# attack-bilinear-transplant — can this frame host Type-II information? The tile side cannot, the zone side can express one, and the estimate it needs is the one already recorded absent *(Staging note, 2026-08-26. Nothing here is integrated into a live document. Question asked: the parity obstruction is defeated in practice only by bilinear/Type-II arithmetic information (Vinogradov; Chen's switching; Friedlander–Iwaniec's asymptotic sieve; Harman's method). Does this programme's frame have anywhere to put such information? No repo file is edited by this note; every proposed addition to a live file is marked PROPOSED and left unapplied. Calibration marked per claim: PROVEN, MEASURED, HEURISTIC, SCRATCHPAD, ABSENT-PER-CONVENTION, NOT REACHED.)* **HELD-document rider, carried at the top because three of this note's premises rest on it.** `quadpoint-identity-01.md`, `quadpoint-prior-art.md`, `stretch-01.md` and `destroyer-census-01.md` are HELD and have never had an adversarial pass (`TODO.md` Z0). Everything below that leans on the capture identity `floor_K = T − X(K)`, on the crossing law `u ω(u) = 2`, or on the stretch's structure is therefore built on an **UNVERIFIED PREMISE**, and the note says so at each use rather than once. --- ## 0. The disconfirming half first Three things go against this run, and they are stated before anything it found. 1. **The question is not new to the corpus, and the corpus already answered the version it asked.** `research/sift-limit-attack.md` §4.1 prices Friedlander–Iwaniec's asymptotic sieve for primes explicitly — "Both are DP4 inputs. Our DP4 is already saturated free of charge, and what these sieves buy (detecting primes inside a sifted sequence, breaking the parity factor 2) is not our bottleneck at 4.2665. Verdict: … no direct application." `research/THE-DIALS.md` prices the almost-prime axis (dial 5, "proven, off the target"), the choice-of-`d` axis (dial 6, "the only breach ever made", "not ours to redo") and the level-of-distribution axis (dial 7, "inert on the target legs"). `research/bv-import-survey.md` §4 already writes Chen's theorem in tile vocabulary and calls a tile-native reproof "feasible and worth doing", with the distance to twins named as the parity wall. **What was not asked before is the Z2/zone-coordinate version and the short-interval version**, and that is the whole of this note's licence. 2. **The route collides in machinery with eight closed rows of `research/REFUTED.md`** (34, 36, 37, 54, 63, 78, 79, 80: Lemma V's mean-square form, Brüdern–Fouvry's left factor and its re-split, the ℓ¹→ℓ²√log conversion, generic chaining, the smooth-profile manufacture, the Kowalski–Michel–Sawin branch, the DI/Pascadi coefficient axis). It does not collide in *target*: every one of those rows aims bilinear machinery at the **level of distribution**, and this note aims it at **parity**. That is a real distinction and §2 argues it is also the distinction that kills the route, so the collision is worth stating twice rather than once. 3. **The brief's own premise (a) is wrong, and correcting it is part of the deliverable.** The brief reasoned that because our capture identity is "Buchstab bookkeeping plus Chen's switching principle", Chen's parity-relevant machinery is already inside it. Switching is not the parity-relevant ingredient: Fouvry and Grupp reached `p + 2 = P₂` **without switching**, Iwaniec's ICM names `P₂` as the *ceiling* of the classical system, and whether `p + 2` has an odd number of prime factors infinitely often is still open. Sources and quotations in §2.6. 4. **The expected answer is the one that arrived.** The frame does not host parity-breaking information; the estimate it would need is already on the corpus's own ABSENT list; and the short-interval literature does not come near the interval length a zone is. Nothing in this note moves an exponent, a constant, or a certificate. **What is new, and it is five flat statements.** - **(N1)** The tile side of the frame is Type-I *by construction*, and the mechanism is periodicity plus the absence of any signed prime-detecting weight. Every bilinear object the corpus owns is bilinear **over moduli**, never over elements. §3.1. - **(N2)** The zone side is not: the capture identity's own candidate map `v = r·m` is a genuine product with both factors ranging, and it spans the Friedlander–Iwaniec Type-II window `(x^{1/3}, x^{1/2})` that their (B1) forces. Measured share of the stretch's composite mass with `lpf(v) > v^{1/3}`: 0.4328, 0.2415, 0.1753, 0.1339 at `Q = 101, 1009, 10007, 99991` (SCRATCHPAD, §4.2). The product structure is there and it carries no signs, which is why it counts and does not cancel. §3.2. - **(N3, and this is the hardest thing in the note)** The Friedlander–Iwaniec machine **cannot be posed** for a zone-supported sequence, for a reason prior to parity. Their (R1) forces the level `D > x^{2/3}`; their own remark forbids `D(x) > A(x)`; a stretch has `A(x) ⩽ x^{θ}` with `θ → 1/2`. Measured fraction of anchors whose stretch even reaches `θ > 2/3`: 0.4053 → 0.1678 → 0.0339 → **0.0023** over the decades `10³` to `10⁷` (SCRATCHPAD, §4.3). Deficit `x^{1/6}` — a power, not a log. §2.3. - **(N4)** The short-interval sieve's own arithmetic input **vanishes at exactly a zone's exponent**. The level in the short-interval Bombieri–Vinogradov lemma that every `p + 2 = P₂` short-interval result runs on is `D ⩽ x^{θ−1/2}(log x)^{−B}`, asserted only for `7/12 < θ ⩽ 1`, and **the bound does not depend on `r`** — so relaxing `P₂ → P₃` buys weighted- sieve slack, not arithmetic. At `θ = 1/2` the level is 1: no moduli. §5.3. - **(N5)** Two numeric locations the corpus did not have. The certificate's usable depth range is `u ∈ [2, u*)` with `u* = 3.565845`, and the proven dimension-2 positivity range is `u > β₂ = 4.26645`: **disjoint**, missing each other by **0.70060 in `u`** (§2.5). And the twin short-interval exponent has moved `0.98 → 0.971` since 1976 — **0.009 in half a century** (§5.2) — against a zone's `θ → 1/2` with an unconditional ceiling of `0.7625` on any zone at any anchor (§5.1). --- ## 1. What the certificate is, in one place *(UNVERIFIED PREMISE: `quadpoint-identity-01.md`, HELD, never red-teamed.)* Anchor `Q`, stretch `S_Q = [Q², Q′²)`, height `h = Q²`, width `w = Q′² − Q²`. `C` = channel positions (pairs admissible mod 30), `T` = twins, `X(y)` = pairs with both members composite and both `y`-rough, `M` = mixed prime/composite pairs. The capture identity is `floor_K = T − X(K)`, so the certificate at depth `y` is exactly > **`X(y) < T` inside `S_Q`.** Its classical restatement (identity note §4.2, main-term only, inheriting the product-form hypothesis of `quadpoint-prior-art.md` §2.3): `S(A, y) < 4T`, where `S(A, y)` is the dimension-2 twin sifting function for `A_p = {n : p | n(n+2)}`. Its precision class: the margin is `(T − X)/C ~ 1/ln²h`, on an interval as short as `w`. Two coordinates matter below. - **`u = ln h / ln y`**, the sieve's sifting parameter. Full depth `y = Q` gives `u = 2`; `y ≤ Q` forces `u ≥ 2`. The crossing law `u ω(u) = 2` has root `u* = 3.565845` (`quadpoint-prior-art.md` §2.2, SCRATCHPAD there, never re-derived inside an embedded producer), so `X(y) < T` holds heuristically for `u < u*`. The certificate's usable range is therefore `u ∈ [2, u*)`. - **`θ = ln w / ln h`**, the interval exponent, which is the coordinate the short-interval literature is indexed by. `w = Q′² − Q² ≈ 2Q·g_Q`, so `θ → 1/2` and the finite-level values are §5.1's. --- ## 2. Question 1 — the minimal bilinear input, written down and priced ### 2.1 The distinction that decides the whole note Two different things are called "bilinear" in this corpus and in the literature, and conflating them is the failure mode this section exists to avoid. | | bilinear **over moduli** | bilinear **over elements** (Type-II) | |---|---|---| | shape | `Σ_{d₁,d₂ ≤ D} λ_{d₁} λ_{d₂} r_{d₁d₂}(x)`, or `Σ_q γ_q (π(x;q,a) − …)` | `Σ_{m∼M} Σ_{n∼N} α_m β_n a_{mn}`, `M, N ≥ x^δ`, `α, β` arbitrary bounded | | what it buys | **level of distribution** `θ` | **parity information** | | in print | BV `θ = 1/2`; BFI; DI; Zhang/Polymath8; Maynard; Lichtman; Pascadi | Vinogradov's minor arcs; FI's asymptotic sieve; Harman's asymptotic pieces | | does it defeat parity? | **No.** Selberg's parity example has remainder `O(1)`, i.e. perfect level; and EH/GEH-strength input still yields only the `d ∈ {2,4,6}` disjunction (Polymath8b, via `bv-import-survey.md` §2) | **Yes**, where it is available | | where the corpus's objects sit | **all of them** | none | Every bilinear object in this corpus is in the left column: Lemma V's remainder is a sum over `(d₁, d₂)` (`sift-limit-attack.md` §4.5), reduced in `attack-sqrt-cancellation.md` to a trilinear Kloosterman-fraction form over `(d₁, d₂, h)`; `Θ_e(a)` (import rows 5 and 6) is indexed by modulus and frequency; the BFI/DI/Pascadi imports are all statements about moduli. **The corpus has never formed a sum over a product of two ranging ELEMENTS with signed coefficients.** That is (N1) and §3 gives its mechanism. ### 2.2 The hypothesis, taken from the source that states it Friedlander and Iwaniec, *Asymptotic sieve for primes*, **Ann. of Math. (2) 148** (1998) **1041–1065**, read in full at source this session (arXiv:math/9811186v1, sha256 in §6). For a non-negative sequence `A = (a_n)`, `A(x) = Σ_{n≤x} a_n`, `A_d(x) = Σ_{n≤x, d|n} a_n = g(d)A(x) + r_d(x)`, their two load-bearing axioms are, **verbatim, p. 1043**: > **(R)** `Σ_{d⩽D} μ²(d)|r_d(t)| ⩽ A(x)(log x)^{−2^{22}}` > for all `t ⩽ x` with some `D = D(x)` such that > **(R1)** `x^{2/3} < D(x) < x`. > We assume an estimate for bilinear forms of the following type: > **(B)** `Σ_m |Σ_{N for every `N` with > **(B1)** `Δ^{−1}√D < N < δ^{−1}√x` > for some `δ = δ(x) ⩾ 2` and `Δ = Δ(x) ⩾ 2`, and where the coefficients are > given by > **(B2)** `γ(n) = γ(n, C) = Σ_{d|n, d⩽C} μ(d)`. > This is required for every `C` with **(B3)** `1 ⩽ C ⩽ xD^{−1}`. Three of their own remarks fix the shape of the answer, all **verbatim, p. 1045**: > "the source of cancellation in the bilinear form in (B) comes from the sign > changes of the Möbius function `μ(mn)` in the inner sum"; > "our stipulation of the lower bound restriction `N > Δ^{−1}√D` in (B1) is > essential; indeed by narrowing this slightly to `N > √D` we would not be > able to break the parity problem"; > "Actually, for thin sequences `A` one cannot expect (R) to hold with > `D(x) > A(x)` and for such sequences the classical sieve does > correspondingly worse." And, **verbatim, p. 1046**, the sentence that answers the question this note was sent to ask: > "we certainly expect that the sieve given here is quite capable of settling, > for instance, the twin prime problem. **The stumbling block is that in this > case we have no idea how to prove that the relevant sequence satifies the > condition (B). Even proving (R) to such a high level is currently beyond > reach.**" *(their typo, reproduced)* **Written in the repo's notation, then.** Take the stretch's sequence > `a_v = 1` if `v ∈ S_Q = [Q², Q′²)` and `v + 2` is `y`-rough; `0` otherwise, so `A(x) = Σ_v a_v` is the count the certificate must bound, and `Σ_p a_p log p` is the twin count with a log weight. The minimal bilinear input is **(B) for this sequence**, i.e. > **(B_Q)** `Σ_m |Σ_{N A(x)(ln x)^{−2^{22}}` for every `N` with `Δ^{−1}√D < N < δ^{−1}√x` and > every `C ⩽ xD^{−1}`, uniformly in the anchor `Q`. ### 2.3 The hypothesis is not merely unproven for our sequence — it cannot be posed **This is the run's hardest result and it does not go through parity at all.** FI's (R1) demands `D(x) > x^{2/3}`. Their own remark (quoted above) says (R) cannot be expected once `D(x) > A(x)`. Our sequence is supported on a stretch, so > `A(x) ⩽ w = x^{θ}` with `θ → 1/2`, and `θ > 2/3` is exactly the condition for `A(x)` to reach the level (R1) forces. **Measured** (SCRATCHPAD, §4.3), the fraction of anchors whose stretch is even that wide: | decade of `Q` | 10–10² | 10²–10³ | 10³–10⁴ | 10⁴–10⁵ | 10⁵–10⁶ | 10⁶–10⁷ | |---|---|---|---|---|---|---| | fraction with `θ > 2/3` | 0.9524 | 0.7133 | 0.4053 | 0.1678 | 0.0339 | **0.0023** | `θ > 2/3` needs a prime gap `g_Q > Q^{1/3}`, against gaps that are `≍ ln²Q` on every model and on every measurement. So the eligible fraction is not merely falling; it is heading to zero faster than any power. **The Friedlander–Iwaniec machine is structurally unavailable to a zone-supported sequence, for a reason that has nothing to do with parity and nothing to do with whether (B) is provable: the stretch does not carry enough mass to support the level its own Type-I axiom requires.** Deficit at the exponent: `x^{2/3}` against `x^{1/2+o(1)}`, i.e. a factor `x^{1/6}` — a power, not a log. ### 2.4 What is proven on each axis, and why the ladder is the wrong ladder **The level-of-distribution ladder, priced as the brief asked.** Every entry below is an (R)-axis statement. None is a (B)-axis statement. | result | level `θ` | class of moduli / weights | residue | |---|---|---|---| | Bombieri–Vinogradov 1965 | 1/2 | arbitrary, absolute values | **max over `a`** | | Bombieri–Friedlander–Iwaniec, *Acta Math.* **156** (1986), Thm 10 p. 209 | **4/7 ≈ 0.5714** | well-factorable `λ(q)` | **fixed `a ≠ 0`**, no max, no absolute values | | Zhang, *Ann. of Math.* **179** (2014), Thm 2 | 1/2 + 1/584 | squarefree, `x^{1/1168}`-smooth | root classes | | Polymath8a, *ANT* **8**:9, Thm 1.1 | 1/2 + 7/300 | squarefree, smooth / densely divisible | CRT-coherent | | Maynard, Mem. AMS **306**(1543), Thm 1.2 / 1.1 | 7/12; **3/5** | well-factorable linear-sieve; **triply** well-factorable | fixed `a` | | Lichtman, *ANT* **19**:1 (2025) 1–38, Thm 1.1 | **10/17 ≈ 0.5882** | his modified linear sieve | fixed `a` — **the twin-usable one** | | Lichtman, arXiv:2309.08522, Cor 1.5 | 66/107 ≈ 0.6168 | **triply** well-factorable | fixed `a` — **not usable for twins** | | Pascadi, arXiv:2505.00653, Thm 1.3 | **5/8 − ε** unconditional; 3/5 − ε | triply well-factorable; well-factorable linear-sieve | fixed `a ≠ 0` | The reason the ladder is the wrong ladder is one sentence with a number in it, and Lichtman states the conversion himself (arXiv:2109.02851 p. 3, via the agent's read): **level `x^{θ−ε}` gives `π₂(x)/Π(x) ≲ 2/θ`, which bottoms out at `2` as `θ → 1`.** That is `natal-cap-10-sieve-cap.md` §1.5's parity floor, re-derived from the top of the ladder rather than from Selberg's example. Every twin output in the entire (R)-axis literature is an **upper** bound. The certificate needs a **lower** bound on `T`, at relative precision `1/ln²h`. The corpus's own numeric form of the same statement is `natal-cap-10-sieve-cap.md` §4's ordering, and no (R)-axis input moves any term of it: > `C*(x) < 1` (truth) `< 2` (parity floor) `< 3.29956` (record) > `< 4` (BV wall) `< 8` (interval-uniform). *(Record-column rider: `3.29956` is the refereed record, Lichtman ANT 19:1 (2025) Thm 1.2. Two preprints stand ahead of it — Lichtman arXiv:2309.08522 at 3.2290 and Pascadi arXiv:2505.00653 Cor 1.4 at `3.203 + o(1)` — and the corpus already carries both in `sift-limit-attack.md` §4.2. Neither crosses the parity floor 2, which is the only thing the ordering is being used for here. Pascadi's paper is a preprint: `Forum Math. Pi` **14** (2026) e8 is his **other** paper, arXiv:2404.04239, and the two must not be conflated.)* **The (B) axis, priced.** Nothing has moved on it since 1998, and the object is posed as an open question in the field's own survey: - **Maynard, "Counting primes", Proc. ICM 2022, Vol. 1, §6 p. 15**, verbatim (agent's read at `mathunion.org`): "**Question 17 (Type II estimates for twin primes).** Can one estimate a Type II sum associated to Twin Primes, such as `Σ_{n∼N} Σ_{m∼M} α_n β_m Λ(nm + 2)` for arbitrary 1-bounded sequences `α_n, β_m`?" — preceded by "we currently do not know how to estimate Type II sums for most of the outstanding open problems on primes." - **Polymath8b §8**, arXiv:1407.4897, verbatim: if one could bound bilinear expressions `Σ_d Σ_m α(d)β(m) 1_{[x,2x]}(dm) Λ(dm+2)` then "one would very likely obtain non-trivial bounds … which would soon lead to a proof of the twin prime conjecture. **Unfortunately, we do not know of any plausible way to control such bilinear expressions.**" - **Tao, "Notes on the Bombieri asymptotic sieve" (2016)**, verbatim: no bound on the relevant `δ_x` beyond the trivial `[0, 2]` "is known, even if one assumes all other major conjectures in number theory than the prime tuples conjecture and its variants (e.g. GRH, GEH, GUE, abc, Chowla, …)." **So `(B_Q)` is Maynard's Question 17 restricted to a short interval and made uniform in the interval's position.** It is strictly harder than an open question, and the short-interval, position-uniform layer on top of it is the object `attack-sqrt-cancellation.md` §6 already records as **ABSENT** — searched in the owning convention "bilinear/trilinear forms with Kloosterman fractions" per `research/SEARCH-CONVENTIONS.md` §1, on WebSearch, the arXiv API and OpenAlex, each calibrated in-session. The uniformity price is quoted at source there: Bettin–Chandee's Remark 1 charges `(1 + hx/MN)^{1/2}`, `O(1)` only for `x ≪ H^{1.212157}` while our `x` reaches `exp(H^{0.2344})`. **`(B_Q)` is TPC-STRENGTH**, and by the import map's circularity rule it is legal to pursue only with an independent reason to think it softly provable. There is none, and the chain is three steps long in the wrong direction: `(B_Q)` ⟹ short-interval position-uniform Type-II ⟹ Maynard's Question 17 ⟹ the twin prime conjecture, with the last implication asserted by Polymath8b in the words quoted above. ### 2.5 Against the sifting limit, and this is the new number The certificate's usable depth range is `u ∈ [2, u*)`, `u* = 3.565845`. Proven dimension-2 positivity — the region where a lower-bound sieve gives anything at all — is `u > β₂ = 4.26645028`. The two ranges are **disjoint**, and the miss is > `β₂ − u* = 4.26645 − 3.56585 = ` **`0.70060`** in `u`, equivalently a factor `h^{1/u* − 1/β₂} = h^{0.046052}` in the depth `y` (2.336 at `h = 10⁸`; 5.455 at `h = 10¹⁶`; unbounded). `quadpoint-prior-art.md` §2.4 recorded the *inclusion* `u* ∈ (2, β₂)` as an observation and made no claim; the miss is computed here for the first time. **Calibration: HEURISTIC on both ends.** `u*` comes from the product-form depth law, validated at exactly one point (`u = 2`), and `β₂` bounds axiom-class sieves, not this particular comparison. The two numbers are not statements about the same object and the note does not treat them as one; what the arithmetic shows is that the operating point the heuristic picks out sits **outside** the region any proven lower-bound sieve reaches, so no strengthening of the sieve axioms puts it inside. ### 2.6 Chen's switching principle is not the parity-relevant ingredient — correcting the premise this run was sent with The brief that commissioned this run reasoned that because `quadpoint-prior-art.md` identifies the capture identity as "standard Buchstab bookkeeping plus Chen's switching principle", Chen's parity-relevant machinery is already inside our identity. **That inference is wrong, and the literature says so directly.** Switching re-organises which variable gets sieved so that an *upper*-bound sieve can remove the `P₃` contribution; it does not supply Type-II information and it does not break parity. - **Matomäki and Zuniga-Alterman, arXiv:2405.19063 pp. 1–2** (agent's read): "Later Fouvry and Grupp managed to show that `p + 2 = P₂` infinitely often **without using switching**, but a better level of distribution coming from deep estimates for sums of Kloosterman sums." Chen's conclusion is reachable without switching, so switching cannot be the ingredient that gets it. - **Iwaniec, ICM 2006 plenary, Vol. I p. 294** (agent's read): "we can also obtain numbers having either one or two prime divisors, but we are not able to determine which of these numbers are there … in order to distinguish primes from numbers having two prime divisors it is necessary to extend the system of sieve conditions by adding a new condition." Chen's `P₂` is named as the ceiling of the classical system, not as a breach of it. - **Pintz, arXiv:1004.1065 p. 1** (agent's read): "it is not known whether there are infinitely many primes `p` such that `p + 2` has an odd number of prime factors." Chen's theorem leaves the parity of `p+2` open, which is the operational test. - **Ford–Maynard, arXiv:2407.14368 p. 2** (agent's read), on the machine that *does* consume Type-II: Selberg's example satisfies Type I for arbitrary `γ < 1` with `Σ_p a_p = 0`, "so **non-trivial Type II information is necessary to detect primes**. Harman's sieve is a technique developed to get non-trivial lower bounds … with weaker assumptions on the Type I and Type II ranges" — weaker, not absent. **Consequence for the corpus.** `quadpoint-prior-art.md` §3's identification stands as prior art and is unaffected; what must not be attached to it is the reading that our frame therefore already carries parity-breaking machinery. It carries the bookkeeping and none of the input. `research/THE-DIALS.md` dial 5 ("almost-primes: proven, off the target") had this right and this note does not move it. --- ## 3. Question 3 — is anything in the frame bilinear? The mechanism, split by side ### 3.1 The tile side: Type-I by construction The tile `T_x` is periodic mod `W = x#`. A periodic set's entire arithmetic is its congruence data: `|A_d|` for `d | W`, equivalently its finite Fourier transform. That is the sieve's own input vector, and it is `DP1` — the corpus's master discard, "from the first line the sieve sees only the vector" (`sift-limit-attack.md` §1, DP1), carrying 71% of the loss budget at `x ≤ 43` (same file §7c). Type-II information is not a function of congruence data. The reason the classical setting has it at all is that `Λ = μ * log` admits Vaughan / Heath-Brown decompositions into Type-I and Type-II pieces **with signed coefficients**; the signs are the whole content. In this frame there is no `Λ` and no `μ` on the element variable. The frame's only prime-detecting device is **full-depth sifting** — "a survivor of the full Scour is a prime" (crystallization, `GLOSSARY.md`) — and `1_{survivor}` admits no such decomposition. That is what the parity problem says, stated in the frame's own vocabulary. The one candidate that looks like an exception is Door 2 (the Fourier budget), because the circle method is where Vinogradov's parity-defeating minor-arc cancellation lives. It is not an exception: Door 2 expands the **tile indicator**, which is periodic and exactly computable, so there are no minor arcs and no prime-detecting weight to put on them. Its measured budget is the price of that: pointwise `2ⁿ`, `L₁` mass per support class `(4/π)ⁿ`, effective base 1.165 / 1.185 / 1.195 / 1.204 at `x = 7, 11, 13, 17`, and the aggregation dies at `x = 11` (`wall-note.md` §1 Door 2). The frame's genuinely signed objects are the Rosser–Iwaniec weights `λ_d`, and they are bilinear over **moduli** (§2.1). `smoothness-front.md` §4 measured what they carry: Möbius-signed weights are Fourier-flat, every mode's share of the factor's mass `≤ 2.8e−3` against the `≳ 0.4` a usable smooth component [Rider 2026-08-29: the "≳ 0.4" share has no derivation at any record; the record's criterion is a ratio, C̃/K ≫ 100, and REFUTED.md's row now states it that way; refuted-audit-0829-4.md row 60.] needs (REFUTED row 78). So the one signed object in the frame is signed in the wrong variable and flat in the one place it was tested. **Verdict, tile side: structurally Type-I.** Falsification test, stated so the claim is not unfalsifiable: exhibit a corpus object that is a **signed** sum over a product of two **element** ranges, each of size `≥ h^δ`. The note searched the corpus for one (grep over `*.md` for Type I/Type II, bilinear, parity, and a read of the Lemma V family, `Θ_e(a)`, the Scour, the staircase injection and the capture identity) and found none. ### 3.2 The zone side: a product structure exists, and it is unsigned The zone side is not periodic and the verdict does not transfer. Three places in the corpus carry the same map `v = r·m`, `r` prime, `m` rough beyond `r`: - **the Scour** — "`⋃_q q × (the tile's own holes)`" (`GLOSSARY.md`); - **the staircase injection** — "for a prime-regime scour prime `q`, every non-self fresh victim satisfies `v = qm` with `m` prime" (`bv-import-survey.md` §3.1, S1); - **the capture identity's candidate map** — "a candidate of `r` is `v = r·m` in the window with `lpf(m) ≥ r`; … `lpf(m) ≥ r` holds iff `r = lpf(v)`" (`quadpoint-identity-01.md` §1, proof step (i), UNVERIFIED PREMISE). In the stretch this map ranges over `r ∈ [7, Q] = [7, v^{1/2}]` and `m ∈ [v^{1/2}, v/7]`, so its sub-range `r ∈ (v^{1/3}, v^{1/2})` is exactly the window where neither factor is small. That window is not chosen for convenience: FI's **(B1)** is `Δ^{−1}√D < N < δ^{−1}√x`, and their **(R1)** forces `D > x^{2/3}`, hence `√D > x^{1/3}` — so **(B1) always sits inside `(x^{1/3}, x^{1/2})` and narrows as `D` grows.** The mass in it is measured in §4.2 and is a positive, slowly falling fraction of the stretch's composites. **So the frame is not Type-I only in the naive sense: it does own a product decomposition with both factors ranging.** What it does not own is a reason for that sum to cancel. Every appearance of `v = r·m` above is inside a non-negative count with coefficient 1. A product decomposition of a counting problem is a *classification*, not Type-II information; Type-II information is cancellation in a signed sum, and there is nothing here to sign. ### 3.3 The one place the shift kills it, stated separately Even granting signs, the decomposition is on the **wrong member**. `v = r·m` factorises one side of the pair; the certificate's content is the joint condition on `(v, v+2)`, and `v + 2 = rm + 2` has no product structure at all. Any Type-II estimate for the pair has to control `rm + 2` in progressions, uniformly, over a short interval — which is the twin problem's own distribution question, not an input to it. This is precisely the move Chen's switching principle makes and precisely how far it goes: switch to the partner's side, where the bad events are semiprime-slot counts an upper sieve can afford, and buy `P₂` rather than prime. `quadpoint-prior-art.md` §3 (HELD, UNVERIFIED PREMISE) already identifies our capture identity as "Buchstab's identity for the dimension-2 twin sieve plus the switching condition", with no verbatim carrier and no novelty. **Read against this section, that identification says something sharper than it was written to say: the frame's switching move is already made, and it lands where switching always lands.** --- ## 4. The measurements this note ran All SCRATCHPAD-grade: session scripts in the agent scratchpad, **not embedded producers, not `qc`-gated**. None of these numbers may be quoted outside this file until re-derived inside an embedded producer, per the precedent of `quadpoint-prior-art.md` §2.2. ### 4.1 Stretch-width exponent `θ = ln(Q′² − Q²)/ln(Q²)` | decade of `Q` | anchors | mean `θ` | min `θ` | max `θ` | |---|---|---|---|---| | 10–10² | 21 | 0.7878 | 0.6642 | 0.9333 | | 10²–10³ | 143 | 0.6959 | 0.6023 | 0.8588 | | 10³–10⁴ | 1061 | 0.6539 | 0.5753 | 0.7943 | | 10⁴–10⁵ | 8363 | 0.6302 | 0.5602 | 0.7400 | | 10⁵–10⁶ | 68906 | 0.6133 | 0.5502 | 0.7153 | | 10⁶–10⁷ | 586081 | 0.6007 | 0.5430 | 0.6974 | The mean falls monotonically toward `1/2`; the **minimum** is the binding column, because the postulate has to hold at every anchor and the narrowest stretch is the hardest one. `w ≈ √h · ln h` is the asymptotic shape. ### 4.2 Type-II mass in a stretch Share of the stretch's **composite** channel members with `lpf(v) > v^{1/3}`, i.e. sitting in the range where `v = r·m` has both factors `≥ v^{1/3}`: | `Q` | `Q′` | width | channel positions | primes | `lpf ≤ v^{1/3}` | `lpf > v^{1/3}` | Type-II share of composites | |---|---|---|---|---|---|---|---| | 101 | 103 | 408 | 109 | 42 | 38 | 29 | 0.4328 | | 1009 | 1013 | 8088 | 2157 | 596 | 1184 | 377 | 0.2415 | | 10007 | 10009 | 40032 | 10675 | 2166 | 7017 | 1492 | 0.1753 | | 99991 | 100003 | 2399928 | 639981 | 104133 | 464091 | 71757 | 0.1339 | Control on the same run: the total share with `lpf > v^{1/3}` (primes included) reads 0.2748 at `Q = 99991` against the Buchstab prediction `3ω(3)·(30/8)/ln v = 0.2757`, a 0.3% agreement, so the counter is doing what it says. ### 4.3 How many stretches are wide enough for the FI machine to be posed The condition is `θ > 2/3`, i.e. `A(x) > x^{2/3}`, i.e. a prime gap `g_Q > Q^{1/3}` (§2.3). The `θ > 0.7625` column is the unconditional ceiling of §5.1 and is carried as a consistency check on the sieve: it must go to zero and it does. | decade of `Q` | anchors | fraction `θ > 2/3` | fraction `θ > 0.7625` | mean `θ` | max `θ` | |---|---|---|---|---|---| | 10–10² | 21 | 0.9524 | 0.6190 | 0.7878 | 0.9333 | | 10²–10³ | 143 | 0.7133 | 0.0769 | 0.6959 | 0.8588 | | 10³–10⁴ | 1061 | 0.4053 | 0.0019 | 0.6539 | 0.7943 | | 10⁴–10⁵ | 8363 | 0.1678 | 0.0000 | 0.6302 | 0.7400 | | 10⁵–10⁶ | 68906 | 0.0339 | 0.0000 | 0.6133 | 0.7153 | | 10⁶–10⁷ | 586081 | 0.0023 | 0.0000 | 0.6007 | 0.6974 | The `θ > 0.7625` column is empty from `Q = 10⁴` up, as the BHP ceiling requires, which is the only calibration available for that argument. --- ## 5. Question 2 — Chen in short intervals, against a zone's width *(Literature block; see §6 for what was reached at source and what was not.)* ### 5.1 What a zone requires `θ → 1/2` (§4.1). In words: **the Zone Postulate asks for a twin pair in an interval of length `≈ √x · ln x` around `x`.** That length is, to within the constant, the interval in which RH guarantees a *single* prime. The zone question is therefore the twin analogue of Cramér-under-RH, one dimension up, unconditionally. **And the width is capped from above, unconditionally, which is what makes the comparison decisive rather than merely unfavourable.** A stretch is as wide as its prime gap: `w = (Q′ + Q)·g_Q` with `g_Q = Q′ − Q`. Any unconditional prime-gap bound `g_Q ≪ Q^{c}` therefore caps the interval exponent at > `θ = ln w / ln h ≤ (1 + c)/2 + o(1)`. At the standing unconditional `c = 0.525` this is `θ ≤ 0.7625 + o(1)`; under RH's `g ≪ √Q ln Q` it is `θ ≤ 0.75 + o(1)`; on average it is `1/2`. The measured maximum over `Q ∈ [10⁵, 10⁶)` is 0.7153 (§4.1), inside the ceiling. **So if the best short-interval exponent in print exceeds 0.7625, no short-interval theorem of that family can cover a stretch at any anchor whatsoever — not typically, not eventually, not ever — and the failure is structural rather than a matter of margin.** ### 5.2 What the literature reaches The owning convention is **"Chen's theorem in short intervals"**, and the object it is indexed by is `D_{1,r}(N, c, θ) = #{p : p ⩽ N, N − p = P_r, p ∈ [cN, cN + N^θ]}` — a Goldbach-form count, with `c` locating the interval inside `[0, N]` and `θ` its length exponent. `c = 1/2` is a genuine short interval; `c = 0` is a *small-primes* restriction, not a short interval, and the two chains must not be tabled together. Our house words — zone, stretch, quadratic point — reach none of this (`research/SEARCH-CONVENTIONS.md` has no row for the object; §7.3 proposes one). **The twin form is in print, and it is stated as a corollary of the Goldbach form.** Cai and Lu, *Chen's theorem in short intervals*, **Acta Arith. 91** (1999) **311–323**, read at page image (§6), Theorem with `U = N^{0.972}` and then, verbatim: > "**Corollary.** For sufficiently large `x` and `y = x^{0.972}`, we have > `Σ_{x ⩽ p < x+y, p+2 = P₂} 1 ≫ Cy/log²x`, where > `C = 2 ∏_{p>2}(1 − 1/(p−1)²)`." That is the exact statement this question asked for: **a Chen-type theorem in `[x, x + x^{0.972}]`, for all sufficiently large `x`, with the Hardy–Littlewood order.** Wu then sharpened it with an explicit constant, also read at source this session — Acta Arith. **114** (2004) **215–273**, p. 5, where `π_{1,2}(x, θ) := |{x ⩽ p ⩽ x + x^θ : Ω(p+2) ⩽ 2}|` is defined and Theorem 5 reads "For every `θ ⩾ 0.971`, `N ⩾ N₀(θ)` and `x ⩾ x₀(θ)`, we have `D_{1,2}(N, θ) ⩾ 0.012 Ξ(N, θ)`, `π_{1,2}(x, θ) ⩾ 0.006 Π(x, θ)`." **`0.971` is the best twin short-interval exponent this note can assert at source.** **The chain.** Taken from Li, *On Chen's theorem, Goldbach's conjecture and almost prime twins III* §1 (read at page image, §6) — the only place the corpus has found the chronology assembled — and cross-checked row by row on zbMATH reviews. | object | chain of `θ` | attributions, with the twin-form ones in bold | |---|---|---| | `p + 2 = P₂` in `[x, x + x^θ]` — **the twin form, and the one this question is about** | 0.98 → 0.973 → 0.9729 → 0.972 → **0.971** | Ross (Ph.D., London 1976), `θ ⩾ 0.98`; **Wu, *J. London Math. Soc.* (2) 49 (1994) 61–72** (`θ ⩾ 0.973`, review-level); **Salerno–Vitolo, *p + 2 = P₂ in short intervals*, Note Mat. 13 (1993) 309–328** (`θ > 0.9729`, review-level); **Cai–Lu, Acta Arith. 91 (1999) 311–323, Corollary** (`θ = 0.972`, page-level); **Wu, Acta Arith. 114 (2004) 215–273, Theorem 5** — `π_{1,2}(x, θ) := \|{x ⩽ p ⩽ x + x^θ : Ω(p+2) ⩽ 2}\| ⩾ 0.006 Π(x, θ)` for every `θ ⩾ 0.971`, `x ⩾ x₀(θ)`, **read at source this session, definition on p. 5 and Theorem 5 on p. 5** | | `D_{1,2}(N, 1/2, θ)` — the Goldbach short-interval form | 0.98 → 0.974 → 0.973 → 0.9729 → 0.972 → 0.971 → **0.97** | Ross (Ph.D., London 1976); Wu; Salerno–Vitolo; Cai–Lu; **Wu, Acta Arith. 114 (2004) 215–273, Thm 5** (0.971, page-level); **Cai, *On Chen's theorem. II*, J. Number Theory 128 (2008) 1336–1357** (`\|p − N/2\| ⩽ N^{0.97}`, review-level, Heath-Brown's zbMATH review) | | `D_{1,2}(N, 0, θ)` — **small primes, NOT a short interval**; carried only so it is not mistabled | 0.959 → … → 0.9409 | Ross; Cai; Cai–Li; Cai; Li | | `D_{1,3}(N, 1/2, θ)` — the `P₃` relaxation | 0.919 → **0.87125** | Li, arXiv:2403.09691 (**math.GM — arXiv declined it from math.NT**); Li, paper III Thm 1.1 | | single primes in `[x, x+x^θ]`, right order | `13/23 → … → 21/40 = **0.525** → 13/25 = 0.52` | Iwaniec–Jutila (1979) … **Baker–Harman–Pintz, Proc. LMS 83 (2001) 532–562** = 0.525, the refereed record; 0.52 is Li, arXiv:2308.04458v8, **unrefereed** | | single primes, asymptotic formula | `7/12 → 17/30 ≈ 0.5667` | Huxley (1972); **Guth–Maynard**, arXiv:2405.20552, Ann. of Math. to appear | | one `P₂` (not a pair) in a short interval, for contrast | 0.45 → … → **101/232 = 0.435345** | Iwaniec–Laborde (1981) … Wu, Sci. China Math. 53 (2010) 2511–2524; no improvement since 2010 | **Four things this table settles that the corpus did not have.** 1. **The twin form exists in print and its best exponent confirmed at source is `0.971`** (Wu 2004, Theorem 5, with the explicit constant 0.006). `0.97` is the Goldbach chain's best and **whether Cai's 2008 paper carries a twin corollary was NOT REACHED** (§6). The twin chain has moved `0.98 → 0.971` in the twenty-eight years 1976–2004, and has not moved since: **0.009 of exponent in half a century.** 2. **`c = 0` is a small-primes restriction, not a short interval.** Its chain (0.959 … 0.9409) must never be tabled beside the `c = 1/2` one, and the `0.817` figure that goes with it is not a short-interval exponent. 3. **`0.87125` is Goldbach-only, and it is fragile.** No `p + 2 = P₃` short-interval result exists in any form, and nothing at all exists for `r ⩾ 4`. The `0.87125` itself has margin 0.003 on a sum of five numerically evaluated Buchstab integrals, is unrefereed, and rests on Lou–Yao, *J. Shandong Univ. (Nat. Sci.)* 2 (1979) 1–19 (in Chinese) — a paper **Li himself states contains errors** and which neither channel could obtain (§6). Quote it as a claim, never as a theorem. 4. **The pair problem and the single-almost-prime problem are not the same problem.** One `P₂` in a short interval reaches `0.4353`, below `1/2`; a `P₂` *pair* reaches only `0.971`. §5.3 gives the mechanism, and it is the run's cleanest statement of the wall. **Negative sweeps, recorded so they are not redone.** No almost-prime-*pair* result in a short interval exists beyond the `p + 2 = P₂` chain above: zbMATH `ti:"short intervals" & cc:11P32` (30 hits) and the same with `cc:11N36` (40 hits) return zero on pairs; `ti:"almost prime" & ti:twins` returns three, none short-interval. The global pair object is J. Kan, *On the problem of Goldbach's type*, Math. Ann. **292** (1992) 31–42 (`D_{s,r}(N)` for any `s ⩾ 1, r ⩾ 2`) and Heath-Brown, *Almost-prime k-tuples*, Mathematika **44** (1997) 245–266 at `k = 2` — **neither has a short-interval version.** Searched in the owning convention "Chen's theorem in short intervals" per §7.3's proposed `research/SEARCH-CONVENTIONS.md` row. One false lead, recorded so nobody re-chases it: W. Luo, *Lower bound for number of B-twins in short intervals* (Hua Loo Keng memorial, Springer 1991, 193–208) — `B` is **sums of two squares**, not almost primes. ### 5.3 The comparison **The decisive line, and it is not about margins: the arithmetic input vanishes at exactly our exponent.** The engine under every entry in the `p + 2 = P₂` chain is a Bombieri–Vinogradov theorem *in short intervals*. Its level, read at page image from Li's paper III Lemma 3.2 (attributed there to Lou–Yao 1979, Lemma 3): > `D ⩽ min( x^{θ − 1/2}(log x)^{−B}, x^{(12θ − 7)/5} )`, for `7/12 < θ ⩽ 1`. Two readings, and both matter. 1. **The level is `x^{θ − 1/2}`, so it is `1` at `θ = 1/2` and the lemma is not asserted at all below `θ = 7/12 = 0.58333`.** A zone has `θ → 1/2`. **The Zone Postulate asks for twins in an interval of exactly the length at which the sieve's own arithmetic input degenerates to no moduli.** That is not a gap to be closed by a better exponent; it is the endpoint of the coordinate. 2. **Neither bound depends on `r`.** Relaxing `P₂ → P₃` buys slack in the *weighted sieve* (Li III: `k₁` drops 7.726 → 5.4644), not in the arithmetic. That is why one almost-prime in a short interval reaches `0.4353` — a single sequence, Fouvry–Iwaniec sums suffice — while a *correlated pair* sits above `0.87`, and why the `P₂ → P₃` step is worth only about 0.10 before stopping. Going to `P₄` and beyond buys nothing and is not attempted anywhere in print. **The unconditional exclusion, as a second and independent argument.** The ceiling on a zone's interval exponent (§5.1) is `θ ⩽ (1 + 0.525)/2 = 0.7625 + o(1)`. The cheapest short-interval almost-prime-pair exponent in print is `0.87125`, and the actual `P₂` figure is `0.971`. **So no member of the Chen-in-short-intervals family can cover a stretch at any anchor for large `Q` — not typically, not for infinitely many anchors, not ever.** The exclusion survives even if the published exponent were improved all the way to the ceiling, because reading 1 above says the family cannot reach `θ = 1/2` at any `r`. The finite-scale sizes, for calibration rather than for the argument (SCRATCHPAD, same session): | `Q` | `x = Q²` | zone width `w` | `x^{0.972}` | `w / x^{0.972}` | `x^{0.87125}` | `w / x^{0.87125}` | |---|---|---|---|---|---|---| | 101 | 1.020e4 | 408 | 7.878e3 | 5.18e−2 | 3.108e3 | 1.31e−1 | | 1009 | 1.018e6 | 8088 | 6.911e5 | 1.17e−2 | 1.715e5 | 4.72e−2 | | 10007 | 1.001e8 | 40032 | 5.978e7 | 6.70e−4 | 9.344e6 | 4.28e−3 | | 99991 | 9.998e9 | 2399928 | 5.247e9 | 4.57e−4 | 5.157e8 | 4.65e−3 | | 1000003 | 1.000e12 | 6.000e7 | 4.613e11 | 1.30e−4 | 2.851e10 | 2.10e−3 | *(The table is computed at `0.972`, Cai–Lu's page-level twin figure, rather than at Wu's `0.971`; the difference is 0.001 of exponent and moves no cell visibly.)* The ratio falls like `x^{θ_lit − 1/2}`, so it diverges; the table's job is only to show it is already four orders of magnitude at `x = 10¹²`. **One comparison in the other direction, stated so the note is not read as uniformly pessimistic.** A zone is *wider* than a BHP interval at every height the corpus computes: `x^{0.525}` is 1.99e6 at `x = 10¹²` against a zone width of 6.00e7, a factor 30. The two cross at `0.025 ln x = ln ln x`, i.e. around `x ≈ e^{215} ≈ 10^{93}` (SCRATCHPAD arithmetic). **Asymptotically, therefore, a zone is narrower than the shortest interval in which unconditional methods can place a single prime** — which is a fair summary of what the Zone Postulate is asking for, and it is asking for two. **Chen's theorem restricted to a zone, per the question as asked: no per-zone content.** Cai–Lu's corollary holds for all large `x` at `y = x^{0.972}`, so it says something about every interval of that length — including intervals that *contain* many zones. It says nothing about any one zone, because a zone is `x^{0.5+o(1)}` and the corollary's own admissible length is bounded below by `x^{0.97}`. The corpus's existing tile-native Chen proposal (`bv-import-survey.md` §4, never executed) is a `[1, x]` statement and inherits the same limitation: `β_Chen(x) ≥ c` is a whole-tile bias, not a per-zone occupancy statement. --- ## 6. Sources, and what was not reached ### 6.1 Read at page image this session, with hashes Fetched to the session scratchpad, extracted with `pdftotext -layout`, and read in extraction. Every quotation above marked "page-level" or "at source" comes from these files, not from a search snippet or a review. | artifact | source | sha256 | |---|---|---| | Friedlander–Iwaniec, *Asymptotic sieve for primes*, Ann. of Math. (2) **148** (1998) 1041–1065 | `arxiv.org/pdf/math/9811186v1` | `d39f5249d9003a8a7ce121c93e785dcffa2dfb5ccc9e728820a705826d953d1f` | | Cai–Lu, *Chen's theorem in short intervals*, Acta Arith. **91** (1999) 311–323 | `matwbn.icm.edu.pl/ksiazki/aa/aa91/aa9142.pdf` | `59205bc24e6275bc3f90f702d4c0436b858749942c2a00579be57f4955b44af6` | | Wu, *Chen's double sieve, Goldbach's conjecture and the twin prime problem*, Acta Arith. **114** (2004) 215–273 | `arxiv.org/pdf/0705.1652` | `41d432dd63da6d1fe7836ba3beda8b601ce6e64420e50501d26d79f7d971043e` | | Runbo Li, *On Chen's theorem, Goldbach's conjecture and almost prime twins III* | `runbolicarey.com/assets/downloads/…III.pdf` | `6ee0d3d6d40eb8683d73c73b252c76b8b6fe10554115f98a297814d4312e2e33` | | Runbo Li, *… almost prime twins II* (v4) | `arxiv.org/pdf/2405.05727v4` | `e62adef510c1e0a695769e03be1f7fe787cc2b30ea1f8761e87a9039efdf3f2c` | **Channel calibration, run in the same session** per `research/SEARCH-CONVENTIONS.md`: WebSearch on `"Baker Harman Pintz difference between consecutive primes II 2001"` returned the paper on the first query; WebSearch on `Cai Yingchun "Chen's theorem in short intervals" Acta Arithmetica` returned EUDML 207358 and the Springer record on the first query. Reached through the two literature agents with their own in-session calibrations: annals.math.princeton.edu, msp.org, mathunion.org, zbMATH open API (`_search`, **not** `_structured_search`, which 400s), MathSciNet `mrlookup`, terrytao.wordpress.com (via `curl` with a browser UA — WebFetch 403s there). ### 6.2 NOT REACHED Recorded so the next wave spends its budget on what is left. - **Cai, *On Chen's theorem. II*, J. Number Theory 128 (2008) 1336–1357** — review-level only (Heath-Brown's zbMATH review, `|p − N/2| ⩽ N^{0.97}`); ScienceDirect 403s. **Whether it carries a twin corollary is unknown**, so the best twin exponent this note asserts is Wu's `0.971`, not `0.97`. - **Wu, JLMS (2) 49 (1994) 61–72** and **Salerno–Vitolo, Note Mat. 13 (1993) 309–328** — review-level only (Zbl 0790.11065, Zbl 0831.11055). Both are twin-form, both above `0.972`, so neither changes any conclusion. - **Lou–Yao, *J. Shandong Univ. (Nat. Sci.)* 2 (1979) 1–19 (in Chinese)** — not obtainable on any channel. It is the sole arithmetic input to Li's `0.87125`, and **Li himself states it contains errors** invalidating its own claimed `θ = 0.92`. Li's Lemma 3.2, which is what §5.3 quotes, is attributed to it. - **Opera de Cribro Ch. 16 (parity) and Ch. 25 (almost-prime sieve, §25.6 twin almost-primes, switching trick indexed at p. 484)** — chapter titles, page numbers and Preface reached; body text not. What Friedlander and Iwaniec say about switching *in the book* is unverified. Their Preface does say of Ch. 18 (the 1998 paper's book form): "the bilinear form part is the new axiom by means of which one breaks the parity barrier." - **Harman, *Prime-Detecting Sieves*, PUP 2007, body text** — TOC verified from the Library of Congress MARC record; the attribution of the Type-II input to Chs. 2, 3 and 5 is inferred from Ford–Maynard and Kumchev citing it, not read off the book. - **Bombieri, *The asymptotic sieve*, Mem. Acad. Naz. dei XL 1/2 (1976) 243–269** — not reached; carried through FI 1998, Iwaniec's ICM and Tao 2016. - **BFI *Acta Math.* 156 Theorems 1–7 (the bilinear engine, pp. 214–244)** — located, not transcribed; only Theorem 10 (p. 209) is quoted. - **Maynard, Mem. AMS 306 (1542)/(1543)** — the 3/5 and 7/12 figures are abstract-level plus Pascadi's and Lichtman's citations, except Theorem 1.2, transcribed from arXiv:2006.07088 p. 2 by an agent. - **Tao's 2007 parity post says nothing about the switching principle in its body** — only Chen's *theorem*, on the parity-permitted side. The mechanism appears in the comment thread (Kowalski), not in Tao's text. - **Every scratchpad number in §4 and §5.3's finite-scale table.** No embedded producer, no `qc` gate. They must be re-derived inside one before being quoted anywhere outside this file. ### 6.3 The conditional negative **No result places any twin-type problem in an interval of length `x^{1/2+ε}` under RH, GRH, or Elliott–Halberstam.** Eleven distinct queries across four calibrated channels (agent-run, list held in its report), searched in the owning convention "Chen's theorem in short intervals" and in "the difference between consecutive primes". The nearest object is **Alweiss–Luo, *Bounded gaps between primes in short intervals*, Res. Number Theory 4 (2018), Paper 15**, which gets *bounded* gaps inside `[x, x+x^δ]` for `δ ∈ [0.525, 1]` — it inherits the BHP floor verbatim because it consumes BHP as an input, and its gap `d` is unspecified rather than 2. That is dial 6 again (`THE-DIALS.md`), the axis this programme does not own. --- ## 7. PROPOSED, not applied ### 7.1 `research/IMPORT-MAP.md` §2, one new row (row 15) Per the map's standing rule ("a future import gets a graded row here, with its circularity pre-check, BEFORE it runs — never a run out of inspiration"), this row is written retroactively and marked as such; it is the grade this run *should* have carried at its start. | # | field | the importable theorem | moiré object | target hole | fit | circularity | payoff | cost | status | |---|---|---|---|---|---|---|---|---|---| | 15 | bilinear (Type-II) prime detection | Friedlander–Iwaniec, *Asymptotic sieve for primes*, Ann. of Math. (2) **148** (1998) 1041–1065, axioms (R)/(R1)/(B)/(B1–B3) **[SOURCED, verbatim]**; Bombieri's asymptotic sieve **[MEMORY, carried through FI]**; Chen 1973's switching principle; Harman, *Prime-Detecting Sieves*, PUP 2007 **[SOURCED-BIB]** | the capture identity's candidate map `v = r·m` on the stretch (`quadpoint-identity-01.md` §1 proof step (i)) | Z2's certificate `X(y) < T` | **VOCABULARY-ONLY on the tile side** (periodic ⟹ congruence data only, §3.1); **STRONG-ANALOGY on the zone side** (the product structure is real and spans FI's own (B1) window, §3.2) | **TPC-STRENGTH** — `(B_Q)` is Maynard's ICM Question 17 restricted to a short interval and made position-uniform, and Polymath8b §8 states in print that controlling such forms "would soon lead to a proof of the twin prime conjecture" | WALL-ADDRESS | 4 h | **LANDED 2026-08-26, route closed, and closed twice over** — (i) FI's machine cannot be POSED for a zone-supported sequence: (R1) forces `D > x^{2/3}` while FI's own remark forbids `D(x) > A(x)` and a stretch has `A(x) ⩽ x^{1/2+o(1)}` (deficit `x^{1/6}`; only 0.0023 of anchors at `Q ∈ [10⁶,10⁷)` are even wide enough); (ii) the estimate is a position-uniform short-interval bilinear form, the object `attack-sqrt-cancellation.md` §6 already records ABSENT. Banked: the tile-side Type-I mechanism (N1), the (R)-axis/(B)-axis distinction that reprices eight REFUTED rows' machinery, `β₂ − u* = 0.70060`, the short-interval-BV level `x^{θ−1/2}` vanishing at a zone's own exponent, and the correction that Chen's switching is not parity-relevant | ### 7.2 `research/REFUTED.md`, one line | route | verdict | why, in one clause | closed | record | |---|---|---|---|---| | transplanting Type-II (bilinear) parity-breaking information into the frame | CLOSED | Friedlander–Iwaniec's machine cannot be posed on a stretch — (R1) needs `D > x^{2/3}` where their own remark caps `D` at `A(x) ⩽ x^{1/2+o(1)}` — and the input it would need is Maynard's ICM Question 17 made position-uniform on a short interval, already recorded ABSENT; the tile side carries no element-side signed sum at all | 2026-08-26 | `history/staging/attack-bilinear-transplant.md` | | Chen-in-short-intervals as a source of per-zone occupancy | CLOSED | the short-interval BV level is `x^{θ−1/2}`, asserted only for `θ > 7/12` and independent of `r`, so the family degenerates at exactly a zone's `θ → 1/2`; the twin exponent in print is 0.971 and has moved 0.009 since 1976, against an unconditional ceiling of 0.7625 on any stretch | 2026-08-26 | `history/staging/attack-bilinear-transplant.md` §5 | ### 7.3 `research/SEARCH-CONVENTIONS.md` §1, one candidate row Offered with the file's own warning attached — a row added to make a claim pass is a suppression wearing a table's clothes. This one is proposed because it is the wording the literature uses, and because the corpus searched the short-interval axis in no convention at all before today. | object | our name | canonical | **OWNING convention — search THIS** | where it lives | |---|---|---|---|---| | a twin-type or almost-prime-pair statement confined to `[x, x + x^θ]` | the Zone Postulate's width, the stretch | — | **"Chen's theorem in short intervals"**, and the count it is indexed by, `D_{1,r}(N, c, θ) = #{p ⩽ N : N − p = P_r, p ∈ [cN, cN + N^θ]}` — note `c = 0` is *small primes*, not a short interval, and its chain must never be tabled beside the `c = 1/2` one. The twin form is a corollary of the Goldbach form. For the single-prime baseline: **"the difference between consecutive primes"**. The house words zone, stretch, quadratic point reach nothing | Cai–Lu, *Chen's theorem in short intervals*, Acta Arith. **91** (1999) 311–323 (Corollary, `θ = 0.972`); Salerno–Vitolo, *p + 2 = P₂ in short intervals*, Note Mat. **13** (1993) 309–328; Cai, Chinese Ann. Math. Ser. B **29** (2008) 687–698 (`θ = 0.97`); Runbo Li, *On Chen's theorem, Goldbach's conjecture and almost prime twins III*, §1 carries the whole chronology; Baker–Harman–Pintz, Proc. LMS **83** (2001) 532–562 | --- ## 8. The fifth naming of the wall, with its coordinates `quadpoint-identity-01.md` §4 counts the corpus's framings of the same obstruction: zones, the origin, the square anchor, the rough-pair census — "fourth frame of the same wall". This note adds a fifth, and unlike the first four it is stated in the convention the *breach* literature uses rather than in one of ours. > **The wall, in Type-II coordinates.** Parity is defeated only by > cancellation in `Σ_{m∼M, n∼N} α_m β_n a_{mn}` with both ranges large. The > tile side of this frame cannot form such a sum at all, because a periodic > set's arithmetic is its congruence data and the frame has no signed > prime-detecting weight on the element variable (§3.1). The zone side can > form one — `v = r·m` spans FI's own `(B1)` window and carries 13–43% of the > stretch's composite mass (§4.2) — but carries no signs, so it classifies > and does not cancel (§3.2). And before parity is even reached, a stretch is > too thin to carry the level FI's Type-I axiom demands (§2.3), while the > short-interval sieve's arithmetic input degenerates at exactly a stretch's > own interval exponent (§5.3). **The zone is the one place in the frame where > Type-II information could live, and it is the place where the two machines > that consume it both run out of input.** Its coordinates, all five new to the corpus: | coordinate | value | calibration | |---|---|---| | mass: what FI's own Type-I axiom needs against what a stretch carries | needs `A(x) > D > x^{2/3}`; a stretch has `A(x) ⩽ x^{1/2+o(1)}` — **deficit `x^{1/6}`**; only **0.0023** of anchors at `Q ∈ [10⁶,10⁷)` even reach `θ > 2/3` | PROVEN from FI's (R1) + their own thin-sequence remark, both quoted; the fraction is SCRATCHPAD, §2.3 and §4.3 | | arithmetic: the short-interval BV level at a stretch's own exponent | `D ⩽ x^{θ−1/2}(log x)^{−B}`, asserted only for `θ > 7/12`, **independent of `r`**; at `θ = 1/2` it is **1** | PROVEN in print, read at page image, §5.3 | | depth: the certificate's usable range against the proven positivity range | `u ∈ [2, 3.565845)` vs `u > 4.26645`; **disjoint, miss 0.70060** | HEURISTIC (`u*` from the one-point-validated product form) + PROVEN (`β₂`); the two bound different objects, §2.5 | | interval exponent: what a zone is, and its unconditional ceiling | `θ → 1/2`; mean 0.6302 and min 0.5602 over `Q ∈ [10⁴,10⁵)`; **no stretch exceeds `(1+0.525)/2 = 0.7625 + o(1)`** | SCRATCHPAD (§4.1) for the values; PROVEN given the prime-gap bound for the ceiling (§5.1) | | interval exponent: what the literature reaches, and how fast it moves | `0.971` for `p + 2 = P₂` (Wu 2004 Thm 5, at source); `0.97` Goldbach chain best; `0.87125` after relaxing to `P₃`, Goldbach-only and fragile. Movement since 1976: **0.009** | PROVEN in print, §5.2 | ## 9. What would change this verdict Stated so the closure is falsifiable rather than rhetorical, in the order a future attempt should test them. 1. **A corpus object that is a signed sum over two ranging element ranges.** §3.1's falsification test, and the cheapest of the four. One exists nowhere in the corpus today; if one is exhibited, §3's mechanism fails and the row reopens. 2. **A version of the Friedlander–Iwaniec machine that does not require `D > x^{2/3}`.** §2.3's obstruction is the only one that is prior to parity, so it is the only one a purely technical improvement could touch. FI say their (B) "can be realistic only if `D` is somewhat larger than `x^{2/3+ε}`", which makes this a hypothesis of the method rather than of the sequence — and therefore, in principle, negotiable. 3. **A position-uniform short-interval bilinear estimate of any strength.** `attack-sqrt-cancellation.md` §6's ABSENT row becoming a PRICED row. Necessary, and by §2.4 nowhere near sufficient. 4. **A `p + 2 = P_r` short-interval result at any `r` with `θ < 0.7625`.** That would breach §5.1's ceiling argument. §5.3 says why it should not exist below `7/12`, and nothing at any `r ⩾ 3` exists in the twin form at all — so this is the one place where the literature is genuinely thin rather than genuinely blocked, and where a reader who knows the field should be asked before the row is treated as final. --- *History layer: process record, staging. See `research/history/CHANGELOG.md` for the corpus rule. Nothing committed, nothing pushed, no existing file edited.*