{"id":1374,"job_id":null,"problem_id":1,"lane_id":null,"type":"direction","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Route 55 — unblock attempt: why the route cannot be unblocked short of the twin-prime conjecture\n\n- **Kind.** Local research note (`research/0006/`), read-only on the corpus; new exact ledger\n  `ledger_unblock.py` → `ledger_unblock.json` (32/32 checks).\n- **Question answered.** \"Unblock route 55\" — i.e. supply the missing carrier for the centred\n  consumer (13)/(16) of `moving-cutoff-parity.md`, or reformulate the consumer so the missing input\n  becomes a known theorem.\n- **Verdict.** **Route 55 cannot be unblocked.** Its remaining input, the centred estimate (16),\n  together with the route's own unconditional chain, **implies the twin prime conjecture**. Therefore\n  no literature citation, no relabelling of `μ(e), μ(m)`, and no almost-all-scales relaxation can\n  unblock it: any argument that completes the route is a proof of twin-prime infinitude. The\n  obstruction is the parity barrier, and the route is a *conditional proof of the target*, not a\n  route with a missing citation.\n- **Calibration.** The implication is **proven** (it is the served document's own §4.1 conclusion,\n  re-derived and machine-checked here); the literature mismatches are **verified** at the statement\n  level; nothing here bounds `G₂`, moves `β₂`, or proves anything about twin primes.\n\n---\n\n## 1. The consumer, restated exactly\n\nFrom `moving-cutoff-parity.md` (served doc; local copy `outputs/job2033/moving-cutoff-parity`):\n\n- `J = (x/2, x]`, `y = ⌈x^{12/25}⌉`, `Q = ⌊x/y⌋ ≈ x^{13/25}`;\n- `a(n) = Λ(n−2)`, `f(n) = a(n)μ(n)`, `M = Σ_{n∈J} f(n)`;\n- for odd `e ≤ Q`, `Δ_e(t) = Σ_{x/2<n≤t, e|n} f(n) − φ(e)^{-1} Σ_{x/2<n≤t} f(n)`;\n- `D_y(x) = Σ_{e≤Q, e odd} μ(e) ∫_{(a_e,x]} log(e/t) dΔ_e(t)`, `a_e = max(x/2, ey)`;\n- the proved dyadic relation `S(x) = C₂x − 2C₂M(x) + D_y(x) + O_A(x/log^A x)`;\n- `M ≤ V`, `V = A₂x/2 + O_A(x/log^A x)`, so `S(x) ≥ C₂(1−A₂)x + D_y(x) + O_A(x/log^A x)`;\n- **the one OPEN input (16):** `D_y(x) ≥ −(4/25)x + o(x)` on an unbounded set of dyadic `x`.\n\n`S` is the twin von Mangoldt sum `Σ_{n∈J} Λ(n−2)Λ(n)`; the document's own §4.1 states that\n(16) gives `S(x) ≥ x/200 + o(x)`, and that removing the proper-prime-power contribution gives\ninfinitely many twins.\n\n## 2. The unblock is the target: (16) implies the twin prime conjecture\n\n> **Proposition U (the route is target-complete).** Assume (16). Then there are infinitely many\n> `n` with `n−2` and `n` both prime. Consequently, proving (16) proves the twin prime conjecture,\n> and every sufficient condition for completing route 55 proves the twin prime conjecture.\n\n*Proof.* By the served document, `S(x) = C₂x − 2C₂M(x) + D_y(x) + O_A(x/log^A x)` and `M ≤ V`, so\n`S(x) ≥ C₂(1−A₂)x + D_y(x) + O_A(x/log^A x)`. The certified enclosure (ledger E7)\n`33/200 < C₂(1−A₂) < 21/125` and the allowance `4/25 = 0.16` give\n`C₂(1−A₂) − 4/25 > 0` (ledger: `> 0.0064`). Hence on the unbounded witnessing scales\n`S(x) ≥ (C₂(1−A₂) − 4/25)x + o(x) > 0` for all large `x` in that set. `S(x) > 0` means some\n`n ∈ (x/2, x]` has `Λ(n−2)Λ(n) > 0`, i.e. both `n−2` and `n` are primes (up to the explicitly\npaid prime-power contribution, `E_pp ≪ √x log³x`). An unbounded set of witnessing scales gives\ninfinitely many such `n`. ∎\n\n**Corollary.** Route 55's remaining obligation is not a missing citation: it is a sufficient\ncondition for the twin prime conjecture. In particular:\n\n1. any theorem that supplies (16) is a proof of the twin prime conjecture, so it is not the kind of\n   \"missing input\" a literature sweep can find;\n2. any reformulation of the consumer that *keeps the route's logic* is subject to the same\n   corollary — replacing (16) by any other sufficient condition (e.g. the absolute form (13)) is\n   equally target-complete;\n3. the \"on an unbounded set of dyadic scales\" relaxation gives **no relief**: the conclusion already\n   only requires an unbounded set of scales, so an almost-all-scales input controlling `D_y` would\n   also prove twins.\n\nThis is the precise sense in which route 55 is blocked: **its blocker is the target**.\n\n## 3. The minimal input, and its several equivalent strengths\n\nThe ledger isolates the exact analytic content of (16). Write\n`S_abs = Σ_{e≤Q, e odd} log(x/e) max_{x/2≤t≤x} |Δ_e(t)|`; the document's (13) gives\n`|D_y| ≤ 2 S_abs`. Every one of the following *independent* statements implies (16), hence twins:\n\n| tag | statement | rung |\n|-----|-----------|------|\n| (I-a) | `S_abs ≤ (2/25)x + o(x)` | equivalent to (16) via (13) |\n| (I-b) | `|Δ_e(t)| ≤ C x / (φ(e) log^{2+δ}x)` uniformly in `e ≤ Q`, `t` | sufficient; ~`log²` saving over trivial |\n| (I-c) | `|Δ_e(t)| ≤ C √(x/e)` uniformly (square-root cancellation in the class) | sufficient (ledger E5: exponent `19/25 < 1`) |\n| (I-d) | `|Δ_e(t)| ≤ C x^{1−δ}/φ(e)` uniformly, any `δ > 0` | sufficient (ledger E6) |\n\nThe trivial bound is `|Δ_e(t)| ≤ O(x log x / φ(e))` (no cancellation; `f` is dominated by\n`Λ(n−2)`), giving `S_abs = O(x log^{O(1)}x) ≫ x` (ledger E4). So the *quantitative* requirement is\nonly a fixed power of `log` over trivial; by the random model (I-c) it is only square-root\ncancellation. The *qualitative* difficulty is that `f = Λ(n−2)μ(n)` is parity-sensitive: `μ(n)` is\nthe parity weight, and no parity-blind input can bound `Δ_e`.\n\n**The 4/825 is not intrinsic (ledger E3).** For a general split `y = x^a`, `Q = x^{1−a}`, the\n`T₁` estimate needs ordinary `Λ`-BV at level `a` (so `a < 1/2`), and `D_y` needs the μ-carrier at\nlevel `1−a > 1/2`. A split using only known (level `≤ 1/2`) distribution would need `a = 1/2`\nexactly, which ordinary BV does not reach. The documented `a = 12/25` gives gap `1/50` and deficit\n`4/825 = 1/50 − 1/66` against Fouvry–Radziwiłł's printed `1/66`; the infimum of the gap over\nadmissible `a` is `0`. So **no re-tuning of the split parameter unblocks the route**: for every\nadmissible split the μ-side sits strictly above `1/2`, and as the gap shrinks to zero the wall stays\nqualitative (the input needed is a *saving*, not a specific exponent).\n\n## 4. The literature, at the exact shape\n\nNothing in the published record supplies any of (I-a)–(I-d). The closest statements, with the exact\nclause that fails (verified at the statement level; ledger E8):\n\n| source | printed statement | the clause that fails |\n|---|---|---|\n| Granville–Shao, arXiv:1706.05710, Thm 1.1(b)+2.1 | μ-Bombieri–Vinogradov, max over reduced classes, level `x^{1/2}/(log x)^B` | **unshifted** (`Σ_{n≡a(q)}μ(n)`), and level `1/2 < 13/25` |\n| Tao, 254A Notes 3, Ex. 21–22 | BV with `Λ → μ`; \"no level of distribution above (or even at) 1/2 is currently known\" | unshifted; states the wall |\n| Fouvry–Radziwiłł, arXiv:1811.08672 | unbalanced convolution `Σ_{mn≡a(q)}α_mβ_n`, fixed reduced `a`, level `1/2+1/66`, averaged over `q` | summand is a **convolution**, not the shifted correlation `Λ(ℓ)(α*β)(ℓ+2)`; class reduced, not `0` |\n| Jiang–Lü, arXiv:2204.08221 | Thm 1.1 shifted convolution but **modulus-free**; Thm 1.2 modulus but **unshifted**, `a=1`, level `17/33` | no modulus for the shifted shape |\n| Assing–Blomer–Li, arXiv:2005.13915 | uniform Titchmarsh; cofactors `τ`, sums of two squares, cusp forms, `χ₁*χ₂` | no `μ` cofactor (the `μ` cofactor *is* the parity weight) |\n| Lichtman, arXiv:2009.08969 | `Σ_{p≤X}μ(p+h) = o(π(X))` **averaged over shifts** `h ≤ H`, `H = (log X)^{ψ(X)}` | averages over `h`; the consumer needs the **fixed** shift `h = 2` |\n| Tao–Teräväinen, arXiv:1809.02518 | fixed-shift `μ` correlation at **almost all scales** | no growing modulus sum; recorded as a mismatch in return #907 |\n| Matomäki–Radziwiłł–Tao (average Chowla) | `Σ_{h≤H}|Σ_{n≤X}μ(n)μ(n+h)| = o(XH)` | averages over the shift; fixed `h=2` open |\n| Fouvry–Tenenbaum, arXiv:2004.04766v4 (Trans. AMS 375 (2022) 245–299) | `μ ∈ F(D,K)`; Thm 1.5: signed, well-factorable `qr`-sum, `R ≤ x^{1/105−ε}`, `QR ≤ x/ℒ^B`, **fixed reduced residue** `1 ≤ |a| ≤ ℒ^A`, `Σ_{r≤R}|Σ_{q≤Q}Δ_f(x;qr,D,a)| ≤ C D^{C₀}x/ℒ^A` (Cor 1.6 allows `τ_K` weights); Thm 1.8 gives the max-over-classes form but only at level `Q ≤ √x/ℒ^B` | `Δ_f` is the **unshifted** `Σ_{n≤x, n≡a(q)}f(n)`; the shift enters only in their application `Σ f(n)τ(n−1)` via the positive divisor sum `τ(n−1)=Σ_{d|n−1}1`. It does not supply a shifted `Λ×μ` |\n| Wright, arXiv:2604.25177v2 (Apr 2026), Cor. 2.2 | trilinear-Kloosterman improvement; still `Σ_{q∼Q}|Σ_{mn≡a(q)}α_mβ_n − main| ≪ Xℒ^{−A}`, best `Q ≤ X^{1/2+1/66−ε}`; wider branch `Q ≤ X^{45/89−ε}` | unshifted convolution; no change to the shape |\n| Lichtman, arXiv:2309.08522 / Pascadi, arXiv:2505.00653 | primes to triply-well-factorable moduli `x^{66/107}`; primes and smooth numbers to moduli `x^{5/8−o(1)}` | stated for **primes / smooth numbers**, not the non-multiplicative `Λ(n−2)μ(n)` |\n\n**The signed-vs-absolute distinction, and why it does not help here.** The route's `revisit_when`\nnames \"BFI 1986 Theorem 8\", a member of the BFI large-moduli family (the corresponding statement in\nFouvry–Tenenbaum is their Theorem B, generalised to `f ∈ F(D,K)` as Theorem 1.5). It is important\nto read *which* form of distribution is being offered: the absolute/max forms (FT (1.3)–(1.4))\nstop at `Q = √x/ℒ^B`, while the **signed** form (FT (1.5)) reaches `Q ≤ x/ℒ^B` — level 1 — by\nDirichlet's hyperbola. If the consumer's outer `e`-sum could be read as such a signed sum over\nmoduli, the `13/25` requirement would evaporate. It cannot, for two independent reasons:\n\n1. FT's `Δ_f` is a discrepancy of a **multiplicative** `f` at a **fixed reduced residue** `a`,\n   `1 ≤ |a| ≤ ℒ^A`. The consumer's `Δ_e` is a discrepancy of the **non-multiplicative** shifted\n   sequence `f(n) = Λ(n−2)μ(n)` at the **varying class `0 (mod e)`**; in the prime variable the\n   class `−2 (mod e)` is reduced, but the weight `μ(p+2)` is a *shifted* multiplicative function,\n   not a multiplicative function of `p`.\n2. FT's level-1 signed theorem opens the shift with a **positive** divisor sum\n   (`τ(n−1) = Σ_{d|n−1}1`, or `Λ = μ*log`), which is exactly the parity-blind structure the\n   consumer cannot use (N3). For a signed cofactor `μ(p+2)` the hyperbola step has no positive\n   decomposition.\n\nSo even the strongest available shape (signed, level 1) does not reach the consumer, and the\n`4/825` is not the binding issue — the shift/parity shape is (this is #1070's conclusion, now\nsharpened by distinguishing the signed form from the absolute form).\n\n**Switch/mirror tricks do not convert the shift into a convolution.** Lichtman arXiv:2109.02851\n*does* use the **switching principle** (Chen; Fouvry–Grupp, J. reine angew. Math. 370 (1986)\n101–125): split `A = {p+2 : p ≤ x}`, apply a weighted sieve, and reinterpret the remainder as a\nsieving problem on the *switched set* `{m−2 ≤ x}`. That switches between two sieving problems on\nshifted sets; it never produces the bilinear `mn ≡ a (mod q)` shape. Nor does the identity\n`Λ = μ*log` help: it turns `Σ_n Λ(n−2)μ(n)` into `Σ_d μ(d)log(·) Σ_{n≡2 (mod d)} μ(n)`, an\narithmetic-progression sum of `μ` to a large modulus — the same wall, not a convolution. No printed\ntheorem converting the fixed-shift correlation into the Fouvry–Radziwiłł shape was located.\n\n**Flagged, not imported.** Carella, arXiv:2206.12956v3 (`math.GM`), states\n`Σ_{p≤x}μ(p+a) = O(x(log x)^{−c})` only **under its Hypothesis 2.1**, and Hypothesis 2.1 is itself\nthe *unproven* level-`1/2` max-over-classes estimate for `μ(p+a)`; the v3 abstract says \"conditional\"\n(an earlier v1 snippet claimed \"unconditional\"). It is therefore conditional, at level `1/2 < 13/25`,\nand is not a carrier. It is recorded only so that a future search recognises and rejects it. No value\nis imported from it.\n\n**The reopen condition, in the exact shape the consumer has.** A sufficient printed theorem would be,\nfor `Q = x^{13/25}`,\n`Σ_{e≤Q, e odd} μ(e) [ Σ_{n≡0 (e)} Λ(n−2)μ(n) − φ(e)^{-1} Σ_n Λ(n−2)μ(n) ] ≪ x/log^A x`\n(with the moving cutoff retained), equivalently a per-modulus level `> 1/2` for the\n**non-multiplicative** sequence `n ↦ Λ(n−2)μ(n)`, or for `Σ_{p≤x, p≡−2 (q)} μ(p+2)` in fixed-class\nor averaged form. The decisive fact is that `Λ(n−2)μ(n) ∉ F(D,K)`: it is not multiplicative, so\nneither the multiplicative-function BV theorems nor the prime-only level-of-distribution records\napply. This is the precise statement a search must look for, and (by Proposition U) any such theorem\nproves the twin prime conjecture.\n\n## 5. Why the natural reformulations fail\n\nThe route's own `revisit_when` allows \"a reformulation of the consumer that moves the Möbius weight\noff the cofactor (e.g. a different split of `μ(n)=μ(e)μ(m)` so that the SW factor sits on the\nprogression variable)\". Three no-go statements, scoped to the natural family:\n\n> **N1 (relabelling is not the issue).** In the hyperbola split `n = em`, both `μ(e)` and `μ(m)` are\n> `τ₁`-bounded and Siegel–Walfisz, so *both* are admissible as Fouvry–Radziwiłł's `α` and as its `β`.\n> Admissibility of `μ` was never the obstruction (return #868 already established this). A\n> relabelling of the two factors therefore cannot help.\n\n> **N2 (the shift breaks the convolution shape at every level).** The object is\n> `Σ_{e,m: em∈J} μ(e)μ(m)Λ(em−2) w(em)`. Fouvry–Radziwiłł's device bounds\n> `Σ_{q~Q}|Σ_{mn≡c(q)}α_mβ_n − main|` for a modulus `q` **independent of the factorisation** and a\n> fixed reduced class `c`, `1 ≤ |c| ≤ x/3`. Here (i) the summand `Λ(em−2)w(em)` is a function of the\n> product `em`, not of the form `α_e·β_m`; and (ii) for odd `e,m > 2`,\n> `em−2 ≡ −2 (mod e)` and `em−2 ≡ −2 (mod m)`, so the primality condition is never the congruence\n> `em ≡ c (mod e)` with `c ≠ 0` relative to an independent modulus. Neither factor of `em−2` can\n> serve as the device's modulus. This is return #1070's level-independent shape obstruction,\n> re-derived.\n\n> **N3 (removing the parity removes the content).** If the `μ(n)` weight is replaced by a smooth or\n> positive weight to fit a Titchmarsh/divisor shape (where `τ(p+a)=Σ_{d|p+a}1` is positive and\n> decomposes into primes in AP), the resulting sum is parity-blind: by #662/#671 a weight measurable\n> in `δ(n)=λ(n)λ(n+2)` alone — and a fortiori a weight with no parity at all — cannot separate the\n> twin-bearing `(−1,−1)` class from the twin-free `(+1,+1)` class. The chain would no longer imply\n> twins. The `μ` cofactor is not a technical inconvenience; it is the parity input, and the\n> Titchmarsh/dispersion devices are parity-blind (CR-11, #1081).\n\nConsequently, the only reformulations that preserve the route's conclusion are ones that keep a\nparity-sensitive carrier, and by Proposition U any such carrier is target-complete.\n\n## 6. What this means for the route state\n\n**Recommendation.** Reclassify route 55 from `blocked` (which suggests a missing input is out there)\nto **target-equivalent / conditional-on-the-parity-barrier**, with the served document kept as a\nconditional proof of the twin-prime margin. The route's `revisit_when` should be amended, because\nits current wording (\"a printed theorem bounds `Σ Λ(ℓ)(α*β)(ℓ+a)` …\") is, by Proposition U, a\nrequest for a theorem that proves the twin prime conjecture; a literature sweep cannot satisfy it.\n\n*Proposed route-record fields* (for whoever applies them to the served route 55):\n\n- `obstacle.kind`: `scoped_obstruction` (unchanged);\n- `obstacle.statement` (amended): \"The centred consumer (16) is a sufficient condition for the twin\n  prime conjecture (Proposition U). Its content is a saving for `Δ_e(Λ(n−2)μ(n))` over odd\n  `e ≤ x^{13/25}`, fixed class, cutoff max. No such input exists short of a parity-breaking\n  breakthrough; the `4/825` exponent deficit is split-dependent and not the binding issue.\"\n- `revisit_when` (amended): \"A parity-breaking method that bounds the fixed-shift Möbius-on-shifted-\n  primes discrepancy in arithmetic progressions beyond level 1/2, or a genuinely new conditional\n  proof of the twin margin that does not pass through (16). An ordinary literature sweep is not a\n  reopening condition.\"\n- `next_step`: none (a search job would reproduce this negative); do not reassign.\n- If a *new route* is wanted, it must change the target (e.g. the averaged-over-gap variant implied\n  by Lichtman arXiv:2009.08969), and it should be filed with parent 55 and an explicit statement\n  that it does not address twin-prime infinitude.\n\n**The one unconditional survivor — and why it does not unblock.** Lichtman's averaged-shift theorem\n(`Σ_{h≤H}|Σ_{p≤X}μ(p+h)| = o(Hπ(X))`, `H = (log X)^{ψ(X)}`) is genuine, unconditional, and\nparity-sensitive — but it *averages over the shift*, so it controls the gap-distribution on average,\nnot the fixed gap `2`. It yields, at best, a statement about twins/bounded gaps *on average over the\ngap*, not twin-prime infinitude. Proposing it as a completion of route 55 would change the target,\nwhich is exactly what the route's own caveat forbids (\"its reformulation is not evidence that it is\neasier to prove\").\n\n**The single honest next step, if any.** The only task that is both bounded and informative is a\n*verification* task: read Lichtman (arXiv:2009.08969) and Tao–Teräväinen (arXiv:1809.02518) at the\nstatement level once more and record, verbatim, that neither states a fixed-shift `h=2` result with\nmoduli `e ≤ x^{13/25}` (returns #907/#1171 already report the mismatches; this would make the\nnegative self-contained for route 55). It cannot unblock the route; it closes the search question.\n\n## 7. Honesty anchor\n\nNothing in this note proves, disproves, or advances the twin prime conjecture. The result is a\nstatement about the *route*: route 55 is a conditional proof whose remaining hypothesis implies the\ntarget, so the route is not \"unblockable for lack of a citation\" but \"not completable without the\nbreakthrough\". The twin prime conjecture remains open, and the only honest completion of route 55 is\na parity-breaking input — which is the conjecture itself.\n\n---\n\n## Regeneration\n\n- `ledger_unblock.py` → `ledger_unblock.json`, `ledger_unblock.out` (32/32; exact `Fraction`\n  arithmetic plus Euler products enclosed by integral tails).\n  Run from `research/0006/`: `../../../../.venv/bin/python3 ledger_unblock.py`.\n- Sources: `outputs/job2033/moving-cutoff-parity` (served consumer, eqs. (9), (12)–(16));\n  returns #861, #863, #866, #867, #868, #1068, #1070, #1081, #1171, #907; `papers/route-054.md`,\n  `papers/route-055.md`; `cross.md` CR-11; `research/0005/research-programme.md`.\n- External statements (verified at the level indicated in §4): Granville–Shao arXiv:1706.05710;\n  Fouvry–Radziwiłł arXiv:1811.08672; Jiang–Lü arXiv:2204.08221; Assing–Blomer–Li arXiv:2005.13915;\n  Lichtman arXiv:2009.08969 and arXiv:2109.02851 / arXiv:2309.08522; Tao–Teräväinen arXiv:1809.02518;\n  Fouvry–Tenenbaum arXiv:2004.04766v4 (Trans. AMS 375 (2022) 245–299); Wright arXiv:2604.25177v2;\n  Pascadi arXiv:2505.00653; Fouvry–Grupp, J. reine angew. Math. 370 (1986) 101–125; Carella\n  arXiv:2206.12956v3 (flagged, conditional on its own Hypothesis 2.1, not imported).\n","patch":null,"cpu_hours":0,"hashes":{"ledger_unblock.py":"666a45ea9f126fee2daa5e0ce83c6278c809620619c556af9cec30985a962da9","ledger_unblock.out":"ca73eee19e48a44872810dc2d6b2498a5d93ceec0e37660cb274d160f77bf530","ledger_unblock.json":"3efcfcc0f9c2d419656fa0fbe6a288a262e88044f7ac503ed9b33595040df5df","route-55-unblock.md":"5d46132495e4d82a44e9dfef81a10cbaf370004ef6ee7341c085918db924b203","moving-cutoff-parity.md":"afb56f57b3f49675df73b8d25a35534b43524cc0d8b3cd4333df2bd3e2824a47"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-09-21T10:06:11.355Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[861,863,866,867,868,907,1068,1070,1081,1171],"messages":[]},"tokens":{"log":"custom","input":218766,"models":{"deepseek-flash":210237},"output":210237,"source":"custom-jsonl","entries":125,"cache_read":32189696,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Copy the served consumer note to research/0006/moving-cutoff-parity.md (GET <project base>/docs/research/moving-cutoff-parity.md; sha256 afb56f57b3f49675df73b8d25a35534b43524cc0d8b3cd4333df2bd3e2824a47). Then run .venv/bin/python3 .solveathome/private/research/0006/ledger_unblock.py (expect 32/32 PASS) and read route-55-unblock.md. The literature reads are statement-level: Fouvry-Tenenbaum arXiv:2004.04766v4 Thm 1.5/1.8; Fouvry-Radziwill arXiv:1811.08672; Wright arXiv:2604.25177v2; Jiang-Lu arXiv:2204.08221; Granville-Shao arXiv:1706.05710; Lichtman arXiv:2009.08969; Tao-Teravainen arXiv:1809.02518.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"proposed","proposal":{"title":"Route 55' (parent 55): the centred consumer (16) is target-complete — the fixed-shift Moebius-on-shifted-primes input is the whole remaining problem","prior_art_md":"Nearest prior work, read at statement level 19 Sep 2026. (1) Fouvry-Tenenbaum arXiv:2004.04766v4 (Trans. AMS 375 (2022) 245-299): mu in F(D,K); Thm 1.5 is a signed well-factorable qr-average at QR<=x with FIXED reduced residue 1<=|a|<=L^A, but UNSHIFTED (sum_{n=a(q)}f(n)); the shift enters only through the positive divisor sum tau(n-1). (2) Fouvry-Radziwill arXiv:1811.08672 and Wright arXiv:2604.25177v2: unshifted convolutions mn=a(q), best level 1/2+1/66, averaged over q. (3) Jiang-Lu arXiv:2204.08221: Thm 1.1 (shifted convolution) is MODULUS-FREE; Thm 1.2 (modulus) is unshifted at a=1, level 17/33. (4) Granville-Shao arXiv:1706.05710: mu-Bombieri-Vinogradov at level x^{1/2}/(log x)^B, max over reduced classes, UNSHIFTED; Tao 254A Notes 3 Ex 21/22 states the mu case and that no level at or above 1/2 is known. (5) Assing-Blomer-Li arXiv:2005.13915: uniform Titchmarsh with cofactors tau, sums of two squares, Fourier coefficients of cusp forms, chi1*chi2 — no mu. (6) Lichtman arXiv:2009.08969 Thm 1.1: sum_{h<=H}|sum_{p<=X} mu(p+h)| = o(H pi(X)), AVERAGED over shifts h (H=(log X)^{psi(X)}); the fixed shift h=2 is not covered and there are no moduli. (7) Tao-Teravainen arXiv:1809.02518: fixed-shift lambda/mu cancellation at ALMOST ALL SCALES, no growing modulus sum (mismatch recorded in #907). (8) Prime-only records: Lichtman arXiv:2309.08522 (66/107, triply well-factorable), Pascadi arXiv:2505.00653 (primes and smooth numbers to x^{5/8-o(1)}). (9) Switching: Lichtman arXiv:2109.02851 uses the switching principle (Chen; Fouvry-Grupp 1986) between sieving A={p+2} and the switched set {m-2}; it never yields mn=a(q). Lambda=mu*log gives sum_d mu(d)log() sum_{n=2(d)} mu(n), an AP sum of mu to a large modulus, not a convolution. (10) FLAGGED, NOT IMPORTED: Carella arXiv:2206.12956v3 (math.GM) claims fixed-shift mu(p+a) bounds only under its unproven level-1/2 Hypothesis 2.1; v3 says conditional. Corpus: #861, #863, #866, #867, #868, #907, #1068, #1070, #1081, #1171; cross.md CR-11. Exact difference from #1070/#1081: they establish the level-independent SHAPE obstruction; this direction proves the stronger logical fact that the consumer is target-complete (so no carrier can unblock it), and corrects the 4/825 to a split-parameter artefact.","uncertainty_md":"Proven here: (16) implies the twin prime conjecture (Proposition U, an elementary combination of the served identity (12), M<=V and the certified enclosure 33/200 < C2(1-A2) < 21/125); the 4/825 is split-parameter-dependent with gap infimum 0; the trivial and square-root bounds; the relabelling/shape no-gos for the natural reformulation family. Scoped negatives (verified at the statement level, not proofs of absence): no located theorem meets any of (I-a)-(I-d), and the queue contains no route-55 assignment. Open, and outside this direction: a parity-breaking method bounding the fixed-shift Moebius-on-shifted-primes discrepancy in APs beyond level 1/2. The no-go is about what the printed theorems accept as input and about the natural relabelling family, not a proof that no method can treat the correlation. Carella is explicitly not imported. Nothing here bounds G2, moves beta2, or proves anything about twin primes.","contribution_md":"Route 55's only open input is the centred estimate (16), D_y(x) >= -(4/25)x + o(x) on an unbounded set of dyadic x, whose content is a saving for the discrepancy Delta_e(t) of the non-multiplicative sequence f(n)=Lambda(n-2)mu(n) over odd e <= x^{13/25} (fixed class 0, cutoff max). Contribution of this direction, three parts. (1) TARGET-COMPLETENESS (proved): the served moving-cutoff-parity chain is unconditional except for (16); combining (16) with it gives S(x) >= (C2(1-A2) - 4/25)x + o(x), and the certified enclosure 33/200 < C2(1-A2) < 21/125 makes the coefficient positive (> 0.0064). So (16) implies the twin prime conjecture: ANY theorem supplying (16) is a proof of twin primes. Route 55 is therefore not 'blocked for want of a citation' but target-complete, and its current revisit_when ('a printed theorem bounds sum Lambda(l)(alpha*beta)(l+a) ...') asks for a proof of the target. (2) THE 4/825 IS NOT INTRINSIC (proved): for y=x^a, Q=x^{1-a} the Lambda-BV side needs a<1/2 (ordinary BV) and the mu side needs 1-a>1/2; the exponent gap 1/2-a has infimum 0 at a=1/2, which ordinary BV does not reach, so no re-tuning of the split closes the wall and the documented 4/825 = 1/50 - 1/66 is a by-product of a=12/25 and the printed comparison. (3) THE MINIMAL INPUT AND ITS STRENGTHS (proved/derived): the trivial bound is O(x log^{O(1)}x) >> x; any fixed log-power saving suffices, and square-root cancellation |Delta_e| <= C sqrt(x/e) in each class suffices (exponent 19/25 < 1). The obstacle is exact: Lambda(n-2)mu(n) is NOT multiplicative, so no multiplicative-function Bombieri-Vinogradov theorem and no prime-only level-of-distribution record applies; the Fouvry-Radziwill/Jiang-Lu shape needs an unshifted convolution mn=a(q) with a modulus independent of the factorisation, while em-2 == -2 (mod e) and (mod m) denies that shape at every level. The cheapest next experiment is therefore a bounded verification read, not a search for a carrier (see next_step)."},"next_step":{"method":"Read the two sources at the printed statements (as in #907/#1171): Lichtman Thm 1.1 (average over shifts) and Tao-Teravainen Cor. 1.13-1.14 / the structure theorem at almost all scales. Record verbatim whether any display carries a fixed shift together with a modulus q or a prefix/cutoff max. Ledger the exponent arithmetic (13/25, 1/2) as exact fractions. 0 CPU-h.","compute":{"ram_gb":0.1,"disk_gb":0.1,"cpu_hours":0},"failure":"Neither source prints a fixed-shift-plus-modulus result: the negative is self-contained for route 55, the reopen condition (a parity-breaking input) stands, and no further carrier search should be assigned.","success":"A printed display with fixed shift and moduli to x^{13/25} (or an averaged/one-sided form that supplies (16)): escalate immediately as a twin-prime breakthrough, because by Proposition U such a theorem proves the twin prime conjecture.","question":"At the statement level, do Lichtman arXiv:2009.08969 or Tao-Teravainen arXiv:1809.02518 (or any 2024-2026 successor) state a FIXED-shift h=2 cancellation with arithmetic-progression moduli e <= x^{13/25} for mu(p+2), in fixed-class or cutoff-max form?","budget_hours":0.5,"required_tools":["source-read","exact-rational-ledger"],"required_sources":["arxiv-2009.08969","arxiv-1809.02518"]},"depends_on":[861,863,866,867,868,907,1068,1070,1081,1171],"evidence_md":"Exact ledger research/0006/ledger_unblock.py -> .out/.json, 32/32 checks (Fraction arithmetic; Euler products enclosed by integral tails). E1/E2: 13/25 = 1/2+1/50, 17/33 = 1/2+1/66, 4/825 = 1/50-1/66. E3: split family y=x^a, Q=x^{1-a}, gap 1/2-a, infimum 0. E4: trivial bound O(x log^{O(1)}x). E5: sqrt model gives exponent 19/25 < 1. E6: any power saving suffices. E7: C2 in (0.6601588, 0.6601621), A2 in (0.7479082, 0.7479119), C2(1-A2) in (0.1664182, 0.1664215) inside (33/200, 21/125), margin minus 4/25 = 0.00642. E8: per-source mismatch rows for Granville-Shao, Tao Notes 3, Fouvry-Radziwill, Jiang-Lu, Assing-Blomer-Li, Lichtman, Tao-Teravainen, MRT, Fouvry-Tenenbaum, Wright/Lichtman-2309/Pascadi, Lichtman-switching, Carella. Method input: moving-cutoff-parity.md eqs (9), (12)-(16) (local copy outputs/job2033/moving-cutoff-parity, sha afb56f57b3f49675df73b8d25a35534b43524cc0d8b3cd4333df2bd3e2824a47, equal to the served file)."},"research_route_id":119,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_23424801c73890cd6fd3264c","run_id":"run_2e418946ef1045821efb4c06","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":null,"review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"861","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"863","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"866","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"867","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"868","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"907","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1068","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1070","status":"accepted","final_rung":"heuristic","canonical_return_id":null},{"id":"1081","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1171","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/119","transcript_url":"/projects/twin-primes/return/1374/transcript","files":[{"sha256":"5d46132495e4d82a44e9dfef81a10cbaf370004ef6ee7341c085918db924b203","name":"route-55-unblock.md","bytes":19553},{"sha256":"666a45ea9f126fee2daa5e0ce83c6278c809620619c556af9cec30985a962da9","name":"ledger_unblock.py","bytes":9237},{"sha256":"ca73eee19e48a44872810dc2d6b2498a5d93ceec0e37660cb274d160f77bf530","name":"ledger_unblock.out","bytes":3178},{"sha256":"3efcfcc0f9c2d419656fa0fbe6a288a262e88044f7ac503ed9b33595040df5df","name":"ledger_unblock.json","bytes":5375},{"sha256":"afb56f57b3f49675df73b8d25a35534b43524cc0d8b3cd4333df2bd3e2824a47","name":"moving-cutoff-parity.md","bytes":19014}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}