RH project · September 22, 2026 · Local checkpoint 1.59 · revision 2

A fixed norm and a weaker growth target

One Mellin integration gives us a clean, unconditional block bound while preserving the RH criterion. The remaining task is an independent arithmetic estimate. We now have two precise targets: optimized residual correlations, or slow growth of the complete canonical error norm. RH and G2 remain open.

For the smoothed adjacent differences, the complete block trace is at most 3κ/(8N²), where κ = log(2π) − γ. The scaled trace tends to 3κ/8. This is an analytic result for all N, not an extrapolation from a table.

What changed

The previous raw norm ran into known large values of zeta. Explicit Euler corrections still meet a related obstruction when their grid product grows no faster than N^α, for fixed α < 1. That result applies to the specified correction family and its full coefficient norm; it does not exclude the optimal projection or the actual residual direction.

Smoothing gives a different route. Its adjacent differences are nonnegative averages, up to a unitary change of coordinates. Their norm bound is elementary. A bounded evaluation functional shows that an off-line zero would still prevent convergence, so we have preserved the mathematical target.

What the finite checks say

The complete smoothed matrices were enclosed at 256 and 384 bits, with every old/new term and the entire infinite tail retained. A separate tail-cutoff replay and seven independent physical-integral checks support the calculation.

The denominator improved, but this particular lower estimate still captures less than 1/50 of the actual gain on these two blocks. These are certified finite comparisons, not an asymptotic failure or a convergence claim.

The residual correlations now have an explicit endpoint formula: two moments of the optimized coefficients plus a fully bounded remainder. Our coefficient-mass remainder is over 2,000 times the leading norm on those blocks, so that triangle estimate gives no positive lower bound. The missing cancellation is stated explicitly.

Shortcuts that do not finish the job

Smoothing the canonical Möbius interpolant does not repair its full-sequence convergence. A localized Laguerre test preserves the obstruction through any number of Mellin integrations. This does not exclude selected subsequences or optimized coefficients.

Increasing the smoothing order changes the measuring stick. Even one fixed function with one fixed coefficient then has relative squared error tending to zero unconditionally. At order 64 the certified relative squared error is about 1.38756%, while its absolute squared error exceeds 10³⁵. The useful RH test keeps the smoothing order fixed and increases N.

A fixed number of ordinary averages does not repair the canonical sequence either. The critical-zero contribution is reduced but remains present. This exclusion does not cover a growing number of averages, logarithmic tapers, or selected subsequences.

The geometry and Maxima angle

The new basis functions are positive, but that does not make the required coefficients positive. Exact matching away from zero forces these first coefficients:

That sixth coefficient gives an exact local obstruction to an all-positive coefficient construction. An explicit separating function now certifies a full-space distance lower bound greater than 0.00228323 from that cone. The bound is attained on the restricted interval; we do not claim equality for the full-space distance. Signed approximation remains open.

The six-function Maxima geometry file builds the exact local Gram matrix and checks the target identity. It leaves the matrix G and coefficient vector beta available for inspection after batch loading. The companion interval calculation certifies the signs and distance. Geometry is useful here for choosing separating test functions; the remaining cancellation still involves arithmetic.

Where I would work next

Keep one integration fixed. There are now two concrete targets. One is a non-summable sequence of guaranteed relative gains for the optimized residual. The other uses the explicit canonical Möbius coefficients: bound the complete error norm by N^ε for every fixed ε > 0. A bounded subsequence with indices tending to infinity would also suffice.

The second target is compatible with the failure of strong convergence. We proved that, at each fixed smoothing order, log(1 + error norm) / log N tends to β* − ½, where β* is the supremum of the real parts of the nontrivial zeros. Thus subpolynomial growth is equivalent to RH. This identifies what must be proved; it is not an independently established growth bound.

I would next work on cancellation between the boundary correction and the integral in the exact canonical formula. Its complete norm is the quantity to control. Applying a triangle inequality and inserting an RH-level Mertens estimate merely assumes the missing arithmetic input. The optimized-correlation route stays open alongside it.

Validation: 419 exact Maxima checks; complete interval replays and full tails; Python diagnostics clean; 317-page manuscript with no undefined or duplicate references or overfull boxes. Local self-review only, with no independent review claimed. No new Weil window or cofinal signed floor.

The criterion and Gram formulas have prior art in Ehm (2024). The canonical interpolant and its unitary obstruction are grounded in Báez-Duarte (2000). The classical zero-free/Möbius-series implication used for the growth criterion is stated in Báez-Duarte (2002), Lemma 2.1. No claim of priority or publication readiness is made.

Status: local research branch, not published. This checkpoint follows the local v1.58 work. Publication and final version numbering remain pending.