RIEMANN PROJECT · SEPTEMBER 23, 2026 · AUDIT VERSION 1
Yes, the central obstacle was already there.
NS96 does not establish a new fundamental barrier or advance the missing arithmetic lower bound. It gives exact results for a different coefficient class. That distinction should have been clearer in the progress report.
The central target remains open, rather than disproved: show that the actual optimized residual must lose enough error as the approximation grows. The tail estimates are available. The required estimate for the signed arithmetic sums is not.
What was already known, and what NS96 added
| Item | Audit finding |
|---|---|
| Positive coefficients can fail | NS62/64 already prove this for adjacent differences. NS96 treats a different LCM coefficient class; the two cones must not be identified. |
| The complete rule c=+q cannot converge | Its c₁≥0 already conflicts with NS78’s necessary limit c₁→−1. NS96 adds an explicit uniform constant, not a new qualitative obstruction. |
| The complete rule c=−q has an error floor | NS74 already displays the same Mellin separator and floor function; 3087/4096 is that function at 4/7. NS96 applies it to a different sign-constrained LCM class. The hypothesis changes; the tool and constant formula were already present. |
| Refitting defeats automatic positivity | NS95 already exhibits the sign problem for ordinary-product pairings. NS96 proves a distinct exact equality for projected LCM mixtures, using elementary rank-one pairs. |
| Actual arithmetic lower gain | Unchanged. Neither NS95 nor NS96 proves or refutes it. No worldwide novelty claim is made by this comparison. |
One proposed escape also repeats the problem
A trace normalization on the positive matrix is ineffective if we still optimize a free step size. Dividing the matrix by its trace shrinks the direction; multiplying the step by that trace restores exactly the same update.
Best line gain = ⟨residual, direction⟩² / ‖direction‖²
This ratio is unchanged by scaling the direction. Trace-one rank-one pairs still give both signs of each available individual LCM observation. A bound on the actual total update, with no scale subsequently restored, is a different restriction. No estimate for that restricted construction is proved here.
What changes in the work plan
Stop treating positivity rephrasings and free-scale normalizations as new directions. Before pursuing a variation, compare it with the nearest recorded result and identify the particular arithmetic term for which a stronger estimate is proposed.
The existing primary target is a lower bound on the complete joint observation energy of the actual optimized residual. A gain comparable to a/j after j doublings is a possible sufficient scale; an eventual fixed fraction at every doubling is already excluded. Restating this target is not itself progress.
This audit supplies no new cofinal estimate. It corrects the description and screens out a redundant continuation. It adds no theorem node or manuscript version, and preserves the earlier proof records.
A wall check on every turn
At Alex’s request, each future research turn must identify itself as Known wall, Same open gap, or Distinct test, cite the closest recorded result and state what changes. The check appears before extending the research and in the final update, including when the instruction is simply “continue.”
This turn: Same open gap. The PR42/NS87 lower-bound input is unchanged. The work is an overlap audit and a correction to future route selection.
Review scope
All 104 registered conclusions, current checkpoints, the evidence-gap ledger and relevant proofs were reviewed at main 6c50142. This was a scope and dependency review, not a numerical replay of every historical result. The scale identity has nine exact Maxima checks. Author review only.
RH, G2 and both missing-original evidence groups remain open.
Claim-by-claim findings · Scaling argument · Audit scope and checks