# Cross-campaign review: coefficients, proposed papers and the proof target **Twin-prime infinitude and every sufficient positive margin in the current handoff remain OPEN.** This review finds a usable one-point estimate and a more specific next question. It does not find a proof concealed in the corpus. The new derivations are in [corner-coefficient-energy.md](corner-coefficient-energy.md). The first follow-up is completed in [sharp-corner-transition.md](sharp-corner-transition.md): sharp squared norms are O_eta(x log^2 x), and for sufficiently small fixed eta the sharp and transition squared norms have that order. The norm transfer fails in that range; the signed transition correlation remains open. The subsequent [global cutoff assessment](global-cutoff-averaging.md) rechecks the next priority using the complete upstream identities. Averaging their initial cutoffs gives a full smooth remainder with no separately unpaid corner transition, and O(x log x) squared coefficient norms. It also derives same-input cancellation between the sharp corner and the coefficient complement. It does not transfer that cancellation to shift 2 or supply a lower twin margin. Its bounded global signed attempt now precedes an automatic extension of the small-cofactor lemma. ## 1. Scope and the standard used Baseline: local commit `382d830`, 2026-09-06. The review compares the question and outcome registers, the current arithmetic chain and its recent reviews, the fold/anchor alternatives, the import map, and all eight entries in [the proposal registry](../paper/proposals/PROPOSALS.md). The paper-suite architecture in [PAPERS.md](../paper/PAPERS.md) is context, not evidence of a theorem or a renewed novelty assessment. The decisive arguments checked at their owning steps include: - the full coefficient and prime-power split in corner-correlation; the lift and sampling consequence in prime-band-transfer and round-review; - the common-factor sign identity in structural-literature-audit, used below in a different, one-point energy; - the signed consumer in TWIN-REDUCTION and consumer-comparison, the coverage cuts and their fixed-margin qualification, and the distinction between a moment diagonal and an upper budget; - the anchored survivor lemma and conditional argument in anchored-note; the product-ensemble versus anchored hypotheses in the Suen proposal and import-shearer's actual correction and tightness statements; - the exact fold transport, alternation and thinning claims and their stated missing recurrence or correlation inputs, through the proposals and their owning U-FRAME, kappa-not-L and outcome entries; - the named source statements and selected proof steps listed in §3. This is a targeted mathematical cross-review, not independent certification of every archived script, every proposed manuscript or the full proofs of the imported analytic theorems. Earlier finite data are reused at their recorded scope; the new algebra has its own reproducible checks in §7. ## 2. What the comparisons recover ### 2.1 Regrouping is useful, but the coefficient being bounded matters The singleton audit has large masses of both signs before grouping. The signed-divisor argument nevertheless controls a nontrivial region after grouping. The later grouped moment also removes a Cauchy loss without using the signs of arbitrary bounded coefficients. These are concrete reasons to check the point at which absolute values were taken; neither result proves that all other groupings work. The full corner has coefficients C(n) = sum_(dk=n, Dlo0 and unbounded dyadic x with C2*x+E_dagger(x)>=c*x/log^K x. It is sufficient and quantitatively stronger than bare infinitude. Its cumulative version is equivalent at the same fixed K up to constants, as the recent review proves. Neither a uniform maximal-gap theorem nor an every-dyadic asymptotic is necessary. Conversely, replacing the target by an equivalent positivity statement does not prove that statement. ## 3. Proposed external papers: what was actually checked Primary texts below were opened during the cross-review. “Checked” means the specified statement, parameter range or selected proof step was read and matched, not that the whole paper was independently verified. Existing broader source trails remain in structural-literature-audit, consumer-comparison and next-correlation-source-map. | source and reading depth | usable content or method | first unpaid obligation here | |---|---|---| | [Chen An, 2206.10104v1](https://arxiv.org/html/2206.10104v1), (1.1), weight definition and Theorem 1.1; integer case of Graham's estimate | Uniform mean square for the actual logarithmic cutoff, including long divisor support. Imported in the owning energy note, with a fixed PDF hash. | Its range is upper cutoff <= summation endpoint. The N=x^2 does not give exact independence, the atomicity premise was not proved, and Shearer closes a graph/marginal interface rather than every use of arithmetic structure. These corrections were already in the owning review but missing here. | | [Anchored note](../paper/proposals/prop-anchored-note.md) | Its self-strike lemma explains why unbounded nonannihilation would suffice. | HELD retained. Lemma 1's reversed elementary inequality is replaced by the correct positive Euler-product factorization. Mean divergence survives; anchored positivity is still open. | | [Staircase note](../paper/proposals/prop-staircase-note.md) | Prime-by-prime caps provide reproducible finite certificates and locate their slack. | Finite floors and an upper bound on kills do not establish a positive infinite-scale margin. HELD retained; no new cap or publication trigger. | In particular, the local-lemma statements cannot be used to rule out a new arithmetic conditional bound. The simple verified conclusion is about the information retained: for a complete graph the independence polynomial is 1-sum p_i. Extra arithmetic information would be a new input and must be proved. Pairwise near-independence does not supply all the required conditional probabilities. ## 5. Quantifiers and losses that do not compose for free ### 5.1 Independent unbounded good scales need not intersect Two good sets of dyadic indices can be the even and odd indices. Each is unbounded, but there is no common scale. Thus separate subsequence results for residual components cannot simply be added. A usable replacement is elementary. On a finite common set of indices J, suppose nonnegative normalized errors e_i(j) satisfy (1/|J|) sum_(j in J) e_i(j) <= epsilon_i. For positive thresholds t_i, the fraction of indices failing any bound e_i(j)<=t_i is at most sum_i epsilon_i/t_i, by Markov and a union bound. If a favorable main-term event occupies fraction delta and that sum is less than delta, at least one common index survives. The error thresholds must also sum to less than the main-term margin. This proves a combination rule; it supplies no arithmetic epsilon_i or favorable event. The existing prime-subfamily average is normalized by N log^2 X. Its relative saving log^(-d) X, d=c_*/2, gives only log^(2-d) X after normalization by N. At a margin N/log^K X, the corresponding Markov cost is log^(K+2-d) X times a fixed threshold factor, which the present rate does not make small. This is the cost of this particular absolute-value conversion, not a necessary exponent for every one-sided proof. ### 5.2 Fixed margins, translations and coefficient norms - Fixed eta estimates do not allow eta to shrink with x without tracking every eta-dependent constant and every x^epsilon loss. Smoothing leaves the actual transition contribution to be estimated. - An averaged translation statement does not locate the anchor. A periodic configuration may be translated to place an empty interval at the chosen origin while leaving translation-averaged moments unchanged. This example concerns those statistics, not the complete residue data. - Bounds for distinct coefficient families cannot be multiplied. The bounded-multiplicative subfamily saving and the full smoothed L2 bound are useful separate facts, not a full-coefficient correlation theorem. - A smaller geometric residual is not a monotone decrease of its signed sum. Regional suprema and area percentages do not bound its mass. - A finite sign-control ratio cannot rule out an asymptotic mechanism. In kernel-sign-control the accessible prime threshold and common-divisor range are explicitly degenerate; that qualification remains decisive. ## 6. The next bounded work, in order 1. **Attempt the complete global signed pair.** The follow-up global-cutoff-averaging derives its uniform complete reduction and O(x log x) squared coefficient norms. These steps are completed work. Analyze the negative factor configurations while retaining positive contributions when needed, and seek the stated one-sided lower margin on common scales. Same-input cancellation does not supply the shift-2 estimate. 2. **Match any proposed correlation input to these actual coefficients.** The small-cofactor transfer can be an ingredient, but its bounded multiplicative theorem does not automatically apply to the global divisor coefficients. Price the representation, every range and the final rate. An O(x log x) norm budget is not an improved signed bound; the global signed O(x) bound was already available. 3. **Require a complete consumer before enlarging computation.** The global identity retains outside arithmetic directly. If returning to the sharp-corner decomposition, retain its transition, mixed terms and E_out. A signed average is admissible when its rate implies the positive margin; full o(x) cancellation is stronger than necessary. No result ranks this direction above every possible proof method. The retained factor inputs already suffice for algebraic falsifiers. Their proxy corner is not the asymptotic fixed-eta corner. A longer census needs a named new question whose outcome can change one of the estimates above. No six-hour run is justified merely to increase a finite twin count. ## 7. Reproduction and falsification [cross-campaign-validation.js](cross-campaign-validation.js) is embedded by the standard custody tool. It checks complex sharp energies, the full prime-log Gram matrix, radical regrouping, smoothing/singleton identities, the short-endpoint rescaling and small-prime dilation. Its deletion controls must change the answer. It also checks the elementary Euler factor and a finite common-scale counterexample. The mean-square theorem and its uniformity are named imported dependencies. A missing hypothesis there would invalidate the smoothed bound. A claimed sharp bound must control the displayed transition; a claimed twin result must additionally establish the global consumer. No finite test supplies either. Proposal corrections and the reusable/failed deductions are recorded under these questions in [OUTCOMES.md](OUTCOMES.md). Reproduction commands: ```sh node research/qc/embed.js --check research/cross-campaign-validation.js node research/qc.js --index --strict node research/qc/selftest.js node research/audit-numbers.js git diff --check ``` The QC and regression gates check their documented finite and syntactic contracts; passing them is not verification of the imported analytic proofs.