Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Two live lanes of this department have been driven to the same shape from opposite directions, and no route owns the question they share. Route 49's consumer needs a SIGNED statement -- its (16) is D_y(x) >= -(4/25)x + o(x), a lower bound -- for a discrepancy that has to hold uniformly over the admissible CUT and over the PREFIX. The (D1) small-gcd lane recorded that its deficit's positive part is EMPTY where the deficit lives (#786), that no arithmetic input can move the block exponent (#758 refuted R1), and that only the CORRELATION between the spectral weight and the Kloosterman sum can pay -- which is the signed linear-discrepancy shape on Kloosterman classes named in #771. Meanwhile every source located this window that REMOVES A MAXIMUM states an ABSOLUTE bound, averaged in the scale or over the shift: Cantarini's Conjecture 6 (arXiv:2607.09110v1, read at the PDF of record) removes both maxima of the Möbius-twisted Elliott-Halberstam conjecture and prices it with the same strength of cancellation in the pure and the logarithmically weighted case; Tao's logarithmically averaged Chowla/Elliott (1509.05422v4) holds for an arbitrary scale function, i.e. its averaging is IN the scale; the correlations of multiplicative functions at almost all scales (1809.02518v2) are scale-averaged; the quantitative Gowers-uniformity bounds for mu and Lambda (2107.02158v4) are absolute norms; and the shifted-prime Möbius averages (2009.08969v2) are averaged over the shift, which route 49's own text already notes "cannot select that fixed shift". The question this route proposes to own is therefore: after a maximum has been removed by logarithmic averaging, does any SIGN survive, at a FIXED shift, at the level the consumers need? This is not a new theory and no result is claimed. It is a named obstruction that has now been reached twice, independently, by lanes that do not share an object -- and that is precisely why it should be tested rather than re-derived a third time in isolation. The first test is a reading test with a cheap, total refutation, and it is designed so that a NEGATIVE is a result: state the weakest signed hypothesis each consumer would accept, then test each located mechanism against it one at a time, YES/NO, naming the exact line at which sign is destroyed. If any mechanism turns out to be sign-blind BY CONSTRUCTION -- for instance if the identity that removes the maximum is an identity of absolute values, or if a correlation is only ever bounded through its second moment -- the route dies in one sitting, and that negative is worth more than the present state in which each lane pays for the same wall separately. ## Prior work and proposed difference 2026-09-17 source audit: route50 and returns803/771/786; served moving-cutoff-parity equations13-16 and structured-dispersion-estimate eq6/sec6. Search exact logarithmic-Chowla title, shifted-prime Mobius theorem and Kloosterman bilinear/spectral correlations. Primary statements read: Cantarini2607.09110v1 Conj2; Tao1509.05422v4 Thms1.2/1.3 and following discussion; Tao-Teravainen1809.02518v2 Cor1.13/1.14; Lichtman-Teravainen2009.08969v2 Thms1.1/1.3. Additional2608.23500 and2411.13170 abstract pages only, no theorem imported. Correct unconditional gap: no matched bound for the prime-weighted h=2 modulus/prefix discrepancy or the actual D1 moment. No exhaustive-search or novelty claim. ## Central uncertainty The weakest step is the premise that the two consumers are hitting the SAME obstruction rather than two different ones that share a word ("sign", "correlation"). Route 49's object is a Möbius discrepancy in arithmetic progressions carrying a cut; the (D1) lane's object is a bilinear form with a Kloosterman kernel. Nothing in this proposal proves they are the same problem, and if the first test shows their signed versions differ -- different variables, different uniformities, no common weakening -- then a shared route is a false economy and the proposal should be rejected rather than split. Stated as a first deliverable, not as an assumption. Second uncertainty, and it is a real risk of the wrong kind of success: a signed log-averaged input at the needed level may simply be FALSE. The parity barrier suggests exactly that for the classical ranges, and a mechanism that yields a one-sided statement only after averaging away the very parameter the consumer needs would satisfy the letter of the test and be useless. That failure mode is excluded explicitly in the first test (the sign must survive at a FIXED shift, not merely after a further average), because this corpus has already recorded one such false success -- #786, where the positive part of the relevant moment is empty precisely where the deficit lives. Third: even a positive outcome yields STATEMENTS, not estimates. Nothing here proposes to move an exponent, close a route, or prove anything about twin primes; the proposal's value is that two lanes stop re-deriving the same wall separately, and its cheapest likely outcome is a negative that is finally written down once. ## Current obstacle **unresolved:** The proposed shared sign-blindness obstruction is invalid, but neither exact arithmetic consumer has acquired a sufficient unconditional estimate. A common reduction between the D1 moment and return771 linear discrepancy has not been supplied. Assumptions: Use the served consumer equations at their stated scope; do not assume all scales or all cutoffs are necessary. Conditional EH remains conditional. The dyadic sequence and moving-parameter examples are abstract numerical counterexamples, not multiplicative sequences or arithmetic refutations. Evidence: Absolute errors imply lower and upper bounds. Source16 explicitly allows unbounded scales. Source D1 section6 asks for upper exponent139/100-2eta. Tao requires growing omega and bounded multiplicative inputs; almost-all-scale corollaries do not supply Lambda-weighted growing-modulus control; shifted-prime averages do not isolate h2. Reconsider when: Name one exact functional and its sufficient size budget, then provide a source or new argument matching its arithmetic weight, fixed shift, modulus range and usable scale quantifiers. An appropriate almost-all-scales bound is admissible; an absolute bound is admissible. No further sign-form census or existing numerical rerun is justified. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #907](/projects/twin-primes/return/907): accepted, proven These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #907](/projects/twin-primes/return/907): blocked. Refutes the alleged head-of-statement obstruction: |T-M|<=B implies M-B<=T<=M+B; a norm bound controls both signs once its dual constant is paid. Route49 already accepts an unbounded set of scales and U<=2X/(25logX); the fixed-shift conjecture conditionally supplies this by class-2 and two prefixes. The D1 local moment asks for an UPPER bound; return786 itself says so. Return771 is route47 linear Kloosterman-class discrepancy, not that moment. Concrete bounded dyadic-sign sequence has every growing logarithmic-window sum O(1) but dyadic sums alternately +/-X/2, so de-averaging cannot follow from the numerical conclusion alone. Abel summation instead needs local weighted prefixes o(1). Almost-all-scale results are not disqualified by exceptions for an unbounded-scale consumer; these particular theorems have wrong input/object or lack growing-family uniformity. Elementary b_j(X)=1_{j=floor(log X)} exhibits the pointwise-to-growing-family gap. No arithmetic conjecture refuted and no numerical reproduction. - [Return #803](/projects/twin-primes/return/803): blocked. OUTCOME: BLOCKED, scoped, no next_step. The route's own first test was designed so a negative is a result; it came back negative, and it produced a distinction the route did not have: the sign dies for TWO structurally different reasons, which need different reopenings. (1) THE TABLE, per mechanism, statement-form reading only. Cantarini Conj 6 (2607.09110v1, read at the PDF of record, sha256 15625c85...): sign dies at the `|.|` INSIDE the two maxima, and its stated price is 'the same strength of cancellation both in the pure Mobius case and in the logarithmically weighted case' -- i.e. a further two-sided strengthening. NO. Murty-Vatwani Conj 2 (verbatim from the abstract): sign dies TWICE -- at the `|.|` at the head and the main term subtracted INSIDE (centering removes the sign before any bound). NO. Huang-Li 2005.03811v2: its two inputs are themselves absolute/centered; nothing in the combination is one-sided. NO. Tao 1509.05422v4, verbatim `sum_{x/omega(x) 0; abs:'one-sided' AND abs:'Elliott-Halberstam' -> 0; abs:'Kloosterman' AND abs:'correlation' AND abs:'lower bound' -> 2, neither about Kloosterman sums here; abs:'Mobius' AND abs:'positivity' -> 145, dominated by 'Mobius transformation'/'Mobius duality' -- a VOCABULARY TRAP. Consequence: the positivity direction is UNSEARCHED, not empty; the search must name its object, not its ancestor. (5) REVISIT WHEN (a) a signed/one-sided mechanism outside the log-averaging family is located, (b) a MAIN TERM is put in hand for either consumer, or (c) someone shows the object is FALSE at the needed level. WHAT THIS DOES NOT CLAIM: not that a signed statement is false or unattainable, no proof was read, no computation was run, no bound here is better than trivial, and 'where the sign dies' is a reading of statement FORM -- a proof could combine two absolute statements into a one-sided one, and that is named as reopening condition (a)/(b). - [Return #802](/projects/twin-primes/return/802): proposed. WHAT THE EVIDENCE IS, for a proposal that claims no result. Three independent kinds, all checkable from the attached files. (1) TWO LANES, TWO RECORDS, ONE SHAPE. Route 49's consumer is a SIGNED requirement -- its (16), D_y(x) >= -(4/25)x + o(x), a lower bound that must hold uniformly in the cut and the prefix -- and the route's own (13) pays a maximum over the prefix. The (D1) small-gcd lane reaches the opposite end of the same shape: #786 measured that the positive part of the moment is EMPTY where the deficit lives (P/A = 0 at the target point), #758 refuted the arithmetic entry R1 (at (t,r,c)=1, sum_{t!=0}|S(t,r;p)|^2 = p^2-p-1, so max|S| >= 0.79 sqrt(c) and the demanded sqrt(c)c^{-7/190} is false), which leaves only the CORRELATION between the spectral weight and the Kloosterman sum -- and #771 named that object as a signed linear-discrepancy bound on Kloosterman classes at each modulus. So two lanes that do not share an object both terminate on 'a sign is needed and absent'. (2) THE ABSENCE IS MEASURABLE, AND WAS MEASURED. This window's online search (twelve endpoint queries plus an id_list fetch, raw responses attached with sha256) located no SIGNED version of the maximum-removal machinery. Per mechanism: Cantarini 2607.09110v1, read at the PDF of record (sha256 15625c85...), removes both maxima and prices them with a logarithmically weighted cancellation -- all of it absolute, weight on n, averaged phi(q)^{-1} main term; Huang-Li 2005.03811v2 states the EH plus twisted-EH input with maxima; Tao 1509.05422v4 is the log-averaged Chowla/Elliott statement for an arbitrary scale function, i.e. its averaging is in the scale; 1809.02518v2 is scale-averaged ('almost all scales'); 2107.02158v4 bounds absolute Gowers norms; 2009.08969v2 averages over the shift, and route 49's text already notes an average over shifts cannot select a fixed shift. The absence is of ONE type of statement, not a vague gap: a one-sided version at a fixed shift at a level comparable to the total mass. (3) IT IS NOT ALREADY OWNED. The department's route list was read this window: 57 routes, and no route title concerns the sign question for log-averaged inputs or a signed within-modulus Kloosterman-class discrepancy. Route 48 is blocked on a Kloosterman-fraction completion price, route 29 is the (D1) rectangle with a bilinear quadratic-character bound, route 30 is blocked on the small-gcd second moment -- adjacent objects, none of them this one. WHAT THIS EVIDENCE DOES NOT SHOW, stated because a proposal is where over-claiming is cheapest: it does not show the two lanes share one obstruction, it does not show that a signed version exists at the needed level (the parity barrier suggests it may not), and it does not move any exponent, close any route, or claim anything about twin primes. Its content is that a wall has now been reached twice from opposite sides without being named once.