# The three killers as obstruction shapes, and where each shape has been beaten in print *(Staging note, 2026-08-29. HELD. Recon only: no script was run, no exponent moves, no existing file is edited. Publication moratorium in force.)* --- ## 0. Verdict, disconfirming half first, with the base rate **The disconfirming half, and it is the whole result.** Nineteen escapes were read. Fourteen fail their hypothesis check outright, and in twelve of those fourteen the fact they fail on was already in this corpus before the pass started. Of the remaining five, one is OPEN and is the programme's own named road (the decoupled vector sieve), one has no escape in print at all in either direction (the `kappa = 2` band), one runs in the opposite direction and is already banked (Erdos-Rankin), one is adverse-direction evidence already cited twice here (Granville-Soundararajan), and one has a hypothesis that holds trivially and a conclusion that is therefore empty (pretentious localisation). Nothing here opens a route, moves an exponent, or produces a number. The pass is a re-indexing of the wall in the literature's own vocabulary, and its honest payoff is nineteen wall addresses with a named mechanism and a named failing hypothesis, plus one name the corpus did not have (Hoheisel, absent from every markdown file, checked by grep across `research/` and `paper/`). **The base rate, stated as the brief requires.** About one hundred imports and recon angles have run in this repository and every one closed. `REFUTED.md` carries 67 rows. `IMPORT-MAP.md` §1's calibration set records five landed imports, none of which moved the exponent. The prior that this pass opens a route is accordingly near zero, and nothing below is stated above OPEN. **The one structural observation, at its lowest rung.** Five of the rows beat their own killer in print and then land inside a different one: Chen's switching principle and the bilinear remainder beat the dimension-2 cap and land on the position quantifier; the Maynard-Tao weights beat the cap and land on the quantifier; Iwaniec's every-interval bound beats the quantifier and lands on the dimension; Maier's matrix beats the ensemble average and lands on the quantifier. This is a pattern over nineteen rows, not a theorem, and it is not novel here: `attack-wrongdirection-audit.md` §0 and `README.md` §Status already record that the three killers were stated together because independent attacks converged. [HEURISTIC, and novel to this note only in tabulated form.] **The brief's one direct question, answered.** Granville and Soundararajan's uncertainty principle is **not** killer 3 in print, and the corpus **already cites it**, in two places. Their theorem asserts that every arithmetic sequence is irregularly distributed *somewhere*, which is an existence statement over intervals and progressions, at relative amplitude `1/u^{cu}`, two-sided (`special-levels-recon.md` §1d, read at source). Killer 3 asserts that no ensemble statement determines the value at the anchor. Those are different statements, and the first does not prove the second. The direction is adverse to this programme rather than helpful, and the corpus withdrew the companion Corollary 1.4 as an obstruction because its `eta` is capped at `1/100`, firing only past `log x >= (5e6)^200` (`REFUTED.md`; `maier-matrix.md` section 8). **What is nevertheless worth carrying forward.** Two hypotheses did not fail, and neither is new: the decoupled vector-sieve lemma (OPEN, already the programme's named road, buys an exponent not TPC) and whether Pomykala's `beta < 3` is essential (OPEN, priced below 2% by `recon-0828-sieve.md` section 4.3). Both are in section 7 with their circularity pre-checks written out. --- ## 1. Channel calibration Calibrated before any negative was counted, per the campaign rule that a clean negative in the wrong convention is the default failure mode (`research/SEARCH-CONVENTIONS.md`). | channel | what it can decide | what a negative from it means | used for | |---|---|---|---| | corpus grep, full-text over `research/`, `paper/` | whether a name is already priced here | a name absent is absent from THIS repo, not from the literature | prior-coverage check on all nineteen rows | | corpus custody, artifacts read at source in an earlier pass | the hypothesis of a published theorem | nothing; a custody quote is as good as the pass that took it | Green-Tao complexity, Granville on Iwaniec, GS Cor 1.2, Pomykala review, Bettin-Chandee | | arXiv abstract page, fetched this pass | title, authors, abstract wording | an abstract does not carry a theorem's hypotheses | GS `math/0406018` only | | memory of standard results | the shape of a classical mechanism | a memory claim is not a source and is flagged MEMORY | Hoheisel's exponent, Linnik's `L`, the three Linnik principles | **The standing compute rule applies** (`primeoire-campaign-lessons`, Chris, 2026-08-19): cite embedded artifacts, do not re-calculate. No script was run in this pass, and every number below is quoted from a file that owns it. **What this pass did not do.** No literature sweep in any owning convention was run; the nineteen candidates are the brief's list plus Hoheisel, and the list is not claimed to be complete. No PDF was opened. Three rows rest on MEMORY for their numerals and say so. --- ## 2. The three killers restated as shapes, with coordinates Restated as obstruction shapes so that an escape in another problem can be matched to one. Grades are taken from `object-bridge-read-0829.md` section 4 and are not re-derived here. **K1, the class-blind cap. Shape: a method whose entire input is a count of forbidden classes per modulus has a positivity threshold fixed by the count alone, and the threshold is above the target.** Coordinates: the input is `|A_d| = g(d)X + r_d` with `g(p) = 2/p`; the dimension is `kappa = 2`; the threshold in print is `beta_2 = 4.26645` (Diamond-Halberstam 2008, unimproved, `paper/beta2-note.md`); the target is exponent 2; the distance 2 enters the proof only to certify there are two classes. **Grade: HEURISTIC with a PROVEN number inside it.** What is proven is the bound and the closed search; what is not proven is that no two-class-only argument goes below it, since `beta(kappa)` is unknown for `kappa > 1/2` except `kappa = 1` and no `kappa = 2` extremal example exists in print (`sift-limit-attack.md` section 2, `SEARCH-CONVENTIONS.md` section 3: "the band is a proof gap, not a truth gap"). **K2, the quantifier. Shape: a theorem holding for almost all members of an ensemble decides a prescribed member only if the exceptional count is below one.** Coordinates: the ensemble has `W = x#` members, the anchor is one of them, so an exceptional fraction `eps` decides the anchor only when `eps W < 1`, i.e. `eps = e^{-(1+o(1))w}` and **not** any rate polynomial in `w` (`anchored-note.md` section 3, Proposition 1 part (i), exact counting). **Grade: this is the killer with a THEOREM in it.** The measured distance: the second-moment bound gives `eps ~ ln^2 W / W`, measured `8.38e-6` at @19 against the decision threshold `1.03e-7`, a factor 81 short and growing like `ln^2 W`. One line generalises it: a single empty window contributes `(H P_y)^2 / X` to a variance of size `M(H,y)`, so it sits below the error term of any variance asymptotic by a factor growing with `X` (`recon-0828-rough.md` section 3.3). **K3, equivariance. Shape: a statement quantified over a group that acts transitively on the ensemble carries no information about any single orbit point, because the group moves every point to every other.** Coordinates: the rotation ensemble is `Z/W` acting simply transitively on phases, and the anchor is the phase `t = 0`. Two components are PROVEN. (a) The tile's transform on `Z/W` **vanishes nowhere**: each local factor is `1 + e(2v/q)` up to a phase, zero only when `4v = q (mod 2q)`, which parity forbids for odd `q`, so `supp That = W` and the whole Donoho-Stark uncertainty family holds with maximum slack (`recon-0828-farfields.md` section 2a, checked numerically at twelve primes). (b) The dependency graph is **complete**, on which Shearer's exact criterion is the union bound (`import-shearer.md` section 4). The operational form the corpus settled on is not Fourier at all: an interval of length `L` is a diagonal segment of `L` points in a CRT box of `x#` cells, and any theorem that counts, covers or optimises over the box sees the segment only once `L` approaches `x#` (`recon-0828-farfields.md` section 2b). **Grade: MEASURED pattern with two PROVEN components**, and three tests at the anchor all null (mirror symmetrisation gains exactly zero, the QR-immune classes deliver exactly the guardrail, the anchored `delta` exceeds the forced scale at 2 of 7 levels). **A coordinate the three share, and it is the one that makes the escapes collide.** The corpus's target window has length about `x'^2`, which is about `(ln x#)^2`. On the scale the short-interval literature measures in, where the window is `X^theta` with `X = x#`, the corpus needs `theta = 0`. Every K2 escape in print is stated at `theta` bounded away from zero. Section 4 quantifies this. --- ## 3. K1 escapes: beating a sieve dimension cap Six escapes, each stated as mechanism, hypothesis, check, verdict, and labelled (i) mechanism named in print, (ii) hypothesis identified, (iii) hypothesis checked against a corpus fact. ### 3a. Bilinear and Type II information in the sieve remainder **Mechanism (i).** Replace the axiom-only input `|A_d| = g(d)X + r_d`, whose remainder is handled in absolute value, with a bilinear form `sum_{m 1*, Bull. Polish Acad. Sci. Math. 38 (1990) 167-171, and Greaves's zbMATH review, read at the record and transcribed in `recon-0828-sieve.md` section 4.1, states the constraint verbatim: *"The construction requires that the sifting limit `beta` satisfies `beta > 2`, because the level of distribution `Delta` is expressed as `Delta = MNK^{beta-2}`. There is also a more technical requirement `beta < 3`, the need for which appears in the proof of a key lemma."* Since the DHR ladder runs at about `2.44 kappa`, `beta < 3` confines the construction to `kappa < 1.23` (`recon-0828-sieve.md` section 4.2, ARITHMETIC). This programme's dimension is 2 and its `beta_2` is 4.26645, above the constraint by a factor of 1.42 on `beta` alone. Part (b) fails independently, and it fails with a number. A trilinear form for the comb's remainder is a trilinear sum of sawtooths over a periodic set, which is the Kloosterman-fraction object of `SEARCH-CONVENTIONS.md` section 1 row 34. The position-uniform case is priced there: Bettin-Chandee's Remark 1 charges `(1 + hx/MN)^{1/2}`, which is `O(1)` only for `x << H^{1.212157}`, while this programme's `x` reaches `exp(H^{0.2344})` (`sift-limit-attack.md` section 4.5 row). The smooth-modulus branch is unavailable because well-factorability supplies 1-bounded factors, not smooth ones (`smoothness-front.md`; `REFUTED.md`). **Verdict.** The hypothesis fails because the only published construction above dimension 1 stops at `kappa < 1.23` against this object's `kappa = 2`, by a constraint its own reviewer flags as needed for a key lemma, and because the position-uniform bilinear estimate is short by the exponent comparison `H^{1.212157}` against `exp(H^{0.2344})`. Both facts are already in the corpus. ### 3b. The Friedlander-Iwaniec asymptotic sieve for primes **Mechanism (i).** The asymptotic sieve for primes produces an asymptotic count of primes in a sequence, defeating the parity obstruction, for sequences that supply a Type II bilinear estimate over a wide range of `M`. The in-print payoff is `a^2 + b^4` (Friedlander-Iwaniec 1998), and the extra structure there is a quadratic or quartic form, not a class count. **Hypothesis (ii).** A Type II sum `sum_{m ~ M} sum_n a_m b_n` controlled for `M` across a range like `[x^eps, x^{1/2-eps}]`, together with a well-behaved density function. Friedlander and Iwaniec state in their own paper that the twin-prime sequence does not supply it, which is why the method does not settle twins. [MEMORY for the wording of their remark; the corpus does not carry the sentence, and this pass did not open the paper.] **Check (iii).** The corpus already prices the input side, and the price is saturation rather than absence. `sift-limit-attack.md` section 3 records "Trivial remainders at level `H^{1-eps}`, uniform in `x` (PROVEN, `beta2-note` section 3)" against discard point DP4 with the verdict *"saturated; blocks all conditional improvement"*. The tile's remainder for a window is a sawtooth over a periodic set of period `x#`, and what makes it trivially bounded is what leaves nothing for a Type II decomposition to cancel: there is no multiplicative variable to split. Second, the conclusion FI delivers is an asymptotic over a long range, not a bound at one window, so even granting the hypothesis the output crosses into K2. **Verdict.** The hypothesis fails because the object supplies no Type II range, its remainder being already trivially bounded and position-uniform at level `H^{1-eps}`, which the corpus records as saturated; and the conclusion would carry the wrong quantifier in any case. ### 3c. The switching principle (Chen 1973) **Mechanism (i), and this is a genuine K1 escape in print.** Chen's theorem gives a positive lower bound for the count of `p` with `p + 2 = P_2`, a positivity statement at a dimension-2 problem where the dimension-2 cap would forbid it. It escapes the cap not by improving the sieve but by counting the bad set from the other tooth, where it becomes a semiprime-slot count an upper sieve can afford. In this repository's vocabulary that is a change of tooth on the comb (`bv-import-survey.md` section 4, which writes the lenient Scour and states Chen's theorem as a lower bound on the anchored bias of the lenient comb). **Hypothesis (ii).** Two classical components: a lower-bound linear sieve on the alive set, and Bombieri-Vinogradov equidistribution of primes in arithmetic progressions over `[1, x]` for the switched count (`bv-import-survey.md` sections 4, 3.2). **Check (iii).** The BV input is not available uniformly in the window position. `natal-cap-10-sieve-cap.md` section 1.4 states it flatly: *"Everything below 8 in the table uses equidistribution of primes in APs (BV or beyond) over `[1, x]`, those inputs are not available uniformly in the window position, so none of the sub-8 constants transfer to a bound for `pi_2(x+l) - pi_2(x)` valid for all x"* [INFERRED from the proofs' structure; standard]. The corpus's target is position-uniform by construction, `G_2` being a maximum over positions. Second, the conclusion is `P_2` rather than a prime pair, so the output does not reach `Z_2` even if the input were available. **Verdict.** The hypothesis fails on the position quantifier: an escape that beats K1 by importing an input K3 forbids. This is one of the five collisions section 0 records. The corpus already judges a tile-native Chen feasible and worth doing as a demonstration, explicitly not as a strengthening (`bv-import-survey.md` section 4, "it would not strengthen Chen"). ### 3d. The vector sieve (Brudern-Fouvry 1996) **Mechanism (i).** The two-class condition is a product of two one-class conditions, and the vector sieve converts exactly that into two linear sieves, through the pointwise inequality `theta_1(r) theta_2(r+2) >= L1- L2+ + L1+ L2- - L1+ L2+`. It is the one asset in `sift-limit-attack.md` section 3's discard table with a named consumer. **Hypothesis (ii).** Decoupled component levels `D_1`, `D_2`. With the levels coupled, absolute-value remainder accounting forces `D_1 D_2 <= H^{1-eps}`. **Check (iii).** Coupled, the route lands at `2(1 + sqrt e) = 5.2974`, worse than `beta_2 = 4.26645`, and `paper/wall-note.md` section 1 records that this is why the route is absent from the problem's record. Decoupling needs one new lemma: signed cancellation of the bilinear interval remainder, uniformly in window position, the two-dimensional analogue of Iwaniec 1980. That is precisely the lemma 3a shows is missing above dimension 1. **Verdict.** The hypothesis is **OPEN, not failed**, and this row is carried to section 7 with its circularity pre-check. Its price is already measured and is not a route to the conjecture: `research/theta-ladder.md` measures the window exponent at which the certificate is positive at every position, and the requirement sits inside the zone budget at every exactly-measured level (`need/z^2 = 0.3550..0.6119` at `z = 13..31`, all exact), which makes the route an exponent-improvement programme (`paper/wall-note.md` section 1). ### 3e. Maynard-Tao multidimensional Selberg weights **Mechanism (i).** Optimise over a multivariable weight `F` rather than a one-variable `lambda`, and the bounded-gaps conclusion follows: at least 2 of `k` admissible linear forms are prime for infinitely many `n`. **Hypothesis (ii), read as what the conclusion is allowed to say.** The mechanism buys positivity of a difference of two sums, and what it purchases is the existence of **some** pair among `k` forms at **infinitely many** `n`. It never forces a prescribed pair, and it never gives a statement at a prescribed position. **Check (iii).** The target here is `G_2(x#)`, a maximum over all positions of `Z/x#`, and `Z_2`, a statement about one prescribed window. `object-bridge-read-0829.md` section 4 records that K2 bites on every "almost all intervals" import and that infinitely-many is already fatal, since the anchor is one point at each level (`attack-wrongdirection-audit.md` section 1, Axis B). Bounded gaps is already priced in `IMPORT-MAP.md` and in `recon-0828-sieve.md`. **Verdict.** The hypothesis fails on the conclusion's shape: a K1 escape whose output is exactly the quantifier K2 forbids. Another collision. ### 3f. Sieve escapes at kappa > 1 in any problem, read for a floor **Mechanism (i).** Across the literature the escapes available at dimension above 1 deliver almost-primes: Iwaniec's `x^2 + 1` and the `P_r` results for polynomials are the standard shape. None delivers positivity at the sifting limit for a dimension-2 problem. **Hypothesis (ii).** That a `kappa = 2` construction below `beta_2 = 4.26645` exists at all. **Check (iii).** `SEARCH-CONVENTIONS.md` section 3, searched in the owning convention: *"Published `kappa = 2` extremal example, a set that blocks an exponent in (2, 4.2665]? None found. The band is a proof gap, not a truth gap."* Ford's notes, quoted in `sift-limit-attack.md` section 2: *"The exact value of `beta(kappa)` is known only for `kappa` in `[0, 1/2]` union `{1}`."* **Verdict.** No escape in print, and no barrier theorem either. This row does not fail a hypothesis; it records that the band `(2, 4.26645]` is unoccupied in both directions, which is what makes K1 a heuristic rather than a theorem. [ABSENT-PER-CONVENTION, per `SEARCH-CONVENTIONS.md`.] --- ## 4. K2 escapes: almost-all to every Seven escapes. The scale fact that decides most of them is stated once here and pointed at afterwards. **The scale fact, stated once.** Write `X = x#` for the tile's period. Every almost-all-to-every theorem for primes in print is stated for windows of length `H = X^theta` with `theta` bounded away from zero: Hoheisel's 1930 result at `theta = 1 - 1/33000` [MEMORY for the numeral], Huxley's 1972 asymptotic at `theta = 7/12 = 0.5833`, and the current record `theta = 17/30 = 0.5667` (Guth-Maynard, arXiv:2405.20552; the corpus's own table at `attack-bilinear-transplant.md`, row "single primes, asymptotic formula"). The window this programme must decide has length about `x'^2`, and `ln X = theta(x)` is about `x`, so the window is about `(ln X)^2`: **on the short-interval literature's own scale the corpus needs `theta = 0`.** The gap between 0.5667 and 0 is not a constant to be improved; the zero-density mechanism produces nothing at any polylogarithmic length, which is exactly the regime where Granville-Soundararajan record that the Montgomery-Vaughan Gaussian heuristic breaks for many highly composite `q` (`special-levels-recon.md` section 1d, at source, GS preprint p. 4). [ARITHMETIC on corpus-held numerals; the reading is the corpus's own diagonal-segment statement restated with a number.] ### 4a. Hoheisel and Huxley via zero density **Mechanism (i).** Convert an almost-all-intervals statement for `pi(x+H)-pi(x)` into an every-interval one by writing the count through the explicit formula over zeros of `zeta` and controlling the exceptional zeros with a zero-density estimate `N(sigma, T) << T^{A(1-sigma)} L^B`. This is the cleanest almost-all-to-every upgrade in analytic number theory. **Hypothesis (ii).** (a) The counted object is expressible by an explicit formula over the zeros of an L-function. (b) The window length is a power of the main variable, with the exponent above the zero-density threshold. **Check (iii).** (a) fails outright: the tile's hole count in a window is a sum of sawtooths over a periodic set on `Z/x#`, with no L-function attached, and the corpus's target is a maximum over positions, which the explicit formula does not deliver in any form. (b) fails by the scale fact above, between `theta = 0.5667` and `theta = 0`. **Verdict.** The hypothesis fails on both halves. **Novelty check: Hoheisel appears in no file of this repository** (grep over `research/` and `paper/`, this pass), and Huxley appears only in three tables. The ROW is new here; the FACT is the corpus's diagonal-segment statement (`recon-0828-farfields.md` section 2b) restated with a number, so this is independent rediscovery inside the corpus, not a discovery. ### 4b. Iwaniec's every-interval rough-number bound **Mechanism (i).** The linear sieve applied with the trivial interval remainder `|r(A,d)| <= 1`, which holds uniformly in `x`, gives an every-position lower bound: `S(x,y,z) >= (4y/(log y)^2)(log(y/z^2) - O(1))` when `y >> z^2`, uniformly in `x` (Granville, *Sieving intervals and Siegel zeros*, Acta Arith. 205 (2022) p. 3, read at page image, `recon-0828-rough.md` section 2, attributing to Iwaniec 1978). Consequence in Granville's own words: `J(P(z)) << z^2`. This is a real escape from the quantifier, and it is the strongest one in print for this object class. **Hypothesis (ii).** Sieve dimension `kappa = 1`, so that the lower-bound function `f(u) = 2 e^gamma log(u-1)/u` is positive for `u > 2`. **Check (iii).** Substituting `Omega_p = {a_p, a_p - 2}` moves the dimension from 1 to 2, `f` and `F` change with it, and the positivity threshold moves from `beta_1 = 2` to `beta_2 = 4.26645`, which is this corpus's own bound (`recon-0828-rough.md` section 2, "Exact at the object and worthless at the exponent"). **Verdict.** The hypothesis fails on the dimension. **This is the sharpest of the collisions**: the only theorem in print that beats K2 for this exact object beats it at `kappa = 1`, and moving to the two classes the target needs lands squarely on K1. `covering-dive.md` section 1.3 names the reason: the sifting limit (2) equals the twin-critical exponent (2) only at one class, and the corpus calls that the one-class miracle. [PROVEN, cited.] ### 4c. Erdos-Rankin and FGKMT long gaps **Mechanism (i).** Construct a long interval covered by choosing one residue class per prime, giving `G(x) >> x log x logloglog x / loglog x`. **Check (iii).** The direction is opposite: this produces a lower bound on the gap, not an upper bound, and the corpus already consumes it. `README.md` §Status records that twin slots are a subset of the tile's holes, so `G_2 >= g` pointwise and Rankin/Pintz/FGKMT give the free lower bound, and a stronger one was obtained by carrying a two-class kill set through Kalmynin-Konyagin (`paper/proposals/draft-kk-lower-bound.md`, checked twice, not refereed). **Verdict.** Not an escape from K2; an adverse-direction theorem already banked. ### 4d. Linnik's theorem on the least prime in a progression **Mechanism (i).** An every-class conclusion, `p(a,q) << q^L`, from ensemble inputs: a log-free zero-density estimate, the Deuring-Heilbronn zero-repulsion phenomenon, and Page's theorem isolating the single exceptional character. The exceptional class is handled separately rather than averaged away, which is what makes this a genuine almost-all-to-every mechanism. **Hypothesis (ii).** (a) The carrier is the zeros of Dirichlet L-functions mod `q`. (b) The range searched is a **power** of the modulus, `q^L`. **Check (iii).** (a) fails for the same reason as 4a: no L-function is attached to the tile. (b) fails on the scale fact: the corpus's window is about `(ln x#)^2` against a modulus `x#`, so the range is polylogarithmic in the modulus, not a power of it. **And the implication runs the wrong way, which is the decisive point.** `covering-dive.md` section 1.1 records Vaughan's observation that a Jacobsthal exponent below 2 would give Linnik's theorem easily, and `recon-0828-rough.md` section 2 records Mercer, INTEGERS 18 (2018) #A26 p. 3, read at page image: a bound `h(n) <= C p_n^{2-eps}` for all `n` "would lead to a short proof of Linnik's theorem and Dirichlet's theorem", and even `h(n) = o(p_n^2)` gives a short proof of Dirichlet. So Linnik sits **downstream** of the one-class target, not upstream of it. **Verdict.** The hypothesis fails on both halves, and the direction is wrong-way: importing Linnik as an escape would be importing a consequence of a weaker form of the target. The corpus already records the false alarm this generates and explicitly declines it: `G_2 >= g` pointwise, so a two-class exponent of 2 implies nothing below 2 at one class, and no Linnik price attaches (`recon-0828-rough.md` section 2, "Recording it so the false alarm is not raised later"). ### 4e. Gallagher's large sieve and the exceptional-set method **Mechanism (i).** Bound the second moment of the error across the ensemble (Barban-Davenport-Halberstam from Bombieri-Vinogradov; Gallagher's large sieve for the variance), then conclude that the exceptional set has density tending to zero, and take a member outside it. **Hypothesis (ii).** The exceptional fraction `eps` must be small enough that `eps` times the ensemble size is below one at the decision point. **Check (iii).** PROVEN counting, and it is K2's core: the ensemble has `W = x#` members and the anchor is one, so the boundary sits at `eps = 1/W`, i.e. `e^{-(1+o(1))w}`, and **not** at any rate polynomial in `w` (`anchored-note.md` section 3, Proposition 1 part (i)). The measured distance: the second-moment bound gives `eps ~ (Var/E)/E ~ ln^2 W / W`, measured `8.38e-6` at @19 against the decision threshold `1.03e-7`, a factor 81 short and growing like `ln^2 W` (`object-bridge-read-0829.md` section 4). The general form, which closes the whole branch at once: a window that is empty rather than carrying its expected `H P_y` survivors contributes `(H P_y)^2 / X` to the variance, which is below the error term of any variance asymptotic by a factor growing with `X`, so a single exceptional window is invisible to a second moment at every level and for every sharpness (`recon-0828-rough.md` section 3.3, and `variance-note.md` section 4 in its own words). **Verdict.** The hypothesis fails exactly, with the number: the method's exceptional density is polynomial in `w` and the decision needs exponential. ### 4f. The combinatorial family: local lemma, second moment, Kim-Vu **Mechanism (i).** Turn "typical" into "some member is good" by showing the bad events have positive probability of simultaneous failure (Lovasz local lemma and its algorithmic form), or by concentration (second moment, Janson, Kim-Vu polynomial concentration). **Hypothesis (ii).** For the local-lemma family, a **sparse** dependency graph. For the concentration family, slack between the worst-case single-coordinate effect and the theorem's own deviation scale. **Check (iii).** All CLOSED here, and the closures are cited rather than re-derived. The dependency graph of the tile's kill events is **complete**, on which Shearer's exact criterion **is** the union bound, Moser-Tardos gains nothing on the resulting chordal graph, and every member that beats Shearer buys it with structure costing `H >= x#` (`REFUTED.md` row on the local-lemma family, `import-shearer.md` sections 4, 6, 8, 2026-08-19). The bounded-differences family including Kim-Vu is closed because the first scour prime's worst-case effect `d(q_1)` misses the theorem's own slack by a factor growing like `sqrt N`, 174 rising to 70,576 over `x = 11..23`, and the published repair's exponent is about `ln^2 x / x`, below 1 at every computable level and falling (`REFUTED.md`, `row7-recon.md`, 2026-08-20). Generic chaining is closed because its entropy integral already exceeds the union bound by 1.04 to 1.10 at every level (`REFUTED.md`, `import-chaining.md`). **Verdict.** The hypothesis fails for the local-lemma family on graph sparsity and for the concentration family on slack, both PROVEN or CLOSED in the corpus. There is a second, independent failure worth one line: even a successful local lemma delivers existence of a good configuration, and the corpus's target is a statement at a prescribed position, so the output would carry the wrong quantifier regardless. ### 4g. Matomaki-Radziwill in short intervals **Mechanism (i).** Control a bounded multiplicative function in almost all intervals of length `h` with `h -> infinity`, with power-saving uniformity in part II. **Hypothesis (ii) and check (iii).** The input is admissible: the `y`-rough indicator is `chi_0 mod y#`, the principal character, hence maximally pretentious. The failure is the conclusion, which is an `L^1` average over starting points, and no averaging statement produces a bound at a prescribed start. Dead by 4e's mechanism, one rung weaker, at the first moment rather than the second (`recon-0828-rough.md` section 3.4, priced 2026-08-28). **Verdict.** Already priced and CLOSED; recorded so the row is not re-run. --- ## 5. K3 escapes: ensemble to point Six escapes. The first is the strongest row in the note, because its hypothesis is refuted by a printed sentence naming this programme's exact system. ### 5a. The circle method: Vinogradov's three primes at a fixed N **Mechanism (i), and it is the canonical ensemble-to-point escape.** Write the count as an integral over all frequencies, split into major arcs where the singular series localises the answer to the specific `N` under consideration, and minor arcs killed by bilinear cancellation. The output is a statement about one prescribed odd `N`, obtained from an object defined over the whole ensemble of frequencies. That is precisely the shape K3 says is unavailable here. **Hypothesis (ii).** The pattern must be Fourier-detectable, which in the modern formulation is Gowers complexity 1 or lower: the linear forms system must be one whose count is controlled by the `U^2` norm, so that the minor-arc contribution is genuinely small. **Check (iii), and the check is a printed sentence about this exact system.** Green and Tao, *Annals* 171 (2010) p. 1760, read at the page image and transcribed in `import-transference.md` section 0, verbatim: *"The system `Psi(n_1) := (n_1, n_1+2)`, which counts twin primes, has infinite complexity,"* and in the same paragraph *"More generally, any system with `d = 1` and `t > 1` has infinite complexity."* Vinogradov's three-primes system has `d = 3` independent variables and one form; the twin system has `d = 1` and `t = 2`. The three-primes escape works because its system is at the easy end of the complexity scale, and this programme's system is at the other end, by a statement in print that names it. **A second, independent failure, PROVEN here.** There are no minor arcs to exploit. The tile's transform on `Z/W` **vanishes nowhere**: the local factor is `1 + e(2v/q)` up to a phase, zero only when `4v = q (mod 2q)`, impossible for odd `q`, and at `v = 0` the local value is `q - 2 >= 1`. The worst-case local modulus has the closed form `min |fhat_q(v)| = 2 sin(pi d_q / q) >= 2 sin(pi/(2q))`, measured at twelve primes as `1.000000, 0.618034, 0.445042, 0.284630, 0.241073, 0.184537, 0.165159, 0.136485, 0.108278, 0.101298, 0.084882, 0.076605` for `q = 3..41`, with `0.076605 = 2 sin(pi/82)` to six places (`recon-0828-farfields.md` section 2a, SCRATCHPAD-GRADE for the numerals, PROVEN for the non-vanishing). **Verdict.** The hypothesis fails by a printed statement about the exact system, and fails again on the transform's full support, which removes the minor-arc / major-arc dichotomy the method needs. This is the cleanest wall address in the note. ### 5b. Siegel-Walfisz at a fixed modulus **Mechanism (i).** An average over characters, uniform enough to give a statement at one fixed modulus rather than for almost all moduli. **Hypothesis (ii).** The modulus is bounded by a fixed power of the logarithm, `q <= (log x)^A`, and the carrier is Dirichlet L-functions. **Check (iii).** This programme's modulus is `x#`, and `ln x#` is about `x`, so `(log x#)^A` is nowhere near `x#`. The corpus already priced what Siegel-Walfisz does buy here, which is the fixed-modulus tail refinements (`bv-import-survey.md` section 3.1, "Provable now by Siegel-Walfisz"). **Verdict.** The mechanism applies, at a modulus range this programme's object does not live in, and buys nothing at the anchor. Already priced. ### 5c. Maier's matrix method: distinguished levels chosen by the argument **Mechanism (i).** Escape an equidistribution average by choosing a row of a matrix whose column sums are known, which produces rows above and rows below the average. This is the closest thing in print to selecting a favourable level rather than accepting the ensemble's verdict. **Hypothesis (ii).** The ability to select a level, plus the conclusion being allowed to be an existence statement over levels. **Check (iii), two parts, both already in the corpus.** (a) On the axis this programme would need it, `THE-DIALS.md` dial 4's fourth channel was surveyed and closed: 23 instruments, 19 eligible, 11 rules each, a 20,000-permutation null with a best-of-family statistic, smallest per-instrument `p_FWE = 0.1082` and Sidak over the nineteen instruments at `0.8865`, so not one instrument reaches 0.05 even before the outer correction. The two signatures with a named mechanism select **finite** sets: `maxsum_{L+1}` is sharp exactly when no gap of the old tile is `= 0, +-2 (mod p)`, whose qualifying-gap counts run `2, 4, 4, 72, 60, 20, 380, 380, 64, 11784, 9452, 9500, 243816, 248058, 95896` and grow like `3D/p` with `D = prod_{3<=q<=x}(q-2)`, so the condition is already unsatisfiable at `T_11`; the greedy oracle's sharp levels are the initial segment `x <= 43` (`ioslack-survey.md` sections 0 and 5, MEASURED at 15 cells, PROVEN-shaped). (b) Even if a signature selected an infinite set, infinitely-many is already fatal, since the anchor is one point at each level (`attack-wrongdirection-audit.md` section 1, Axis B). **Verdict.** The hypothesis is available and the conclusion is on the wrong axis. Dial 4 is FINISHED and this row is a re-statement of that closure in the literature's vocabulary rather than a new check. ### 5d. Granville-Soundararajan, the uncertainty principle This is the row the brief singles out, so it is answered in the brief's own two parts. **Is it killer 3 in print for sieve-weighted sequences? No.** The paper is Granville and Soundararajan, *An uncertainty principle for arithmetic sequences*, Ann. of Math. 165 (2007) 593-635, preprint arXiv:math/0406018; title, authors and abstract fetched at the arXiv abstract page this pass [SOURCED for the abstract only]. The abstract's own statement of scope: *"In various discrepancy problems, combinatorics researchers have analyzed limitations to equi-distribution, as have Fourier analysts when working with the 'uncertainty principle'. In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples."* The theorem asserts that any arithmetic sequence **is** irregularly distributed somewhere. Killer 3 asserts that no ensemble statement **determines** the value at the anchor. The first is an existence statement over intervals and progressions; the second is a non-transfer statement about a fixed point. Neither implies the other, and GS's amplitude is `1/u^{cu}`, tiny and two-sided. **Does the corpus already cite it? Yes, in two places, and one of them is a withdrawal.** (a) `special-levels-recon.md` section 1d carries Corollary 1.2 at source (GS preprint pp. 1-4): for `q` large squarefree with `sum_{p | q} (log p)/p >= 60 log_3 q`, a bound *"attained when q is the product of the primes up to some large y"*, there are intervals `I+-` of length at least `z^u` whose reduced-residue counts read `{1 +- 1/u^{c_2 u}} (phi(q)/q) |I+-|`, and GS note this breaks the Montgomery-Vaughan Gaussian heuristic at `h = log^A q` for many highly composite `q`. **The extremal level is the primorial**, which is exactly this programme's level. (b) `REFUTED.md` and `maier-matrix.md` section 8 record that the corpus **withdrew** GS Corollary 1.4 as an obstruction, because its `eta` is capped at `1/100`, firing only past `log x >= (5e6)^200`. **Verdict.** Not an escape from K3 and not K3 in print. It is adverse-direction evidence, already held at source, already cited twice, and its one obstruction-shaped corollary is already withdrawn here for a quantified reason. The one thing it adds to the wall address is the identification of the primorial as the extremal level for the irregularity it certifies, which strengthens the scale fact of section 4 and weakens nothing. ### 5e. Pretentious localisation (Granville-Soundararajan, the pretentious large sieve) **Mechanism (i).** Replace equidistribution by a distance to a single character, and localise a mean value by identifying which character the function pretends to be. **Hypothesis (ii).** The function is multiplicative and 1-bounded, and the conclusion is a mean over a long interval. **Check (iii).** The hypothesis holds trivially and the conclusion is therefore empty: the `y`-rough indicator **is** `chi_0 mod y#`, the principal character, so the input is maximally pretentious and the theory identifies it correctly and says nothing further (`recon-0828-rough.md` section 3.4, which records exactly this for the Matomaki-Radziwill branch of the same theory). The quantifier is still an average. **Verdict.** The hypothesis holds and the conclusion carries no information, the one failure mode that is not a hypothesis failure. Recorded as VOCABULARY-ONLY in the sense of `IMPORT-MAP.md` section 0. ### 5f. Duke and the Linnik ergodic method: equidistribution of a fixed orbit **Mechanism (i).** For a **fixed** discriminant, Heegner points equidistribute on the modular surface. This is an honest single-orbit result rather than an average, obtained through subconvexity for the associated L-functions, and it is the strongest ensemble-to-point escape available in the homogeneous-dynamics family. **Hypothesis (ii).** (a) The point of interest lies on a positive-dimensional orbit of a group action with a spectral gap. (b) An L-function is attached, with subconvexity available. **Check (iii).** (a) fails structurally. The ensemble here is `Z/W` acting simply transitively on the phases, so every orbit is the whole ensemble and the anchor's stabiliser is trivial: the "orbit" of `t = 0` under anything smaller is a single point, the degenerate case the method excludes. There is no spectral gap to exploit, because the transform has full support with no eigenvalue separation, `min |fhat_q(v)| >= 2 sin(pi/(2q))` at every frequency (`recon-0828-farfields.md` section 2a). (b) fails for the same reason as 4a and 4d: no L-function is attached to the tile. **Verdict.** The hypothesis fails on both halves, the first for a reason that is K3's own shape restated: transitivity of the ensemble action is exactly what makes a single phase invisible. --- ## 6. The wall-address table One row per escape. "Lands in" names the killer the escape falls into after its own killer is beaten, where that happens. "Prior" records whether the corpus already priced the row before this pass. | # | escape, in print | killer it beats elsewhere | the hypothesis that must hold here | the fact it fails on, cited | lands in | prior | |---|---|---|---|---|---|---| | 3a | bilinear / Type II remainder (Iwaniec 1980; Vinogradov; Heath-Brown identity) | K1 | a bilinear construction at `kappa = 2`, cancelling uniformly in window position | only construction above `kappa = 1` is Pomykala 1990, capped at `beta < 3` i.e. `kappa < 1.23` (`recon-0828-sieve.md` 4.1-4.2); position-uniform branch short, `H^{1.212157}` vs `exp(H^{0.2344})` (`sift-limit-attack.md` 4.5) | K3 | priced 2026-08-28 | | 3b | Friedlander-Iwaniec asymptotic sieve (`a^2+b^4`) | K1 | a Type II range for the sequence | remainder already trivial and position-uniform at `H^{1-eps}`, recorded **saturated** at DP4 (`sift-limit-attack.md` 3) | K2 | input side priced; FI row new | | 3c | switching principle (Chen 1973) | K1 | lower linear sieve plus BV over `[1,x]` | BV is not available uniformly in window position (`natal-cap-10-sieve-cap.md` 1.4); output is `P_2`, not a pair | K3 | priced | | 3d | vector sieve (Brudern-Fouvry 1996) | K1 | **decoupled** component levels | coupled gives `2(1+sqrt e) = 5.2974 > 4.26645` (`wall-note.md` 1); decoupling needs the lemma 3a shows is missing | **OPEN**, see 7a | the programme's named road | | 3e | Maynard-Tao multidimensional weights | K1 | that an i.o.-and-some-pair conclusion suffices | target is a max over all positions; i.o. already fatal (`attack-wrongdirection-audit.md` 1, Axis B) | K2 | priced | | 3f | `kappa > 1` almost-prime escapes (Iwaniec `x^2+1`, `P_r`) | K1 | a `kappa = 2` construction below 4.26645 | none in print, and no extremal example either (`SEARCH-CONVENTIONS.md` 3; Ford via `sift-limit-attack.md` 2) | none; the band is unoccupied both ways | priced | | 4a | Hoheisel / Huxley via zero density | K2 | explicit formula over L-function zeros; window `X^theta`, `theta` above the density threshold | no L-function on `Z/x#`; window is `(ln X)^2`, i.e. `theta = 0` against 0.5667 (`attack-bilinear-transplant.md` table; `recon-0828-farfields.md` 2b) | K3 | **Hoheisel absent from the repo** | | 4b | Iwaniec's every-interval rough bound | K2 | `kappa = 1`, so `f(u) > 0` for `u > 2` | two classes move the threshold `2 -> 4.26645` (`recon-0828-rough.md` 2; `covering-dive.md` 1.3) | **K1** | priced | | 4c | Erdos-Rankin / FGKMT long gaps | direction reversed | n/a | lower bound, not upper; already consumed (`README.md` §Status; `draft-kk-lower-bound.md`) | n/a | banked | | 4d | Linnik's least prime in an AP | K2 | L-function zeros; range a **power** of the modulus | no L-function; range is polylog in the modulus; and the implication runs the other way, `h(n) <= C p_n^{2-eps}` gives Linnik (Mercer 2018 p. 3 via `recon-0828-rough.md` 2) | wrong direction | priced | | 4e | Gallagher large sieve / BDH exceptional set | K2 | `eps W < 1` at the decision point | boundary is `eps = 1/W = e^{-(1+o(1))w}`, not polynomial in `w` (`anchored-note.md` 3 Prop 1(i), exact); measured `8.38e-6` vs `1.03e-7` at @19, 81x short | n/a, K2 itself | priced | | 4f | local lemma / Shearer / Moser-Tardos; second moment, Janson; Kim-Vu | K2 | sparse dependency graph; slack over the single-coordinate effect | graph is **complete**, Shearer = union bound (`import-shearer.md` 4); `d(q_1)` misses slack by `sqrt N`, 174 to 70,576 over `x = 11..23` (`row7-recon.md`) | n/a | CLOSED | | 4g | Matomaki-Radziwill short intervals | K2 | none; input admissible | conclusion is an `L^1` average over starting points (`recon-0828-rough.md` 3.4) | n/a | CLOSED | | 5a | circle method, Vinogradov three primes at fixed `N` | K3 | pattern of Gowers complexity `<= 1` | Green-Tao, Annals 171 (2010) p. 1760, verbatim: the twin system *"has infinite complexity"*, and *"any system with `d = 1` and `t > 1`"* does (`import-transference.md` 0); and no minor arcs exist, `supp That = W` (`recon-0828-farfields.md` 2a, PROVEN) | n/a | complexity fact held 2026-08-27 | | 5b | Siegel-Walfisz at a fixed modulus | K3 | `q <= (log x)^A` | modulus is `x#` with `ln x# ~ x` (`bv-import-survey.md` 3.1) | n/a | priced | | 5c | Maier's matrix, distinguished levels | K3 | a recognisable signature selecting an infinite set | the two signatures with a mechanism select **finite** sets; `p_FWE >= 0.1082`, Sidak `0.8865` over 19 instruments (`ioslack-survey.md` 0, 5); and i.o. is already fatal | K2 | dial 4 CLOSED | | 5d | Granville-Soundararajan uncertainty principle | neither; adverse direction | n/a | an existence-of-irregularity statement, amplitude `1/u^{cu}`, not a non-transfer statement; Cor 1.4 already withdrawn here, `eta <= 1/100` (`REFUTED.md`; `maier-matrix.md` 8) | n/a | cited twice already | | 5e | pretentious localisation | K3 | multiplicative, 1-bounded, mean over a long interval | hypothesis holds trivially, `chi_0 mod y#` is maximally pretentious, conclusion empty (`recon-0828-rough.md` 3.4) | n/a | priced | | 5f | Duke / Linnik ergodic method, fixed orbit | K3 | positive-dimensional orbit with a spectral gap; L-function subconvexity | `Z/W` acts simply transitively, the anchor's orbit is a point; no spectral gap, `min \|fhat_q(v)\| >= 2 sin(pi/(2q))` at every frequency; no L-function | n/a | new row, old facts | **The audit label, where the audit settles it.** The table above grades every row by killer and by landing killer and carries no `(i)/(ii)/(iii)` label, which red team C records as a gap rather than an error: "two rows turn on conclusions the audit has already labelled, and a reader moving between the two 0829 recon notes will find one that labels every angle and one that labels none". Three rows are settled by citation to `attack-wrongdirection-audit.md` section 2: | row | audit label | what settles it | |---|---|---| | 3d | **(i)** strictly weaker | item 10: the band `(2, 4.2665]` is legal, and the decoupled vector sieve's payout sits inside it | | 3e | **(i)** strictly weaker | item 3's quantifier distinction plus section 1 Axis B: an i.o.-and-some-pair conclusion is decidable short of the all-window target, which is why beating K1 here buys nothing | | 4c | **(i)** strictly weaker | item 10: a lower bound on long gaps runs inside the legal band and never reaches `min_x N(x) >= 1` | **The other sixteen rows, graded 2026-08-29, and the debt is paid at the lower rung.** The audit's section 2 grades this programme's own targets, not the literature's escapes, so most of what follows is a fresh judgement and not a citation, and the caption says which is which per row. The rule is the audit's own, section 0's last bullet: **(i)** the escape's conclusion, if it held in this object, would still not imply TPC through any bridge this corpus holds; **(ii)** it would; **(iii)** the note cannot tell. | row | audit label | reason, one clause | |---|---|---| | 3a | **(i)** | the only consumer section 3a names for a `kappa = 2` bilinear construction is 3d's vector sieve, whose payout item 10 places inside the legal band `(2, 4.2665]`, and no landing below exponent 2 is claimed anywhere in the row; a construction delivering an asymptotic rather than a lower bound would be (ii), and section 3a claims none [FRESH] | | 3b | **(ii)** | the asymptotic sieve's output is an asymptotic count of primes in the sequence, and on the twin sequence such a count is the Hardy-Littlewood asymptotic, which gives TPC outright with no bridge, which is why the row's own hypothesis is the one Friedlander and Iwaniec say the sequence does not supply [FRESH] | | 3c | **(i)** | Chen's conclusion is a `P_2` and not a prime pair, and Chen's theorem is PROVEN in print while TPC is open, which is the audit section 0 witness form for strictly weaker [FRESH, on the audit's stated notion] | | 3f | **(i)** | the hypothesis is a `kappa = 2` construction below `beta_2 = 4.26645`, and item 10 makes the whole band `(2, 4.2665]` legal, with the separation exhibited rather than modelled [CITED, item 10] | | 4a | **(i)** | the every-interval conclusion arrives at window length `X^theta` with `theta` bounded away from zero, and Axis A makes an every-window statement TPC-implying only at `H <= w^2 - w` [CITED, Axis A] | | 4b | **(i)** | granted at two classes the positivity threshold is `beta_2 = 4.26645`, so the every-position conclusion arrives at window exponent 4.26645, above Axis A's line at 2 [CITED, Axis A] | | 4d | **(i)** | Linnik's theorem is PROVEN while TPC is open, and section 4d establishes it sits downstream of a one-class exponent below 2 rather than upstream of anything here, so its conclusion presupposes more than it delivers [FRESH, on the corpus facts the row cites] | | 4e | **(ii)** | the hypothesis granted is `eps W < 1` at the anchored decision point, which certifies the anchored window itself, and audit item 8 grades that certification TPC-strength and stronger, the error term's only use being to certify `X(y*) < T` [CITED, item 8] | | 4f | **(i)** | the family's output is existence of a good configuration somewhere in the ensemble, and Axis D admits only `min_x N(x) >= 1` at a prescribed position, which the row states as its own second failure [CITED, Axis D] | | 4g | **(i)** | the conclusion is an `L^1` average over starting points, which is Axis C's almost-all side and never a value at a prescribed start [CITED, Axis C] | | 5a | **(ii)** | the major-arc output at a prescribed point is, on the twin system, the Hardy-Littlewood asymptotic, which implies TPC; Green-Tao's infinite-complexity sentence is exactly the statement that the mechanism cannot deliver it here [FRESH] | | 5b | **(i) as the theorem reads, (ii) at the modulus the escape would need** | at `q <= (log x)^A` the conclusion already holds here and buys only the fixed-modulus tail refinements the corpus priced, which is legal; the same conclusion at modulus `x#` over a window of about `x'^2` is the Zone Postulate, hence TPC-implying by `G2-STATE.md` section 1b [FRESH, and the split is the honest reading rather than one label] | | 5c | **(i)** | the mechanism's output is existence of favourable rows, and the anchor is fixed by arithmetic rather than chosen by the argument, so Axis D's prescribed-position requirement is untouched and Axis B disposes of the level quantifier [CITED, Axis D and Axis B] | | 5d | **(i)** | a two-sided existence-of-irregularity statement of amplitude `1/u^{cu}`, PROVEN in print while TPC is open, and adverse-direction here in any case [FRESH] | | 5e | **(i)** | the conclusion is empty in this object, the hypothesis holding trivially at `chi_0 mod y#` and the theory identifying it and saying nothing further, and an empty conclusion implies nothing [FRESH] | | 5f | **(i)** | the anchor's orbit under a simply transitive action is a single point, so the equidistribution conclusion is degenerate here and carries no statement about the anchor [FRESH] | **Caption, and what this table is not.** Twelve rows come out (i), three come out (ii) (3b, 4e, 5a), one is split (5b), and none is (iii). Seven reasons are citations to `attack-wrongdirection-audit.md` and name the axis or item, marked [CITED]; the other nine are **fresh judgements made 2026-08-29 against that audit's axes**, marked [FRESH], and they have had no second reader. A **(ii)** here opens nothing and is not a route: it says that if the escape's conclusion held in this object it would settle the conjecture, which is a reason the escape is expensive rather than a reason to spend a session on it, and all three fail their own hypothesis check in sections 3 to 5 above. The labels change no verdict in the table above, no killer, no landing killer and no number in this note. **Reading of the table, at its lowest rung.** Nineteen rows. Fourteen fail a hypothesis check (3a, 3b, 3c, 3e, 4a, 4b, 4d, 4e, 4f, 4g, 5a, 5b, 5c, 5f); one is OPEN (3d); and four fail in some other way, either by having no escape in print (3f), running in the opposite direction (4c), not being an escape at all (5d), or having a hypothesis that holds while the conclusion is empty (5e). Of the fourteen, twelve fail on a fact that was in this corpus before the pass, one fails on a comparison the corpus holds in pieces across two files (4a's scale comparison), and one fails on a printed sentence naming the twin system (5a). Five rows land in a different killer after beating their own: 3a into K3, 3c into K3, 3e into K2, 4b into K1, 5c into K2. That is the collision pattern of section 0, and it stays at HEURISTIC. --- ## 7. What did not fail the hypothesis check Three items. Each is stated at its lowest rung with the circularity pre-check written out before the item, per `IMPORT-MAP.md` section 0's gate. ### 7a. The decoupled vector-sieve lemma **Circularity pre-check, written first.** The lemma is: signed cancellation of the bilinear interval remainder for the two-class comb, uniformly in window position, the two-dimensional analogue of Iwaniec's 1980 error term. Does it, if proven, imply the Zone Postulate or TPC, which would make it a wrong-direction target of the kind `attack-wrongdirection-audit.md` catalogues? **No, and the reason is quantitative:** the corpus has already measured what the decoupled route buys and it is an exponent, not the conjecture. `research/theta-ladder.md` measures the window exponent at which the certificate is positive at every position, and that requirement sits inside the zone budget at every exactly-measured level, `need/z^2 = 0.3550..0.6119` at `z = 13..31`, all exact (`paper/wall-note.md` section 1). A lemma about cancellation in a bilinear sum is an analytic estimate about a fixed lattice of divisor pairs and carries no positivity hypothesis, so the circularity grade is **CLEAN**, the same grade `sift-limit-attack.md` section 3 already assigns the Brudern-Fouvry target ("Parity does not forbid it: the target 2.649 sits above 2"). **Status: OPEN.** Not novel: this is the programme's named road and has been since before this pass. The pilot measures the cancellation and finds it strong at toy scale, including exhaustively over full periods (`paper/wall-note.md` section 1), which is a MEASURED input at small `n` and not evidence about the uniform statement. What section 3a adds to it is the adverse context: the only published attempt at a bilinear remainder above dimension 1 carries `beta < 3`, so the analogue this lemma asks for has been attempted once in thirty-six years and stopped below the needed dimension. ### 7b. Whether Pomykala's `beta < 3` is essential **Circularity pre-check.** Relaxing a technical constraint in someone else's lemma is a literature question, not a statement about the tile, so it cannot be circular. It is also not a route on its own: even granted, section 3a's part (b) remains, and that is an exponent comparison rather than a technicality. **Status: OPEN, priced below 2%** by `recon-0828-sieve.md` section 4.3, on the grounds that the announcement is five pages in a bulletin, its OpenAlex forward graph returned one citing work and that one unrelated, and nobody appears to have taken the construction past `kappa = 1.23` since 1990. Not novel; carried here only so the row is not re-opened. ### 7c. The collision pattern **Circularity pre-check.** The observation is about the literature's escapes, not about the tile, so it is not circular. It is also not falsifiable by any computation in this repository, which is the honest limit on it. **Statement, at HEURISTIC.** Across the nineteen rows of section 6, every escape that beats one killer in print and can be transplanted at all lands inside another killer: 3a and 3c beat K1 and land on K3's position quantifier, 3e beats K1 and lands on K2's, 4b beats K2 and lands on K1's dimension, 5c beats K3 and lands on K2's. **Novelty: none.** `attack-wrongdirection-audit.md` section 0 and `README.md` §Status already record that the three killers were stated together because independent attacks converged, and `object-bridge-read-0829.md` section 4 already records that killers 2 and 3 bite on the same link, `E(x) -> S(x)`. What this note adds is the tabulated form and the count. A pattern over nineteen rows of a non-exhaustive candidate list is not evidence of impossibility, and this note does not claim it as such. **What it would take to raise this above HEURISTIC.** A theorem of the form "any method whose input is a class count and whose conclusion is position-uniform is capped at `beta_2`", which is exactly the barrier theorem `sift-limit-attack.md` section 2 records as absent for `kappa = 2` and `natal-cap-10-sieve-cap.md` section 1.5 flags as "rigorous folklore" for the floor. No such theorem was found in this pass, and none was searched for in an owning convention. --- ## 8. Provenance ledger **What was read at source this pass:** one item. arXiv abstract page for `math/0406018`, Granville and Soundararajan, *An uncertainty principle for arithmetic sequences*; title, authors and abstract confirmed verbatim. That is an abstract, not a theorem statement, and section 5d says so. **What is CORPUS CUSTODY**, meaning read at source in an earlier pass of this repository and quoted from the file that owns it, per the standing compute rule: | fact | owning file | grade there | |---|---|---| | Greaves's zbMATH review of Pomykala 1990, `Delta = MNK^{beta-2}`, `2 < beta < 3` | `recon-0828-sieve.md` 4.1 | SOURCED at the zbMATH record | | Granville on Iwaniec's every-interval bound, `f(2) = 0`, `J(P(z)) << z^2` | `recon-0828-rough.md` 2; `covering-dive.md` 1.2 | SOURCED, page image | | Mercer 2018 p. 3, `h(n) <= C p_n^{2-eps}` gives Linnik and Dirichlet | `recon-0828-rough.md` 2 | SOURCED, page image | | Green-Tao, Annals 171 (2010) p. 1760, twin system has infinite complexity | `import-transference.md` 0 | AT SOURCE, Annals page image | | GS Corollary 1.2, the primorial as extremal level | `special-levels-recon.md` 1d | AT SOURCE, GS preprint pp. 1-4 | | GS Corollary 1.4 withdrawn, `eta <= 1/100` | `REFUTED.md`; `maier-matrix.md` 8 | corpus closure | | Ford's notes, `beta(kappa)` known only on `[0,1/2] u {1}` | `sift-limit-attack.md` 2 | SOURCED | | Bettin-Chandee Remark 1, `(1 + hx/MN)^{1/2}` | `SEARCH-CONVENTIONS.md` 1 row 34; `sift-limit-attack.md` 4.5 | corpus custody | | Huxley 7/12 and Guth-Maynard 17/30 | `attack-bilinear-transplant.md` short-interval table | corpus custody | | transform full support, twelve measured minima | `recon-0828-farfields.md` 2a | PROVEN + SCRATCHPAD-GRADE numerals | | complete dependency graph, Shearer = union bound | `import-shearer.md` 4 | PROVEN | | `eps W < 1` boundary; `8.38e-6` vs `1.03e-7` at @19 | `anchored-note.md` 3 Prop 1(i); `object-bridge-read-0829.md` 4 | PROVEN + MEASURED | | dial 4 permutation test, `p_FWE >= 0.1082`, Sidak `0.8865`; the 15 miss cells | `ioslack-survey.md` 0, 3, 5 | MEASURED | | `d(q_1)` slack miss, 174 to 70,576 | `REFUTED.md`; `row7-recon.md` | CLOSED | | BV not position-uniform | `natal-cap-10-sieve-cap.md` 1.4 | INFERRED from proofs' structure | | `2(1+sqrt e) = 5.2974`; `need/z^2 = 0.3550..0.6119` | `paper/wall-note.md` 1; `theta-ladder.md` | corpus custody | **What is MEMORY, flagged and not load-bearing:** Hoheisel's 1930 exponent `1 - 1/33000`; the value of Linnik's `L` and the names of the three Linnik principles; the wording of Friedlander and Iwaniec's own remark that the twin sequence supplies no Type II range. Each appears in a sentence whose verdict does not turn on the numeral. **What was not done.** No script was run. No PDF was opened. No owning-convention sweep was performed, so the eighteen rows are the brief's candidate list plus Hoheisel and are not claimed complete. No existing file was edited and no git command was run. --- ## 9. What would falsify this, and whether that check has run | claim in this note | what would falsify it | has that check run? | |---|---|---| | no escape read here reaches the wall | any one of the nineteen rows whose hypothesis in fact holds for the tile or zone objects | partially: each row was checked against one or two corpus facts, none against a fresh reading of the escape's own paper | | the corpus needs `theta = 0` on the short-interval scale, against 0.5667 in print | a short-interval prime theorem at polylogarithmic window length | not searched in an owning convention this pass; `special-levels-recon.md` 1d is adverse evidence, GS breaking the Gaussian heuristic at `h = log^A q` | | Hoheisel appears nowhere in this repository | a hit under a different spelling or inside a PDF or a non-markdown file | grep over `research/` and `paper/` markdown only; PDFs and scripts not scanned | | the circle method fails because the twin system has infinite complexity | a printed complexity assignment putting the system at finite `s` | yes, and the opposite is printed: Green-Tao p. 1760, read at the page image in `import-transference.md` | | GS is not killer 3 in print | a GS theorem asserting non-transfer of an ensemble bound to a fixed point | partial: abstract read this pass, Cor 1.2 held at source in `special-levels-recon.md`, the full paper NOT REACHED | | the collision pattern (7c) | one escape that beats a killer and lands in none | no; the pattern is over a non-exhaustive list and the check is not defined sharply enough to run | | 3d's lemma is CLEAN rather than TPC-strength | a derivation of the Zone Postulate from the decoupled bilinear lemma alone | no; the grade is inherited from `sift-limit-attack.md` 3 and was not re-derived here | | Pomykala's `beta < 3` cannot be relaxed | a construction above `kappa = 1.23` in print or in preprint | partial: forward citation graph checked in `recon-0828-sieve.md` 4.3, one citing work, unrelated | **The honest summary of this table.** The strongest falsification checks that have actually run are the Green-Tao complexity assignment (5a) and the exact `eps W < 1` counting (4e); those two rows are the ones this note would defend hardest. The weakest are 7c, which is a description rather than a claim, and the `theta = 0` scale comparison, which is arithmetic on corpus-held numerals and has not been checked against a literature sweep for polylogarithmic short-interval results. Nothing in this note has been through an adversarial pass, and the campaign's own rule is that four of five same-day integrations needed correcting within twelve hours (`primeoire-campaign-lessons`).