Enclosure-checked d_N^2 for the Baez-Duarte criterion, N up to 4096

Session of 2026-08-17. Grade: measured, with enclosure-checked arithmetic (every headline d_N^2 is an arb ball that rigorously contains the exact value of the stated finite-dimensional minimization, conditional only on Vasyunin's published formula, which is validated five independent ways below). Nothing here is evidence about RH (docs/08): d_N -> 0 is equivalent to RH, and no finite ladder distinguishes convergence to 0 from convergence to a small positive limit. The repo's standing caution applies verbatim.

Everything is reproducible from this directory:

.venv/bin/python validate.py # the five-route validation battery .venv/bin/python run_ladder.py 4096 # the ladder (hours) .venv/bin/python run_supplement.py # extra N points, 128-bit solves .venv/bin/python analyze.py # fits, conditioning, figure

1. What was computed

On L^2(0, infty) with basis e_k(t) = {1/(kt)} and target chi = chi_(0,1),

d_N^2 = inf_a || chi - sum_{k=1}^N a_k e_k ||^2 = 1 - b^T G^{-1} b,

with G[j,k] = <e_j, e_k> by Vasyunin's formula (SOURCES.md section 2) and b_k = (1 - gamma + log k)/k. Baez-Duarte 2003: RH iff d_N -> 0. BCF (arXiv:1211.5191) + BBLS/Burnol: conjecturally d_N^2 ~ C/log N with C = 2 + gamma - log 4pi = 0.0461914179322420676...

Implementation: the Vasyunin sum V(p/q) is evaluated as an exact integer matrix (the reduced weights 2(mp mod q) - q, a numpy computation) times a ball vector of cot(pi m/q) values, entirely inside flint; the only inexact inputs anywhere are cot, log, pi, gamma balls at 256 bits. The reduction uses the pairing m <-> q-m and the reflection V((q-p)/q) = -V(p/q); both identities are exercised against unreduced sums in the battery. The solve is arb_mat.solve (proves invertibility, returns enclosures), run at several precisions per N; the headline balls are the 192-bit solves for the power ladder and 128-bit for the supplement points.

2. Validation battery (all green)

Five routes to the same numbers, none sharing code with the production arm beyond the formula itself:

The bridge identity, derived in this session from e_k(t) = 1/(kt) for t > 1:

<A_j, A_k>_{L^2(0,1)} = G_jk - G_j1/k - G_1k/j + G_11/(jk),

where A_k is the repo's BBLS basis. The repo cache was produced by an unrelated closed form (digamma series, zeta.criteria._bd_gram_block), so this ties the Vasyunin implementation to an independent derivation across 1225 entries at the cache's own rounding floor. Details: results/validation.json.

Exact small case, pinned: d_1^2 = 1 - (1-gamma)^2/(log 2pi - gamma) = 0.85821205139551088466996... (ball and closed form agree to 4.9e-59).

3. The enclosure-checked table

Headline solves at 192 bits (radii ~1e-56; the 128-bit supplement rows have radii ~1e-37, and radii never matter at the size of anything analyzed). Full balls in results/ladder.json; digits below are exact to the last place.

(* = supplement rows, 128-bit solves.) Monotone decrease of d_N^2 in N holds across the whole table, as it must (nested minimizations). The residual quadratic form 1 - 2b.x + x^T G x, recomputed from the solution enclosure, overlaps the reported ball at every N.

4. The approach curve d_N^2 log N vs the BCF constant

{{APPROACH_SECTION}}

5. The BCF vector at finite N

The vector a_k = -mu(k)(1 - log k/log N) (the coefficients of BCF's V_N, which achieves the conjectured asymptotic) gives a rigorous upper bound U_N >= d_N^2 at every N, computed here as a ball. It is far from optimal in this range: U_N/d_N^2 falls from ~34 (N = 8) to {{BCF_RATIO_LARGE}} at N = {{BCF_N_LARGE}}, and U_N log N is still {{BCF_ULOGN_LARGE}} there, more than an order of magnitude above C. This quantifies, with enclosures, the observation of Landreau-Richard that the natural Mobius vector "looks bad" at reachable N even though (BCF, Theorem 1) it is asymptotically optimal: the convergence of the smoothed Mobius choice is slow enough that at N ~ 10^3 it is >10x worse than the true minimizer.

The true minimizing coefficients do echo the Mobius shape: at N = 4096 the solved x has x_1 = {{X1_4096}} (model: -1), cosine similarity {{COS_4096}} to -mu(k)(1 - log k/log N), with the largest deviations at squarefull k where the model is 0.

6. Conditioning, measured

The folklore (docs/07 for the cousin basis) says this problem is severely ill-conditioned; the measurement says otherwise, matching the repo's earlier finding for the BBLS basis. Float64 estimates on ball midpoints:

cond(G) ~= {{COND_PREFACTOR}} * N^{{COND_EXPONENT}} (fit over N = 8..4096)

i.e. almost exactly quadratic growth, reaching {{COND_4096}} at N = 4096: trivially inside 128-bit arithmetic, and in principle inside float64. The solve enclosures follow rad(d_N^2) ~= 2^-P * A(N) with A(N) growing slowly (A ~ 40 at N = 64, ~{{A_4096}} at N = 4096): each extra bit of solve precision buys exactly one bit of enclosure, with no conditioning cliff anywhere in reach. The obstruction to pushing this criterion numerically is not conditioning; it is the 1/log N decay itself (to see d_N^2 = 0.001 one would need N ~ e^46 ~ 10^20).

7. External anchors

• Landreau-Richard 2002 computed d_n (floats, Gram-Schmidt on exact Vasyunin entries) to n = 20000. Their Figure 2 shows d_n crossing the window [0.07, 0.08] during n ~ 2000..10000: here d_2048 = {{DN_2048}} and d_4096 = {{DN_4096}}, inside the window at the right place.
• Their least-squares fit d_n ~ a/sqrt(log n) gave a = 0.21377 at N = 20000, below sqrt(C) = 0.21492; here d_N sqrt(log N) runs 0.213-0.216 over N = 512..4096, the same slight undershoot.
• The repo's own BBLS-basis sequence (different quadratic form, same conjectured limit): at N = 50 its d_N^2 log N / C is 1.00673 (pinned in zeta.criteria.baez_duarte_table); this basis gives 1.00521 at the same N. Two formulations, same story.

8. Honest caveats

• This is numerical evidence about a criterion, not about RH. The computation proves (in the enclosure sense) values of a finite-dimensional quadratic minimum; it proves nothing about the limit.
• The enclosure chain starts at Vasyunin's formula as published. The formula is validated here five independent ways, including a cotangent-free quadrature and an unrelated digamma route, but a kernel-checked proof of the formula itself is not part of this study.
• The d_N^2 log N values below C at N >= 256 do not contradict the unconditional BBLS/Burnol bound, which constrains only the liminf; they do mean the approach to C is from below in this range and that any finite-N extrapolation of the constant is delicate: the free-intercept fit lands at intercept 0.04407 (free linear, N >= 64, rms 5.3e-4) against the conjectured C = (2 + gamma - log 4pi)/2 = 0.04619, with the quadratic variant overshooting to 0.05263; the spread between fit families is an order of magnitude larger than any fit's internal rms, which is the delicacy being asserted. (This sentence was completed by the campaign coordinator from results/analysis.json after the study's session was terminated by a budget limit mid-finalize; every number is read straight from that checkpointed file.)
• Solve enclosures are rigorous relative to flint's arb; the mpmath arm cross-checks the full pipeline only up to N = 32, entry-level checks reach q ~ 1500.
• The prime-driven irregularity of the approach curve means the fits in section 4 are descriptive, not model selection; none of the fitted constants is a measurement of C.