riemann

v1.50 — NS-34: the positive arithmetic weight and critical capacity budget

Exact identities and proved scoped implications, plus named open input CAE. No numerical experiment or certificate. G2 and RH remain open.

The task follows the recommendation to try an explicit positive comparison weight in the existing physical prime-shift/Picone identity, allowing the uniform finite error of v1.36. The selected weight is the archived Gaussian arithmetic radical, h(x)=E(H)(exp x). It is explicit, even, strictly positive, and independent of the unknown ground state. It is not identified with the actual repaired finite-window source.

What was established

• Positivity and explicit upper/lower double-exponential bounds for h. Its pole mass is exactly integral cosh(x/2) h(x) dx = 1/sqrt(3), by the Mellin transform. The associated completed-zeta kernel is classical.
• The exact global identity q[hg] = J_h[g] - (2/3) Var_nu(g)
  • 2 |integral sinh(x/2) h(x) g(x) dx|^2. Both J_h and the subtracted expression are nonnegative; the last square is essential in the odd sector. All expressions are finite on compact tests, and extend to each fixed-window form domain. No positive global realization is assumed.
• The exact finite-window exterior term T_a is retained. The local potential is r_a = (T_a - 2 I_h cosh(x/2))/h. This is an identity, not a tail metric.

• For this weight, the sufficient capacity form is B_(a,delta) = q_a - integral (r_a)+ |f|^2 - delta J_(a,h)[f/h]. Every valid capacity error on either parity must obey

  eta_a >= [delta_a*kappa*exp(gamma*lambda^2) - B_a]+,
  B_a = A + (8a+1)*exp(a) + 2a,
  kappa = (2*pi-3)*log(2)/(128*pi),
  gamma = (3*pi/4)*exp(-4*log(2)-2*b), b=log(2)/16.
  

  The finite A depends only on two fixed compact smooth bumps, never on the window. A fixed positive energy fraction cannot be discarded with uniformly bounded eta. Necessarily delta is at most O(lambdalog(lambda)exp(-gamma*lambda^2)). Deleting prime-2 transformed energy alone gives the same obstruction with delta=1.

• Source admissibility is exact: a two-dimensional even translated-bump space contains a unit vector orthogonal to the actual source for every window. The odd space is already source-orthogonal. The bounds apply to all mixtures in these spaces, so no uncontrolled cross term is omitted.

What had to change; what remains missing

A generic strictly contractive absorption estimate cannot work for this weight, even with the weakened bounded-error target. Its transformed edge ratios amplify a fixed energy loss like exp(gamma*lambda^2). This closes only the stated fixed-slack/prime-2-deletion comparisons.

The full coefficient-one signed comparison remains the named open input Critical Arithmetic Energy comparison (CAE). In these coordinates it is equivalent to the original uniform complement-floor target; writing it has not supplied an independent arithmetic proof. The local capacity test with delta=0 is stronger because it also drops the positive potential. Neither its uniform error nor CAE is proved. A different weight is not excluded. This is a failure of the proposed lower comparisons, not of the physical Weil form or the full capacity route.

Record and provenance

• capacity_weight.tex: complete integrated proof and open-input statement.
• scope_review.md: internal claim, domain, parity and algebra checks.
• provenance.json: source hashes and reviewed mathematical ancestry.
• build_report.json: actual manuscript build and reference counts.
• research-report-2026-09-21-v1.html: user-facing report.
• NS-37: audits/ns37-capacity-weight-2026-09-21-v2.html records no unresolved MAJOR/MINOR; the reviewed fragment hash matches exactly.
• Actual manuscript build: 246 pages, 0 undefined/duplicate references, 48 bibliography items, no duplicate keys or overfull boxes.
• The audited goals sentence about the symbol amplitude now agrees with the weighted strict-cutoff formula already proved in v1.46; historical diagnostic artifacts are untouched.

No new window, tail metric, numerical optimizer, source projection run, or zero enclosure. NS-1’s two evidence groups remain OPEN. Existing missing v1.29/v1.30 candidate artifacts are not claimed recovered; this work relies on the complete manuscript identities and its archived v1.15/v1.35 analytic radical inputs.

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