Independent check, 2026-09-12 (attempt a-0078). Scope: the five items listed in the assignment, covering sections 1, 2 (arcs), 3 (major arcs), 4 (minor arcs, including 4.1 in full), and 5 (budget) of ARC_SPLIT_BUDGET.md at commit dbe1c98.
Read: ARC_SPLIT_BUDGET.md in full; UPPER_BOUND.md sections 2-6; LOCALIZED_MIXED_ENERGY.md sections 1 and 6; SIEGEL_UNIFORMITY.md in full; ENDPOINT_SHARP.md sections 1, 4, 5, 6; ENDPOINT_BOUND.md sections 1-7.
Item 1 was checked with a script written from scratch for this review, review_a0078_identity_check.py, results in results_review_a0078_identity_check.json; it does not import or reread arc_split_probe.py. Items 2-5 are hand recomputations against the cited source equations, reproduced below with the intermediate numbers.
Item 1: the identities of section 1 (equations (2) and (3))
Verdict: confirmed.
Both identities are algebraic consequences of the definitions and hold exactly, not just asymptotically.
Equation (2). Writing the coefficient of $|F|^2$ at shift $h\ge1$ as $\psi_2(N,h)$ and of $V_y$ as $(N-h)\mathfrak S_y(h)$ (both standard, from the Ramanujan expansion of $|K_N(\alpha-a/q)|^2$ and orthogonality), and unwinding $X=\mathcal C(2\mathrm{Re}(\overline HW))$, $Y=\mathcal C(|W|^2)$, $R_{\rm mod}=\mathcal C(|H|^2-V_y)-\widehat C_N$ from LOCALIZED_MIXED_ENERGY.md section 6, gives $[X+Y]_h=\psi_2(N,h)-\psi_2^a(N,h)$, $[R_{\rm mod}]_h=\psi_2^a(N,h)-(N-h)\mathfrak S_y(h)-C_N(h)$, and their sum $[G_{\rm corr}]_h=\psi_2(N,h)-(N-h)\mathfrak S_y(h)-C_N(h)$, matching the document exactly. The toy-model script confirms this in coefficient space at $N=600$: the maximum absolute difference between $[G_{\rm corr}]_h$ and $[X+Y]h+[R{\rm mod}]_h$, computed two independent ways (direct windowed sums $\psi_2,\psi_2^a,\mathfrak S_y$ versus the definitional sum), is $4.4\times10^{-16}$.
Equation (3). $\mathcal C(|F|^2)-\mathcal C(|H|^2)=|F|^2-|H|^2-\kappa$ since the constant Fourier coefficients are $d_N$ and $\sum a(n)^2$ respectively, and $G_{\rm corr}=\mathcal C(|F|^2)-\mathcal C(V_y)-\widehat C_N$ splits as $[\mathcal C(|F|^2)-\mathcal C(|H|^2)]+R_{\rm mod}$, giving $G_{\rm corr}=(|F|^2-|H|^2)+R_{\rm mod}-\kappa$ exactly as stated. Checked pointwise on an independent toy model, $\nu(n)=b\,1_{(n,30)=1}$ built from scratch (not the document's own toy model or script) at $N=600$, $Z=10$, on a grid of $4096$ points, with $F,H$ evaluated directly and $V_y$ via the closed form of $K_N$: the maximum of $|G_y-(|F|^2-|H|^2)-R_{\rm mod}+\kappa|$ is $9.1\times10^{-13}$ against $\max|G_y|=2.24\times10^4$, relative defect $4.1\times10^{-17}$, with no exceptional character ($C_N=0$), matching the document's own check (1) in character but on an independently built model.
Item 2: major arcs, section 3
Verdict: confirmed, including the account of where $R^7$ became $R^2$.
Recomputing each step: (5) follows from ENDPOINT_SHARP.md (1$'$) applied to each residue class mod $r$ ($r\le R$ classes) plus Abel summation against $e(n\theta)$ (cost $1+2\pi N|\theta|$), then $N|\theta|\le R/r$ on $I_{r,a}$ gives $r(1+2\pi R/r)=r+2\pi R\le(1+2\pi)R<8R$ since $r\le R$; this reproduces $|W(\alpha)|\le8C_1RNe^{-\gamma\sqrt\ell}$ exactly. For (6): $||F|^2-|H|^2|\le|W|(|F|+|H|)$ is the standard factorization $F-H=W$, so $(|F|^2-|H|^2)^2\le|W|^2(|F|+|H|)^2$, and bounding $\sup_{\mathfrak M}|W|^2$ times $\int_{\mathbb T}(|F|+|H|)^2\le 2(d_N+\sum a(n)^2)\ll N\ell$ (Parseval, $d_N=\sum\Lambda(n)^2$, $\sum a(n)^2\ll Nb^2\ll N\ell$) gives $64C_1^2R^2N^2e^{-2\gamma\sqrt\ell}\cdot O(N\ell)=N^3\ell R^2 e^{-2\gamma\sqrt\ell}$, matching.
The comparison with ENDPOINT_BOUND.md (9), $N^3\ell^2R^7e^{-2\gamma t}$: tracing that derivation, $\sup_{\mathfrak M}|W|\ll NR^2e^{-\gamma t}$ there (radius $2R/N$ gives $N|\theta|\le2R$, so $r(1+N|\theta|)\le R(1+4R)\ll R^2$, one power of $R$ worse than here), $|\mathfrak M|\ll R^3/N$ there (radius $2R/N$ instead of $R/(rN)$, one more power of $R$ in the measure, absent entirely from the new derivation which uses no measure factor at all), and the pointwise $(|F|+|H|)^2\ll N^2\ell^2$ there in place of the integrated $\int(|F|+|H|)^2\ll N\ell$ (one power each of $N$ and $\ell$). Multiplying out: old $=(R^3/N)\cdot(N^2R^4e^{-2\gamma t})\cdot(N^2\ell^2) =N^3R^7\ell^2e^{-2\gamma t}$; new $=(N^2R^2e^{-2\gamma\sqrt\ell})\cdot(N\ell) =N^3R^2\ell e^{-2\gamma\sqrt\ell}$. Ratio old/new $=R^5\ell$, splitting as claimed into $R^3\ell$ from dropping the arc-measure factor via Parseval and $R^2$ from the arc radius. The document's accounting of the $R^7\to R^2$ change is correct in every factor.
(7) and (8): the split $[R_{\rm mod}]_h=[\psi_2^a-(N-h)\mathfrak S-C_N]+(N-h)(\mathfrak S(h)-\mathfrak S_y(h))$ is exact algebra from (2); squaring and summing with $(a+b)^2\le2a^2+2b^2$ gives $\|R_{\rm mod}\|_2^2\ll N\cdot N^2e^{-2c_m\sqrt\ell}+D_{\rm tail}(N,y)\ll N^3e^{-2c_m\sqrt\ell}+N^2$ (using $D_{\rm tail}(N,y)\ll N^2$ from UPPER_BOUND.md section 2 at $y=\lfloor\sqrt N\rfloor$). The constant term $\kappa^2|\mathfrak M|\ll(N\ell)^2\cdot R^2/N=NR^2\ell^2$ follows directly from $|\kappa|\ll N\ell$ (section 1) and $|\mathfrak M|\le2R^2/N$. Assembling the three parts with $(a+b+c)^2\le3(a^2+b^2+c^2)$ reproduces (8) exactly.
Item 3: minor arcs, section 4
Verdict: confirmed.
The identity $G_{\rm corr}|_{\mathfrak m}=|F|^2-H_R-a_0(N,R)-T_{y,R} -\widehat C_N$ is UPPER_BOUND.md (10), $G_y=B_Q-H_Q-a_0(N,Q)-T_{y,Q}$, restricted to $Q=R$ and to $\mathfrak m$, where the arc model $A_R$ vanishes by construction (it is supported on indicator functions of the arcs comprising $\mathfrak M_R$), so $B_R=|F|^2-A_R=|F|^2$ there, and $G_{\rm corr}=G_y-\widehat C_N$. A point worth recording: UPPER_BOUND.md (7) defines its arcs with radius $\delta_q=Q/(qN)$, which at $Q=R$ is exactly the radius $R/(rN)$ that ARC_SPLIT_BUDGET.md section 2 adopts (rather than ENDPOINT_BOUND.md's $2R/N$). The two arc systems therefore coincide exactly when $Q=R$, which is what makes citing UPPER_BOUND.md's estimates directly, without re-deriving them for a different arc geometry, valid.
On the four cited bounds holding for arbitrary $R$ with $2R^2<N$, not only for the specific $Q$ used in UPPER_BOUND.md's own sections:
- (20), $I_Q\ll(N^3/Q+N^{13/5})L^6$: derived in
UPPER_BOUND.mdsection 5 from Dirichlet approximation with $\lceil N/Q\rceil$, which uses only the general arc setup (7) (needing $2Q^2<N$ for disjointness), not the specific choice $Q=\lfloor L^B\rfloor$ used earlier in that section. Valid for any $R$ with $2R^2<N$. - (13), $\|H_Q\|_2^2\ll(N^3/Q^2)\log(2Q)w(Q)^2$:
UPPER_BOUND.mdstates this "uniformly for the parameters in (7)", i.e. for any $Q$ meeting $2Q^2<N$. Valid. - $a_0(N,Q)=N\log(N/Q)+O(N)$: a CHHL asymptotic identity for the Ramanujan-sum truncation at any level $Q\le N$, not tied to a specific $Q$. Valid.
- (9), $\|T_{y,Q}\|2\ll N^{3/2}/Q$: this uses $D{\rm tail}(N,Q)\ll N^3/Q^2$, which
UPPER_BOUND.mdsection 2 proves only for $1\le z\le \sqrt N$. The constraint $2R^2<N$ inARC_SPLIT_BUDGET.mdsection 2 gives $R<\sqrt{N/2}<\sqrt N$, so this is exactly the range where the bound is available; the constraint is not incidental, it is what this step needs.
The assembly (9) is Cauchy-Schwarz on the five-term pointwise identity ($(a_1+\cdots+a_5)^2\le5\sum a_i^2$), and (11) follows by substituting the four bounds above plus (10) (checked in item 4) and dropping subdominant terms ($N^3/R^2$ terms are $\le N^3/R$ for $R\ge1$, absorbed into the $\ell^{O(1)}R^{-1}$ notation). Recomputed and correct.
Item 4: the correction polynomial on the minor arcs, section 4.1
Verdict: confirmed, with one auxiliary estimate (identified below) accepted on order-of-magnitude grounds rather than fully re-derived, and noted because it is not the binding term either way.
The $C_N(h)$ formula. Substituting $u_q(h)=\mu(q)\chi(-h)/q$, $v_q(h)=\mu(q)\chi(h)/q$ (odd $q$, SIEGEL_UNIFORMITY.md (17)) into $C_{q,\beta,Z}(h)=\mathcal L_h[-u_q(h)J_1(h)-v_q(h)J_2(h)+(c_q(h)/q) J_{12}(h)]$ (its (19)) reproduces the displayed formula exactly, with the $1_{q\ \rm odd}$ correctly gating the linear part (at $4\mid q$, $u_q=v_q=0$, only the quadratic term survives, matching (19) directly).
The quadratic piece. Recomputed the full three-range split from scratch.
Range $R<m\le\sqrt N$. Writing $A(h)=\sum_{R<qd\le\sqrt N}(q/\phi(qd)^2) c_{qd}(h)$, expanding $c_{qd}(h)=\sum_a^*e(ah/qd)$ turns $\sum_h|A(h)|^2$ into a dual-large-sieve sum over $\sim1/N$-spaced points (fractions with denominator $\le\sqrt N$ are at least $1/N$ apart), giving $\sum_h|A(h)|^2 \le2N\sum_{R<qd\le\sqrt N}\phi(qd)q^2/\phi(qd)^4$ (the $2N$ is $N+\delta^{-1}$ at $\delta=1/N$, matching the additive large sieve (LS) of UPPER_BOUND.md used in dual form). With $\sup J_{12}(h)^2\le4N^2$, the total is $8N^3(q^2/\phi(q)^3)\sum_{d>R/q}\mu(d)^2/\phi(d)^3$, exactly the displayed chain. Using $\sum_{d>D}\mu(d)^2/\phi(d)^3\ll D^{-2}\ell^{O(1)}$ ($D\ge1$) or $O(1)$ ($D<1$) and $q/\phi(q)\ll\ell$: at $q\le R$ this gives $N^3\ell^{O(1)}q/R^2$ (using $q^4/\phi(q)^3=q(q/\phi(q))^3\ll q\ell^3$); at $q>R$ it gives $N^3\ell^{O(1)}/q$ (using $q^2/\phi(q)^3=(1/q)(q/\phi(q))^3 \ll\ell^3/q$). Both recombine exactly into $N^3\ell^{O(1)}\min(1/q, q/R^2)$, and both branches of the minimum are indeed $\le1/R$ as stated ($q\le R\Rightarrow q/R^2\le1/R$; $q>R\Rightarrow1/q<1/R$); the direction is not reversed.
Range $m>\sqrt N$. The final order $N^{2+o(1)}$ was checked dimensionally: with the stated per-$h$ coefficient bound $\ll Nq\tau(h)\ell^{O(1)}/\sqrt N$, $\sum_h\tau(h)^2\ll N\ell^3$ (standard), and $q<Z=e^{\sqrt\ell}$, the mean square is $\ll(N^2q^2/N)\cdot N\ell^3\cdot\ell^{O(1)}=N^2q^2\ell^{O(1)}\ll N^2e^{2\sqrt\ell}\ell^{O(1)}=N^{2+o(1)}$, matching. The auxiliary bound $\sum_{m>X,g\mid m}\phi(m)^{-2}\ll g\phi(g)^{-2}X^{-1}\ell^{O(1)}$ feeding the per-$h$ coefficient estimate was not independently re-derived here; it is a plausible sieve-type tail bound and, since $N^{2+o(1)}\ll N^3/R$ for any $R=N^{o(1)}$ regardless of its precise constant, an error here of a power of $\ell$ would not change the final budget, which never binds on this range.
Range $m\le R$ (present only when $q\le R$). Recomputed: centers $a/m$ for $m\le R$ lie in $\mathfrak M$ by construction of the arcs in section 2, so on $\mathfrak m$, $\|\alpha-a/m\|>R/(mN)$, giving $\sum_a^*\|\alpha-a/m\|^{-1}\le mN/R+O(m\log m)\ll mN/R$ for large $N$. Chaining this through the Abel bound $|\Phi_{J_{12}}(x)|\le J_{12}(1)/\|x\| \le2N/\|x\|$ and summing over $d\le R/q$ reproduces both displayed bounds, $\sup_{\mathfrak m}|\cdot|\ll N^2\ell^{O(1)}/R$ and $\int_{\mathfrak m}|\cdot|\ll N\ell^{O(1)}$, and hence $\ll N^3\ell^{O(1)} /R$ via $\int f^2\le(\sup f)(\int f)$.
The linear pieces. The Gauss-sum expansion $\chi(h)\tau(\bar\chi) =\sum_a\bar\chi(a)e(ah/q)$ combined with $S_*(h)=\sum_{d\mid P,(d,q)=1} c_d(h)/\phi(d)^2$ correctly produces a sum over fractions with denominator dividing $qd$ and coefficient modulus $\le q^{-1/2}\phi(d)^{-2}$; combined with the front weight $(q/\phi(q))^2q^{-1}=q/\phi(q)^2$, the net per-fraction weight is $q^{1/2}/(\phi(q)^2\phi(d)^2)$, exactly a factor $q^{-1/2}$ smaller than the quadratic piece's weight $q/(\phi(q)^2\phi(d)^2)$, and this is the weight the document uses later for the $qd\le R$ range. The three sub-ranges were not each re-derived to the same constant-tracking level as the quadratic piece, but the claimed final orders, $N^3\ell^{O(1)}/R^2$ for $R<qd\le\sqrt N$ (an extra $q^{-1}$ from the squared weight relative to the quadratic piece's $1/R$, consistent), $N^{2 +o(1)}$ for $qd>\sqrt N$ (same argument as the quadratic case), and $N^3\ell^{O(1)}/R$ for $qd\le R$ (same single power of $R$ as the quadratic case, since that range's $R$-dependence comes from one reciprocal-distance factor, not the large-sieve tail), are structurally consistent with the mechanism verified in full for the quadratic piece, and none of them exceeds $N^3\ell^{O(1)}/R$, so (10) follows.
The consistency check against SIEGEL_UNIFORMITY.md (23). Recomputed independently: at $q>R$, only the first two ranges apply (the third requires $q\le R$), giving $N^3\ell^{O(1)}/q$ from the quadratic piece (dominant over the linear piece's $N^3\ell^{O(1)}/q^2$ there) plus the subdominant $N^{2+o(1)}$. Taking the crude bound $J_1,J_2,J_{12}\ll N$ (the $\beta\to1$ worst case implicit in the document's generic bounds) in SIEGEL_UNIFORMITY.md (23), $N^{4\beta-1}q^2/\phi(q)^3\to N^3q^2/\phi(q)^3=N^3(1/q)(q/\phi(q))^3\ll N^3\ell^3/q$, exactly matching. The cross-check holds.
No lost factor of $q/\phi(q)$, $\phi(qd)$, or $\sqrt q$ was found in the quadratic-piece derivation (fully re-derived), no reversed direction in $\min(1/q,q/R^2)\le1/R$, and the $q>R$ case is explicitly and correctly handled (range 3 is explicitly gated on $q\le R$, and the consistency check confirms the $q>R$ regime separately).
Item 5: the budget, section 5
Verdict: confirmed. Recomputed both optima independently.
General balance. For a budget of the shape $N^3R^ke^{-2\gamma\sqrt\ell} +N^3R^{-j}$, balancing $2\gamma-k\sigma=j\sigma$ at $R=e^{\sigma\sqrt\ell}$ gives $\sigma^=2\gamma/(j+k)$ and exponent $j\sigma^=2\gamma j/(j+k)$.
This budget, $k=2$, $j=1$: $\sigma^*=2\gamma/3$, exponent $2\gamma/3$. Matches the document. Checked directly: at $\sigma=2\gamma/3$, $R^2e^{-2\gamma\sqrt\ell}=\exp((4\gamma/3-2\gamma)\sqrt\ell) =\exp(-(2\gamma/3)\sqrt\ell)$ and $R^{-1}=\exp(-(2\gamma/3)\sqrt\ell)$, equal as required.
Previous budget (ENDPOINT_SHARP.md (18$'$)), $k=7$, $j=1/6$, but with an extra factor $\tfrac12$ in that document's own definition $c'=\tfrac12\min(2\gamma-7\sigma,\sigma/6,c,2c_m)$: balancing $2\gamma-7\sigma=\sigma/6$ gives $12\gamma=43\sigma$, i.e. $\sigma^=12\gamma/43$, value at optimum $\sigma^/6=2\gamma/43$, and with the document's own $\tfrac12$ factor, exponent $=\gamma/43$. Matches the document's claim exactly. At the document's actually stated $\sigma=\min(\gamma/20,1/20)=\gamma/20$ (since $\gamma<1/2$ throughout, established in ENDPOINT_SHARP.md section 4's own range for $\gamma$): $2\gamma-7\sigma=33\gamma/20$, $\sigma/6=\gamma/120$, so $c'=\tfrac12\min(33\gamma/20,\gamma/120,\ldots)=\tfrac12\cdot\gamma/120 =\gamma/240$, exactly reproducing the stated $\gamma/240$ and confirming that $\gamma/43$ is a genuinely better exponent available from the same budget structure that was not the one used.
Constraints. $\sigma=2\gamma/3<1/3$ since $\gamma<1/2$, and $1/3< 1/\sqrt2$, so $\sigma\le1/\sqrt2$ holds with room. $2R^2<N$ holds for sufficiently large $N$ for any fixed $\sigma$. $2c_m>2\gamma/3$: with $c_m=1/4-o(1)$ (fundamental lemma at level $N^{1/4}$, giving sieve parameter $s=\sqrt\ell/4$ and hence $c_m\approx1/4$) and $\gamma<1/2$, $2c_m\to1/2-o(1)>1/3>2\gamma/3$. All confirmed.
Section 7's cross-term claim. $2c_m=1/2-o(1)$ exceeds both $\sigma/6=\gamma/120$ (this budget's binding rival in the old chain) and $2\gamma/3$ (this budget's own binding term) since $\gamma<1/2$ makes both $\gamma/120$ and $2\gamma/3<1/3$ strictly below $1/2-o(1)$. So in both budgets the cross term with $R_{\rm mod}$, bounded by $2\|R_{\rm mod}\|^2\ll N^3e^{-2c_m\sqrt\ell}$, is dominated by a strictly smaller exponent elsewhere and cannot move either budget's binding constant. Confirmed.
Section 6's list. Scanned ARC_SPLIT_BUDGET.md for the named objects: $D_0$, $A(N)$, the properties (P1)-(P4), and the term $N^{2+A(N)}e^{\sqrt\ell/2}$ do not appear anywhere in the document, nor do ENDPOINT_BOUND.md's Type I estimate (10), character estimate (12), Cases A/B (13)-(14), or cubic moment (16). What is cited instead is UPPER_BOUND.md (7), (9), (13), (15), (20), which is a different source document's estimates entirely, matching the claim that the divisor approximant is genuinely unused rather than silently smuggled back in. Input (1$'$) and input (2) of ENDPOINT_SHARP.md, cited directly in sections 3(a) and 3(b), are the two of the original four inputs that remain, plus the singular-series truncation (also via input (2)/$c_m$); the fundamental lemma inside the proof of (1$'$) is inherited implicitly through citing (1$'$) as a black box. Consistent with the document's own accounting.
Summary
No defect found in any of the five items. Item 1 was verified numerically to floating-point precision on an independently written toy model. Items 2, 3, and 5 were fully recomputed by hand against the cited source equations and match exactly, including the specific accounting of how $R^7$ became $R^2$ on the major arcs and the numerical optimum of both budgets. Item 4, the document's own most novel and highest-risk estimate, was recomputed in full for the quadratic piece and the three-range split technique (dual large sieve, gcd/tail bound, minor-arc reciprocal-distance bound), all of which check out exactly including the self-consistency test against SIEGEL_UNIFORMITY.md (23); the linear-piece constants and one auxiliary sieve-tail lemma in the $m>\sqrt N$ range were checked only at the order-of-magnitude level rather than re-derived symbol by symbol, but neither is the binding term in the final bound, so an error there confined to logarithmic factors would not change (S$'$).