# Adversarial review 19: prime-power-dispersion §§4-5 and singleton-fiber-audit §3 Read-only pass, 2026-09-06. Nothing in the repository was created, edited or deleted; `git status --porcelain` is clean apart from another agent's untracked `research/corner-log-average-validation.js`, which I did not touch. The two validators write deterministic artifacts (`data-reuse/factor-windows.json`, `data-reuse/prime-power-dispersion.json`) that reproduce byte-identically, so running them left no diff. ## Verdicts, one line each - **PART 1 (prime-power-dispersion.md §§4-5, (8)-(10), (12)-(13), and residual-coverage §3's consumption): NO DEFECT FOUND.** Every decisive step in (8), (9), (10), the assembly into (12) and (13), the k=1 / k>=2 table, the region (14), the supremum (17)-(18), the §7 budget table, and the `D -> D/P` substitution into residual-coverage (5)-(6) and the §3 budget table was re-derived independently and reproduced exactly. - **PART 2 (singleton-fiber-audit.md §3, (6)-(8)): CHECKED, and the counting argument does transfer to the corner boxes as corner-correlation.md uses it.** One defect of wording, not of mathematics, at `research/reachability-coverage.md:216-217`: the sentence claiming its S_0 count "localises" the *per-sign* `c*x*log x` lower bounds over-states what is derived — neither note establishes a per-sign lower bound on S_0, and deriving one would require the two-point Möbius correlation that the same wave records as OPEN. Neither part changes the OPEN status of the signed endpoint remainder or the twin margin. Nothing below is an estimate. --- # PART 1 — prime-power-dispersion.md §§4-5 ## 1.1 (8), the zero-numerator count — CHECKED Two classes, and they are exhaustive: *Identical prime, `h1=h2`.* `O(P)` bases in the band times `O(A)` harmonics, weight `(log p)^2 |c_h|^2 << log^2 x / A^2`. Total `<< (P/A) log^2 x`. Matches the first monomial. *Distinct primes.* `r = theta(h1 p2^k - h2 p1^k) = 0` with `(p1^k, p2^k) = 1` forces `p1^k | h1`, hence `h1 = j p1^k`, `h2 = j p2^k` with the same positive `j` — I re-derived this rather than accepting it, and it is exactly right. Since `p_i^k` lies in `[R, 2R)`, the count of admissible `j` with both `j p_i^k ~ A` is `O(A/R)`, and a solution requires `j >= 1`, hence `R <= 2A`, so `A/R >> 1/2` and the `+1` from interval endpoints is absorbed — the note's "absorbs the endpoint constant" is correct and load-bearing. `O(P^2)` prime pairs and weight `1/A^2` give `P^2/(AR)`. `P = R^{1/k} <= R` collapses this into `P/A`, with equality of order at `k=1`. The subsequent use is consistent: the `r=0` inner `n`-sum is bounded by `N f^2`, so the zero class contributes `f^2 D N P / A` to (12) — the first bracket term. Note: `|c_h| << 1/|h|` for Vaaler coefficients, giving `sum_{h~A}|c_h| = O(1)` and `sum_{h~A}|c_h|^2 = O(1/A)`; both are used and both hold, including in the clipped `A = 1` block. ## 1.2 (9), the prime-power gcd factor — CHECKED, and the refuted shortcut is genuinely refuted *Same prime.* `c = d p^k` with `p ∤ d`, so `gcd(r, c) = gcd(r,d) gcd(r, p^k)`; `p` odd and `theta ∈ {1,2}` give `gcd(theta(h1-h2), p^k) = gcd(h1-h2, p^k)`. *Distinct primes.* `c = d p1^k p2^k` with three pairwise-coprime factors. `r ≡ theta h1 p2^k (mod p1^k)` and `gcd(theta p2^k, p1^k) = 1`, so `gcd(r, p1^k) = gcd(h1, p1^k)`; symmetrically for `p2`. Hence `H = gcd(h1,p1^k) gcd(h2,p2^k)`. Both branches need `p_i` odd and `p_i ∤ d`; both hypotheses are in force (`B_k` is defined over odd primes only, and `p ∤ d` sits inside `P_d`). The refuted shortcut is correctly refuted and the full factor is kept: the note retains `gcd(h_i, p_i^k)` and `gcd(h1-h2, p^k)` everywhere downstream, never a radical. `gcd(25, 5^2) = 25 != 5`. The validator asserts the identity exactly — `assert.equal(bgcd(r,c), bgcd(r,d)*H)` — over the full pair loop, with 217,616 nonzero cases carrying a nontrivial `H` and 10,612 distinct-prime zero terms, so the "omit distinct-prime zero pairs" shortcut is also actively falsified rather than merely asserted. The example `(25,25)`, `(49,49)` does have equal frequency. ## 1.3 (10), the harmonic averages with exponent alpha — CHECKED The majorization `gcd(h,p^k)^alpha <= sum_{0<=j<=k, p^j | h} p^{j alpha}` is valid: the term at `j0 = min(v_p(h), k)` is present and equals the left side. Each `j` contributes at most `p^{j alpha} * 2A/p^j = 2A p^{j(alpha-1)} <= 2A` because `alpha <= 1`, and only `p^j <= 2A` can divide a positive `h <= 2A`. `k+1` values of `j` give `A(k+1)`. For the difference sum, each nonzero value `n = |h1-h2| < A` is realised by `O(A)` ordered pairs, and the first bound applied on `[1,A)` gives `A^2(k+1)`. Normalized: for a distinct prime pair the two `h`-sums separate and give `O((k+1)^2)`; for the same prime `O(k+1)`. `k = O(log x)` since `p^k <= V` and `p >= 3`, so these are fixed powers of `log x`, uniformly in `k`. The note's "up to fixed logarithms uniform in k" is the correct reading of that. Validator coverage note: `prime-power-dispersion-validation.js` tests (10) on the **half-open** band `[A, 2A)` (`for(let h=A;h<2*A;h++)`), whereas the note's `h ~ A` reads as `(A, 2A]`. Same class of gap the chain review found in `grouped-divisor-validation.js`. I re-ran (10) on `(A,2A]` and `[A,2A]` independently (2,538 cases; `p ∈ {3,5,7,17,31,101}`, `k <= 8`, `A <= 16384`, `alpha ∈ {1/2, 1}`): maximum ratio to `A(k+1)` is 1.4688 and to `A^2(k+1)` is 1.0000, both inside the validator's own constants 2 and 4. **Not a defect** — a validator coverage gap that the closed-band re-run closes. The `d`-average `sum_{d~D} gcd(r,d)^alpha << D tau(|r|)` (dispersion-range (11)) was re-derived: `gcd(r,d)^alpha <= sum_{s | (r,d)} s^alpha`, and each divisor `s` of `|r|` contributes `D_0 s^{alpha-1} + s^alpha <= 3 D_0` for `alpha <= 1`. No hypothesis `r < D` and no coprimality is used, and `r = 0` is correctly excluded. ## 1.4 The assembly into (12) — CHECKED I reconstructed each of the four bracket terms independently, with the harmonic and prime sums put back in: | class | reconstruction | (12) | |---|---|---| | zero numerators | `D * N * (P/A) * f^2` | `f^2 D N P / A` | | cross-prime Weil | `sqrt(p1^k p2^k) * sqrt(D) * D tau(r) * (k+1)^2 * P^2` = `x^eps D^{3/2} R P^2` | `D^{3/2} R P^2` | | same-prime Weil | `sqrt(p^k) * D^{3/2} * (k+1) * P` = `x^eps D^{3/2} R^{1/2} P` | `D^{3/2} R^{1/2} P` | | complete periods | `(N H / (p1^k p2^k)) * sum_d gcd(r,d)/d` = `N P^2/R^2 <= N` (same prime: `N P/R <= N`) | `N` | The `f^2` prefactor is right in both directions: the zero class has `|Phi_1 conj(Phi_2)| << f^2` with no variation cost, and the nonzero classes carry `f^2 (1 + r_phase) << f^2 x^{3 tau}` from the sparse-dispersion (9) derivative argument, which transfers because differentiation is in `n` and `d p^k ~ M ≍ DR`. `r_phase = Ax/(DRN) << x^{3 tau}` on the retained boxes `MN > x^{1-tau}` with `A <= T = ceil(x^{2 tau} max(1, MN/x))`. The completion input (11) is `prime-dispersion.md` (11), which I checked against its cited source (see §1.8): complete periods cost `(M/c)G` by the Ramanujan bound, the residual interval costs `sqrt(cG) log c` by Fourier inversion plus the composite-modulus Weil bound. No squarefree-modulus hypothesis anywhere. ## 1.5 (13), including the envelope — CHECKED `|B_k|^2 << F^2 D * (12)` is the two Cauchy steps (in `n`, then in `d`), and I checked the enlargement of the nonnegative moment to all `d` in the interval is the only place a restriction is dropped — `p ∤ d` and `(n, dp^k)=1` stay inside `Y_d`, as claimed. The envelope `f/sqrt(A) <= sqrt(x/(DRN))` is `min(u,v) <= sqrt(uv)` with `u = 1/sqrt(A)`, `v = sqrt(A) x/(MN)`, `M ≍ DR`. Applying it to the zero term gives exactly `sqrt(DxP/R)`; the three nonzero terms use `f <= 1` and give `D^{5/4}R^{1/2}P`, `D^{5/4}R^{1/4}P^{1/2}`, `sqrt(DN)` after the outer `F sqrt(D)`. All four match (13). The `k=1` / `k>=2` table follows from `P = R` and `P <= R^{1/2}` respectively, with `R <= V`; each of the eight cells reproduces. ## 1.6 The claim that the first-power case removes the small-prime cut and the unpaired diagonal — CHECKED *Small-prime cut.* `dispersion-range.md` §3 needed `p > x^kappa > 2T` precisely so that `gcd(r,c) = gcd(r,d)` with no distinguished-prime factor and no distinct-prime zero numerators. (8)-(10) price both of those objects directly, so the cut is no longer needed for odd `p`. The prime 2 is still removed separately as `Q_2` by the paired bound (support `O(D)`), which the note states. So: "the small-prime cut is removed" is true **for the odd primes**, which is what `B_1 = Q_odd` contains. Correct as written. *Unpaired diagonal.* In `dispersion-range` the diagonal used `A >= 1` and gave exponent `a+b-w/2`, which needed its own condition in (2). Here the zero class is paired, giving `J = 1/2 + (a+b)/2 - w/2`. That is strictly smaller whenever `a+b > 1`, which holds throughout the region. I verified the two implications: - `W_L < 1` and `a > 2w` give `a+b < 2 - 7w/2 = 29/25`, hence `J < 24/25`. - `P_0 = b + a/2 - w/2 = 2 W_L - w - 2a < 2 - 5w = 4/5`. Both are exact; the validator checks the same two rational identities. So (14) legitimately drops the diagonal condition. ## 1.7 §6-§7 region arithmetic — CHECKED in exact rationals, by hand and against the validator - `C' = C - w/2` and `G' = G - w/2` for `B_exc` follow from `dispersion-range (4)` with `a' = a`, `rho = (a-w)/2`, `b' = b`, `sigma = b/2`: the `(a+b)` coefficients cancel to zero (`1/2 + 1/5 - 7/10 = 0`) leaving `-w/2`. Cross-check: `dispersion-range` gives `C - w/4` for the whole `B` sector, and `B_exc`'s norm is smaller by exactly `w/4`, so `C - w/2` is consistent. - `G <= (25/22)(C - 3/20)` reduces to `7(t - 2 max(a,b))/88 <= 0`, i.e. to `max(a,b) >= (a+b)/2`. True. Then `C < 28/25` gives `G - w/2 < 2161/2200`. - Low terms save `7w/10` in the first exponent and `7w/8` in the second, relative to `C` and `G` — I confirm the comparison is against `C`, `G`, not against `C - w/2`, and it holds because `max(a-w,b) <= max(a,b)`. - §7 table at `a = 0.487`, `b = 0.662` (`delta = 0.247`, `nu = 0.612`, product `0.859`): `C = 1.1198`, so `C - w/2 = 0.9998`; `W_L = 0.99975`; `J = 0.9545`; `P_0 = 0.7855`; `W_L - w/2 = 0.87975`; `W_L - 3w/4 = 0.81975`; `C - 7w/10 = 0.9518`. All seven reproduce; the validator's rational output `4999/5000, 3999/4000, 1909/2000, 1571/2000, 3519/4000, 3279/4000, 4759/5000` is the same list. - The two control budgets are correctly reported as insufficient here: `C - (w+v)/4 = 1.0473` (BB) and `a+b-w/2 = 1.029` (unpaired diagonal). - Supremum (17): `max(a,b) >= b` turns `C < 28/25` into `14a + 19b < 97/5`; `W_L < 1` gives `5a + 2b < 94/25`; the multipliers `3/67` and `5/67` give coefficient 1 on both `a` and `b` and right side `77/67`, hence `delta + nu < 77/67 - 6/25 - 1/20 = 5757/6700`. The intersection `(816/1675, 1109/1675)` sits on both boundary lines, has `b* > a*` (needed for `max(a,b) = b`), `a* = 0.48716 > 2w = 0.48`, and `J`, `P_0` strictly below 1. Reproduced exactly. ## 1.8 Consumption in residual-coverage §3 — CHECKED *The `D -> D/P` substitution (residual-coverage (5)).* The first branch's original divisor is `pm = p g d ∈ I`, so `gd` lies in a single interval of length `≍ D/p ≍ D/P`, and the union over the band `p ∈ [P, 2^{1/k}P)` is one interval of length `≍ D/P`. Both the size of `d` and the length of the `d`-range scale as `D/P`, and `M ≍ (D/P) R`, so the substitution is consistent in all three roles `D` plays in (13). Substituting term by term: `sqrt((D/P)xP/R) = sqrt(Dx/R)`, `(D/P)^{5/4}R^{1/2}P = D^{5/4}R^{1/2}P^{-1/4}`, `(D/P)^{5/4}R^{1/4}P^{1/2} = D^{5/4}R^{1/4}P^{-3/4}`, `sqrt(DN/P)`. All four match (5) exactly. *The reduction (5) -> (6).* Uses `R <= 2PW` (from `p^k = p * p^{k-1} <= 2P W`), `P <= W` (from `p <= p^{k-1} <= W`, valid because `k >= 2` on this branch) and `P >= 1`. Gives `D^{5/4}W^{3/4}`, `D^{5/4}W^{1/4}`, `sqrt(Dx)`, `sqrt(DN)`. The "smaller by `3w/4` than the first-power cross budget" claim is `W^{3/2} -> W^{3/4}`, i.e. `3w/4`. Correct. The note's own caveat that (6) supplies no extra saving on the zero term is honest and correct. *The prime-free moment (8).* I re-derived it from scratch: zero weight `O(1/A)`, Weil `D^{3/2}`, periods `N`, then `f/sqrt(A) <= sqrt(x/(DN))` gives `F[sqrt(Dx) + x^{3tau/2}(D^{5/4} + sqrt(DN))]`. Matches. *Coefficient coverage.* The seven pieces — `A_0`, low `A_1`, `Q_2` (prime-free (8)); `Q_odd = B_1` ((13), `k=1`); odd `B_k`, `k>=2` ((13), `P <= sqrt(R)`); odd first branch ((6)); `p=2` first and negative branches ((7)) — exhaust `A_0 log n + A_1` via `coefficient-structure.md` (1)-(3), which I checked matches `prime-power (1)` and `residual-coverage (3)` term for term. `(9)` dominates all seven envelopes. "There is no remaining `B_exc` norm condition" is therefore correct. *The §3 budget table.* Substituting `F << sqrt(N)`, `D = x^{a-w}`, `W = x^w`, `N = x^b` into (9) gives left `W = 5a/4 + b/2 + w/4`, `J = 1/2 + (a+b)/2 - w/2`, `P = b + a/2 - w/2`; the right row is the same with `a <-> b`, `w <-> v`. Both rows reproduce. *(11)-(12), the simplification.* `J_R = 31/50 + theta/2` is exact, and `J_R < 1 <=> theta < 19/25` is an equivalence, so the `19/25` edge is genuinely set by the right zero term as claimed. `W_R <= 5 theta/4 + 3/200` and `P_R <= theta + 43/200` **use `delta >= w` and `nu >= v`** respectively — the crude bound `delta/2 + 5nu/4 <= 5 theta/4` would give the constant `39/200`, not `3/200`, and the claimed `193/200` at `theta = 19/25` would fail. The domain (1) does supply `delta >= w`, so the note is right, but the dependence on the domain constraint is implicit in the display and worth a word if that display is ever reused outside (1). Not a defect. `W_L = (5 delta + 2 nu)/4 + 77/200 < 1 <=> 5 delta + 2 nu < 123/50` reproduces. The frontier crossing `(47/150, 67/150)`, the regional supremum `87/100` approached at `(6/25, 63/100)`, the containment of the generic paired region (`C < 1 => a+b < 34/33`, so `theta < 0.7403 < 19/25`), the concrete `theta = 3/4 -> 199/200` and `lambda = 49/20 -> 399/400`, and all four budgets at `(delta,nu) = (2/5,2/5)` (`179/200`, `51/50`, `21/25`, `217/200`) all reproduce in exact rationals. ## 1.9 Imported theorems, with primary source and hypotheses - **Ramanujan and Weil bounds.** `link.springer.com/article/10.1007/s00039-026-00746-0` fetched successfully (HTTP 200, 849 KB). **Lemma 3.2 (Ramanujan bound):** "For `c ∈ Z_+` and `n ∈ Z`, one has `|S(0,n;c)| <= (n,c)`." **Lemma 3.3 (Weil bound):** "For `c ∈ Z_+` and `m, n ∈ Z`, one has `S(m,n;c) << c^{o(1)} sqrt((m,n,c) c)`", proved from Iwaniec-Kowalski Corollary 11.12 plus the divisor bound. Numbering, statement and quantifiers confirmed verbatim; **no squarefree-modulus hypothesis, `n = 0` and nonprimitive numerators admitted**, which is exactly how the notes use them. This **repairs** the `chain-review-0906.md` §5 row "numbering not confirmed by the fetcher" and the §7 entry listing these as unchecked. - **Bettin-Chandee, Theorem 1 and Remark 1**, `arxiv.org/pdf/1502.00769` (HTTP 200). Theorem 1 reads `B(M,N,A) << ||alpha|| ||beta|| ||nu|| (1 + |theta|A/(MN))^{1/2} [ (AMN)^{7/20+eps}(M+N)^{1/4} + (AMN)^{3/8+eps}(AN+AM)^{1/8} ]`, for arbitrary coefficients on dyadic `[M/2,M]`, `[N/2,N]`, `[A/2,A]`, with `(m,n)=1` and `theta != 0`. With `||nu|| ≍ A^{-1/2}` (Vaaler) and `A ≍ MN/x` this collapses to `x^{3/20}||alpha|| ||beta|| (MN)^{1/5}(M+N)^{1/4}` and `||alpha|| ||beta|| (MN)^{3/8}(M+N)^{1/8}` — **exactly** the two monomials of `dispersion-range.md` (4) that `prime-power §6` uses for `B_exc`. Remark 1 permits the endpoint perturbation with the factor replaced by `(1 + (|theta|A + X)/(MN))^{1/2}`, as the notes assume. Hypotheses met as used. ## 1.10 Do the validators test what the notes say? Mostly yes, with the scope the notes themselves state. Directly tested: the exact decomposition (1)-(2) against an independently built truncated convolution (1,883,736 identities); the support and norm majorants of (3); the gcd identity (9) as an exact BigInt equality over every pair, with active nontrivial-`H` and distinct-prime-zero counters; the `O(A/R)` collision count of (8) as `count <= 3A/R` (15 configurations); the two harmonic bounds of (10) with constants 2 and 4; the two Cauchy inequalities of (6); the shifted reciprocity identity with an active wrong-shift control (`0.11336` error); and the whole §6-§7 rational budget list including the supremum certificate. Ran in 3.13 s here; `embed.js --check` reports code, body, input and out hashes all matching for both scripts. Not tested, and correctly declared not tested: the assembly (12)-(13) itself — the summation of the completion bound over `d`, `h` and the prime pairs is a written derivation, and the note says so ("The uniform all-harmonic bound is the derivation (8)-(13)"). Also untested: any asymptotic rate; the note says that too. `singleton-fiber-validation.js` checks only **finite nonemptiness** of the two sign classes (`assert(pos && neg)`, with the comment "Finite nonemptiness, not the asymptotic density claim") — it does not test (6), (7) or (8) as inequalities. Honest, but it means §3 carries no numerical backstop in the repo; see §2.3 below for one I ran. --- # PART 2 — singleton-fiber-audit.md §3 ## 2.1 (6)-(8) — CHECKED *Odd pairs:* `UY/4 + O(U+Y+1)`. *Union bound (7):* each of `p^2|d`, `p^2|e`, `p | (d,e)` contributes main term `UY/(4p^2)` for odd `p`, hence `(3UY/4) sum_{p>2} p^{-2}`. The chain `sum_{p>2} p^{-2} <= sum_{j>=1}(2j+1)^{-2} < sum_{j>=1} 1/(4j(j+1)) = 1/4` is correct because `(2j+1)^2 > 4j(j+1)`, giving `< 3UY/16` and `|G| >= UY/16 + o(UY)`. The stated error terms `O(Y sqrt U + U sqrt Y + ...)` and `O((U+Y)log(2Y) + Y)` are the right shapes (the second comes from `sum_{p<=2Y}(U/(2p) + 1)(Y/(2p) + 1)`), and all are `o(UY)` at `U = x^{6/25}`, `Y = x^{1/20}`: `x^{0.17}`, `x^{0.265}`, `x^{0.24} log x` against `x^{0.29}`. *The sign split.* `T = sum_{d,e odd, (d,e)=1} mu(d) mu(e) = |G_+| - |G_-|` because nonsquarefree pairs weigh zero. The fixed-`P` argument is sound: for fixed `e`, the inner `d`-sum is `sum_{d~U, (d,m)=1} mu(d)` for one of at most `2^{pi(P)+1}` fixed moduli `m`, each `o_P(U)` by §2's fixed-`m` Möbius mean; summing `O(Y)` values of `e` gives `o_P(UY)`. Removing pairs sharing a prime `> P` costs `O(UY/P) + o(UY)` since `sum_{p>P} p^{-2} << 1/P`. Letting `x -> ∞` then `P -> ∞` is the standard limsup argument and is valid: `limsup |T|/(UY) <= C/P` for every fixed `P`. **No growing-modulus Möbius correlation is used**, and the note's own disclaimer to that effect is accurate. *(8).* `|G| >= UY/32` and `|G_+| - |G_-| = o(UY)` give each sign `>= UY/128` eventually; `d phi(e) <= 4UY` gives `sum_{G_sigma} 1/(d phi(e)) >= 1/512`. I also re-derived §4 (which is what turns (8) into (4)) since (4) is the cited result: the singleton criterion (`kv > x` and `g | 2` force at most `xg/(2kv) + 1 < 2` solutions), `kv ≍ x^{171/100} > x`, `beta_V(k) = log k`, the primitive progression `k ≡ 2 d^{-1} (mod e l)`, the uniqueness of the representation `q = e l` for fixed `d` (two primes `> 2Y` would force `l' | e <= 2Y`), the BV parameters (`q << x^{1/4}` against `sqrt K ≍ x^{19/50}`), the loose prime-power subtraction `x^{6/25 + 1/4 + 19/50} = x^{87/100}`, and the Mertens/PNT evaluation `sum_l log l/(l-1) = (1/10 + o(1)) log x`. All reproduce. `c = 1/40960` is conservative against `(1/2)(1/512)(1/20) = 1/20480`. Imported inputs for (4), with hypotheses: **Bombieri-Vinogradov for `Lambda`** (Tao, 254A Notes 3, Theorem 17) at level `q <= K^{1/2}/log^{B_A} K` with the maximum over coprime classes — the note's `q << x^{1/4}` against `x^{19/50}` meets it with a fixed power margin, and the one-representation lemma is what licenses summing the maximum without multiplicity; **quantitative PNT**; and **`sum_{n<=t} mu(n) = o(t)`** (Tao, 254A Notes 2, Exercise 66, `q=1` case), extended to fixed coprimality by the convolution argument in §2. I did not independently re-open the two Tao pages this pass; `chain-review-0906.md` §5 records both as source-confirmed. ## 2.2 Does §3 transfer to the corner boxes? — CHECKED, as used **What is actually claimed.** `corner-correlation.md:344` says only that "singleton-fiber-audit §3 derives that both signs occupy a positive proportion of odd squarefree coprime divisor pairs on a box, and that counting argument transfers verbatim to the corner's boxes." That is a claim about §3's *counting* argument, not about §4's `c x log x` bound. **It does transfer.** §3 uses only: (i) `D, E -> ∞` with intervals of length `≍ D`, `≍ E`; (ii) `sum_{p>2} p^{-2} < 1/4`; (iii) the fixed-`m` Möbius mean at modulus `-> ∞`. None of these involves `x`, the relation of `D, E` to `x`, Bombieri-Vinogradov, or any ordering of `D` and `E`. At the corner scales `D ≍ x^{19/25}` (interval length `≍ D`, since `D_0/D_1 = x^{2 eta_0}`) and `E ≍ x^{19/20}`, I redid the error bookkeeping: `E sqrt D = x^{1.33}`, `D sqrt E = x^{1.235}`, `(D+E) loglog x + D/log D = O(x^{0.95} loglog x)`, all `o(DE) = o(x^{1.71})`. One phrase does not transfer literally — "a common prime is at most `2Y`" becomes "at most `2 min(D,E)`", since at the corner `E > D` — but the resulting bound has the same form and the same order. So "verbatim" is one word stronger than the truth; "with `U, Y` replaced by the box's `D, E` and the same error bookkeeping redone" is exact, and the conclusion is unchanged. **What corner-correlation uses it for is legitimate.** The row is arguing that the support restrictions (`(d,e)|2`, oddness, squarefreeness) do not bias `mu(d) mu(e)`. §3 gives precisely that: both signs at positive density in counting measure over the box. It does **not** claim, and corner-correlation does not need it to claim, anything about the sign distribution once the determinant constraint `dr - er' = 2` and the `beta` weights are imposed. **§4 does not transfer, and corner-correlation says so correctly.** At the corner `K = x/d ≍ x^{6/25}`, `sqrt K ≍ x^{3/25}`, while `q >= e ≍ x^{19/20}` — BV is unusable by a factor `x^{0.83}`. Independently, `kt ≍ x^{29/100} < x`, so the corner fibers have `≍ x^{71/100}` points and are the opposite extreme to the singleton family, which is exactly what `corner-correlation.md:348` states. That row is correct. ## 2.3 The one defect: reachability-coverage.md:216-217 `research/reachability-coverage.md:216-217` reads: > "This bounds absolute term mass and says nothing about the signed value. It > localises [singleton-fiber-audit.md]'s `c*x*log x` lower bounds on each sign's > ungrouped mass." **DEFECTIVE as literally read.** §2.6 derives a lower bound `>> eta_0^2 x log^2 x` on the sum of **absolute values** of the terms of `R` restricted to `S_0` — both signs together. It does not split that mass by the sign of `mu(d) mu(e)`, and `singleton-fiber-audit §3` cannot supply the split here: §3 gives equidistribution of the sign in **counting measure over a raw divisor box**, whereas a per-sign lower bound on `S_0` would need the sign of `mu(n/r) mu((n-2)/r')` along the CRT progression `r | n`, `r' | n-2` — that is the two-point Möbius correlation at shift 2 which `corner-correlation.md` records as the OPEN object. So the sentence asserts, as an established localisation, something whose derivation would require the open input. Severity: **wording, not load-bearing.** §2.6's actual conclusion (a triangle- inequality treatment of `S_0` cannot reach `x/log^K x`) needs only the absolute mass, which is derived independently there and does not use singleton-fiber at all. Suggested repair, if the note is edited: drop "on each sign's ungrouped mass", or restate as "it is the `S_0`-local analogue of singleton-fiber-audit's global `c x log x` absolute-mass lower bound, without the per-sign split". Adjacent, outside my scope and non-load-bearing: "the same in all three readings ... by a factor `log^{K+2}`" at :214-216 is exact only for the fixed-`eta_0` reading; with `eta_0 = C loglog x/log x` from §2.5 the factor is `(loglog x)^2 log^K x`, and for the single top dyadic box it is `log^K x`. The qualitative conclusion (misses by at least `log^K x`) holds in all three. ## 2.4 An independent numeric probe of §3 Not in the repository; run in the scratchpad only, as a measurement, on 10 box shapes including four with `E >> D` (the corner's orientation, the reverse of the note's `U > Y`). Counted odd squarefree coprime pairs `(d,e)` with `d ∈ (D,2D]`, `e ∈ (E,2E]`: | D | E | `|G|/(DE)` | `(|G_+|-|G_-|)/(DE)` | `min/(DE)` | `min_sigma sum 1/(d phi(e))` | |---|---|---|---|---|---| | 512 | 64 | 0.1425 | -1.25e-3 | 0.0706 | 0.0409 | | 16384 | 1024 | 0.1434 | 7.51e-6 | 0.0717 | 0.0412 | | 1024 | 16384 | 0.1434 | 7.51e-6 | 0.0717 | 0.0412 | | 4096 | 65536 | 0.1434 | 8.65e-6 | 0.0717 | 0.0412 | | 8192 | 131072 | 0.1433 | 3.60e-6 | 0.0717 | 0.0411 | | 64 | 262144 | 0.1433 | 2.44e-5 | 0.0716 | 0.0419 | Note's thresholds: `|G|/(DE) >= 1/32 = 0.03125`, `min/(DE) >= 1/128 = 0.0078`, `sum >= 1/512 = 0.00195`. Measured values clear all three by factors of 4-20, the sign imbalance decays, and the `E >> D` shapes behave identically to the `D >> E` shapes. This is a **measurement at `D, E <= 3e5`**, not a proof, and it does not reach corner scales; it is consistent with the derivation in §2.1-§2.2 and with the note's constants being conservative. --- # What I did not check - `endpoint-pairing.md` §5 (the divisor-count bound on the full positive Vaaler majorant) and its (9), which both notes import for the majorant and for the paired budget shape. I confirmed that `dispersion-range (4)`'s two monomials are the specialisation of Bettin-Chandee Theorem 1 (§1.9), but I did not read endpoint-pairing's derivation of (9). - `prime-dispersion.md` (11)'s derivation line by line — I checked its cited inputs verbatim and the completion sketch, not the full text. - The **right-orientation** rewrite of the prime-power and first-branch moments (`residual-coverage §3`). It is asserted from sign-agnostic ingredients and the native phase (10) rather than written out. **ASSUMED-UNVERIFIED**, same as the earlier review's flag. Since the `19/25` product edge is set entirely by `J_R`, this is the largest unwritten dependency in the part I reviewed. - `residual-coverage §4`'s Perron separation and §5's `E_*`; the prior chain review adjudicated §§1-5 with no defect and I did not repeat it. - `signed-divisor-grouping §2`'s excluded-prime Möbius lemma (used by residual-coverage §4's density step). - Tao, 254A Notes 2 Exercise 66 / Corollary 39 and Notes 3 Theorem 17 as fetched pages this pass. - `coefficient-structure.md` §2's norm lower bound, and everything in `grouped-divisor-moment.md`. - Whether the OPEN signed margin holds. Nothing here bears on it. A checked derivation is not a refereed theorem. The full endpoint remainder, the sufficient twin margin, and the corner correlation all remain OPEN, and this review moves none of them.