RH PROJECT · THEOREM-APPLICABILITY AUDIT · 2026-09-29 · V1

Cancellation is available.
The required transfer is not.

Neither of the two examined theorem families directly supplies NS98’s bounded projected cost. The mismatch is in the quantifiers, weights and required energy scale. This audit does not prove that an adaptation is impossible.

Wall check: Same open gap. Closest results: NS92–93 and NS98. MOBIUS-COST and the separate MOBIUS-MARGIN input remain open. No thaw, research row, manuscript version or new RH bound.

The quantity that must survive the transfer

Keep the prescribed coefficients d_n = −μ(n) log(2N/n) for N < n ≤ 2N, without rescaling. With the complete old/new Gram blocks,

H_N = D − C^TG⁻¹C,
K_N = d^TH_Nd.

The sufficient cost target is a constant upper bound for

(1/J) ∑_J≤j<2J K_2^j

on all sufficiently large dyadic J. The diagonal contribution is already bounded. The unresolved part is the signed cross sum containing the full matrix H_N. Removing its old-space subtraction replaces the problem with a larger sufficient budget. NS98: exact target and stop record.

What the source theorems provide

Source scope: these named statements and their accompanying remarks were checked, including the short-interval paper’s Proposition 1. This is not an exhaustive survey of subsequent cancellation results.

Tao: fixed shifts do not settle the growing sum

Write the projected cross term as a sum over shifts:

C_N^proj = 2 ∑_h=1^N−1 ∑_m=N+1^2N−h μ(m)μ(m+h)L_N(m)L_N(m+h)H_N(m,m+h),
where L_N(m) = log(2N/m).

Elementary scope check: positivity of H_N and NS98’s existing diagonal bound imply

|H_N(m,n)| ≤ 4κ/√(mn),
∑_1≤h≤L |C_N,h^proj| ≤ 8κ(log 2)²L.

Here κ = log(2π) − γ, and each fixed-shift sum has at most N terms, each bounded using m,n > N. Thus every fixed finite collection of shifts already has bounded cost, without a Möbius-cancellation theorem. This is bookkeeping from the inherited bound, not a new arithmetic estimate. Letting L grow destroys that constant bound.

The missing transfer must therefore handle growing shifts and the changing projected weights together. Summation by parts would require control of the variation of 2mL_N(m)L_N(m+h)H_N(m,m+h), with errors summable over the shifts. No such control is supplied here.

Averaging in j does create a long logarithmic range of integers, so the fixed-ratio mismatch alone does not dispose of that possibility. But each doubling changes the projection and taper, and the number of shifts grows. A uniform weighted transfer and a total error budget would still have to be established. Applying a fixed-shift limit and then summing a growing set of shifts is not justified.

Short intervals: the scale is too weak for the proposed substitution

NS92’s larger scalar budget is

K_N ≤ κ log(2) S_N,
S_N = ∫_N^2N |∑_x<n≤2N μ(n) log(2N/n)/n|² dx.

To expose the normalization, put v(u) = log(2/u)/u and R_N(u) = ∑_Nu<n≤2N μ(n)v(n/N). A change of variable gives the exact bookkeeping identity

S_N = (1/N) ∫₁² |R_N(u)|² du = N e_N,
e_N := ∫₁² |R_N(u)/N|² du.

Pointwise bounded scalar cost would follow from e_N = O(1/N). The actual averaged target only needs bounded means of 2^je_2^j, which is weaker than that pointwise demand. Qualitative cancellation e_N → 0 does not supply either bound.

Even an optimistic transfer giving e_N ≤ C/(log N)^c would leave an upper allowance CN/(log N)^c. Averaging that allowance over J ≤ j < 2J still grows. This is a limitation of that upper bound; it is not evidence that the actual cost grows.

There is also a domain mismatch before any rate comparison: R_N uses varying-length suffixes with a common endpoint. An almost-everywhere statement for each fixed interval length cannot automatically be evaluated along the variable length 2N−x. Nor does a small exceptional set guarantee that selected dyadic endpoints avoid it. The paper’s long-average term and, for Theorem 3, the excluded integers would also need accounting. None is discarded in this audit. NS92 already records the stronger square-root oscillation requirement for its supremum bound.

Controls, companion input and disposition

This is a theorem-applicability audit, exempt from claiming a new research row. It proposes no new sufficient RH inequality to certify or scan. Davenport–Heilbronn and the NS100/101 theta controls do not carry the stipulated Möbius reciprocal-Euler coefficients together with this exact fractional-part Gram family; importing their coefficients would change the problem. They are marked not applicable to the proposed literal transfer, not counted as passed tests. No unchanged example screen was run.

The existing NS74 unitary control preserves the whole Gram, hence every projected cost, while changing the target-approximation behavior. It reinforces the existing requirement to prove a compatible lower numerator separately. NS93’s sufficient pair remains: bounded averaged cost and a positive averaged normalized lower numerator. A cost theorem alone would not complete that argument.

Decision: stop direct substitution of these two theorem families. A follow-up would need a concrete uniform estimate for the growing-shift sum with the actual projection retained, or a justified interval-energy transfer with a sufficient total error budget. Neither has been obtained. That requirement restates the same open input; it is not a newly opened route.