RIEMANN PROJECT · SEPTEMBER 23, 2026 · VERSION 1

What positivity gives us—and what refitting removes

The positive divisor-square proposal does not supply the missing lower gain. Used as the entire coefficient rule, it leaves a fixed error floor. Once the old coefficients are free to change, its positive mixtures generate exactly a signed linear space. This identifies where the proposed positivity argument stops.

The result concerns one precisely defined construction. The actual optimized approximation, the arithmetic lower-gain target, G2 and RH remain open.

A genuine square uses least common multiples

If a divisor sum is squared pointwise, its coefficients group at lcm(i,j). This differs from the ordinary-product convolution square tested earlier. For a positive semidefinite matrix Γ, define q_ℓ by summing Γ_ij over lcm(i,j)=ℓ. Then every divisor sum ∑_ℓ|mq_ℓ is nonnegative, although individual coefficients can be signed.

This is classical Selberg-sieve algebra. The new project test asks what happens when it is used in our fixed approximation space.

Complete coefficient rules have fixed error floors

Both bounds hold at every size. The first uses a complete real Mellin moment, including the exterior interval. The second follows from the exact residual on the first two physical cells. These are bounds on the two restricted constructions, not on the unrestricted optimized error.

After refitting, both signs return

For indices i,j already in the old fit, the two rank-one positive matrices made from e_i+e_j and e_i−e_j give the functions

a_i + a_j ± 2a_lcm(i,j).

The old-space projection removes the first two terms. Positive mixtures can therefore realize either sign of every available new direction. With unrestricted old refitting, their projected cone is exactly the signed span of those directions.

Consequently the matrix of pairings with an old-orthogonal residual is either zero or indefinite. Positivity of Γ alone forces no sign for the numerator. A fixed trace budget, a normalized square, or a single rank-one square is an additional restriction and is not covered by the equality of cones.

Where the project should go next

Return to the joint signed arithmetic estimate for the actual optimized residual: a lower bound on its available improvement relative to its current error. Keep the complete projected energy and the old-coefficient cancellation in the same estimate. The tail is already controlled at the required scale.

A square-based alternative would need a quantified restriction that survives refitting and still gives the required numerator and cost bounds. Positivity by itself, or another finite efficiency table, does not meet that requirement.

Checks and limits

All 103 prior conclusions were reviewed for scope and dependencies. The new argument has 17 Maxima identity checks, exact finite divisor checks, and scalar replays at 256 and 384 bits. No new finite optimization or physical cutoff was used. Author review only; no independent proof audit is claimed.

The unchanged v1.66 manuscript builds to 358 pages with zero undefined or duplicate references; its 16-file manifest verifies. Both groups of missing historical originals remain open.

Complete derivation · Replay instructions and scope · Author review