Riemann project / October 5, 2026 / Next-step review v1

Where to go next
after the theta audit

None of the twelve reviewers supplied a new estimating method that meets the project’s gate. My recommendation is to finish this record and stop automatic variations of the theta construction. The original route remains open; this review does not establish its impossibility.

The new moving 4D model explains the missing sign. It does not provide the arithmetic information needed to force that sign. No actual theta trajectory is plotted.

Why “just a lower bound” is misleading

The identity and boundary repair tell us what to estimate. They do not supply an estimating mechanism. The new reviews separate three things that had been easy to conflate:

• Accuracy is not a sign margin. A small approximation error only proves positivity if the same comparison has a lower margin large enough to absorb it.
• A distinguishing identity is not an inequality. The original Jacobi evolution differs from the NS101 control, but uniqueness of that evolution does not bound its signed transform.
• This first target is not RH by itself. Proving the first Laguerre inequality would settle that partial target. An additional real-zero argument would still be needed.

For an elementary scope example, F(r) = 1 − r^4 satisfies F′² − F·F″ = 4r^6 + 12r² ≥ 0 on the real line, yet it has the nonreal zeros +i and −i. This polynomial is not an original-theta control; it shows why the single inequality alone is insufficient in general.

What the reviews change

They reject a few specific constructions and sharpen the admission requirements. The original slope’s divided-difference matrix fails a two-point test in its large-coordinate tail. Direct half-line convexity and positive Gaussian scale mixtures also cannot represent the complete associated kernel. These are narrow failures, not a closure of all geometric or kernel methods.

The prime-side review retains the exact residues in the Gonek–Hughes–Keating smoothing formula. Distant residues can be bounded; the local signed contribution remains. Its hypothetical off-line pair calculation tests the proposed error mechanism, not the zeros of the actual zeta function.

The Bessel review also finds a concrete mismatch of scales: the available crude lower allowance is of order exp(−r²)/r². Merely beating an exponential in |r| does not meet that allowance. A larger cutoff could improve the approximation error, but would still leave the comparison to the reference function unestimated.

The next decision

No scan or new proof task is admitted now. Another rotating model, finite table, shifted clock or generic agent round would not supply the missing input. The next research proposal must arrive with a concrete mechanism for estimating an expression, not merely an exact formula for it.

A proposal can be smaller than RH. It must name its arithmetic expression, coefficient hypotheses, uniform range, estimating step and complete dependency on the intended target. For this theta lane, that means an original-coefficient signed estimate or an explicit arithmetic factorization with justified boundaries. The matched control must fail at the actual estimating step.

Returning to full NB-GAIN is a separate choice, not an automatic escape. That lane has the clearest existing specialist brief, but still lacks the actual-target nonsummable gain estimate. It retains the full projected Gram, corrected loads, old-space refit and N^6 tail; NS74 and NS83 remain in force. The brief is already prepared and has not been sent. This review gives no evidence that NB is mathematically more promising than the other open routes.

Budget: this twelve-review synthesis and one publication PR; no follow-on computation. A future concrete proposal can receive a bounded one-to-two-hour admission check, with the exact time adjusted to its scope. Without that proposal, no automatic repeat is scheduled.

Twelve focused reviews

Nine method reviews, two scope/strategy reviews and an independent admission review. Their agreement is a bounded assessment of the supplied methods, not a proof that no other method exists.