# V3 report: independent reading of lane D's held block bound (D1) and Lemma H Lane / stable question id: V3 (reader for lane D) / Q-structured-dispersion-estimate Starting commit / report commit or shared-checkout paths: 1285d47, shared checkout; candidate research/structured-dispersion-estimate.md and research/structured-dispersion-estimate-validation.js (uncommitted, not edited); this report and scratch script under scratchpad/round-0908b/V3/ Disposition / exact claim / unproved hypotheses: (D1) block bound (3): VERIFIED WITHIN STATED SCOPE, conditional on two named inputs it inherits and does not prove: the completion bound grouped-divisor-moment (7) (Pascadi Lemmas 3.2-3.3, reused at recorded scope, not re-read by me either) and the unreviewed block shape sum_m A_left(gm) Y(m). Lemma H: VERIFIED, own proof reconstructed and checked. Section 6's proposed cut C': CORRECTION REQUIRED (substituting C' for the existing cut C loses control of boxes with 71/100 <= delta < 151/200, delta+3nu < 321/200; the cut must be added as a fourth simultaneous condition, not substituted). Changed step compared with the reviewed baseline: none by me; I read the author's changed step (first Cauchy over (m,q), moment over e-pairs sharing q) and reconstructed it. Source theorem and first unmatched hypothesis, if any: no new import. (7) reused; its recorded hypotheses (composite modulus, any integer numerator, polynomially bounded c) match the new modulus c=q j l_1 l_2 and numerator sigma theta R. Nothing unmatched at the recorded scope. Validation command, falsifier, result and compute used: node research/structured-dispersion-estimate-validation.js (author's; ALL CHECKS PASSED, 1.1 s); node scratchpad/round-0908b/V3/v3-check.js (mine; 2.3 s, one core, independent seed and grid: Lemma H on 132866 open-interval and 137503 closed-interval configurations, reconstructed majorant on 16372, exact exponent assembly, strip grid, section 6 defect witness). Zero enumeration; embed.js not run. Independent reviewer / disposition (PENDING until actually reviewed): this report is the independent reading; the handler decides integration. Full-consumer payoff and unpaid complement: none for the global margin; (D1) is an upper bound for a block and says nothing about the sign of E_dagger. Regional only, and only after the section 6 correction. Proposed shared-record changes / next bounded obligation: section 8 below. ## 1. What remains open, then what this reading establishes Open, unchanged: twin-prime infinitude; every sufficient signed margin of RESEARCH-HANDOFF section 3; target (21) at (delta,nu)=(8/25,9/20) (worst sector 407/400 under (D1), box not controlled); the corner a=b=1 ((D1) is worse there, zero budget 41/40); the uniform product threshold (witness (8/25,11/25) at 401/400). Established by this reading, at the calibration stated: - Lemma H (harmonic gcd average with a fixed prime-power factor): PROVEN as stated. I reconstructed the proof independently (section 3 below) and the constant (k+1)tau(l_1)tau(l_2)[4A^2+2^{3/2}A^{3/2}q^{1/2}] survives a direct count. The largest ratio found in my adversarial search is 0.155 (open interval) and 0.341 (closed interval), so the constant is loose by a factor of at least 3; nothing depends on that. - Block bound (D1), eq. (3): DERIVED correctly from (7), Lemma H, the elementary averages (9) and the collision count (4) of structural-literature-audit. Each step reconstructed; no summation-order error; no sign of mu or Lambda used except through absolute values and nonnegativity. Conditional on (7) and on the inherited block shape, both named by the author. - Exponents (4) and the strip (5): exact, recomputed independently; 161/200, 407/400, 61/100 at the target box; strip delta<71/100, delta+3nu<327/200; 245 new grid boxes; non-prime-power sectors controlled by the grouped bound on all 245. - Section 6 application: DEFECTIVE as written. The sentence "This would replace the third condition of W_dagger by (d>floor(x^kappa') or de^3>floor(x^lambda')) with kappa', lambda' slightly below 71/100, 327/200 in place of 151/200, 321/200" is wrong: kappa' < 71/100 < 151/200, so the candidate cut C' is not a superset of the existing cut C. Witness (delta,nu)=(73/100,29/100): inside C (d x^{kappa'}), outside cuts A and B. Replacing C by C' would re-admit this box to E_dagger's domain with no estimate controlling it. On the author's 191 x 191 grid, 928 boxes are controlled only by the existing third condition and not by the strip. The correct application is a fourth simultaneous condition in W_dagger, i.e. the cut set A u B u C u C'; the Perron machinery of grouped-divisor-moment section 5 handles finitely many intersections at logarithmic cost, so this is bookkeeping, not a new estimate. The author did not apply the cut, so no record is damaged; the proposed OUTCOMES text "would permit a new exact cut" is correct only with "added" in place of "replace". Disposition for (D1): verified within stated scope, conditional on named inputs ((7) as recorded; block shape). Disposition for Lemma H: verified. Disposition for the regional candidate in section 6: correction required before any application. ## 2. What was checked and what was not Checked (by reading and reconstruction, with finite checks where stated): 1. Lemma H statement, the role of (0,q):=q, both cases p | l_1 and p | l_2, the case p not dividing l_1 l_2, i=0, d_i > 2A, the closed-interval variant, and the constant. Own proof in section 3. 2. The combined phase e_c(sigma theta R mbar) with c = q j l_1 l_2 = lcm(qe_1,qe_2), c/(qe_1)=l_2, c/(qe_2)=l_1, coprimality 1_{(m,qe_1)=1}1_{(m,qe_2)=1}=1_{(m,c)=1}, and the gcd bound G <= 2(R,q)(R,j)(h_1,l_1)(h_2,l_2), including (R,l_i)=(h_i,l_i) from (l_1,l_2)=1 and (theta R,c) <= 2(R,c). 3. Step 1's Cauchy: the coefficient on (m,q) is the product a_m lambda(q); weighted Cauchy with lambda >= 0 is legitimate; first factor sum_m|a_m|^2 sum_q lambda(q) <= M log^2 x times Q log(2Q) (at most Q prime powers in [Q,2Q), lambda <= log 2Q). The note's "4MQ log^2 x" drops one logarithm; absorbed in x^epsilon. 4. Step 4's summation order: for fixed q and fixed (l_1,l_2,h_1,h_2) the innermost j-sum of sqrt(j(R,j)) over j~J uses (9) at alpha=1/2 with R nonzero and independent of j, giving 2^{3/2}J^{3/2}tau(|R|) <= x^epsilon J^{3/2}; then the h-double-sum with (q,l_1,l_2) fixed by Lemma H; then the l-sum of sqrt(l_1 l_2) over l_i <= 2L, O(L^3). No average is taken over a variable that a remaining factor still depends on. Dropping (l_1,l_2)=1 and the interval restrictions happens after Lemma H has been applied to each coprime pair, on a sum of nonnegative bounds; that order is correct. Lemma H includes the R=0 pairs, so it bounds the R != 0 restriction. 5. Step 3 (R=0): count (4) with E in place of N; each pair at most (C/A)^2 f^2 M; total f^2 M E log/A per q; after sum_q lambda(q) <= Q log, MN/A. Matches the zero term. 6. Step 5 (periods): (R,q) <= q cancels the q in MG/c; then grouped-divisor-moment (11) verbatim; O(M x^epsilon) per q; QM after the q-sum. 7. Step 6 assembly: sum_q lambda(q) q^{1/2}(1+(q/A)^{1/2}) << x^epsilon(Q^{3/2}+Q^2A^{-1/2}); E^3Q^{5/2}=N^3Q^{-1/2}; E^3Q^3A^{-1/2}=N^3A^{-1/2}; multiplication by MQ and square root give (3). The top-band monotonicity (f=v=A/A_top below A_top; bands above A_top cost (1+v)^{1/2} <= x^tau) is as in the grouped moment. 8. Exponents (4) and the numbers 161/200, 407/400, 61/100, the strip (5), the count 245, the closed-form equivalence outside the existing region (alpha = delta+nu-71/100 >= 1/20 = sigma there), and the worst-sector value 407/400 at (rho,sigma)=(6/25,1/20) with the sector formulae of left-divisor-signs section 4 taken as recorded (monotone in rho and sigma, so the top corner is the maximiser of the minimum). 9. The reuse of (7): modulus q j l_1 l_2 composite and polynomially bounded; numerator sigma theta R any integer; (7) holds for every subinterval; the coefficients do not enter (7). Same use as in grouped-divisor-moment section 3. 10. Signs: (3) sees a_m through |a_m|^2, beta through |beta| <= 1, lambda through nonnegativity and sum_q lambda(q). The Mobius signs of d and e and the left Lambda(r) are not used; confirmed. The g=2 branch matches left-divisor-signs section 1 item 3 (odd r gives u=d'r with |beta(d')|=|mu(2d')(2d')^{-s}| <= 1; r=2^k gives q=2^{k-1}, lambda=log 2, including the sector Q=1 where (D1) reduces to the grouped bound). 11. The non-prime-power sectors (A_0 and the -mu log term, b=nu): grouped budgets (1+a)/2, a/2+3nu/2, a are below one on every one of the 245 new boxes (delta<71/100 implies delta<19/25; delta+3nu<327/200 implies delta+3nu<44/25). 12. Section 6's regional claim: only the controlled region and the domain of E_dagger would change; nothing about the sign of E_dagger; the sufficient signed margins are untouched. The author states this correctly. The application sentence is defective as in section 1. Not checked (stated so the handler does not read more into this report than it carries): - Pascadi Lemmas 3.2-3.3 were not re-read from the source; (7) is accepted at the scope recorded in grouped-divisor-moment section 3 on 2026-09-06. - The block shape sum_m A_left(gm) Y(m) with b_u=A_right(gu) (endpoint-fourier (8)-(9), grouped-divisor-moment section 4) remains unreviewed upstream, as left-divisor-signs and the author both say. I checked only that b_u for g=1 is exactly the convolution shape (1) with beta(e)=-mu(e)e^{-s}1_I(e), lambda=Lambda, from grouped-divisor-moment (13). - The Perron separation of grouped-divisor-moment section 5 was not repeated; the statement that a fourth cut costs only more Perron integrals at logarithmic cost is by inspection of that section, not a re-derivation. - Lemma II's sector exponents E_1, E_2, E_3 were taken from left-divisor-signs section 4 and its validator, not re-derived; they affect only the "worst surviving sector" bookkeeping, not (D1). - No asymptotic rate, onset or lower bound follows from any finite check here. ## 3. Lemma H: own proof and the direct count Statement. q=p^k, (l_1,l_2)=1, H a subset of the integers in (A,2A] (or [A,2A]), R=h_1l_2-h_2l_1, (0,q):=q. Then sum_{h_1,h_2 in H}(R,q)^{1/2}(h_1,l_1)^{1/2}(h_2,l_2)^{1/2} <= (k+1)tau(l_1)tau(l_2)[4A^2+2^{3/2}A^{3/2}q^{1/2}]. Proof (two steps). (i) Majorise each gcd power by a sum over divisors: (R,q)^{1/2} <= sum_{0<=i<=k}p^{i/2}1_{p^i|R} (for R != 0 the term i=v_p(R) alone gives (R,q)^{1/2}; for R=0 every p^i divides 0 and the term i=k gives q^{1/2}), and (h,l)^{1/2} <= sum_{d|l,d|h}d^{1/2}. The left side is then at most sum_{i,d_1|l_1,d_2|l_2}(p^id_1d_2)^{1/2}N(i,d_1,d_2) with N the number of pairs in H^2 with d_1|h_1, d_2|h_2, p^i|R. (ii) Count N. Since (l_1,l_2)=1, p fails to divide one of them; say p does not divide l_1. For fixed h_1 the congruence h_2l_1 ≡ h_1l_2 (mod p^i) has one solution class mod p^i (l_1 invertible), and with d_2|h_2 the solutions lie in one class mod lcm(d_2,p^i) or none, so at most A/lcm+1 <= 2A/max(d_2,p^i)+1 of them lie in an interval of A (or A+1) integers. The number of h_1 divisible by d_1 is at most 2A/d_1 (exact count floor(2A/d_1)-ceil(A/d_1)+1 <= A/d_1+1 <= 2A/d_1 for d_1 <= A; at most 1 for A 2A contributes nothing). Summing d_1^{-1/2} <= 1 over the divisors of l_1, 1 over those of l_2 with d_2 <= 2A, and over the k+1 values of i gives the bound. If p | l_1 then p does not divide l_2 and the same count runs with the indices exchanged (h_1 in one class for fixed h_2), giving the same bound with tau(l_2) from the d_2^{-1/2} sum. QED. Direct count check. My script computes the exact majorant of step (i) (true counts N) on 16372 configurations and confirms LHS <= majorant <= closed-form bound on every one, so both inequalities of the proof hold separately, not only their composition. Adversarial search (seed and grid different from the author's, prime powers to 2^9, 3^4, 31^2, 101, A from 1 to 20, l_i to 36 with p-divisible l_i and q > A over-represented, random subsets H): 132866 open-interval and 137503 closed-interval configurations, no violation; extreme diagonal configuration l_1=l_2=1, H full, q up to 2^20 and 7919 with all A pairs R=0 carrying q^{1/2}: ratio at most 0.174. The author's negative control (dropping the q^{1/2} tail fails for q > A with R=0 pairs) is consistent with the proof: the tail is exactly the p^i > 2A term of the count. For the application at the target box q <= x^{1/20} < A_top = x^{3/50}, so the tail is dominated there and the harmonic average costs A^2 x^epsilon, the same as the old (9); the whole saving in (D1) comes from (R,q)^{1/2} being free where q <= A. Failing configuration: none found; none exists by the proof above. ## 4. The block bound: where the saving comes from, and the checks of the steps nobody else had read The mechanism, restated so the handler can see it is not an accounting artefact. The (m,q) Cauchy equals Cauchy in m followed by Cauchy in q inside the moment; it discards the pairs (q_1,q_2) with q_1 != q_2 at the price of the factor sum_q lambda(q) ~ Q in the first factor. For q_1=q_2=q the completed modulus is q·lcm(e_1,e_2) and the pair count is E^2 instead of N^2=Q^2E^2, so with the Weil bound sqrt(c) the per-q Weil mass is q^{1/2}E^3 (at j=1) and the q-sum gives Q^{3/2}E^3=N^3Q^{-3/2}, against N^3 for arbitrary coefficients on (N,2N]. Net after the factor Q: N^3Q^{-1/2}, i.e. the cross term of (3) gains Q^{-1/4} in the block. This is the standard dispersion trade of a Q loss in the diagonal factor against a Q^{3/2} gain from the sublinear modulus dependence of completion; the author's "technique provenance" paragraph is accurate and makes no novelty claim. Step 4 as written is correct. The one place I looked for the classic error is the (R,j) factor: it is summed innermost with R fixed, so the h-average never sees j and the l-sum never sees h. The author's ordering "j-average, then h-average with q fixed, then l-sum" is the right one and is what the display shows. Step 1's first factor: correct as a product coefficient. If the actual block carried an (m,q)-coupled coefficient this would fail; in the application the coupling sits only in the phase and in Phi through u=eq, which is what Y_q(m) contains. This is exactly the block-shape dependency the author names as inherited. The structure-discarding control (R,q) <= q returning 103/100 and the black-box (12) route giving 417/400 are both correctly computed; I recomputed them. ## 5. Point-by-point answers to the seven reconstruction items (1) Lemma H: verified; own proof in section 3; no failing configuration; the R=0 pairs are covered by (0,q)=q and produce the q^{1/2} tail; p | l_1 and p | l_2 are symmetric; the constant survives the direct count with margin. (2) Combined phase and gcd bound: verified. c=q j l_1 l_2=lcm(qe_1,qe_2); e_{qe_1}(sigma theta h_1 mbar)conj e_{qe_2}(sigma theta h_2 mbar)=e_c(sigma theta R mbar) for (m,c)=1; G=(sigma theta R,c) <= 2(R,q)(R,j)(h_1,l_1)(h_2,l_2). (3) Summation order: verified, no variable overlap (section 4 above). (4) Reuse of (7): at recorded scope; modulus composite and polynomially bounded, numerator any integer, subinterval arbitrary; coefficients enter only through absolute values in step 4. Not re-read at source. (5) Exponent assembly: (1+a+sigma)/2=161/200, a/2+3b/2-min(sigma,alpha)/4=407/400 with alpha=3/50 and min=1/20, a+sigma=61/100. Strip: zero and period both read delta+6/25+1/20<1, i.e. delta<71/100; cross reads delta+6/25+3nu+3/20-1/40<2, i.e. delta+3nu<327/200; alpha>=sigma on every box with delta+nu>=19/25 since alpha=delta+nu-71/100>=1/20. 245 new grid boxes reproduced by an independent implementation. (6) Signs: confirmed; only |a_m|, |beta| <= 1, lambda >= 0 and sum_q lambda(q) are used. (7) Regional strip: it would change only the controlled region and hence the domain of E_dagger through an added cut; the complete signed budget, the sign of E_dagger and the sufficient margins are unchanged. With the correction of section 1 (add C', do not substitute it), the new W_dagger would carry four simultaneous conditions. ## 6. Minor observations (no correction to the bound) - Lemma H is stated for H in (A,2A]; section 2 of the note and grouped-divisor-moment section 1 use [A,2A]. The proof's counts hold on the closed interval with the same constants (checked in section 3 and finitely, max ratio 0.341); a one-word alignment. - The first-factor bound "4MQ log^2 x" omits the factor log(2Q) from lambda <= log 2Q with at most Q prime powers in [Q,2Q); absorbed in x^epsilon. - The validator clips alpha at 0 in the sector formulae. For sectors with a+b<1 the top harmonic band does not exist (MN1 on every band) and the block is handled by the trivial |Delta| <= 1 route of grouped-divisor-moment section 4; the clipping is conservative and no survivor sector at the target box has alpha<0 (survivors have rho+sigma >= 0.245 > 0.23). - The Chebyshev remark is not needed and is not used in a way that matters. ## 7. Files, commands, compute Files written (scratch only, no repo file edited): /private/tmp/claude-501/-Users-benjaminsen-Files-Git-primeoire/9058ffd0-271d-4adc-bdd5-dabb0916b242/scratchpad/round-0908b/V3/v3-check.js and this report V3-report.md. Commands: node research/structured-dispersion-estimate-validation.js (author's; ALL CHECKS PASSED); node scratchpad/round-0908b/V3/v3-check.js (all passed, 2.3 s, one core). Compute: about 4 s total; zero enumeration; embed.js not run. ## 8. Proposed record updates and next move Proposed edit to the candidate note before integration (handler applies; I edit nothing): in section 6 and in the proposed OUTCOMES text, replace "This would replace the third condition of W_dagger by ..." with "This would add a fourth simultaneous condition to W_dagger, (d>floor(x^kappa') or de^3>floor(x^lambda')), alongside the existing third condition; substituting it for the third condition would lose the boxes 71/100<=delta<151/200, delta+3nu<321/200 (witness (73/100,29/100))." Likewise in section 8 ("apply the cut ... and record the new W_dagger" should read "add the cut"). Proposed OUTCOMES amendment (append to the author's proposed entry under Q-structured-dispersion-estimate): "Independent reading (V3, 2026-09-08): Lemma H verified with an independent proof and 270k finite configurations; block bound (3) verified within stated scope, conditional on grouped-divisor-moment (7) at recorded scope and the unreviewed block shape; exponents and strip reproduced; one correction to the regional application: the candidate cut C' must be added to the cut set, not substituted for C (928 of 36481 grid boxes otherwise lose control)." Proposed CHANGELOG line: 2026-09-08, V3: independent reading of structured-dispersion-estimate; Lemma H and (D1) verified at stated scope; section 6 cut application corrected from "replace" to "add". Next move, one, justified: with the correction applied, the bounded documentation task the author names (add C' through the Perron machinery of grouped-divisor-moment section 5, record the four-condition W_dagger, re-derive E_*-E_dagger=O_H(x/log^H x) for the new domain) is admissible; it changes the controlled region only. The target box, the corner, the product threshold and every signed margin remain OPEN, and the reopening condition for the target box is the author's (a saving above x^{17/200} on the coprime e-pair class inside a common q with j_e <= x^{17/300}), which nothing in this reading changes. ## Re-read of D section 6 (four-condition application, displays (10)-(12)) Disposition: VERIFIED WITHIN STATED SCOPE as a domain change, conditional on the same two inherited inputs as (D1) (grouped-divisor-moment (7) at recorded scope; the unreviewed block shape) and on the reused separation and density machinery of residual-coverage section 4, grouped-divisor-moment section 5 and signed-divisor-grouping (8), none re-derived here. One wording clarification required before the handoff text is committed (item 3 below); no mathematical defect found. Validation: node research/structured-dispersion-estimate-validation.js (section E, ALL CHECKS PASSED); .../V3/v3-reread.js (exact rationals, all passed, <1 s). Zero enumeration; embed.js not run. (1) Slack. At (31/100,11/25): a=11/20, b=49/100, alpha=1/25x^{3/4} the strip has no uniform slack near the product line. On C' (delta<=141/200, delta+3nu<=1631/1000, delta+nu>=77/100): alpha-sigma>=77/100-71/100-1/20=1/100, so min(sigma,alpha)=sigma; continuous infima of the slack are cross 327/400-1631/2000=1/500, zero (71/100-141/200)/2=1/400, period 1/200. So the fixed margin on the continuous set is 1/500 (cross term, attained on de^3=x^{lambda'}); the validator's 77/38000 and 273/9500 are grid minima above these infima (reproduced exactly). Non-prime-power sectors under the grouped bound with b=nu: infimum slack 11/400 (zero term binds; cross 129/2000). The note's sentence "at most 1-77/38000, below 1-1/500" should read as a grid statement; the continuous fixed slack is 1/500. Recorded so nobody later cites 77/38000 as the analytic margin. (2) Coverage of boxes meeting C'. A dyadic box meeting C' has lower endpoints with deltax^{mu'}, so its exponents satisfy the C' inequalities up to O(1/log x), absorbed by the 1/500 and 1/100 margins exactly as grouped-divisor-moment section 5 absorbs dyadic and floor constants. Prime-power sectors: (4) is nondecreasing in rho and sigma (cross gains 3/2 per unit sigma and loses at most 1/4 through min(sigma,alpha); zero and period are increasing), checked on sampled lattices, so the top sector (rho,sigma)=(6/25,1/20) binds and the Q=1 and small-Q sectors reduce to the grouped bound; A_0 and -mu log sectors: grouped with b=nu, slack 11/400. Both uniform in the twists as in (D1)'s Uniformity paragraph. Covered. (3) Perron separation. The lower cut 1_{de>K_3}=1-P_{K_3}(de)+O((K_3+1)/T_P) is legitimate, and the twist (de)^{-w} has nonnegative real part on the original divisors. The one point that must be explicit: the "1" term carries no lower cut, so it is only harmless because the separation is applied box by box on boxes meeting C' cap S (as residual-coverage section 4 does: boxes not meeting the cut set contribute zero exactly, and only then is each indicator replaced). Applied to the full domain, the "1" term would run over boxes with de<=x^{77/100} where (D1) has no slack (item 1). D's derivation implies this order (input 1 is stated on boxes meeting C', and "every dyadic box meeting it meets C'"), but input 2 does not say it. Required clarification, one sentence: "the indicators are separated on each dyadic box meeting C' cap S; boxes not meeting it contribute zero." No uncovered term arises once that is stated. Count: A one monomial, B one, C two, C' three (the last as 1 minus an indicator): seven threshold conditions on the full intersection, so at most seven integrals and cost O(log^7 x); the product error uses |P_K|<=1+O((K+1)/T_P) and |1-P_K|<=2+O(...), fine. Pointwise errors O(x^{max cut}/T_P) with the O(x^2 log^C x) absolute mass give O(x^{2+49/20-10}log^C x). Correct. (4) Excluded-prime Mobius lemma. signed-divisor-grouping (8) is uniform for U/2<=a=U>U/2, upper <=D_0<=x. The lemma applies; the r,s,e harmonic sums cost fixed logarithms as in residual-coverage section 4. The note says "finite union of intervals"; at fixed e it is one interval per S, which is stronger. Applies. (5) Inclusion-exclusion. W_4=W_3 minus (W_3 cap C'); W_3 cap C'=C' minus C' cap (A cup B cup C); inclusion-exclusion over A,B,C gives the eight sets C' cap S (S = the empty intersection and the seven nonempty ones), each with sign; nothing is double-subtracted, and E_*-E_dagger^{new}=(E_*-E_dagger^{old})+R(W_3 cap C'). Validator: W_4 contained in W_3, W_4 meets no box of C', W_3 minus W_4 = 239 boxes equals the boxes of W_3 inside C'. Correct. (6) Handoff text. The proposed W_dagger display adds the fourth condition alongside the third (not substituted), the sentence names it as a domain change, and the section 4 region union adds "(delta<71/100 and delta+3nu<327/200)". I checked that union as a region statement: every interior point has fixed margin (for delta+nu>=19/25, alpha>=sigma and the margin is (327/200-delta-3nu)/2; for delta+nu<19/25 the point is in the first region). No sign, margin, target-box or corner claim is made; the uniform product threshold sentence is kept. The 50 sliver boxes (19/25<=delta+nu<77/100, in the strip, outside the old region) are correctly left in W_dagger. Exactly a domain change. Not checked: (7) at source; the block shape; the Perron formula (14) and the excluded-prime lemma were read for hypotheses only, not re-derived. No asymptotic rate follows from the grid. Proposed edits before commit: (a) in section 6 input 2 add the box-restriction sentence of item 3; (b) replace "at most 1-77/38000, below 1-1/500" by "grid maximum 1-77/38000; on the continuous set the fixed slack is 1/500 (cross term) and alpha-sigma>=1/100"; (c) in section 6 input 3 "finite union of intervals" may read "one interval per S at fixed e". OUTCOMES amendment: append to the V3 line "four-condition application (10)-(12) read: verified as a domain change at stated scope, with the box-wise separation made explicit; E_dagger's domain shrinks by C'; no signed claim changes."