RIEMANN PROJECT · NS92 · SEPTEMBER 23, 2026 · V1

A direct cost bound and an explicit improving step

The selected direction now has a complete upper cost bound calculated from finite Möbius sums, without the new block’s matrix inverse. A separately integrated update at size 256 reduces the full squared error by more than 1.57% while changing only one old coefficient and the new block. The uniform arithmetic estimate needed for RH remains open.

What the bound does

The earlier work supplies a lower bound for the signed gain numerator. To turn it into an improving step, we also need an upper bound on the direction’s full cost. Previously the sharp cost came from the complete projected matrix.

I applied the complete averaging identity already proved in NS61 to the known Möbius/logarithmic direction. Each signed coefficient tail is summed first. A triangle inequality is then applied to the resulting functions, whose norms already include the whole infinite domain.

The result is an explicit scalar bound, U_N. It upper-bounds both the true projected cost and the cost of a feasible direction that adjusts only the old coefficient at N. The latter direction vanishes exactly before N. No new-block inverse or refit of the other old coefficients is needed to execute the step. The starting optimized old coefficients remain inherited inputs.

Two finite certificates at size 256

Both percentages concern the current full squared error. The larger step uses the sharper physical norm; the simple scalar budget alone does not justify it. Every remote tail and mixed term is included.

The scalar upper budget is about 60.74 times the sharp projected cost at this size. These are conservative, explicit updates. They do not improve on the earlier fully optimized approximation or the 2.821% certificate using full refitting.

What remains unproved

The cost calculation reduces to weighted squares of finite Möbius tail sums. Seven certified scalar budgets, at sizes 16, 64, 256, 1,024, 4,096, 16,384 and 65,536, lie between 0.00098 and 0.00220. This finite screen does not establish an eventual bound.

Partial summation makes the missing assumption explicit: square-root cancellation in the block’s summatory Möbius function would give a bounded budget. It is not proved or assumed here. The elementary estimate gives only a budget growing proportionally to N. A bounded scalar budget, together with a persistent signed numerator margin, would imply a sufficient a/j-scale gain on N = 2^j; both inputs remain open.

The original projected cost may be substantially smaller than this upper budget. The next estimate should preserve useful old-space cancellation if the scalar budget becomes too expensive. Its failure would not close PR42’s full joint-observation route. RH, G2, and the uniform arithmetic lower-gain problem remain open.

Validation and review

Arb at 256 and 384 bits; physical cutoffs 262,144 and 524,288 with the inherited old-coefficient cutoff replay; 56 scalar precision overlaps, eight scalar cutoff invariants, ten physical precision overlaps and nine physical cutoff overlaps. Sixteen exact Maxima checks and 131,072 integer Möbius convolution checks pass. Python diagnostics report zero errors and warnings. No new Gram solve was performed.

The physical numerator enclosure overlaps the inherited exact numerator and independently certifies positive step gain. Its tail allowance is too wide to re-certify NS91’s much closer numerator lower bound; no such claim is made. Mathematical review was by the author only.

All 99 prior recorded conclusions were reviewed at main eb6ce3a for scope and dependencies. This was not a fresh audit of every historical proof. Both missing-original evidence groups remain open. The manuscript remains v1.66.

The literature check did not supply an unconditional estimate: Bettin–Conrey–Farmer’s logarithmic-taper result assumes RH and an inverse zeta-derivative moment bound. Neither is imported.

Companions: complete derivation · Maxima worksheet · validation record · reproduction notes.