{"id":2120,"job_id":null,"problem_id":1,"lane_id":null,"type":"audit","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Q-corner-measurement: distinguish measured widths from the admitted analytic corner\n\nThis is a prose scope correction. The served note says every reachable actual-exponent row has at most one right-band prime. Its own B-eta40 row at j=36 records [5,7,11], while A records [5]. Accepted #1719/review485 checked these A/B bands independently and left the old ledger wording intact.\n\nCurrent corner-correlation §0 fixes 0<eta_0<1/400 for its analytic regime, while the measured A/B widths are 1/100 and 1/40. Inclusion in W_dagger alone permits the broader eta<13/56; this correction does not invalidate the widened finite identities. It identifies a missing transfer to the currently admitted widths.\n\nAt x=2^36 and any admitted eta, Z=floor(2^(9/5))=3 because 3^5<2^9<4^5. Since e>floor(x^(1-v-2eta)) and e is an integer, r'=(n-2)/e<x^(v+2eta)<2^(99/50)<4. No prime lies in (3,4), so the admitted right weight and K vanish there by support, rather than cancellation. This proof is a hand-checked bound, not a rerun.\n\nThe main revision qualifies the A-versus-B statement and labels widened eta diagnostics. Three companion candidates update both QUESTIONS layouts, the README router and OUTCOMES. The four-base patch applies strictly and yields their exact uploaded bytes. Only these scope descriptions change: status PARTIAL, MEASURED calibration, old tables, producer and OPEN cancellation/twin-margin conclusions are retained. The paired registry candidate is manually bounded to two rows; no full regeneration is claimed. The OUTCOMES large-scale band counts are explicitly the previously reported widened A-band counts.\n\nThe main revision's base is f5ceb46f31259ad80a280ea4c036ef420eb85e0ecb3169ffa3f8140acbb56110; its file is 4cfddab77857c3cb611b7ce5afcb0b6daf2543f0f56431211ff549621ef824bc. Companion base/hash bindings appear in also_fix. Apply the linked patch only against all four named bases; otherwise rebase the bounded edits without replacing intervening work. The candidates remain proposed until independent review and integration.\n\nEvidence and prior-art comparison are in return #2119: exact current definitions and A/B rows, accepted #1719/#595, published unweighted correlations, and the unmet moving-weight transfer. Scientific CPU 0; no measurements rerun. Cheapest check: read current corner §0, source B/A j36 rows, the integer inequalities above, and the exact patch. Transcript removes private source identifiers/instructions and full third-party payloads; actual source calls and native usage remain. Reused usage is deduplicated by the server.\n","patch":"--- a/research/QUESTIONS.md\n+++ b/research/QUESTIONS.md\n@@ -49,7 +49,7 @@\n | C | `Q-corner-coefficient-energy` Can the full corner coefficients be studied before splitting their cofactor branches, and what do exact energy and classical smoothed sieve weights supply? | ANSWERED | The sharp coefficient energy is an explicit signed quadratic form in short cofactors plus O_epsilon(x^(1/2)K^2*x^epsilon), a power below x at the fixed corner. Logarithmically averaging the divisor cutoff gives Barban–Vehov weights and an O_eta(x log x) squared norm on each side and absolute shifted-product bound. The follow-up sharp-corner-transition.md gives sharp O_eta(x log^2 x) norms and proves a non-negligible norm transition for sufficiently small fixed eta. Signed transition saving and the global complement remain OPEN; no exact residual cut or twin margin changes. | [corner-coefficient-energy.md](corner-coefficient-energy.md) |\n | C | `Q-corner-correlation` On the corner S_0 where both cofactors sit just above their own cutoffs, what exactly is the remaining endpoint sum, what does the sufficient consumer need there, and does any reviewed correlation theorem supply it? | PARTIAL | The full corner is exactly sum_n C(n)C'(n-2). Proper-prime-power terms above the cutoffs are negligible at fixed eta by §1.1; the remaining prime-r terms with s>1 or s'>1 and non-squarefree inputs remain OPEN. The s=s'=1 piece has a nonnegative Mobius-product weight with exact moving cuts. The multiplicative lift and round-review sampling give small continuous and dyadic scale-average log savings for that subfamily, with no o(x), full-corner control or twin margin. The complement is still open. | [corner-correlation.md](corner-correlation.md) |\n | C | `Q-corner-log-average` What do logarithmically averaged correlation theorems and their quantitative successors actually supply for the prime-cofactor corner weight, with exact support, rates and scale quantifiers? | PARTIAL | The pure band is invariant under dilation by primes outside it; the exact cofactor window is not, but its support extends to x^(1-2eta) and need not leave the window under a fixed dilation. Uniform-prefix logarithmic bounds imply block bounds by Abel summation; a relative exponent above 3 is one sufficient route to o(x), not a universal necessary threshold. Pilatte's 1/(96e) calculation applies only to his displayed parameter choice. A bounded multiplicative Fourier lift now gives a small continuous scale-average log saving for the prime-cofactor subfamily via Tao-Teravainen v2; see prime-band-transfer.md. No o(x), dyadic pointwise estimate, full-corner estimate or positive twin margin follows. | [corner-log-average.md](corner-log-average.md) |\n-| C | `Q-corner-measurement` At finite dyadic x, how large is the corner two-point correlation K(x)=sum_{x/2<n<=x} mu(n)mu(n-2)L(n)L'(n-2) of corner-correlation (5) relative to C_2 x, to its own absolute mass, and to a matched random-sign control, and does \\|K\\|/mass fall with x? | PARTIAL | MEASURED, on the range and cutoffs recorded in the bound OUTPUT block of corner-measurement.js. The dominant finding is disconfirming for the measurement itself, not for the corner: at every reachable x the actual right cutoff Z=floor(x^(1/20)) admits at most one prime in its band and is empty at several j, so the actual-parameter rows are a one-prime object rather than the asymptotic corner, and the enlarged-cutoff rows are a model of the corner's shape and not the corner. On the rows that exist, \\|K\\|/mass falls with x at a rate that is not faster than the matched random-sign control's own decay, so the runs give no evidence of arithmetic cancellation beyond square-root-of-count. K stays many orders below C_2 x at every j measured and never falls below -C_2 x. No asymptotic rate, saving or cancellation is established, and nothing here changes the OPEN status of the corner or of the sufficient twin margin. | [corner-measurement.md](corner-measurement.md) |\n+| C | `Q-corner-measurement` At finite dyadic x, how large is the corner two-point correlation K(x)=sum_{x/2<n<=x} mu(n)mu(n-2)L(n)L'(n-2) of corner-correlation (5) relative to C_2 x, to its own absolute mass, and to a matched random-sign control, and does \\|K\\|/mass fall with x? | PARTIAL | MEASURED, on the range and cutoffs recorded in the bound OUTPUT block of corner-measurement.js. The dominant finding limits the measurement, not the corner: over j=20..36 the primary A-eta100 right band admits at most one prime and is empty at several j, whereas B-eta40 has three right-band primes at j=36. A and B use the decomposition cutoff exponents but widened eta_0=1/100 and 1/40, outside the current analytic S_0 range 0<eta_0<1/400. C is a dyadic-band diagnostic and D changes the right exponent. These finite diagnostics do not measure the admissible fixed-eta asymptotic corner. On the rows that exist, \\|K\\|/mass falls with x at a rate that is not faster than the matched random-sign control's own decay, so the runs give no evidence of arithmetic cancellation beyond square-root-of-count. K stays many orders below C_2 x at every j measured and never falls below -C_2 x. No asymptotic rate, saving or cancellation is established, and nothing here changes the OPEN status of the corner or of the sufficient twin margin. | [corner-measurement.md](corner-measurement.md) |\n | C | `Q-cross-campaign-synthesis` What can be recovered by comparing the fold, anchored, arithmetic and literature campaigns, including the proposed papers, against the actual twin-prime consumer? | ANSWERED | The full-coefficient viewpoint leads to an exact sharp energy quadratic form, a Graham/Barban–Vehov O_eta(x log x) bound for smoothed full coefficients, and a small-prime cutoff-difference identity. The follow-up sharp-corner-transition.md prices the full sharp norm and rules out negligible L2 transfer for sufficiently small fixed eta. The local proposals supply quantifier and dependency lessons; a reversed anchored inequality, overbroad Suen closure and stale proposal header are repaired. Signed transition correlations and the global margin remain OPEN. The subsequent global-cutoff-averaging.md gives the current complete global formulation and signed research priority; its one-point cancellation is not a shift-2 estimate. | [cross-campaign-synthesis.md](cross-campaign-synthesis.md) |\n | C | `Q-data-reuse-audit` Do existing runs or their embedded inputs retain what the current determinant-2 sum needs, and can useful data be recovered without repeating the old campaigns? | ANSWERED | The fold CSV supplies a verified base-prime list and archived interval boundaries; three old rows are reproduced and their bounded prefixes reconstructed with full factorizations retained. Other inspected outputs are aggregate statistics or twin-gap records. These finite inputs support coefficient and grouping tests, not an asymptotic twin bound. | [data-reuse-audit.md](data-reuse-audit.md) |\n | C | `Q-determinant-corollary` Does Bettin-Chandee Corollary 1, applied to d*k-e*t=2 after moving the non-smooth part of beta_V and beta_Z onto the coefficient side, control any part of the endpoint remainder that the current estimates do not? | ANSWERED | The transfer is legal and exact: the smoothness hypothesis the literature scout recorded as the blocker is removable, at the cost eta=x^kappa and a boundary error O(x^(1-kappa)log^C x). Priced, it adds nothing. Requiring all four pieces and all bands gives 22max(a,b)+17min(a,b)<20, whose supremum of delta+nu is 2869/3900 < 19/25, so the region is strictly inside the already controlled delta+nu<19/25; the added area is exactly 0. A split by the size of the moved prime power also adds nothing, because every per-band budget is nondecreasing in the band exponents and the top band reproduces the full box. The log-log piece alone is controlled on 12.97 percent of the domain outside the current region, which is one of four pieces and changes nothing for R. W_dagger does not shrink, E_dagger is unchanged, and the sufficient twin margin remains OPEN. | [determinant-corollary.md](determinant-corollary.md) |\n@@ -414,7 +414,7 @@\n | `Q-corner-coefficient-energy` | ANSWERED | Can the full corner coefficients be studied before splitting their cofactor branches, and what do exact energy and classical smoothed sieve weights supply? | The sharp coefficient energy is an explicit signed quadratic form in short cofactors plus O_epsilon(x^(1/2)K^2*x^epsilon), a power below x at the fixed corner. Logarithmically averaging the divisor cutoff gives Barban–Vehov weights and an O_eta(x log x) squared norm on each side and absolute shifted-product bound. The follow-up sharp-corner-transition.md gives sharp O_eta(x log^2 x) norms and proves a non-negligible norm transition for sufficiently small fixed eta. Signed transition saving and the global complement remain OPEN; no exact residual cut or twin margin changes. | C | [corner-coefficient-energy.md](corner-coefficient-energy.md) |\n | `Q-corner-correlation` | PARTIAL | On the corner S_0 where both cofactors sit just above their own cutoffs, what exactly is the remaining endpoint sum, what does the sufficient consumer need there, and does any reviewed correlation theorem supply it? | The full corner is exactly sum_n C(n)C'(n-2). Proper-prime-power terms above the cutoffs are negligible at fixed eta by §1.1; the remaining prime-r terms with s>1 or s'>1 and non-squarefree inputs remain OPEN. The s=s'=1 piece has a nonnegative Mobius-product weight with exact moving cuts. The multiplicative lift and round-review sampling give small continuous and dyadic scale-average log savings for that subfamily, with no o(x), full-corner control or twin margin. The complement is still open. | C | [corner-correlation.md](corner-correlation.md) |\n | `Q-corner-log-average` | PARTIAL | What do logarithmically averaged correlation theorems and their quantitative successors actually supply for the prime-cofactor corner weight, with exact support, rates and scale quantifiers? | The pure band is invariant under dilation by primes outside it; the exact cofactor window is not, but its support extends to x^(1-2eta) and need not leave the window under a fixed dilation. Uniform-prefix logarithmic bounds imply block bounds by Abel summation; a relative exponent above 3 is one sufficient route to o(x), not a universal necessary threshold. Pilatte's 1/(96e) calculation applies only to his displayed parameter choice. A bounded multiplicative Fourier lift now gives a small continuous scale-average log saving for the prime-cofactor subfamily via Tao-Teravainen v2; see prime-band-transfer.md. No o(x), dyadic pointwise estimate, full-corner estimate or positive twin margin follows. | C | [corner-log-average.md](corner-log-average.md) |\n-| `Q-corner-measurement` | PARTIAL | At finite dyadic x, how large is the corner two-point correlation K(x)=sum_{x/2<n<=x} mu(n)mu(n-2)L(n)L'(n-2) of corner-correlation (5) relative to C_2 x, to its own absolute mass, and to a matched random-sign control, and does \\|K\\|/mass fall with x? | MEASURED, on the range and cutoffs recorded in the bound OUTPUT block of corner-measurement.js. The dominant finding is disconfirming for the measurement itself, not for the corner: at every reachable x the actual right cutoff Z=floor(x^(1/20)) admits at most one prime in its band and is empty at several j, so the actual-parameter rows are a one-prime object rather than the asymptotic corner, and the enlarged-cutoff rows are a model of the corner's shape and not the corner. On the rows that exist, \\|K\\|/mass falls with x at a rate that is not faster than the matched random-sign control's own decay, so the runs give no evidence of arithmetic cancellation beyond square-root-of-count. K stays many orders below C_2 x at every j measured and never falls below -C_2 x. No asymptotic rate, saving or cancellation is established, and nothing here changes the OPEN status of the corner or of the sufficient twin margin. | C | [corner-measurement.md](corner-measurement.md) |\n+| `Q-corner-measurement` | PARTIAL | At finite dyadic x, how large is the corner two-point correlation K(x)=sum_{x/2<n<=x} mu(n)mu(n-2)L(n)L'(n-2) of corner-correlation (5) relative to C_2 x, to its own absolute mass, and to a matched random-sign control, and does \\|K\\|/mass fall with x? | MEASURED, on the range and cutoffs recorded in the bound OUTPUT block of corner-measurement.js. The dominant finding limits the measurement, not the corner: over j=20..36 the primary A-eta100 right band admits at most one prime and is empty at several j, whereas B-eta40 has three right-band primes at j=36. A and B use the decomposition cutoff exponents but widened eta_0=1/100 and 1/40, outside the current analytic S_0 range 0<eta_0<1/400. C is a dyadic-band diagnostic and D changes the right exponent. These finite diagnostics do not measure the admissible fixed-eta asymptotic corner. On the rows that exist, \\|K\\|/mass falls with x at a rate that is not faster than the matched random-sign control's own decay, so the runs give no evidence of arithmetic cancellation beyond square-root-of-count. K stays many orders below C_2 x at every j measured and never falls below -C_2 x. No asymptotic rate, saving or cancellation is established, and nothing here changes the OPEN status of the corner or of the sufficient twin margin. | C | [corner-measurement.md](corner-measurement.md) |\n | `Q-covadj-sign` | PARTIAL | Is Cov_adj < 0 for every scour prime q at every level? | Uniform-in-q anticorrelation is REFUTED, with six counterexample primes known; the aggregate is negative at all four levels in exact integer arithmetic over 599 scour primes, and the Low-Band Lemma is true but is not the mechanism, so the all-x aggregate theorem stays OPEN. | none | [natal-cap-23-covadj-proof.md](natal-cap-23-covadj-proof.md) |\n | `Q-covering-dive` | ANSWERED | What do the one-class versus two-class Jacobsthal, covering-system and interval-sieving literatures hold for our object? | No two-class upper bound exists in print: Iwaniec's g(q) << (log q)^2 is one class per prime, and four independent passes (including the 82-work citation graph of Iwaniec 1978 and a zbMATH title sweep of 324 records) came back ABSENT; the synthesis names where the proof breaks when the second class enters. | none | [covering-dive.md](covering-dive.md) |\n | `Q-covering-pruning` | CLOSED | Run backwards, does the pruning test inside OEIS A144311's branch-and-bound program give an upper bound on G2? | The pruning test is admissible (PROVEN, confirmed three ways), so A144311's terms above x = 43 are proven maximal, but run backwards it gives nothing from x = 13 onward because its whole content is sum_{5<=p<=x} 2/p < 1 and Mertens crosses 1 between 11 and 13; the cheapest admissible repair is Brun's pure sieve, whose exponent diverges like 7.182 lnln x. | none | [attack-beta2-05-covering-pruning-bound.md](history/staging/attack-beta2-05-covering-pruning-bound.md) |\n--- a/research/corner-measurement.md\n+++ b/research/corner-measurement.md\n@@ -6,7 +6,7 @@\n todo: C\n parity: This note contains no arithmetic estimate and no proof step. Its only inputs are exact integer sieving (Mobius by segmented factorisation, prime bands by direct enumeration) and arithmetic on the resulting finite sums. Nothing is asserted about any method class, and no obstruction is claimed. The measurement can support or undercut a heuristic about the corner's cancellation; it cannot establish or exclude any asymptotic rate, and it supplies no residue, non-residue, bilinear or sieve input to any argument.\n question: At finite dyadic x, how large is the corner two-point correlation K(x)=sum_{x/2<n<=x} mu(n)mu(n-2)L(n)L'(n-2) of corner-correlation (5) relative to C_2 x, to its own absolute mass, and to a matched random-sign control, and does |K|/mass fall with x?\n-verdict: MEASURED, on the range and cutoffs recorded in the bound OUTPUT block of corner-measurement.js. The dominant finding is disconfirming for the measurement itself, not for the corner: at every reachable x the actual right cutoff Z=floor(x^(1/20)) admits at most one prime in its band and is empty at several j, so the actual-parameter rows are a one-prime object rather than the asymptotic corner, and the enlarged-cutoff rows are a model of the corner's shape and not the corner. On the rows that exist, |K|/mass falls with x at a rate that is not faster than the matched random-sign control's own decay, so the runs give no evidence of arithmetic cancellation beyond square-root-of-count. K stays many orders below C_2 x at every j measured and never falls below -C_2 x. No asymptotic rate, saving or cancellation is established, and nothing here changes the OPEN status of the corner or of the sufficient twin margin.\n+verdict: MEASURED, on the range and cutoffs recorded in the bound OUTPUT block of corner-measurement.js. The dominant finding limits the measurement, not the corner: over j=20..36 the primary A-eta100 right band admits at most one prime and is empty at several j, whereas B-eta40 has three right-band primes at j=36. A and B use the decomposition cutoff exponents but widened eta_0=1/100 and 1/40, outside the current analytic S_0 range 0<eta_0<1/400. C is a dyadic-band diagnostic and D changes the right exponent. These finite diagnostics do not measure the admissible fixed-eta asymptotic corner. On the rows that exist, |K|/mass falls with x at a rate that is not faster than the matched random-sign control's own decay, so the runs give no evidence of arithmetic cancellation beyond square-root-of-count. K stays many orders below C_2 x at every j measured and never falls below -C_2 x. No asymptotic rate, saving or cancellation is established, and nothing here changes the OPEN status of the corner or of the sufficient twin margin.\n -->\n \n **Twin-prime infinitude remains OPEN. The sufficient margin\n@@ -43,8 +43,9 @@\n \n with V=floor(x^w), Z=floor(x^v), D_0=floor(x/(V+1)),\n E_0=floor((x-2)/(Z+1)), d_lo=max(V,D_1), e_lo=max(Z,E_1). In *eta mode*\n-D_1=floor(x^(1-w-2eta_0)) and E_1=floor(x^(1-v-2eta_0)), which is exactly\n-the corner cut of corner-correlation §0. In *dyadic mode* D_1=E_1=0 and the\n+D_1=floor(x^(1-w-2eta_0)) and E_1=floor(x^(1-v-2eta_0)), the corner-cut formula of corner-correlation §0. The analytic note fixes\n+0<eta_0<1/400; neither measured eta value is in that range. The finite\n+prime-subfamily identity is still exact with the declared widened cuts. In *dyadic mode* D_1=E_1=0 and the\n bands are capped at r<=2V, r'<=2Z, the \"one dyadic band\" sizing of the\n wave-1 reachability analysis.\n \n@@ -52,9 +53,9 @@\n \n | key | v | mode | eta_0 | what it is |\n |---|---|---|---|---|\n-| `A-eta100` | 1/20 | eta | 1/100 | the actual corner cutoffs, primary |\n-| `B-eta40` | 1/20 | eta | 1/40 | the actual cutoffs, secondary eta_0 |\n-| `C-dyadic` | 1/20 | dyadic | n/a | actual cutoffs, bands (V,2V] and (Z,2Z] |\n+| `A-eta100` | 1/20 | eta | 1/100 | decomposition exponents, widened eta proxy, primary |\n+| `B-eta40` | 1/20 | eta | 1/40 | decomposition exponents, widened eta proxy, secondary |\n+| `C-dyadic` | 1/20 | dyadic | n/a | decomposition exponents, dyadic diagnostic bands (V,2V] and (Z,2Z] |\n | `D-scaled` | 1/8 | eta | 1/100 | **a model of the corner shape, not the corner** |\n | `E-empty` | 1/20 | eta | 1/100 | control (iv): r band forcibly emptied |\n \n@@ -120,14 +121,15 @@\n \n The actual right cutoff is Z=floor(x^(1/20)), so the r' band of\n corner-correlation (5) is the set of primes in\n-(floor(x^(1/20)), floor((x-2)/(E_1+1))]. At every x this machine can reach\n+(floor(x^(1/20)), floor((x-2)/(E_1+1))]. For the primary A-eta100 rows j=20..36,\n that interval contains **at most one prime**, and at several j it contains\n-none, so:\n-\n-- the actual-parameter rows are not a measurement of the asymptotic\n-  corner. They are a measurement of a sum in which the right prime is a\n-  single fixed small prime and L'(n-2) is a fixed multiple of an\n-  indicator of one residue class;\n+none. B-eta40 uses a wider band and contains [5,7,11] at j=36. Both eta\n+values exceed the current analytic S_0 range 0<eta_0<1/400, so:\n+\n+- the primary A-eta100 rows are a sum in which the right prime is a\n+  single fixed small prime and L'(n-2) is a fixed multiple of one\n+  residue-class indicator; B-eta40 is a few-prime, wider-eta diagnostic.\n+  Neither is a measurement in the admissible analytic S_0 range;\n - the s>1 and s'>1 branches of the corner are unpopulated wherever\n   x^(2eta_0) < 2, which for eta_0=1/100 means every j below 50. The\n   s=s'=1 sub-family and the full corner therefore coincide at most of the\n@@ -152,9 +154,10 @@\n no faster than the control's is not evidence of arithmetic structure, and\n that is what the runs show.\n \n-The second disconfirming point, from §3, is prior to the first: the rows\n-carrying the actual cutoffs have one prime in the right band, so what was\n-measured is not the asymptotic corner.\n+The second disconfirming point, from §3, is prior to the first: the primary A-eta100 rows\n+have at most one prime in the right band; B-eta40 has up to three in the\n+reported top row. Both use widened eta outside the analytic S_0 range.\n+These diagnostics do not measure the admissible asymptotic corner.\n \n Nothing in either reading bears on whether the corner cancels\n asymptotically. corner-correlation §2.2 prices the absolute target at\n@@ -268,10 +271,11 @@\n \n ## 5. Limitations\n \n-- **The actual corner is not measured.** §3. The rows carrying the actual\n-  cutoffs carry a one-prime right band; the rows with a populated right\n-  band carry a different cutoff exponent. No argument is given here that\n-  the two have the same shape, and none should be assumed.\n+- **The actual corner is not measured.** §3. The primary A rows carry\n+  at most one right-band prime; B has a few primes with wider eta. Both\n+  eta values are outside the analytic S_0 range. D uses a different\n+  cutoff exponent. No argument is given here that\n+  the widened diagnostics transfer to the admissible widths, and none should be assumed.\n - **The reachable range is short.** The largest j is recorded in the bound\n   block. A least-squares slope over that range has no error bar that\n   distinguishes a power law from a slowly varying prefactor, and log x\n--- a/research/README.md\n+++ b/research/README.md\n@@ -101,7 +101,7 @@\n | do the actual Mobius signs measurably help the kernel moment against random signs | [kernel-sign-control.md](kernel-sign-control.md), [script](kernel-sign-control.js); MEASURED at x<=2^30 on the (8/25,9/20) box: no advantage over random signs in any of four families, slopes within one sd of zero, in a regime where the R=0 class carries about all of the moment and the kernel is invisible; a failure to detect, not a refutation |\n | where does the Mobius Bombieri-Vinogradov input come from, since no published theorem statement was located in the owning convention of SEARCH-CONVENTIONS.md §1 (Mobius function in arithmetic progressions, Bombieri-Vinogradov) | [mobius-bv-derivation.md](mobius-bv-derivation.md), [validator](mobius-bv-validation.js); derived from four numbered results of Koukoulopoulos GSM 203 plus Vaughan's identity for mu, with the maximum over y<=T inside and level Q<=T^(1/2)/(log T)^(A+6); assembled from published theorems, not itself refereed |\n | how does our sufficient consumer compare with the published conditional routes to twins | [consumer-comparison.md](consumer-comparison.md); Murty-Vatwani reduces to BV plus one hypothesis EH_{mu_2}(x^(1/2+eps)); the dyadic dictionary to Tao's delta_x and a strictly weaker block-summed consumer; every integer-side conditional route needs equidistribution in progressions, not a bound; the corner shape has prior art in Tao's 2016 GEH statement and Maynard's ICM 2022 Question 17 |\n-| how large is the corner correlation at finite x against its mass and a random-sign control | [corner-measurement.md](corner-measurement.md), [script](corner-measurement.js); MEASURED at x<=2^36 over 1.3e9 integers: |K|/mass falls no faster than the random-sign null in all four variants (largest deviation 1.4 s.e.), K never below -C_2 x, and the actual right prime band holds at most one prime at every reachable x, so this class of measurement cannot inform the asymptotic corner; a negative, methodological result |\n+| how large is the corner correlation at finite x against its mass and a random-sign control | [corner-measurement.md](corner-measurement.md), [script](corner-measurement.js); MEASURED at x<=2^36 over 1.3e9 integers: |K|/mass falls no faster than the random-sign null in all four variants (largest deviation 1.4 s.e.), K never below -C_2 x, and the primary A-eta100 right band holds at most one prime for j=20..36 (B-eta40 has three at j=36), while both measured eta values exceed the analytic S_0 range eta_0<1/400, these rows supply no transfer to the admissible asymptotic corner; a negative, methodological result |\n | **the whole arithmetic campaign in one document** | [TWIN-REDUCTION.md](TWIN-REDUCTION.md); the reviewed reduction, the three sufficient margins, the controlled region, the ceiling of the moment shape, the corner identification with prior art, every priced interface and measurement, and where each claim lives |\n | can weighted correlation theorems handle the prime-cofactor subfamily | [corner-log-average.md](corner-log-average.md), [prime-band-transfer.md](prime-band-transfer.md), [round-review-0906.md](round-review-0906.md); continuous and weaker dyadic scale averages, no every-dyadic estimate, o(x) or full-corner margin |\n | does the multiplicative band transfer follow at its reviewed scope | [next-transfer-review.md](next-transfer-review.md), [round-review-0906.md](round-review-0906.md), [validator](next-transfer-review-validation.js); the source condition (3.2) is scoped to case (i); the coordinating review adds the weaker dyadic-average consequence and retains the open full margin |\n--- a/research/OUTCOMES.md\n+++ b/research/OUTCOMES.md\n@@ -2452,11 +2452,13 @@\n K=0. A full (d,e,k,t) enumeration at j in {20,22,24} reproduces K exactly\n and verifies mu(d)mu(e) = mu(n)mu(n-2) term by term.\n \n-**Limits, which are larger than the result:** at every reachable x the\n-actual right band (Z, Z x^(2 eta_0)] holds at most one prime and is empty\n-at j=20,21,22,32,33, so the right weight is a fixed residue-class\n-indicator for the prime 3 or 5; populating the band needs 5 primes at\n-x=2^70 and 20 at x=2^100, decided exactly from the band's integer endpoints\n+**Limits, which are larger than the result:** over the primary A-eta100 rows j=20..36 the\n+right band holds at most one prime and is empty at j=20,21,22,32,33, so\n+its right weight is a fixed residue-class indicator for prime 3 or 5.\n+B-eta40 has three primes [5,7,11] at j=36. Both measured eta values\n+(1/100 and 1/40) exceed the analytic S_0 range 0<eta_0<1/400. The\n+primary widened A-eta100 band holds 5 primes at x=2^70 and 20 at x=2^100,\n+as externally reported from that diagnostic band's integer endpoints\n (corner-measurement.md; \"about 4\" and \"17\" were prime-number-theorem estimates,\n which understate pi(29) and pi(127)). This degeneracy is structural, not a compute\n limit. 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