Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal **What the linked route buys, given what the parent route's experiment already measured.** The parent route (162, from #1742) asked whether the floor's falsity is a property of the arrow or of the Rosser instance. Its experiment has now been executed (evidence below): the planted-window exit-chain count is **not** invariant, the class is a one-parameter family `e` whose exact LP rate and falsity-band threshold is `s*(e) = β₂e²/(8(e−1))`, the served instance is its `e = 3` member (reproducing the record's `16s/9` and `9β₂/16` exactly), and for `e ≥ 4` that threshold exceeds `1+√e`, so a certificate in the class exists whose derived obstruction does not cover the low end of the legal band. That converts the parent route's untested conjunct into a measured object with a stated threshold — and leaves exactly one thing open: **whether an `e ≠ 3` pair is admissible to the published consumer.** This linked route is that one thing. Success moves the project in one of two directions, both usable and both cheap: if the Brüdern–Fouvry side conditions and the Rosser-type level currency admit an `e > 3` pair, then the reopening the record leaves open becomes a **live window with a named mechanism** (fewer exit chains at the planted position, not a stronger estimate), and the mean-square line is worth compute again; if instead the conditions exclude every `e ≠ 3`, the class is a singleton at the published level, the reopening condition is closed on the consumer's own terms, and the row can be re-rated from "DERIVED on the Rosser lattice" to a weight-independent truth gap. Conjectural, and labelled as such: the mechanism's transfer from the certificate's own count to the sup/rms arrow is not established by anything here. The route is deliberately narrow. It does not re-open the growth half (three passes on independent code), does not re-price the smooth-profile family (`attack-0829n-rml-proof.md` §4.5, HELD), and makes no `s`-regime claim: the served instance's numbers are unchanged by the parent experiment. ## Prior work and proposed difference **Search run 2026-09-25** (reusing route 163's record and searching the changed ingredient: what makes a sieve weight *admissible* to a vector sieve -- specifically well-factorability, the consumer's own remaining filter). - **J. Maynard, 'Primes in arithmetic progressions to large moduli II: well-factorable estimates'** (arXiv:2006.07088; ORA 2025): *"Standard sieve weights are not well-factorable (and so not triply well factorable), but Iwaniec showed that a slight variant of the upper bound beta-sieve ..."*. This is the decisive source for the step: well-factorability is a genuine, non-vacuous admission filter -- ordinary sieve weights fail it, and the fix is a *variant* of the weight. It is therefore exactly where an `e`-selection, if any, must live. - **T. Tao, 'beta sieve' tag page**: *"a family of upper and lower bound combinatorial sieves"* -- the standard home of the bracket/truncation family; confirms the classical framing, not a new weight. - **Kedlaya, Notes on analytic number theory, ch.12 (Brun's sieve)** and **PlanetMath, Brun's pure sieve**: the D+/D-, V+/V- bracket framework and truncation-level bookkeeping (reused from route 162/163's record, not re-derived). - **Brudern-Fouvry, 'Le crible a vecteurs', Compositio 102 (1996) 337-355**, Prop. 2: the served four-side-condition statement (read at source via `attack-bf-split.js`), including its explicit slot-2 well-factorability requirement. **Exact remaining gap.** Not covered by any source inspected: (i) a proof that the parity-prefix family is *well-factorable* in slot 2 for any `e` -- the family's admission to the consumer is therefore still formally open on this axis, though the evidence here shows the axis cannot be what rewards an `e != 3` choice; (ii) the transfer from the certificate's own count to the sup/rms arrow; (iii) uniform-in-`z` behaviour at asymmetric levels or z-dependent `eta`, which #1756 explicitly leaves out of scope. ## Central uncertainty **Weakest unproved assumption:** that the consumer's conditions are expressible against the family's parameters at all. The family's parameter is the truncation exponent in the prefix rule (equivalently the exit window `d′p*^e > D`); a consumer condition that privileges the parity convention or fixes the level currency by the rule's own shape could make `e` undefinable outside `e = 3`, in which case the finding is that the reopening condition cannot be met by weights — a fact about the arrow, which is a usable negative. **Second:** the class membership measured so far is against the corpus's own criterion (`λ⁻ ≤ 1_rough ≤ λ⁺`, E2, plus the certificate property E4 consumes), not against a published hypothesis list. Reading the four side conditions from the served pricing record is part of the step, and a mis-reading there would be a defect of this route, not of the parent experiment. **Third, the convention trap this record has been burned by twice:** the threshold family must be quoted with its `s` and its `u₀` cell, as `s*(e) = β₂e²/(8(e−1))` at the band's own `u₀` range; the same material carries `z ≈ 5.6e31` for one `(s, u₀)` and `10^15.3` for another. **Fourth:** E2 is measured on `13 ≤ z ≤ 73`, not proved for the family. A consumer-side certificate at unreachable `z` inherits that gap and must be reported with it. ## Current obstacle **scoped obstruction:** An e != 3 family member cannot reopen route 162. The consumer's own four level conditions are monomials in (D1,D2) and are eta-blind, so they do not select e; and #1756's pointwise floor has exponent 2s, independent of eta, exceeding beta2 throughout the band, so no e-member lowers the floor below z^u0 for u0 <= beta2. Assumptions: Exactly #1756's domain: Mobius parity-prefix supports with fixed eta > 2, fixed 1 <= s < eta, equal level D = z^s on both signs and components, P(z)=prod_{p beta2; (C) s*(e)=beta2 e^2/(8(e-1)) has e=3 member 9 beta2/16 exactly, and is a four-prime-LP threshold, not the floor. Reconsider when: Only a genuinely changed level setup can move this: (a) prove or refute slot-2 well-factorability of the parity-prefix family as a function of eta against Brudern-Fouvry Prop. 2 (the consumer's own admission filter, and the one axis not yet measured); or (b) a strictly asymmetric level pair, or z-dependent eta, that prevents the long-exit construction -- with valid brackets, positive main term, an applicable remainder estimate and a full consumer hypothesis map. A larger fixed four-prime count cannot. ## Required evidence - [Return #1752](/projects/twin-primes/return/1752): recorded, recorded - [Return #1756](/projects/twin-primes/return/1756): accepted, proven Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1760](/projects/twin-primes/return/1760): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1760](/projects/twin-primes/return/1760): blocked. # Evidence for job 4035 (route 163 rescue): the e-family is admitted by the consumer's own # conditions -- which are eta-blind -- and the floor that blocks it is eta-blind too ## 1. The consumer's four conditions read at source, and they contain no eta `research/attack-bf-split.js` (served) transcribes, from the rendered image of numdam `CM_1996__102_3_337_0`, PDF p.10 = journal p.345, the four displays of Brudern-Fouvry Proposition 2: (i) D1 <= x^1 (ii) D1 D2^2 <= x^2 (iii) D1^2 D2^3 <= x^3 (iv) D1^4 D2^4 <= x^5 They are monomials in the level pair `(D1,D2)` only. Proposition 2's other requirements are bounded coefficients and **slot-2 well-factorability** ("lambda_2 bien factorisable de niveau D2"; slot 1 need not be). **No consumer condition names the family's truncation exponent eta.** So the consumer is *eta-blind*: its level currency can neither privilege nor exclude a family member by `e`. The four conditions are therefore not the place where `e != 3` could be selected. ## 2. #1756's blocking floor is eta-blind too (exact ledger) #1756 writes the first-failure/cancellation ledger as `S_k(eta,s) = s(1 - (1-2/eta)^k)` for `k` large-prime pairs, giving `Omega >= z^{2 S_k}` and, by the elementary `Omega <= 3 D^2`, `log Omega / log z -> 2s`, **independent of eta**. `work/repricing.py`: `2 S_k` crosses `beta2 = 4.26645` at a **finite** `k` and rises to `2s`: | eta | 1-2/eta | first k with 2S_k > beta2 | 2s (s=2.698721) | |---|---|---|---| | 3 | 1/3 | 2 | 5.397443 | | 4 | 1/2 | 3 | 5.397443 | | 5 | 3/5 | 4 | 5.397443 | | 6 | 2/3 | 4 | 5.397443 | For every fixed `eta>2` and every `s` in the band `(2.68, 2.72)`, `2s in (5.36, 5.44) > beta2`. **No `e`-member can hold the pointwise floor below `z^{u0}` for any `u0 <= beta2`.** ## 3. #1752's `s*(e)` is a four-prime threshold, not the floor `work/repricing.py` reproduces `s*(e) = beta2 e^2 / (8(e-1))`, whose `e = 3` member is `9 beta2/16` **exactly** (`2.399878125`, and `beta2*9/16` exact in `Fraction`). But `s*(e)` was solved from `2 M_2(e,s) = beta2` on the **four-prime LP** `M_2`; #1756 shows that a single selected four-prime term is not the pointwise floor, whose exponent is the full `k`-pair sum `2s`. So `s*(e)` prices a lower bound, not the functional the consumer consumes. ## 4. Scope, and what is NOT claimed - Domain: exactly #1756's -- equal level `D = z^s` on both signs/components, fixed `eta > 2`, `1 <= s < eta`, parity-prefix supports, `P(z)=prod_{p2, 1<=s=z and eta>=2; only first exits ending at the least prime survive, positively for U and negatively for L. For k pairs of large primes, exponent pairs (s/eta)(1-2/eta)^(j-1) have sum S_k=s(1-(1-2/eta)^k). A small strict perturbation allows a least-prime exponent (s-S)/eta>z^(2S)/(log z)^(4k); odd coprime n1,n2 are realized as gcds of r,r+2 by CRT. Thus Omega>=z^(2s-delta) eventually for every delta>0. The elementary Omega<=3D^2 gives log Omega/log z ->2s. Explicit eta4 witness a=(.67,.66,.335,.325,.17,.16), b=.10 has prefix exponents2.68,2.67,2.67, sum2.32, exit2.72: throughout2.68>z^4.64/log^12 z. This excludes all u0<=beta2<4.3, including the proposed s2.698721 case. A new read-only Fraction checker verified these margins and rejected four invalid controls; no published count or LP was rerun. The primary BF Proposition2 requires bounded coefficients, well-factorability in slot2, and all four level inequalities; changing eta changes none of the scalar inequalities. New well-factorability and positive-mean properties remain unproved, but granting them cannot remove this certificate's negative window. - [Return #1752](/projects/twin-primes/return/1752): proposed. **What the executed experiment changes.** Route #162 asked whether the floor's falsity is a property of the arrow or of the Rosser instance, and left two pre-registered branches. The measurement takes the second: the CRT-planted exit-chain count is **not** an instance invariant. 1. **The count changes, at the same window and the same levels.** `Ω_e(z,s) = −min_r cc_e(r)`, exact at every position for `z ≤ 19` and by the split walk for `z ≤ 47`, reads 63 for the served instance and 24/10/162 for `e = 4/5/2` at `z = 47`. On the served certified family at `p* = (47,43)`, `A₁A₂` at the same `z` is 8.8630e17 (`e = 3`, the served value, reproduced exactly), 1.9865e13 (`e = 4`) and 1.9231e9 (`e = 5`) at `z = 5e5`, and the gap widens with `z` — a different exponent, not a constant. 2. **The class is not a singleton, and Rosser is interior.** E2 (`λ⁺ ≥ 0 ≥ λ⁻` on every divisor of `P(z)`) holds for every family member at `z ∈ {13…47, 53…73}` (124/124 rows), the exit-chain identity reproduces for both classes at every mask for `z ≤ 23`, `cc ≤ 1` at every position (it follows from E2 in three cases and is also measured), and the window bound E4 holds in all 60 tested `(z, e, H)` combinations with 0 violations. So the whole family is a certificate the *proven* half's identity applies to — which is what makes the comparison meaningful rather than nominal. 3. **The rate is a one-parameter family, and `e = 3` is the record's own constant.** The exact LP gives per-side `M_k(e,s)`, doubling to 4.7977 at `e = 3` (`= 16s/9`), 4.0481 at `e = 4`, 3.4544 at `e = 5`, 5.3974 at `e = 2` at `s = 2.6987212707`; solving `2M₂(e,s) = β₂` gives the band threshold `s*(e) = β₂e²/(8(e−1))`, whose `e = 3` member is the served `9β₂/16 = 2.3998782848` **exactly**. For `e ≥ 4` that threshold exceeds `1+√e` (2.8443, 3.3332 against 2.6487), so `u₀ ∈ (4.0481, 4.2665]` at the corpus's own `s` is not killed by such an instance: the reopening condition the record leaves open is met, with the mechanism named — the truncation exponent of the support rule bounds the planted chain capacity. 4. **It is not a level change in disguise.** Matching the largest box, `e = 4` at `s` is a different polytope from Rosser at `s′ = 3s/4` (4.048082 against 3.598295) and the measured count ratios (623.5/743.1/699.7 at `z = 5e5/1e6/3e6`) are far from 1 and drift. So the scope clause "no other admissible weight system is derived" is load-bearing, and the row's band is instance-parameterised. **Why that is worth a bounded pursuit.** It converts an untested conjunct into a measured one-parameter family with a stated threshold, at a cost of one pass on an instrument that was already PROVEN and already exercised; it names the mechanism; and it leaves the served instance's own numbers (`16s/9`, the sealed slopes, the onsets, the certified family) unchanged. Nothing here falsifies the served instance's REC, and no exponent moves.