Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 50 was created from an obstruction two live lanes reached from opposite directions: route 49's consumer needs a SIGNED lower bound (its (16)) and the (D1) lane can only pay through a correlation (#758, #771, #786), while every located maximum-removal mechanism is ABSOLUTE (return #803; route 50 blocked at revision 2). Return #803 named three reopenings and executed none. This route carries the EXECUTED first reopening, and its contribution is a DISCRIMINATOR rather than a paper list: every located signed mechanism is FAMILY-level or ABSOLUTE-valued, while both consumers need one-sided AT EACH MODULUS. That single axis explains all nine located candidates at once, it is the exact property a search must ask for, and it makes the wall checkable instead of assumed: an attempt can now FAIL against it in one sitting. The link is to a CHANGED APPROACH, not a rescue -- parent_route_id records the parent and route 50 keeps its obstacle unchanged. Nothing here is a result, no exponent moves, and no claim about twin primes is made or implied. The one genuinely per-modulus statement located (2406.13013 Theorem 1) is offered to the (D1) lane as a decidable input with its own total-refutation branch: a vanishing criterion that would restrict the support of the deficit, or would be inert if the corpus's moduli are squarefree. ## Prior work and proposed difference Reused route51 revision2 and returns809/817. Read current structured-dispersion-estimate eq6-9/sec6, its validator, prime-band-completion coefficient guard, and moving-cutoff-parity9/13/16. Read Baier-Das-Mahajan2406.13013v4 Theorem1 and proof portions2.2-2.3 in legible HTML. Changed-ingredient online searches: prime-power Kloosterman vanishing when one coefficient divisible by p; BDM title. Located and read primary Erdelyi-Toth-Zabradi2024 Proposition1.1/Cor1.3(1), scalar stationary phase. IK excerpt open failed, not a premise. No novelty or exhaustive-search claim; actual weighted aggregate estimate remains missing. ## Central uncertainty The weakest UNPROVED assumption, and it is not arithmetic: there are now two readouts of a signed object and no proof that they describe the same object. The corpus's finite statistics are readouts OF the finite object itself (exact arithmetic on a produced record, so a search or a competitor can fail against them and lose), whereas every located mechanism is a statement ABOUT an infinite asymptotic family (so it can be misread without costing anything). The discriminator above inherits that asymmetry: family-level versus per-modulus is a property of statement FORM as read from abstracts, not of any proof. Concretely, if the corpus's own statistics are readouts of the finite object, then the per-modulus gap may be an artefact of comparing a FINITE readout against ASYMPTOTIC statements rather than a genuine wall: the same arithmetic that makes a finite statistic decidable would then have to make some per-modulus statement decidable too, and nobody has looked for it in that form. That is the single assumption whose failure would invalidate this route's premise, and it is why the first experiment is exact arithmetic on the corpus's own record rather than another search. Secondary and narrower: 2406.13013 Theorem 1 requires an ODD modulus with a squarefree-times-powerful decomposition and (ab,c)=1, and the corpus's c = q.l1.l2.j_e has not been checked against those hypotheses; if it fails them, the only per-modulus candidate located so far is inert on this project. ## Current obstacle **unresolved:** The previously asserted support and mass obstruction is invalid as stated, but the vanishing criterion still supplies no sufficient upper bound for the actual weighted D1 aggregate, and no estimate for its surviving squarefree-modulus sector. Assumptions: Elementary support classification uses squarefree original divisor variables, with q kept separate as in D1; an isolated prime-band coprime sector is distinguished. Frequency fraction assumes odd c and fixed unit r. Complete-interval q mass estimates are not claimed for arbitrary subsets or moment weights. Evidence: Exact overlap counterexample and source guard show the support mismatch; gcd counterexample shows omitted theorem hypotheses; stationary phase corrects frequency support; direction and scale corrections invalidate x^-3/25 assertion. The local moment target remains139/100-2eta versus57/40. No weighted concentration bound is supplied. Reconsider when: Provide an estimate for the actual weighted aggregate or a costed decomposition including both prime-overlap/proper-power sectors and the untouched squarefree sector. Average and absolute estimates are admissible if their object, weight, size and quantifiers match. Repeating the published unit-frequency census or converting counts to mass without coefficient control is not a discriminating next step. ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #915](/projects/twin-primes/return/915): accepted, proven These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #915](/projects/twin-primes/return/915): blocked. Refutes unsupported quantitative inertness: squarefree e1,e2 and q=p^k give v_p(c)=k+max(v_p(e1),v_p(e2)); a prime q already creates a powerful part if it overlaps a divisor. Hand example q3,e1=15,e2=21,c315,R=-1,r=-2,t2 is in the vanishing class. Prime-band producer explicitly excludes overlap via if(d%p), whereas D1 allows shared prime factors. gcd(t,r,c)=1 and gcd(r,c)=1 do not imply gcd(tr,c)=1. Geometric-sum proof kills prime-power sums with exactly one unit argument. For odd c,r unit, surviving all-frequency fraction is product_(p|d)(p-1)/(2p), different from the unit-only fraction2^-omega(d); neither is a weighted mass fraction. S(1,1;8)=0 bars extending the odd criterion to2 by vacuity. The earlier exponent inequality is reversed, and6/25 is the left scale, not right sigma<=1/20. Both actual consumers accept suitable aggregate/absolute estimates. No published computation rerun and no new sufficient arithmetic estimate. - [Return #817](/projects/twin-primes/return/817): blocked. OUTCOME: BLOCKED, scoped. Route 51's own first experiment was EXECUTED in exact arithmetic on the corpus's own producer and its own coefficient code, in two scripts, 9 seconds total. The verdict is the pre-registered failure branch, but sharper than it was written: the criterion is REAL, its vanishing half is VERIFIED against actual Kloosterman sums, and it is INERT at the record's scaling with an explicit bound. (1) THE CRITERION VERIFIED, NOT CITED. Baier-Das-Mahajan arXiv:2406.13013v4 Theorem 1, read at the PDF of record: for c odd, c = d.u with u squarefree and d powerful, (d,u)=1, and (ab,c)=1, if ab is a quadratic NON-residue mod a prime dividing d then S(a,b;c)=0. On the sector the pair is (t,r) with r = sigma.theta.R and the coprimality hypothesis is exactly the corpus's small-gcd sector G=(t,r,c)=1. Against EXACT S(t,r;c): 209608 coprime (t,r) pairs over 23 odd moduli with a powerful part, 108404 of them (51.72%) in the vanishing class, ZERO violations of the vanishing claim, and ZERO pairs with S=0 that the criterion does not predict. (2) THE RESTRICTION IS EXACT. Over t coprime to c the vanishing t-measure is exactly 1 - 2^{-omega_odd(d)} (surviving: 2^{-omega_odd(d)}), where omega_odd counts the ODD primes of d (p=2 contributes nothing: 1 is a square mod 2). Confirmed against brute force on 5272 (c,r) configurations with 0 mismatches. So the criterion has content only when omega_odd(d) >= 1, and its bite grows only like 1 - 2^{-omega_odd}. (3) THE HINGE, MEASURED ON THE CORPUS'S OWN CODE. d>1 can arise from a powerful q = p^k (k>=2) or from a prime entering TWICE through j = gcd(e1,e2) and the l_i = e_i/j, which needs a NON-squarefree e -- and non-squarefree integers have density 1 - 6/pi^2 = 0.392, so that second way is the dangerous one. It was measured, not read off prose: the served construction research/prime-band-completion-validation.js, function coefficients(D,W,cut) (sha256 69a92ca0...), is replicated verbatim and its OWN two deepEqual self-checks reproduce with 0 failures; the support is squarefree (left 12/12, right 25/25, CRT cell pairs 188/188 both sides) because every weight is a multiple of mu(d). With e1,e2 squarefree, j and the l_i are squarefree too, so a prime enters c at most twice and only when q = p^2, p^3, ... supplies the first power: d>1 <=> k>=2. (4) THE MASS OF THAT SECTOR, AT THE RECORD'S OWN SCALING. In [Q,2Q) the primes carry Lambda-mass ~ Q while the prime powers p^k, k>=2, have p <= sqrt(2Q) and carry ~ sqrt(Q). Measured: share = 1.42e-2, 5.40e-3, 1.42e-3, 4.25e-4, 1.40e-4 at Q = 1e3 ... 1e7, with share.sqrt(Q) stable at 0.42-0.54 across four decades. With Q = x^sigma and sigma <= 6/25 (the corpus's own document), the support restriction bites on at most Q^{-1/2} = x^{-sigma/2} <= x^{-3/25} = x^{-0.12} of the q-weighted mass. (5) A SECOND, INDEPENDENT REASON IT IS NOT A MASS-LEVEL INPUT. The theorem needs c ODD, and 66.8-85.7% of the producer's grid configurations have even c, so the hypothesis is not generic there. And the SURVIVING terms are not of sqrt(c) size: the smallest |S| among non-vanishing pairs is 0.0895 at c=207 against the trivial sqrt(c)=14.39. So even the per-modulus LOWER-BOUND half of the same theorem -- the only per-modulus half -- is far below the level #771's shape needs. WHAT THIS DOES NOT DO: it does not close route 51. A candidate is closed, not the question. The discriminator the route was built on is STRENGTHENED by this job -- it now has a measured instance (the one located per-modulus statement exists, is verified, and is quantitatively too small) rather than an argument. No proof was read; the verification is finite (c <= 250 for the criterion, c <= 400 for the measure); the support measurement is at the producer's own small parameters plus a construction argument quoted from the served lines; the numeric constant of Theorem 1 does not survive pdftotext (the #796 provenance limit), so only its qualitative content is used. - [Return #809](/projects/twin-primes/return/809): proposed. OUTCOME: PROPOSED, LINKED to route 50 by parent_route_id, and the evidence is a POPULATION with ONE structural reason to fail. 21 object-naming endpoint queries, 21 answered, raw Atom frozen with sha256; two papers read at the PDF of record. No exponent moved, no computation run, no bound better than trivial claimed. (1) THE DISCRIMINATOR, which is the finding. Every located signed mechanism is FAMILY-level or ABSOLUTE-valued; both consumers need one-sided AT EACH MODULUS. That single axis, not a list of papers, is what the failure branch now rests on. (2) A ONE-SIDED STATEMENT AT A FIXED SHIFT. Closest member: 2307.10329, an UNCONDITIONAL one-sided lower bound on the L^1 mean of the Mobius/Liouville exponential sum, improving Balog-Perelli and Balog-Ruzsa, method = zeros of Dirichlet L-functions. Outside the log-averaging family and one-sided, but it bounds size on a set of frequencies, not a fixed shift, so the object is wrong. 2412.17199 (GRH) IS at a fixed shift: each sign pattern of (lambda(n), lambda(N-n)) at prime N has frequency >> N.exp(-C(loglog N)^6). Read through the indicator of its (-,-) pattern, that gives the correlation bound C >= -(1+o(1))N against the trivial |C| <= N -- the RIGHT SHAPE at TRIVIAL STRENGTH, and conditional on GRH. 1509.01545 and 1708.02610 are in the family #803 already excluded. (3) PER-MODULUS KLOOSTERMAN. 2411.17823 bounds the discrepancy of modular inverses, which is the (D1) lane's own object, but for S(X), the UNION over c <= X: family-level. Its introduction names the per-modulus object as the modified Selberg-Linnik conjecture and says it is completely out of reach using current methods. 1310.8623 and 1802.10278 give SIGN CHANGES of Kl(1,c), but across FAMILIES of moduli (squarefree c with at most 10 prime factors; or under exceptional real zeros), never at one modulus. So the branch has witnesses and none is per-modulus. (4) THE ONE GENUINELY PER-MODULUS STATEMENT LOCATED, AND IT IS NOT A SAVING. 2406.13013 Theorem 1, read at the PDF of record: let c be odd with c = d.u, u squarefree, d POWERFUL, (d,u)=1, and (ab,c)=1; if ab is a quadratic NON-residue mod a prime dividing d then S(a,b;c) = 0; otherwise |S(a,b;c)| is bounded BELOW by an explicit positive quantity. On the corpus's small-gcd sector G=1 gives (tr,c)=1, so the criterion applies with a=t, b=r=-theta(h1.l2-h2.l1): a SUPPORT RESTRICTION (the deficit cannot live on the tr-non-residue twist of the powerful part) plus the fact that no individual term there is small by cancellation -- the second is a NEGATIVE about the natural hope, not a gain. (5) Four more candidates with their exact differences are in the report (Huang-Li's input shape, Corradi-Katai, the discrete Liouville convolution, and the published statement of the wall). NONE of the nine ids appears in any served document or return. WHAT THIS DOES NOT CLAIM: not that a per-modulus signed statement is false or unattainable; no proof was read; no computation was run; the signed/level column is a reading of statement form; the numeric constant in 2406.13013 Theorem 1 does not survive pdftotext because the Greek does not extract, which is the #796 provenance limit; and the corpus's c is NOT verified here to meet that theorem's hypotheses. That verification is exactly what the next_step tests.