0432. Artifact disproof_jensen16x25.json. Honest reading for both: no violation found in range, never support for RH.
Gaps
- The Li rows are measured. The enclosure obligations in AUDIT.md (complex modulus bound, outward rounding, angle coverage, run metadata) are open, and no repaired rerun exists.
- No uniform estimate. Finite N, however enclosed, does not imply lambda_n >= 0 for all n (Littlewood).
- Width grows like n^3 R^{-n}/K^2 under the midpoint-Cauchy scheme (n^2 from the oscillation term in G2). At K = 65536 and R = 0.8 the saved rows stop deciding at n = 59. The saved width grows about 1.31x from n = 58 to 59, so with the 1/K^2 law deciding one more n needs roughly 1.13x the panels (an estimate from the saved rows, not a rerun). A higher-order rule (Simpson via the same Cauchy device) would shift the boundary, not remove it.
- Formal statement check not yet done: any Lean claim must be checked against the original RH statement (zeros of zeta, real part 1/2), not against the encoded Li equivalence alone.
- Weil Fejer defect 1.3e-04 is truncation, not a signal; needs tail accounting before any positivity reading.
Logarithmic-derivative equivalence (2026-10-03)
Classical, recorded here; not a resolution of RH and not new. RH is equivalent to Re(xi'/xi(s)) > 0 for every s with real part greater than 1/2. Main states it in hunts/epp_herglotz/RESULTS.md; see Lagarias, Acta Arith. 89 (1999), 217-234. EQUIVALENCE.md records an ordinary derivation of this known result (unreviewed), its dependencies, and the positive-weight witness. The log-concavity obstruction remains conditional on an unproved uniform margin. Spot checks: check_logderiv.py. The open step is the sign of that real part inside the strip, where the archimedean term and zeta'/zeta nearly cancel.
The doors
- Active constraints: first repair the enclosure soundness gaps in AUDIT.md. The historical calculation reports a boundary at n = 59; its nominal width grows like n^3 R^{-n}/K^2. The n^2 factor comes from the oscillation term in G2 (missing it once cost a full correction round). Shadow price near the boundary: the saved width grows about 1.31x per unit of n at fixed R = 0.8, K = 65536, so each further n costs roughly 1.13x the panels (estimate from the saved rows).
- Frozen-constant inventory: contour radii R = 0.7/0.8 with outer rho = 0.85/0.88 (raising rho toward the 0.92 zero-free edge tightens M2c but risks the branch condition, since zero images start at 0.929); midpoint order (Simpson via the same Cauchy device would change the width law to 1/K^4 and shift the boundary outward at equal cost); prec 128 (widths are discretization-dominated, more digits buy nothing); Gaussian a in {0.5,1,2} and Fejer b = 1 for the Weil probes (arbitrary pair choices, relaxing the pair class is the Weil door, already priced at zero by the outband hunt); heat z in {0,1,2} (illustrative only).
- Information class: all enclosures stay inside values of xi on Re s > 1/2, hence under the Li-configuration ceiling; they cannot close the uniform tail. The tail door requires reading more: a uniform bound on the zero sum or on the Taylor remainder valid for all n at once, which the current family does not contain. Finite K decides finitely many n, no matter how large, so extension is mileage, not approach.