Status: proved obstruction to a previously proposed route. No G2 sign gap was closed. RH is not proved. Delivery is LaTeX only.
The v1.36 reduction made a uniform finite ordinary-norm lower floor sufficient. That relaxed the earlier o(1) target and required retesting the v1.34 primitive mechanism rather than simply retaining its old rejection.
For the complete physical symbol beta_a, define
Delta_a = sup_{x<y} (−integral_x^y beta_a(xi) dxi)_+.
| The previous proved estimate is q_a[f] >= −a Delta_a | f | ². Could a Delta_a stay bounded along a cofinal family? |
Answer: no. The new proof establishes a Delta_a -> infinity unconditionally for the actual arithmetic symbol. It also establishes inf_xi beta_a(xi) -> −infinity. A linear rate is only proved conditionally on RH.
| Complete canonical closed physical Weil form, domain integral log(2+ | xi | ) | F(xi) | ² dxi < infinity, where F is the unitary Fourier transform of zero extension. |
Choose delta=1/100 and a fixed even positive smooth mollifier kappa supported in (−delta,delta), with integral1. Set
w = 1_[pi,3pi] * kappa,
H(t) = integral w(s) exp(−its) ds,
K(u) = integral_[−1,1] H(t) exp(iut) dt.
| Then w>=0, integral w=2pi, | w’ | _1=2, H(±1)=0, K(u)=O((1+u²)^−1), and K(0)<−1/(3pi). The sign follows analytically from |
integral_pi^(3pi) sin(s)/s ds
= −pi integral_0^pi sin(u)/[(pi+u)(2pi+u)] du <= −1/(3pi),
plus the 1/2 Lipschitz bound for sinc and the mollification error pi/100. All constants are fixed before the window grows.
Assume RH temporarily and fix a zero ordinate gamma0 of multiplicity m0. The explicit formula gives the exact identity
J_a = integral beta_a(xi) w(T(xi−gamma0)) dxi
= sum_gamma m_gamma K(T(gamma−gamma0)) + E_T,
|E_T| <= 8pi exp(−5T/2) / [5T(1−exp(−2T))].
| There is no omitted 2pi or T factor. The inverse-transform kernel outside | t | <=T is precisely −exp(−5 | t | /2)/(1−exp(−2 | t | )), from the 5/4 reference multiplier. The transform of the probe is T^−1 exp(−i gamma0 t) H(t/T). Inside the cutoff, the complete geometric distribution and both pole signs are retained. |
| The endpoint zeros make the truncated test continuous and piecewise smooth, with its second derivative a finite measure. Mollification extends the smooth explicit formula to this test with a uniform O((1+ | gamma | )^−2) zero-sum majorant at fixed T. If exp(T) is a prime power, its endpoint atom pairs with zero. Isolation of gamma0 and N(Y)=O(Y log Y) give |
J_a = m0 K(0) + O_gamma0,w(T^−2) + E_T <= −1/(6pi)
eventually. The last inequality is RH-conditional.
| The optimal scalar primitive decomposition beta_a=b_a+U_a’, b_a>=0, | U_a | ∞=Delta_a/2 gives J_a>=−Delta_a, because the probe has variation2. Its mass is pi/a. Thus, under RH, |
Delta_a >= 1/(6pi),
inf beta_a <= −a/(6pi²)
eventually. The threshold is not claimed effective here.
Now suppose a Delta_a were bounded on any cofinal sequence. The old primitive inequality would give a complete ordinary uniform lower floor; v1.36 would imply RH. The preceding RH-conditional estimate would then contradict that same bounded sequence. This proves a Delta_a -> infinity unconditionally. The identical contradiction, using the full symbol identity, excludes any cofinal pointwise lower bound beta_a>=−C and proves inf beta_a -> −infinity.
This is not circular: RH is first a consequence of the hypothesized scalar certificate, and is then used to disprove that certificate. We do not assume RH to obtain an unconditional sign for the physical form.
An independent reviewer checked the explicit-formula extension, all Fourier factors, the reference multiplier’s exterior tail, the strict prime cutoff at resonance, multiplicity and the cofinal contradiction. See adversarial_review.md.
| The nonnegative frequency probe is not asserted to equal | F | ² for a physical Paley–Wiener transform supported in I_a. Under RH its negative pairing therefore does not contradict Weil positivity. This is precisely why a scalar frequency comparison can fail while a support-constrained operator comparison survives. |
The script check_probe.py checks the rational gate 6*(22/7)²<100 and gives independent binary64 quadrature diagnostics of the probe and a single spectral atom at T=2,7,19,100. These T values are abstract scaling tests, not new arithmetic windows. The approximate K(0) is −0.35434715073059375. Numerical quadrature is not an interval certificate and is not used as the proof of the theorem. No lambda=8 calculation was attempted. An initial attempt to use mpmath found that package unavailable; the diagnostic was rewritten using the standard library. The analytic proof is independent of that implementation choice.
No unconditional linear rate for Delta_a or inf beta_a is asserted. The unconditional conclusions are the two divergent limits. The theorem does not rule out all matrix-valued metrics, signed concentration estimates, or symbol modifications with a separately proved physical error bound.
ID: G2.6-PT-01b, retest of PT-01 under the v1.36 finite-floor reduction.
Precise proposed sufficient estimate: sup_j a_j Delta(beta_a_j)<infinity, a_j->infinity. Status: rigorously ruled out. The related scalar pointwise estimate inf_j,xi beta_a_j(xi)>−infinity is also ruled out.
Nearest prior proposal: v1.34 optimal signed-primitive/Bernstein estimate, previously rejected as an o(1) mechanism using a Dirichlet model and finite diagnostics. Those tests did not settle the relaxed O(1) objective. The new ingredient is an endpoint-vanishing positive probe and the absolutely convergent actual zero-sum identity; it proves divergence for the arithmetic symbol rather than fitting a few windows or changing a constant.
Other prior routes: this is not a renamed Schur or structured inverse, scalar prime norm, finite-rank metric repair, source-only small-residual argument, dyadic pair norm, common-Gram off-block factorization, Picone transform, or a new concentration criterion. It closes a scalar simplification of the surviving signed concentration route. It does not revive the negative-only Schatten, commuting-channel, Cotlar, point-value repair, or raw-transfer shortcuts.
Primary-literature screen (September 21, 2026):
The obstruction is new within the inspected project log. The ingredients are classical; worldwide novelty remains unresolved. It is registered as a proved no-go, not as a new live positive mechanism.
Retain the signed source-compressed concentration estimate
−D_a P_a + sum_j w_a,j C_a,u(G_a,j) >= −C_* P_a,
| with one finite C_* along a cofinal family, plus the full odd block and any physical blocks/couplings not subordinate to this projection. Here P_a=I− | u_a><u_a | is the actual ordinary source-complement projection and C_a,u is the physical concentration operator. The new theorem says that the primitive budget and the minimum of the scalar symbol cannot supply C_*; favorable and unfavorable frequency levels must be controlled jointly on the physical transform image. |
The next concrete lemma is a uniform signed estimate for that joint operator, retaining off-diagonal concentration interactions rather than replacing it by a scalar drawdown or pointwise envelope. No arithmetic proof of this lemma is supplied here. No G2 gap was closed; RH remains open.