# The kappa = 2 barrier attempt: Granville's exceptional-character construction transfers, satisfies the two-class axioms at level theta approaching 1, and then stops dead at u = 2, which is below the entire open band *(2026-08-28. Staging note, HELD pending an adversarial pass. No existing file was edited and no git command was run, per the session fence. One companion script, `attack-barrier-kappa2.js`, scratchpad-grade, embedded with `research/qc/embed.js`. Nothing here is a TPC claim, nothing here lowers an exponent, and the one thing that lands is a wall address.)* --- ## 0. Bottom line, disconfirming half first **The attempt fails, and it fails at a point that can be named exactly.** The verdict is **(b)** in the brief's own numbering. The construction reaches `u = 2` and stops there, for every choice of character sign pattern, on both the arithmetic-progression form and the interval form. The open band `(2, 4.26645]` is not touched at any point, and the corpus should not record that this route can be pushed further with more work: section 5 gives the reason the ceiling is exactly 2 and it is an arithmetic identity, not a technical shortfall. **The failure point, in one sentence.** The exceptional character supplies a single arithmetic identity, `lambda(n) = chi(n)` for `z`-rough `n <= x`, which is a one-point statement; it constrains each coordinate of a two-class problem separately; the resulting constrained set is empty exactly when a `lambda = +1` `z`-rough integer below `x` cannot exist, which happens exactly when `z^2 >= x`, that is `u <= 2`; above `u = 2` the surviving question is whether the shifted intersection of two positive-density `lambda`-classes is small, and that is a two-point question the character does not answer. **Whether the missing input is Chowla-strength: yes, and the corpus's stated reason for it is wrong.** `sift-limit-attack.md` section 2 records the mixed-sign twin example `C = {n(n+2) : lambda(n) = -1, lambda(n+2) = +1}` as dying because "its axioms need two-point lambda-equidistribution at full level: Chowla-strength, open". Substituting the exceptional character for `lambda` **pays that off in full**: section 3 below derives the density and remainder axioms for the character version unconditionally, with dimension exactly 2 and level `theta` approaching 1, because the character version is periodic and the `lambda` version is not. The set then dies anyway, at its sifted count rather than at its axioms, above `u = 2`. So section 2's verdict survives and its reason does not. That correction is owed and this note cannot apply it. **A caveat that outranks everything above.** Granville's hypothesis implies the conjecture this programme is attacking. Heath-Brown proved in 1983 that infinitely many Siegel zeros give infinitely many twin primes, which the corpus already carries at `research/bv-import-survey.md` section 2 and which was re-confirmed at zbMATH in this session. A Siegel-zero-conditional barrier at `kappa = 2` therefore lives in a world where the target is already a theorem. It can price a route. It can never be a step along one. Anyone reading this note as progress toward the conjecture has misread it. **What did move, at its rung.** One conditional statement lands on Face 4, where the corpus records none: under a Siegel-zero hypothesis strictly stronger than Granville's Corollary 1 needs, the `kappa = 2` interval sifting problem attains the DHR lower bound value zero at sieve parameter `v` approaching 2, so `beta_interval(2) >= 2` conditionally. That is below the whole open band and lowers nothing. Angle A12 of `history/staging/recon-0828-jacobsthal.md` moves from LIVE to CLOSED. Angle A13, the Siebert 1983 chapter, was answered in parallel this session by a sibling agent and is also closed: Siebert is `kappa` equal to 1 only, on three independent statements of his theorem, so the `kappa = 2` extremal example is not already in print there (`history/staging/lit-siebert-1983.md`). The area's two live angles are therefore both closed today, and neither closure lowers anything. --- ## 1. Granville's theorem, read at the page **Source, with custody.** A. Granville, *Sieving intervals and Siegel zeros*, arXiv:2010.01211v1, Acta Arith. **205** (2022) 1-19. Fetched this session from the author's copy at `https://dms.umontreal.ca/~andrew/PDF/Sieve.Remark.20.09.26.pdf`, 376,359 bytes, sha256 `a5403b56e7aab8dd2f66bbc776e049e1ac470bc8a948abd65d8fe2b6c250281b`, extracted with `pdftotext -layout`. This confirms the hash `recon-0828-jacobsthal.md` recorded, on a differently sized download; the byte count there was 222,744 against 376,359 here, and the sha256 agrees, so the earlier figure is a transfer artefact and not a different file. Every quotation below is from that extraction. The **published** Acta Arith. version was not obtained, and the check that it says nothing above `kappa = 1` has run only on v1. **[SOURCED]** **The hypothesis.** Section 2, verbatim: "for any `kappa > 0` arbitrarily small, there is a sequence `(q_j, chi_j, beta_j)` such that `beta_j >= 1 - kappa/log q_j` for all `j >= 1`", where each `chi_j` is a **primitive real (quadratic) character mod `q_j`** and `beta_j` is a real zero of `L(s, chi_j)`. The quantifier that matters is that `kappa` may be taken arbitrarily small, which Granville justifies in the same section: if the zeros fail for a small enough `kappa` then there are none and the whole discussion is vacuous. The working parameter throughout is `Delta = Delta_beta(x) = (1 - beta) log x`, and the usable range is his (6): `x >= q^A` with `(1 - beta) log x <= Delta`, for a large absolute constant `A`. **The conclusion, Corollary 1, verbatim.** "Assume that there are infinitely many Siegel zeros. For each fixed `v > 1`, there exist arbitrarily large `x, X, y, z` with `y = z^v` such that `S(x, y, z) = (F(v) + o(1)) G(z) y` and `S(X, y, z) = (f(v) + o(1)) G(z) y`." So both Jurkat-Richert bounds are attained by the **interval** problem `S(x,y,z) = #{n in (x, x+y] : (n, P(z)) = 1}`, at every fixed `v > 1`, along a sequence of `x` and `X` that depend on the sequence of Siegel zeros. **The uniformity, stated honestly.** It is not uniform in `v` and it is not uniform in `x`. The extremal intervals occur only at the `x` the construction produces, one per exceptional zero, and Granville says so on his page 6: "It is feasible that the limits ... might not exist; indeed if the extremal examples all come from Siegel zeros (as in this paper), and if Siegel zeros are very spaced out ... then these limits will not exist. Therefore, one needs to work with lim sup and lim inf". So the theorem is a `lim sup` / `lim inf` statement, which is exactly what a barrier needs and no more. **The quantitative machinery, for section 5.** Proposition 1 gives, in the range `y^{1-epsilon} > z > y^{1/2 - o(1)}`, `S(X, y, z) <~ 4y (log y)^{-2} log^+(qy/z^2) + (1 - beta) y`. Proposition 2 gives four regimes of `1 - beta`, of which the second, "if `1 - beta <= (log q)^{-kappa}` for some fixed `kappa > 1`", is the one this note uses. --- ## 2. The mechanism, extracted Three steps, all from the hashed file. **Step 1, the character captures the primes.** Corollary 4: under the hypothesis, for `x >= q^A` in range (6) and `(a,q) = 1`, `pi(x; q, a) = (pi(x)/phi(q)) times O(Delta)` if `chi(a) = 1`, and `(pi(x)/phi(q)) times (2 + O(Delta))` if `chi(a) = -1`. Primes are pushed out of the `chi = +1` classes entirely and doubled in the `chi = -1` classes. Tao's 2007 post states the same thing informally, and was fetched this session (sha256 `7f8042f8b554b80d85602a8a86dbbe48c2775d0b0984bca9c06120fdd87ebf77`): "the primes can be largely recovered from the almost primes as being those almost primes which are quadratic non-residues modulo the conductor `q`". **[SOURCED]** **Step 2, it propagates multiplicatively.** Theorem 2.1: for `k < u` a positive integer, `x = z^u`, `(a,q) = 1`, `pi_k(x, z; q, a) = (1 + (-1)^k chi(a) + O(Delta)) pi_{k,q}(x, z) / phi(q)`, where `pi_k(x,z)` counts `n = p_1 ... p_k` with all `p_i > z`. The weight `1 + (-1)^k chi(a)` is `1 + lambda(n) chi(a)`, and since `n` is congruent to `a` and `chi` is completely multiplicative, `chi(a) = chi(n)`. **So the content of Theorem 2.1 is the single identity `lambda(n) = chi(n)`, valid for every `z`-rough `n <= x` outside an exceptional set of size `O(Delta)` in the natural normalisation.** That identity is the whole gift, and section 5 turns on the fact that it is a statement about one integer at a time. **Step 3, the class becomes an interval.** Corollary 5 converts step 2 into `N(x, z; q, a) = (F(u) + O(Delta)) phi(q)^{-1} prod_{p<=z}(1 - 1/p) x` when `chi(a) = -1`, and the same with `f(u)` when `chi(a) = +1`, by identifying the class with Selberg's set `A_+` or `A_-`. Then the proof of Corollary 1 dilates: let `P_q(z)` be the product of primes up to `z` not dividing `q`, choose `r` with `rq = 1 mod P_q(z)`, and set `b = ra mod P_q(z)`. Then `b + j = r(a + jq) mod P_q(z)`, so `(b+j, P_q(z)) = (a + jq, P_q(z))`, and the interval `j in [0, y-1]` inherits the sifted count of the progression `a mod q` up to `x = qy`. The primes dividing `q` are handled by deleting one class each, at a cost `phi(q)/q`, and `u = v + 1/A` with `z = q^A`, so `F(u) -> F(v)` as `A -> infinity`. The dilation is the reason the construction reaches intervals at all, and section 4 is about what it costs at `kappa = 2`. --- ## 3. The kappa = 2 analogue, and the one thing that works **The problem.** Sift by two classes per prime. The instance the programme cares about is the twin instance, `{0, -2} mod p` for odd `p` and `{0} mod 2`, so `omega(2) = 1` and `omega(p) = 2`, dimension `kappa = 2`, which is the Diamond-Halberstam book's own worked example (`paper/beta2-note.md`, header, read line-level against Cambridge Tracts 177). The lower-bound function is `f_2`, and DHR Theorem 6.1 (6.2) gives `f_kappa(u) = 0` for `0 < u <= beta_kappa`, with `beta_2 = 4.26645028414864191641` (Booker-Browning, `dhr-verification.md` section 1.1). **A point of framing the brief blurs, worth fixing before anything else.** "Making the sifted count sit at the DHR lower bound `f_2(s)`, zero for `s < beta_2`" is not the content of a barrier, because zero is easy: the empty set attains it. The content is **how large `u` can be** while a set that satisfies the axioms still has sifted count `o(G_2(z) |A|)`. That is what `beta(2) >= u` means, and it is what the rest of this note measures the construction against. **The candidate set.** Take a real primitive `chi mod q` with an exceptional zero, `z = q^A`, `x = z^u`, and a residue class `a` coprime to `q`. Put `A = { n <= x : n = a mod q }`, sifted by `{0, -2} mod p` for every `p <= z`. **The axioms hold, unconditionally, at dimension 2 and level theta -> 1.** For `d | P(z)` squarefree with `(d, q) = 1`, the condition `d | n(n+2)` selects `2^{omega(d)}` classes mod `d`, independent of the class mod `q` by CRT, so `#A_d = (2^{omega(d)} / d)(x/q) + O(2^{omega(d)})`, giving `g(d) = 2^{omega(d)}`, `g(p) = 2`, dimension exactly 2, and `|r(A, d)| = O(2^{omega(d)})`. Then `sum_{d <= D, d | P(z)} |r(A,d)| << D log D`, which is `o(|A| / (log x)^B)` as soon as `D <= (x/q)^{1 - epsilon}`. Since `z = q^A` and `x = z^u`, the modulus is `q = x^{1/(Au)}`, which is `x^{o(1)}` as `A` grows, so `D` may be taken to be `|A|^{1 - epsilon}` and the level `theta` approaches 1. The sieve parameter is `log|A| / log z = u - 1/A`, which approaches `u`. **[PROVEN, elementary]** The primes dividing `q` are inside the sieve range, since `q < z`. Removing two classes at each of them costs the constant factor `prod_{p|q}(1 - 2/p)` and changes nothing below. **Why this matters, and it is the one positive result in this note.** The corpus's own candidate, `C = {n(n+2) : lambda(n) = -1, lambda(n+2) = +1}`, is recorded as failing because its axioms need two-point `lambda`-equidistribution in progressions at full level, which is Chowla-strength. The character version above needs nothing of the kind, because a condition on `chi(n)` and `chi(n+2)` is a condition on `n mod q`, hence periodic, hence equidistributed in every progression coprime to `q` by counting. **The axiom obstruction that `sift-limit-attack.md` section 2 names is removed by the substitution.** What follows is that the set dies anyway. --- ## 4. The interval transfer is class-preserving, so intervals see only same-sign pairs Apply Granville's dilation to the two-class problem. With `b = ra mod P_q(z)` and `rq = 1 mod P_q(z)`, `b + j + 2 = r(a + jq) + 2 = r(a + jq + 2q) mod P_q(z)`, since `r(2q) = 2`. So the two coprimality conditions on the interval, `(b+j, P_q(z)) = 1` and `(b+j+2, P_q(z)) = 1`, transfer to roughness of `a + jq` and of `a + (j+2)q`. **Both lie in the class `a mod q`.** The character assigns one value to the pair, not two. This is not an artefact of the choice `d = q` for the progression's common difference. Theorem 2.1's control is a condition modulo `q`, and it is constant along a progression if and only if `q` divides the common difference `d`; and then every fixed shift `h` in the transferred interval maps to the shift `hd`, which is `0 mod q`. So on intervals the construction can only produce **same-sign** pairs, `(lambda(n), lambda(n+2)) = (epsilon, epsilon)`. **Mixed signs are reachable on a progression and not on an interval.** On the progression `A` of section 3, with the actual shift 2, the two coordinates sit in classes `a` and `a+2`, and `chi(a) = -1` with `chi(a+2) = +1` is available: the companion script measures the density of such `a` as 0.249752 at `q = 1009` and 0.249925 at `q = 10007`, converging on `1/4`. That is a positive proportion of residues mod `q`, not a bounded number of classes, so it cannot be imposed on the transferred interval by Granville's own device of deleting one class per prime dividing `q`; and the set of admissible `j` is then not an interval at all. **[MEASURED, exact character sums, `attack-barrier-kappa2.js` reading 2]** Two characters do not help. If `chi_1 mod q_1` and `chi_2 mod q_2` both have exceptional zeros, then on `z`-rough `n` both say `lambda(n) = chi_i(n)`, which is the same statement twice. There is no second, independent bit to be had. --- ## 5. The ceiling is u = 2, at every sign pattern, and it is an identity **The vanishing direction, proven.** Let `n <= x = z^u` be `z`-rough with `lambda(n) = +1`. Then `n` has an even number of prime factors, each greater than `z`, so `n = 1` or `n > z^2`. If `u <= 2` then `z^2 >= x`, so no such `n` exists below `x` apart from `n = 1`. Consequently **any sign pattern with a `+1` coordinate has sifted count at most `1` plus the leakage**, where the leakage is the count of `z`-rough integers in the relevant class that violate the identity `lambda = chi`. By Corollary 4 that leakage is `<< Delta x / (phi(q) log x)`. **The leakage arithmetic, which forces a stronger hypothesis than Corollary 1 needs.** The `kappa = 2` normalisation is `G_2(z)|A| ~ c u^2 x / (q (log x)^2)`, against a leakage `~ Delta x / (phi(q) log x)`. The ratio is `Delta (log x) (q/phi(q)) / (c u^2)`. A fixed `Delta` makes this diverge, so the construction needs `Delta = o(1 / log x)`, that is `(1 - beta)(log x)^2 -> 0`. With `log x = Au log q`, the hypothesis `1 - beta <= (log q)^{-C}` with a fixed `C > 2` suffices. That hypothesis is in Granville's paper, as the second bullet of Proposition 2, but **his Corollary 5 is stated for a fixed `Delta > 0` and the uniformity used here is not stated at the source**. Pushing `Delta` below `1/log x` inside Corollary 5's statement is this note's extension, not a quotation. **[INFERRED, uniformity not stated at the source; the mechanism is Granville's, the uniformity claim is not]** **What lands, therefore.** Under the strengthened hypothesis, both the progression form of section 3 and the interval form of section 4 give sifted count `o(G_2(z)|A|)` at sieve parameter approaching 2. On the interval form, take `chi(a) = +1`: both coordinates are forced to `lambda = +1`, both are therefore absent, and the interval's twin-sifted count is `o(G_2(z) y)` at `v -> 2^-`. So, conditionally, `beta_interval(2) >= 2`. This is the first barrier statement of any kind on Face 4, and it sits below the whole open band. **[CONDITIONAL, on a hypothesis strictly stronger than Corollary 1's, with the uniformity flagged above]** **The ceiling direction, and why it is exactly 2.** For `u > 2` the same arithmetic runs backwards: `z^2 < x`, so `z`-rough integers with `lambda = +1` below `x` exist, namely products of two primes both above `z`, and they exist in quantity. Every sign pattern is then nonempty, and the question becomes whether the **shifted intersection** of two positive-density `lambda`-classes is `o(G_2(z)|A|)`. Nothing in the character speaks to that. The companion script measures the transition directly at `x = 2 times 10^7`: the `(-,+)` count is `0` at `u = 1.6, 1.8, 1.9, 2.0` and `2917, 8902, 29592` at `u = 2.1, 2.2, 2.5`, and the share of the full `z`-rough pair count carried by that pattern is `0.025755, 0.070546, 0.168063` at those three values, rising to `0.248549` at `u = 4.0` and converging on `1/4`. **[MEASURED, `attack-barrier-kappa2.js` readings 4 and 5, with a control at reading 3 reproducing an independently computed 107407 twin pairs. The caveat is large and is stated in reading 5: at this `x` the natural logarithm of `x` is under seventeen, so this is an illustration of the mechanism and not a measurement of the limit, and it cannot rule out that the share tends to zero as `x` grows.]** **The pattern the interval transfer actually delivers is the worst one.** With `chi(a) = -1`, both coordinates are forced to `lambda = -1`, which for `u < 3` means both are prime. The construction then selects **every twin prime pair in range**, which is the largest piece of the sifted set rather than the smallest: reading 6 measures `107374` of `176077` rough pairs at `u = 2.5` and `107389` of `261012` at `u = 3.0`. Choosing the `chi = -1` class is the exact opposite of extremal, and it is the choice Granville uses for his `F` direction, where at `kappa = 1` it is correct. **Why the ceiling is 2 and not some other number, stated once more.** The character forces parity. Parity forbids a `z`-rough integer only when the smallest admissible even factorisation, a product of two primes both above `z`, already exceeds `x`. That is `z^2 >= x`, that is `u <= 2`. At `kappa = 1` this ceiling is not binding, because `beta(1) = 2` and the ceiling is the answer. At `kappa = 2` the ceiling is `2` and the target is `4.26645`, so the method covers none of the gap. This also explains, without needing a search, why Granville's paper contains no statement above `kappa = 1`: there is nothing above `kappa = 1` for the method to say. --- ## 6. What the character supplies about pair correlations, and why it is the wrong sum The brief's conjecture on this point is **correct as stated and does not help**, and both halves are worth recording. **The correlation is available, unconditionally.** For `q` an odd prime and `chi` the Legendre symbol, `sum_{n mod q} chi(n) chi(n+2) = sum_{n mod q} chi(n^2 + 2n) = sum_{m} chi(m^2 - 1) = -1`, exactly. The companion script confirms `-1` at all fourteen prime moduli tried, `11` through `10007` **[MEASURED, reading 1]**. For general squarefree `q` the sum factors over the prime divisors and has absolute value 1. So the two-point input that the `lambda` version of the sketch could not obtain **is** obtainable for `chi`, with no Chowla hypothesis, exactly as the brief guessed. **It is the wrong sum, and the reason is a degeneracy.** Over the residue class `a mod q`, which is where the construction lives, `chi(n) chi(n+2)` is the **constant** `chi(a) chi(a+2)`. A constant carries no information about the class's sifted count. Over an interval the sum is genuinely non-constant, but the identity that would convert it into information about `lambda` holds only on `z`-rough integers, so writing `T = sum_{n in I, n and n+2 both z-rough} lambda(n) lambda(n+2) = chi(a)chi(a+2) times S_2(A, z)` reintroduces the sifted count on the right-hand side. The correlation and the sifted count are the same object up to a sign, and the identity is therefore empty. Detecting roughness by sieve weights instead, to reach the unrestricted character sum where Weil applies, replaces `T` by a bilinear sum over pairs of moduli `d, e` up to the level, and completing each residue class costs `q^{1/2}` per pair; that is affordable, and it delivers a bound on the unrestricted sum, which is the sum that was already known and was already useless. **So the Chowla-strength requirement is relocated, not removed.** It leaves the axioms, where the character pays it off completely (section 3), and reappears at the sifted count, where it is worse: no longer a hypothesis one could assume and proceed from, but the conclusion itself, and where both the standard pair heuristic and the measurement of section 5 say the conclusion is false above `u = 2`. `sift-limit-attack.md` section 2's verdict stands. Its stated reason does not. --- ## 7. The hypothesis implies the target, which is the honest circularity grade D. R. Heath-Brown, *Prime twins and Siegel zeros*, Proc. London Math. Soc. (3) **47** (1983) 193-224, Zbl 0517.10044, keywords "existence of Siegel zeros; existence of infinitely many prime twins", read at the zbMATH record this session; quantified by Tao and Teräväinen, arXiv:2112.11412. The corpus already carries it at `research/bv-import-survey.md` section 2, with the right reading: "a catastrophic failure of equidistribution would also settle our question, in our favor". **[SOURCED-BIB for the paper; SOURCED for the zbMATH record]** Three consequences for this note. 1. **The barrier, had it worked, would still have been a route-pricing device and nothing more.** Under Granville's hypothesis the twin prime conjecture is a theorem, so a conditional `kappa = 2` barrier can never be a step toward it. The logically useful form is the contrapositive, which is the form Granville himself extracts at `kappa = 1` when he reproves Motohashi: an unconditional class-blind improvement below the sifting limit would disprove the existence of Siegel zeros. That shape is real, and this note does not reach it. 2. `recon-0828-jacobsthal.md` grades A12's circularity CLEAN "because the hypothesis is believed false". The stronger and correct reason is that the hypothesis **implies the target**. The grade is unchanged and the reason should be. 3. **Do not generalise the section 5 ceiling into a claim about Siegel zeros.** Heath-Brown uses the same exceptional character to obtain a genuinely two-point conclusion about the twin problem. So "an exceptional character supplies only one-point information" is false. What is true, and all that is true, is that **Granville's transfer** supplies only one-point information. The failure here is a failure of a transfer. Reporting it as a theorem about the hypothesis would be an overstatement of exactly the kind this repository exists to prevent. A speculation, marked as such and load-bearing on nothing: since Heath-Brown's world is the one in which the twin problem gets *easier*, it is plausible that no `kappa = 2` extremal example exists under Siegel zeros at all, which would mean the barrier on Face 4 is not merely unproven by this route but unreachable by it. No check on that has run. **[SPECULATION]** --- ## 8. Verdict, and what it buys **Verdict (b).** The exact failure point is the shifted intersection of two positive-density `lambda`-classes at `u > 2`, and the input needed to bound it is Chowla-strength, so the corpus's section 2 verdict stands and the barrier is not provable this way. **It is not (a).** The band `(2, 4.26645]` is untouched at every point. **It is not (c) either, and the reason is worth one line.** Attaining the upper bound `F_2` requires a *lower* bound on the same two-point count, which is Hardy-Littlewood-strength and therefore strictly harder than the upper bound this route already cannot obtain. Neither DHR bound is attained above `u = 2`. **What lands, priced.** A wall address, and a small one. `beta_interval(2) >= 2`, conditional on a Siegel-zero hypothesis strictly stronger than Corollary 1 needs, with the uniformity flagged in section 5. Face 4 moves from "the programme has no barrier result on this face" to "one conditional barrier result, at exponent 2, below the whole open band". **It lowers nothing.** The constant-shift structure `twin slots = S_1 intersect (S_1 - 2)` is not established as the only remaining input, because the band that would have to be closed for that sentence to be true is exactly the band this attempt failed to close. **Route status.** Angle A12 of `recon-0828-jacobsthal.md`: LIVE to CLOSED, with the qualifier of section 7 point 3. Angle A13, Siebert 1983: closed in parallel by a sibling agent, `kappa` equal to 1 only, chapter text not reached, verdict resting on three third-party statements (`history/staging/lit-siebert-1983.md`). That closure and this one leave the area with no live angle. It also removes the one way the section 5 ceiling could have been checked against a published `kappa = 2` construction, so the ceiling argument stands unaudited against anything in print. --- ## 9. Corrections owed to files this note may not edit 1. **`research/sift-limit-attack.md` section 2.** The mixed-sign twin example is recorded as dying because "its axioms need two-point lambda-equidistribution at full level: Chowla-strength, open". With an exceptional character in place of `lambda` the axioms hold unconditionally at dimension 2 and level `theta` approaching 1 (section 3), and the example dies at its sifted count instead, above `u = 2` (section 5). The sentence names the wrong obstruction. 2. **`research/history/staging/recon-0828-jacobsthal.md` A12.** LIVE to CLOSED; circularity reason to be replaced by Heath-Brown per section 7. 3. **`research/history/staging/recon-0828-jacobsthal.md` section 1.** The Granville PDF's byte count is recorded as 222,744; the file fetched this session at the same URL with the same sha256 is 376,359 bytes. One of the two byte counts is a transfer artefact. The hash is the binding record and it agrees. 4. **`paper/wall-note.md` section 2 Face 4.** "The programme has no barrier result on this face" is accurate today and would need the section 5 qualifier if this note survives an adversarial pass. 5. **`research/REFUTED.md`.** One row: the exceptional-character `kappa = 2` extremal construction, CLOSED, ceiling `u = 2`, this file. 6. **`research/SEARCH-CONVENTIONS.md` section 1.** Heath-Brown 1983 as the owning anchor for "Siegel zeros and the twin problem", which the map does not carry although `bv-import-survey.md` cites the paper. 7. **`research/QUESTIONS.md` TODO item 0** owes a `Ledger: Q-barrier-kappa2` line, which the fence on this session forbids writing, so the gate will say so. --- ## 10. What would falsify this, and whether that check has run - **"The ceiling is exactly `u = 2`."** Falsified by any use of the exceptional character that constrains a *pair* of integers jointly rather than each coordinate separately. The argument in section 5 is a two-line identity about the size of a product of two primes above `z`, and it has been checked against the measurement in reading 4, which finds every `+1`-containing pattern empty at `u <= 2.0` and nonempty at `u = 2.1`. It has **not** been checked against Heath-Brown's method, which is the one known place where Siegel zeros do give two-point conclusions, and which was not read at the page in this session; if his machinery admits an extremal-set reading, section 5's ceiling is a statement about Granville's transfer only, which is how section 7 point 3 already states it. - **"The axioms of the character version hold at dimension 2 and level `theta` approaching 1."** Falsified by an error in the CRT count of section 3. The derivation is elementary and was done by hand here; it has **not** been independently re-derived, and no script checks it. - **"The uniformity in `Delta` used in section 5 is available."** Falsified by showing Granville's Corollary 5 fails when `Delta` shrinks with `q`. This is the note's own extension of his statement, it is flagged as such, and the check has **not** run: the error terms in his Theorem 2.1 induction were read but not re-derived with `Delta` as a function of `q`. If this fails, the small Face-4 gain of section 5 fails with it and the verdict (b) is unaffected. - **"The candidate is not extremal above `u = 2`."** Falsified by showing the share of the sifted set carried by the constrained sign pattern tends to zero as `x` grows. The measurement runs at one `x` only, `2 times 10^7`, where the natural logarithm of `x` is under seventeen; the share is rising toward `1/4` across the table rather than falling, which is evidence against the falsifier, but no larger `x` has been run and no asymptotic has been derived. - **"Granville states nothing above `kappa = 1`."** Falsified by any higher-dimensional statement in the published Acta Arith. version. The check ran on arXiv v1 only, as in `recon-0828-jacobsthal.md`, and the published version was **not** obtained in this session either. - **"Siebert 1983 does not already contain the `kappa = 2` construction."** Checked in parallel by a sibling agent and answered `kappa` equal to 1 only, but **the chapter's text was not reached**, so that answer rests on the publisher abstract plus two third-party accounts and not on primary reading (`history/staging/lit-siebert-1983.md`). If the chapter turns out to carry a general-`kappa` statement, section 5's ceiling becomes a check on it rather than a closure of the angle. - **"The hypothesis implies the target."** Falsified by an error in the reading of Heath-Brown's theorem. The paper itself was **not** opened; the statement rests on the zbMATH record read this session, on Tao's post, and on the corpus's own citation at `bv-import-survey.md`, which is three independent restatements and no primary text. *This note states current understanding, is HELD pending an adversarial pass, and opens no route.*