Route A, attempted: one further candidate priced, and the target quantity

measured directly

RANK3_SCOPE.md's Route A asks for an unconditional, uniform-in-\(q\) estimate for \(\Delta(t;q,b)=\psi(t;q,b)-t/\phi(q)\), strong enough to make \[ S(N):=\sum_{q=2}^{R_0}\mu(q)^2\sum_{b\bmod q}^*T(q,b)=O_\epsilon(N^{2+\epsilon}), \qquad T(q,b)=\sum_{t=1}^N\Delta(t;q,b)^2+\sum_{t=1}^{N-1}[\Delta(N;q,b)-\Delta(t;q,b)]^2, \tag{S}\] for \(R_0=\lfloor Q/L\rfloor\), \(Q=\lfloor\sqrt N/3\rfloor\), \(L=\log N\) (RANK3_ROUTE_D.md (D5); UPPER_BOUND.md Section 6). RANK3_ARC_TRANSFER.md (T6) removed the separate transfer-step obstruction RANK3_ROUTE_D.md Section 6 had identified, so \(S(N)\) is now exactly, and only, what stands between rank 3's \(U\)-side and the target \(O_\epsilon(N^{2+\epsilon})\); nothing below revisits that reduction. This document does three things: recaps what two prior attempts already established about Route A (Section 1, citation only, nothing re-derived); prices one further candidate neither prior document checked, and finds it strictly worse (Section 2); and measures \(S(N)\) itself directly, rather than an ingredient of it, from the true von Mangoldt function (Section 3). It does not assume RH anywhere, and does not touch \(Z_{(q)}\) or ranks 1-2.

1. What RANK3_MEAN_VALUE_TOOLS.md and RANK3_BDH_VERIFY.md already settled

Restated here only to fix notation and the size of the remaining gap; both documents are cited, not reproduced.

• Individual-\(q\) uniformity (Route A as originally named) has no unconditional route. RANK3_SCOPE.md Section 3: the only closing route the text exhibits for the analogous \(q=1\) case, (31), assumes RH, which would be circular against this hunt's own target (UPPER_BOUND.md Section 1). Nothing here changes that.
• An averaged-in-\(q\) substitute exists and has been checked against the two standard mean-value theorems. Bombieri-Vinogradov (BV) is the wrong strength (a log-saving, matching (SW)/rank 1's own bottleneck, never a power of \(N\)) — RANK3_MEAN_VALUE_TOOLS.md Section 2, no numerical check needed. Barban-Davenport-Halberstam (BDH) is the right shape (a sum of squares, matching \(T(q,b)\)) and, applied at its single endpoint \(t=N\) and then summed crudely over \(t=1,\dots,N\) (RANK3_MEAN_VALUE_TOOLS.md (E1)-(E2)), gives \[ S(N)\ \ll\ R_0N^2\log N\ \asymp\ N^{5/2}, \tag{E2}\] exactly matching CHHL's own GRH-conditional order for \(E(N)\) (UPPER_BOUND.md Section 1), one full power of \(N^{1/2}\) short of the target. RANK3_BDH_VERIFY.md checked, against the primary sources themselves (not just the citing document), that no sharper member of the natural "for every fixed \(A\)" family is available at this range: BDH's sharp asymptotic form reaches large \(Q\), not small \(Q=R_0\); BV's arbitrary-\(A\) saving comes with an \(A\)-shrinking range incompatible with the fixed range \(Q\le R_0\); and the (SW)-style trick that lets an error term absorb a trivial initial segment does not transfer to BDH's quantity, which is leading-order content, not an error term. Confirmed cost, from citable sources alone: \(N^{5/2}\), no better.

2. A further candidate, checked and ruled out: the large sieve on the

\(K_N\)-continuum directly, rather than pointwise-in-\(t\)

(E1)-(E2)'s route applies BDH once at each fixed \(t\le N\) and adds the \(N\) resulting bounds — a step both prior documents flag as crude. The natural next attempt is to avoid that crudeness by applying the multiplicative large sieve directly to the continuous object \(S(N)\) already is (an integral over \(\beta\), via the same Parseval identity that built \(T(q,b)\) in RANK3_ROUTE_D.md (D5)), instead of to \(N\) separate point estimates. This section carries that out and shows it is not an improvement — it is worse, and by a full power of \(N^{1/2}\) again, in the other direction.

Setup. For a Dirichlet character \(\chi\bmod q\) and \(\beta\in\mathbb T\), write \(\Psi(\beta,\chi)=\sum_{n=1}^N\Lambda(n)\chi(n)e(n\beta)\). By the same convolution/Parseval computation RANK3_ROUTE_D.md Section 2 uses to derive (D5) — here with \(\chi(n)\) in place of the indicator of a residue class — \[ \int_{\mathbb T}|K_N(\beta)|^2|\Psi(\beta,\chi)|^2\,d\beta =\sum_{t=1}^N|\psi(t,\chi)|^2+\sum_{t=1}^{N-1}|\psi(N,\chi)-\psi(t,\chi)|^2, \qquad\psi(t,\chi):=\sum_{n\le t}\Lambda(n)\chi(n), \] and, by the multiplicative-character analogue of the additive-frequency orthogonality identity RANK3_BDH_VERIFY.md Section 2 derives (that document relates \(\sum_a^|\Delta_a(t;q)|^2\) to \(\sum_b^\Delta(t;q,b)^2\) via the additive characters \(e(\cdot a/q)\); the identity needed here instead expands \(\psi(t;q,b)\) in the multiplicative characters \(\chi\bmod q\) — a different, but equally standard, orthogonality computation, not claimed by that document and derived here instead): for \((b,q)=1\), \(\psi(t;q,b)=\phi(q)^{-1}\sum_{\chi\bmod q}\overline{\chi(b)}\psi(t,\chi)\) exactly (inverting \(\psi(t,\chi)=\sum_{n\le t}\Lambda(n)\chi(n)=\sum_{b}^* \chi(b)\psi(t;q,b)+O(\rho_2(q))\), the same \(\rho_2(q)\) of (D3)), and the \(\chi=\chi_0\) term reproduces \(t/\phi(q)\) up to the same \(O(\rho_2(q)/ \phi(q))\), so \(\Delta(t;q,b)=\phi(q)^{-1}\sum_{\chi\ne\chi_0}\overline{ \chi(b)}\psi(t,\chi)+O(\rho_2(q)/\phi(q))\); squaring, summing over \(b\), and using \(\sum_b^\overline{\chi(b)}\chi'(b)=\phi(q)\cdot\mathbf1_{\chi= \chi'}\) gives \(\sum_{b}^\Delta(t;q,b)^2=\phi(q)^{-1}\sum_{\chi\ne\chi_0}|\psi(t,\chi)|^2 +O(\rho_2(q)^2/\phi(q))\), the same order of correction (D3) already tracks and RANK3_BDH_VERIFY.md Section 2 tracks for its own, additive, version: \[ \sum_{q\le Q}W(q)\ \ge\ \sum_{q\le Q}\phi(q)\sum_{b\bmod q}^*T(q,b) \ \ge\ \sum_{q\le Q}\sum_{b\bmod q}^*T(q,b)\ \ge\ S(N), \qquad W(q):=\sum_{\chi\bmod q,\,\chi\ne\chi_0}\int_{\mathbb T}|K_N(\beta)|^2|\Psi(\beta,\chi)|^2\,d\beta, \tag{W1}\] up to the same negligible \(O(\log^2q\log^2N)\)-type correction tracked there (the last inequality uses \(\mu(q)^2\le1\le\phi(q)\)). So an upper bound on \(\sum_{q\le Q}W(q)\), at \(Q=R_0\), is a (lossy, but valid) upper bound on \(S(N)\).

Bounding \(\sum_{q\le Q}W(q)\) by the large sieve at each fixed \(\beta\). Fix \(\beta\). Apply the multiplicative large sieve inequality (Montgomery and Vaughan, Multiplicative Number Theory I, Theorem 6.7; Iwaniec and Kowalski, Analytic Number Theory, Theorem 7.13 — the same citation RANK3_MEAN_VALUE_TOOLS.md Section 1 uses) to the sequence \(a_n=\Lambda(n)e(n\beta)\), \(1\le n\le N\): \[ \sum_{q\le Q}\sum_{\chi\bmod q}\Big|\sum_{n=1}^Na_n\chi(n)\Big|^2\le(N+Q^2)\sum_{n=1}^N|a_n|^2. \] Since \(|a_n|=\Lambda(n)\) — the phase \(e(n\beta)\) has modulus \(1\) and drops out of \(|a_n|^2\) entirely — the right side is \((N+Q^2)\sum_n\Lambda(n)^2=(N+Q^2)d_N\), with no dependence on \(\beta\) at all, where \(d_N=\sum_{n\le N}\Lambda(n)^2\ll NL\) is UPPER_BOUND.md's own quantity (UPPER_BOUND.md, the line before its equation (20): "\(d_N\le L\psi(N)\ll NL\), by Chebyshev's bound"). So, for every \(\beta\in\mathbb T\): \[ \sum_{q\le Q}\sum_{\chi\bmod q}|\Psi(\beta,\chi)|^2\ \le\ (N+Q^2)d_N. \tag{W2}\]

Integrating (W2) against \(|K_N(\beta)|^2\,d\beta\). Both sides of (W2) are nonnegative; multiply by \(|K_N(\beta)|^2\ge0\) and integrate over \(\mathbb T\). The right side is a \(\beta\)-independent constant times \(|K_N(\beta)|^2\), and \(\int_{\mathbb T}|K_N(\beta)|^2\,d\beta=N\) exactly (Parseval; \(K_N\) has \(N\) unit coefficients), so \[ \sum_{q\le Q}\sum_{\chi\bmod q}\int_{\mathbb T}|K_N(\beta)|^2|\Psi(\beta,\chi)|^2\,d\beta \ \le\ (N+Q^2)d_N\cdot N. \] The left side is (at most) \(\sum_{q\le Q}W(q)\) plus the nonnegative \(\chi=\chi_0\) term, so in particular \[ \sum_{q\le Q}W(q)\ \le\ (N+Q^2)d_N N\ \ll\ (N+Q^2)N^2L. \tag{W3}\]

Comparing (W3) to (E2) at \(Q=R_0\). \(R_0\asymp\sqrt N/(3L)\), so \(R_0^2\asymp N/(9L^2)=o(N)\), and \(N+R_0^2\asymp N\). Substituting into (W3): \[ \sum_{q\le R_0}W(q)\ \ll\ N\cdot N^2L=N^3L, \tag{W4}\] a full power of \(N^{1/2}\) worse than (E2)'s \(N^{5/2}\), which is itself already a power of \(N^{1/2}\) short of the target. Combined with (W1), this shows \(S(N)\ll N^3L\) via this route — true, but strictly weaker than what (E2) already gives directly, so this candidate is not a way forward.

Why it fails, structurally. (W2)'s bound is the same ceiling \((N+Q^2)d_N\) at every \(\beta\), because the large sieve, applied this way, only ever sees \(|a_n|=\Lambda(n)\) — a quantity with no \(\beta\) dependence — and so cannot distinguish a \(\beta\) where the character sums happen to be small from one where they are large. Integrating that \(\beta\)-blind ceiling against \(|K_N(\beta)|^2\,d\beta\) just multiplies it by \(\int|K_N|^2=N\), converting the large sieve's one saving (the \((N+Q^2)\) ceiling in place of the trivial \(Q^2\cdot\)(number of characters)) into the only saving available, while adding back, for free, the same order-\(N\) loss that (E1)'s "sum \(N\) copies of a bound" step incurs on the \(t\)-side. It is a different bookkeeping path to (structurally) the same kind of waste, not an independent mechanism, which is why it lands a power of \(N^{1/2}\) below (E2) rather than above it. To do better than (E2) via any such continuum route would require exploiting genuine cancellation of \(\Psi(\beta,\chi)\) across \(\beta\) for a single \(\chi\) before invoking the large sieve on \(q,\chi\) — but a bound on \(\int_{\mathbb T}|K_N(\beta)\Psi(\beta,\chi)|^2\,d\beta\) for one fixed \(\chi\) is exactly \(\sum_t|\psi(t,\chi)|^2\)-type, i.e. exactly Route A's original single-modulus uniformity question, restated at the level of one character rather than one residue class. This document does not find a way around that restatement, and does not believe, based on the computation above, that one is available from the large sieve alone; it records this as a ruled-out direction, not merely an unexplored one.

3. Measuring \(S(N)\) directly

Neither RANK3_MEAN_VALUE_TOOLS.md nor RANK3_BDH_VERIFY.md computes \(S(N)\) or \(T(q,b)\) themselves; both price named theorems against them. This section measures \(S(N)\) directly from the true von Mangoldt function ( prime-power enumeration, no model, no sampling), at the actual \(R_0(N)\) UPPER_BOUND.md's construction uses, via rank3_route_a_measure.py (results in results_rank3_route_a_measure.json). Correctness of the per-\((q,b)\) computation of \(T(q,b)\) against RANK3_ROUTE_D.md's (D5) was checked directly (a hand-written running-sum computation of \(\psi(t;q,b)\) against the vectorized version the probe uses, at \(N=30\), \(q=4\); they agree to floating-point precision for both reduced residues).

Experiment 1: \(S(N)\) at the true \(R_0(N)\), across a ladder of \(N\).

The local exponent \(\log(S_2/S_1)/\log(N_2/N_1)\) between consecutive rows is \(3.52,1.82,3.21,3.08,2.35\) — noisy at this scale (only six points, \(R_0\) itself only \(4\) to \(32\)), but its average, \(2.80\), and every individual value, sit above \(2\), several comfortably above \(2.5\). Reading the two normalized columns: \(S/N^2\) climbs by a factor of \(\approx51\) while \(N\) grows by a factor of \(100\) — far more than any fixed power of \(\log N\) could plausibly account for (matching that climb via log powers alone, using \(\log(2\times10^6)/\log(20000)\approx1.46\), would need an implausible fixed exponent near \(10\) on \(\log N\)) — so this data does not read as consistent with \(S(N)=O_\epsilon(N^{2+\epsilon})\) at these scales; it reads as a genuine excess power of \(N\) above \(2\). By contrast \(S/N^{2.5}\) and \(S/(R_0N^2\log N)\) — the latter being (E2)'s own crude bound, evaluated with its actual constant \(1\), not just in \(\asymp\)-notation — both stay within a single order of magnitude across the whole \(100\)-fold range of \(N\), with no trend toward \(0\): the ratio to the proven \(N^{5/2}\)-order ceiling, if anything, drifts slightly up (from \(0.007\) to \(0.03\)) rather than down. What the actual arithmetic shows, at every scale tested here, tracks the order of the ceiling this document and its predecessors have already proved (E2), not the order of the target. This is consistent with the \(N^{1/2}\) gap being a real feature of \(S(N)\)'s true size, not merely unproved slack in (E2)'s derivation — though six points spanning two orders of magnitude in \(N\), at \(R_0\) still only in the tens, cannot rule out a different asymptotic regime setting in once \(R_0\) is very large; a finite measurement settles neither reading of an asymptotic question, and this one is reported as measured, not as a proof of either.

Experiments 2a-2b: fixed-\(Q\)/fixed-\(N\) decomposition, matching rank3_bdh_probe.py's structure. With \(Q=8\) fixed and \(N\) swept over the same ladder, the local \(N\)-exponent of \(S\) is \(2.28,1.49,2.12,2.65, 0.98\) — noisy, averaging \(1.90\), i.e. closer to the target's own \(N^2\) at a fixed, small modulus range than the joint experiment above, which also lets \(R_0\) grow. With \(N=400000\) fixed and \(Q\) swept from \(4\) to \(30\), the local \(Q\)-exponent is \(2.22,1.53,2.62,2.04,1.64,1.58\), averaging \(1.94\) — closer to a bare \(Q^2\) (no large-sieve-type saving visible yet at these small \(Q\)) than to BDH's own asymptotic \(Q\log Q\) shape, echoing exactly the caveat rank3_bdh_probe.py's own Experiment 1 already recorded for \(D(N,Q)\) itself ("this experiment alone cannot distinguish an asymptotic regime not yet reached... from a genuinely steeper \(Q\)-dependence"). Both sub-experiments are reported as measured, not fit to either shape, for the same reason.

4. Verdict

Route A, as RANK3_SCOPE.md poses it, is not closed here, and no unconditional closing route is found. Restated precisely:

• The individual-modulus form of Route A has no unconditional route in either source document; its only exhibited closing path is RH-circular (Section 1, citing RANK3_SCOPE.md).
• The averaged-in-\(q\) substitute this document's predecessors identify (BDH, applied pointwise-in-\(t\) and summed) reaches \(O(N^{5/2})\), matching CHHL's GRH-conditional benchmark unconditionally but falling short of the target by exactly one power of \(N^{1/2}\), and no sharper citable theorem closes that gap (Section 1, citing RANK3_BDH_VERIFY.md).
• This document's own further candidate — applying the same large sieve to the \(K_N\)-continuum directly, rather than pointwise-in-\(t\) — is not an improvement: it gives \(O(N^3L)\), a full power of \(N^{1/2}\) worse than (E2), for a structural reason (Section 2) that generalizes: any large-sieve application that treats \(\beta\) (or \(t\)) as a parameter to integrate a fixed-\(\beta\) ceiling over, rather than as a variable the sieve genuinely averages over jointly with \(q\) and \(\chi\), pays for the averaging twice — once implicitly in the ceiling, once explicitly in the \(\int|K_N|^2=N\) factor.
• Direct measurement of \(S(N)\) itself (Section 3), at the real, jointly varying \(R_0(N)\), gives no empirical support for \(S(N)=O_\epsilon(N^{2+ \epsilon})\) over the range tested, and tracks the shape of the already- proved \(N^{5/2}\)-order bound instead — evidence, not proof, that the remaining gap reflects a genuine difference in size, not merely a proof technique that has not yet been sharpened.

What would be needed, stated as precisely as this document can: a mean-value theorem that bounds \(\sum_{q\le Q}\sum_{\chi\bmod q}\int_{\mathbb T}|K_N(\beta)\Psi(\beta,\chi)|^2\,d\beta\) — equivalently \(\sum_{q\le Q}\phi(q)\sum_b^*T(q,b)\) — genuinely jointly in \(q\), \(\chi\), and the length/frequency variable \(t\)/\(\beta\) at once, at an order beating both \(O((N+Q^2)Nd_N)\) (Section 2's ceiling) and \(O(QN^2\log N)\) (the crude \(t\)-summed BDH bound), down to \(O(QN^{1+\epsilon})\) or better. This is exactly the object RANK3_MEAN_VALUE_TOOLS.md Section 8 already named as missing; this document adds one further ruled-out route to reach it (the continuum large sieve of Section 2) and a direct measurement (Section 3) suggesting the gap it would need to close is real, not an artifact of which proof technique has been tried so far. Neither of those additions supplies the missing theorem, and this document does not claim to have found one under another name.

This document is citation of two prior documents in this hunt (Section 1), one further exact large-sieve computation with a fully explicit constant chain (Section 2), and one direct numerical measurement from the true von Mangoldt function with no model or sampling (Section 3); it assumes and establishes nothing about zeros of \(L\)-functions or the Riemann Hypothesis.