RH PROJECT · LOCAL ADMISSION AUDIT · 29 SEPTEMBER 2026 · V1

The finite Jacobi product does not supply the proposed sign

The first finite product, completed evenly with its required boundary mass, gives a negative radial-sign diagnostic inside the critical strip. The value agrees at 50 and 80 decimal digits and with an independent direct integral. This stops the stated finite-approximant positivity attempt. It does not refute the infinite-product target or RH.

Wall check: Same open gap — ZERO-GEOMETRY, NS100–101. Jacobi’s exact product provides original arithmetic data beyond reciprocal symmetry, but this audit obtains no independent signed estimate for the full original kernel. The group remains frozen. The numerical finding is a diagnostic, not an interval certificate or a new impossibility theorem.

What was tested

The original identity is

Θ(r) = ∏_n≥1(1−q²ⁿ)(1+q²ⁿ⁻¹)²,   q=e^−πr, r>0.

This is the z=0 specialization of DLMF 20.5.3. It specifies the original integer-square theta series, not a generic positive reciprocal Gaussian mixture.

The candidate mechanism was: truncate that product after M factors, complete it evenly, and obtain the desired sign at every finite M before passing to a justified limit. Explicitly, define P_M(q) as the finite product and

h_M(t)=e^|t|/2P_M(e^−πe2|t|),
Φ_M=(h_M″−h_M/4)/4,
F_M(z)=∫ e^zt dΦ_M(t).
Candidate: Re(F_M′(z) conjugate(F_M(z))) ≥ 0
for every M≥1, 0<Re z≤½ and Im z∈ℝ.

The derivative is distributional: reflection creates an atom at zero of mass b_M=h_M′(0+)/2. Omitting that atom would change the candidate. The original smooth theta completion has no such atom. We did not screen an alternative construction that discards or otherwise corrects it.

The diagnostic that stops the candidate

Already M=1 gives the exact polynomial P₁(q)=1+2q−2q³−q⁴. Its factors are positive for 0<q<1, but its Gaussian expansion has signed coefficients. At z=¼+14i:

The first-product boundary mass is approximately 0.00160965454270846585. The negative value belongs to the explicitly defined approximant, not to xi. A bounded initial grid used x=¼ and y=0,2,…, stopping at its first negative value, y=14. We did not expand the scan after that finding.

The 50- and 80-digit runs agree in all 35 stored significant digits of the candidate value. Direct integration of the differential density on [0,3], plus the boundary atom, gives the same result. At 80 digits, its transform differs from the incomplete-gamma evaluator by about 4.72×10⁻⁸². An explicit absolute tail majorant beyond t=3 evaluates to about 8.99×10⁻⁵⁴⁷ for F and 1.81×10⁻⁵⁴⁵ for F′. Those tail numbers and the quadrature are floating evaluations; neither rounding nor quadrature errors are interval-certified.

Boundary and domain accounting

Write P_M(q)=1+Σ_k>0c_kq^k. For t>0 the ordinary differential density is

φ_M(t)=e^t/2 Σ c_k[(πk)²e^4t−(3/2)πk e^2t]e^{−πk e2t}.

The complete transform is b_M+2∫₀^∞φ_M(t)cosh(zt)dt. A separate evaluator uses

I_k(z)=½(πk)^−(z+½)/2 Γ((z+½)/2,πk),
F_M(z)=¼+(z²−¼)/4 · Σ c_k[I_k(z)+I_k(−z)].

Here Γ(a,v) is the upper incomplete gamma integral. All sums in these finite-product formulas are finite; their half-line integrals have double-exponential tails. The formula includes the boundary atom and is entire. It gives F_M(½)=¼, matching the original normalization.

There is a separate warning for the unreflected product. Set h̃_M(t)=e^t/2P_M(e^−πe2t) on the whole line. For every finite M≥1,

∫ e^t/2(h̃_M″−h̃_M/4)/4 dt
= ¼[e^t/2(h̃_M′−h̃_M/2)]_−∞^∞ = 0.

Both boundary values vanish: near r=e^2t=0, the finite product has order r^M, while at +∞ the nonconstant Gaussian terms decay rapidly. The original completed transform at z=½ is ¼. Thus pointwise convergence of the raw product and its derivatives does not justify passing this whole-line integral to the limit. This is the same kind of domain issue already recorded in NS101, now visible for product truncations; it is not a new closure of conditional or completed approaches.

What the infinite product still leaves open

For L(r)=log Θ(r), differentiating the product on compact positive-r intervals gives

L′(r)=Σ 2πn/(e^2πnr−1)
−2Σ π(2n−1)/(e^π(2n−1)r+1),
φ(t)/h(t)=r²[L″(r)+L′(r)²]+(3/2)rL′(r),   r=e^2t.

This retains signed terms and their cross products. No estimate for their full Fourier-side comparison was found. Positive real product factors do not assign that Fourier sign. The full NS101 integrated kernel remains unestimated; the original product identity itself remains valid.

Controls, scope and record

The pinned Davenport–Heilbronn interface was used with the actual radial-sign functional. It detects a negative value near its off-line zero, and the NS101 reciprocal-mixture completion also has a negative sampled value. These calibrate the sign evaluator. Neither control satisfies the original finite Jacobi-product premise, so neither is presented as a matched counterexample to that stronger premise. The direct M=1 evaluation is the relevant candidate screen.

Maxima checked eight exact algebra identities, including the first polynomial, differential density, boundary mass and logarithmic derivatives. Python diagnostics report zero errors and zero warnings. These checks validate bookkeeping; they do not certify the numerical sign or prove an asymptotic statement.

Disposition: stop the stated finite-approximant mechanism at its failed screen. Starting at a later M, removing the atom, smoothing the reflection, or using a different completion would be changed hypotheses needing their own specific justification and gate; none is established or prescribed here. No claim that every product-based route is impossible. No new research row, thaw, manuscript version, PR or push. RH, G2, the signed arithmetic estimates and both historical original-evidence gaps remain open.

Reviewed public commit f32cadfb7be523a8920b19be57d40d652e39a2ef. All 104 registered conclusion statuses/scopes and 14 continuation demands were reassessed; historical proofs and certificates were not replayed. Reproduction scripts, admission record and outputs are in the adjacent evidence folder. This local report has not been published.