Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The identification step of the variance limit (lim Var/E = 0.45546, still HEURISTIC) is blocked on one named inequality: uniform o(1) equidistribution of y-friable squarefree integers in progressions to moduli up to y^{4/5} with the lam1 weight (attack-0830-varE-identification.md sec.4, display (*)). This pass shows the nearest literature object is not one range but two unconditional sides of the unweighted, all-classes BDH variance: Harper, JLMS 112 (2025) e70293 Cor. 1 for large y under x^{0.51} <= Q <= x, and the Vaughan personal communication Harper records for moderate y under Q <= Psi(x,y). Mapped through the object's own dictionary both sides touch the obligation (u <= 1.324503 and u >= 3/2), do not meet at u in (1.324503, 1.5), and neither carries the weight, the squarefree restriction, or the L^1-in-a / max-over-t quantifier. If a weighted or sieved version of either side can be read or transferred, the unnamed equidistribution input of sec.4 is discharged and the identification step can be restated unconditionally; if not, the residual reduces to a named hypothesis on each side. Either way the route replaces 'no theorem applies' with a two-sided, source-located claim about exactly what is missing, which is what the obligation was asking for. Conjectural links: that the variance identification alone moves lim Var/E = 0.45546 is unchanged by this route and remains TPC-adjacent only. ## Prior work and proposed difference **Reused, not re-derived.** Job #1745's recon (served with #3700) contains: - `(H_w)` (sec. 3.1); - Lemma A (sec. 3.2), with partial summation, `sup|G| <= 2/M` and `Var G <= 14`; - Lemma B (sec. 3.3), two-branch cells `o(ln^2 y)` given `(H_w)`, proven conditionally and unreviewed; - the dilation outline `(H_w)` to Harper Thm 2 (sec. 3.4), via `g = 1*h` with summable `h(m)/sqrt m` for lam1 (not for lam0's `2p/(p-4)`), written out only for `Q <= X^{1/8}`, `y >= X^{1/3}`. It is an outline, NOT a proof; - the three-branch type left uncovered (sec. 3.5). None of these is claimed here. The sources read in #3700 (Granville 1993, Soundararajan, Harper, Drappeau, Drappeau-Granville-Shao, Pascadi) are reused. The corrected target is from #3658, and the cell band geometry from the attack note served with #3642. **Unweighted sides (record):** - Harper arXiv:1208.5992 Thm 2 (BV/BDH for `Psi(x,y)`, `log^K x <= y <= x`, `Q <= Psi/log^A`) covers the level on blocks `d ~ D <= L^{1/2}/log^{A/2}`. It has weight 1, no max over `x'`, and an ineffective bounded-`u` case. - Harper JLMS 112 (2025) e70293 (arXiv:2412.19644) works on the dyadic all-classes object for `sqrt(2x) < Q <= x`. Its Thm 2 route excludes sieved sets. - Vaughan, Phil. Trans. A 356 (1998) [23]/[24], is closed access (403; S2 CLOSED; OpenAlex is_oa false, #4084). **Online search this pass (changed ingredient: fixed-modulus bias of a multiplicative weight).** - Granville-Shao, Forum Math. Sigma 6 (2018), "When does the Bombieri-Vinogradov theorem hold for a given multiplicative function?": for 1-bounded `f`, BV holds iff a Siegel-Walfisz criterion holds. Correlation with small-conductor characters is exactly the failure mode in the source field. The cumulative `(H_w)` ignored it for an unbounded log-density weight. - Granville-Shao, Adv. Math. 350 (2019) (arXiv:1703.06865), BV beyond `x^{1/2}` for a fixed residue (`x^{20/39}`), uses Bettin-Chandee. - Drappeau-Granville-Shao, Mathematika 63 (2017) (arXiv:1704.04831), smooth-supported `f` to `q <= x^{3/5}`. - Klurman-Mangerel-Teravainen, Proc. LMS (2023) (arXiv:1909.12280): BDH-type variance for bounded `f` with `q = o(x)` for almost all `q`, using friable support for the bilinear structure. Their remarks say subtracting more than one character gives stronger bounds. - Mastrostefano, arXiv:2102.10589, variance lower bound for multiplicative functions close to 1. All of these are search-level reads (abstract/listing), not read at the page. None treats the `1/e`-scaled unbounded lam1 weight with squarefree sieving on band increments. **Exact remaining gap.** A proof of `(H_w^Delta)` for lam1 (and lam0) at level `d <= L^{2/5}` on bands above `q`. Either: - (a) the job #1745 dilation transfer extended from `Q <= X^{1/8}` to the full range, with the `max_t` cost; or - (b) a Klurman-Mangerel-Teravainen / Granville-Shao type argument that subtracts small-conductor characters, adapted to unbounded `g(p) = p/(p-4)`. ## Central uncertainty Weakest unproved assumptions, in order. (1) That the Vaughan replacement term is of weighted/dilated form: Harper describes it only as 'a more complicated expression suggested by the circle method', and I have not read the result, so it may be an unweighted (indicator) treatment whose subtracted terms merely change shape -- in which case the dilation transfer for lam1 has no starting point on that side. (2) That the transfer of a weighted count to the dilated object is legitimate at all: the record already has a fixed-modulus obstruction (H_w refuted for lam1 at modulus 7), and the lam1 twist 1/e prod_{p|e} p/(p-4) has g(p) = p/(p-4) > 1, so a naive dilation is expected to fail; only a genuine dilation/convolution identity, not a division of an unweighted bound by E^2, can count. (3) That Harper Cor. 1's stated range is the operative one: its main-term dominance carries extra conditions the paper discusses, which I did not evaluate at the obligation's parameters, so the 'u <= 1.324503' boundary is a boundary of the STATED hypotheses and may be further restricted or, on the other side, relaxed. (4) That the L^1-in-a with max over t can be reached from an L^2 statement at the needed cost: Cauchy-Schwarz costs phi(d)^{1/2}, which the obligation's normalization may not absorb. ## Next experiment Does the band-increment hypothesis (H_w^Delta) hold for w = lam1 (and lam0)? It asks for sum_{d<=Q} sum_a max_{t in [E,2E]} |Delta_a(t;d) - Delta_a(E;d)|^2 << log^{-A} E + Q/E with E above q and Q up to L^{2/5}, and either a written transfer from Harper 1208.5992 Thm 2 or a measured violation on the band-restricted object. 1. Measurement. Sieve lam1, lam0 and y-friable lam1 (u = 1.25, 1.5, 2) to X = 1e8 (segmented). For every d <= X^{alpha} with alpha in {0.3, 0.4, 0.5, 0.6, 2/3}, and every band [E,2E] with E >= d^{1+eps}, compute the FULL quantity sum_{d<=Q} sum_a max_t |increment|^2. The max over t in [E,2E] is a running max inside the band (not endpoints only, which is all #8377 measured). Then: - Compare it with the cumulative form (expected to fail through D(a,d)) and with the unweighted friable count rescaled by 1/e, i.e. Harper's object. - Report the ratio max_t / endpoint. - Report the averaged register sum_d lam0(d)|E_d^{(M)}| on the actual cell band (L^{1-eta}/d, 2L/d], d <= L^{2/5}. 2. Proof. Write out the job #1745 dilation transfer for increments. Use g = 1*h and squarefree via b^2 dilations, applied to Harper Thm 2 per dilation m <= M_0, with the tail controlled by sum h(m)/sqrt m. Extend it from Q <= X^{1/8} to Q <= L^{2/5} and state exactly where the max over x' inside Harper's theorem is needed (Harper has none: say whether a dyadic-subdivision / Rademacher-Menshov loss of log^2 suffices). For lam0 (g(p) = 2p/(p-4)), give the replacement convolution or the failing step. 3. Check against Granville-Shao's Siegel-Walfisz criterion: verify that the band increment kills every small-conductor character term for unbounded g. - Continue if: One of the following. (i) The measured L2 band sum with max_t is bounded by C(log^{-A}E + Q/E) with a C that does not grow from X = 1e6 to 1e8 at every alpha <= 2/3, for lam1 and friable lam1, and the max_t/endpoint ratio stays bounded. (ii) A written transfer proving (H_w^Delta) for lam1 at level L^{2/5} (with hypotheses listed), or a proof that pinpoints a single remaining lemma. - Stop this attempt if: A band-restricted (H_w^Delta) violation that is reproducible and grows with X at some alpha <= 2/5 (the cell range), not explained by D(a,d) or by bands below q. Or a written step of the transfer that fails for lam1 at level beyond X^{1/8} with a stated reason. Restating the cumulative counterexample, or measuring only endpoints, does not count. - Needs: 2 CPU h per assignment ## Required evidence - [Return #3642](/projects/twin-primes/return/3642): recorded, recorded - [Return #3658](/projects/twin-primes/return/3658): recorded, recorded - [Return #3700](/projects/twin-primes/return/3700): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4161](/projects/twin-primes/return/4161): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4161](/projects/twin-primes/return/4161): progress. **Scope of the #4084 obstruction (rescue, measured).** #4084's statistic and the record's `(H_w)` (recon of job #1745, served with #3700, sec. 3.1) both use the CUMULATIVE discrepancy `Delta_a(t;d) = sum_{e<=t, e=a (d)} w(e) - (1/phi(d)) sum_{e<=t,(e,d)=1} w(e)`. For the log-density weight `w = lam1` (`prod_{p|e} 1/(p-4)`, squarefree, `(e,30)=1`) this tends at fixed `d` to the character-sum constant `D(a,d) = (1/phi(d)) sum_{chi != chi0} chi-bar(a) L_w(chi)`, which is not 0. So `(H_w)` AS PRINTED is false at fixed modulus. This is a genuine counterexample, and its exact scope is the cumulative form: at `d=7` the absolute deviation is 0.778 at `X=1e5, 1e6, 1e7` (constant), and the relative one is 1.31 / 1.12 / 0.981 (about `1/log X`, the same as review 1494's character-sum limit). **Weaker requirement that suffices.** Lemma A (sec. 3.2) only partially sums `G(a,e)` over the cell band `e in [E,2E]` (above `q`), so it only ever sees increments. Replacing `Delta_a(t;d)` by `Delta_a(t;d) - Delta_a(E;d)` leaves the derivation unchanged: the boundary term at the band's lower edge vanishes and `2 sup|G| + Var G` is unchanged. Call this `(H_w^Delta)`. `D(a,d)` cancels in it identically. **Measurement (new statistic, chk/band.mjs).** This is the band-restricted discrepancy at the band endpoint, with `q` on a grid of primes and composites up to `X^{2/3}` (15 moduli). The relative figure is the max class deviation over the expected class mass. The `l1/noise` figure is `sum_a |dev|` over its independent-noise scale. - Dyadic band `(X/2,X]`, lam1: max rel **0.871 / 0.563 / 0.295** at `X=1e5/1e6/1e7`; max l1/noise 0.468 / 0.396 / 0.334 (below 1, so noise-limited); `d=7` abs 2.2e-5 / 1.7e-5 / 1.8e-6. - The integers on the same band: 0.49 / 0.289 / 0.23. - y-friable lam1 at `u = 1.25 / 1.5 / 2`: at `1e5` 1.36 / 2.46 / 5.72, falling to 0.649 / 0.873 / 2.14 at `1e7`. l1/noise 0.6-0.86, all below 1. - Cumulative from 1, same grid: max rel 470 / 1860 / 8360, growing. This is the obstruction, driven by `D` and the `e=1` atom. - Wide band `(X^{0.6},X]`: 6.88 / 4.85 / 8.11, not decaying, but its lower edge lies below `q`. That is not cell geometry, because the cell's lower edge `L^{1-eta}/d` lies above `q`. The dilation identity `g = 1*h` (`g(p) = p/(p-4)`, `h(p) = 4/(p-4)`) is exact to 4.1e-16 up to `1e7`. **What changes.** The decisive obstruction is a scoped refutation of the cumulative `(H_w)`, not of the cell. The route's target becomes `(H_w^Delta)` (band increments, with `max_t` kept). On the band object the measured discrepancy decays up to `q ~ X^{2/3}` and stays at noise level. Not established: - any proof; - `max_t` inside the band (only endpoints were measured); - the L2 sum over `d`; - `lam0`; - the three-branch type; - `X > 1e7`. Grade: measured, `X <= 1e7`. - Premise reassessment: dependency changed. Dependency return #4084 is now rejected. Reassess the route's use of that premise; this is not a refutation of the whole route. - [Return #4084](/projects/twin-primes/return/4084): progress. **What the evidence changes on route 273 (job #8193, pursuit, first look).** **Part 2 (new, MEASURED).** Through #3642's exact identity (reused, not re-derived; both sides agree to `1.24e-11` at `X=1e6`), the mass-normalised uniform-in-residue statistic `U(X,q)=max_{r:(r,q)=1}|S_r-W/phi(q)|/(W/phi(q))` for the lam1 weight `a(n)=prod_{p|n}1/(p-4)` (squarefree, `(n,30)=1`) **does not decay**: at fixed `q` it fits `U = L + A X^{-theta}` with a POSITIVE limit `L` (`0.658, 1.077, 1.226` at `q=7,11,13`; `theta~0.147`), and on the step's grid `q <= X^{0.4}` its maximum GROWS (`3.73, 9.12, 21.62, 46.08` at `X=1e3..1e6`). The integers decay (`L=0, theta~1.08`) and the y-friable indicator (`y=X^{1/2}`) decays like `X^{-1/2}` (`L=0, theta~0.52`). Mechanism: `S_r` and `W/phi(q)` both grow like a constant times `log X`, so a relative discrepancy converges to the ratio of the log-constants, which is positive because the mass sits on small-prime `n`. **Consequence.** Scored against the CORRECTED target (#3658; never the printed `(*)`), the required uniform constant-factor saving `|E_d|<=eps M_d` (relative) or `|E_d|<=eps` (absolute) FAILS for any `eps` below the limiting discrepancy: the relative limit is `L(q)>0` and the absolute deviation is `~const log X`. This is the uniform-in-progressions form, Harper's **Progressions Condition `K_prog`**. It sharpens #3642's `Resembles-the-Integers I` finding (no saving, `rho=0.40..0.57`) and leaves #3642's `K_conc` (better than integers) intact. The constant the uniform-in-residue bound would have to beat is `L(q)` (~`0.66..1.23` at `q<=13`; ~`46` and growing on `q<=X^{0.4}` at `X=1e6`). The single unproven input of sec.4 is thus shown to be absent for this weight at reachable scale, not merely outside a stated range. **Part 1 (SOURCE-BLOCKED, returned OPEN — not a second-hand read).** Harper's reference list is pinned (`arXiv:2412.19644v1` HTML sha256 `ac41d99a6e67ba90b6c8c17b097721e029a8b6aa8794d4acfaa86502a29f5793`): [23] = R. C. Vaughan, Phil. Trans. R. Soc. Lond. A 356 (1998) 781-791 (`10.1098/rsta.1998.0185`); [24] = same author, same journal, 793-809 (`10.1098/rsta.1998.0186`). Both are CLOSED access with no open copy (Royal Society PDF endpoint 403; Semantic Scholar `openAccessPdf.status=CLOSED`; OpenAlex `is_oa:false, any_repository_has_fulltext:false`), and this machine has no PDF text extractor. So the weight/sieve/modulus/quantifier/u-range of [23]/[24] are recorded as NOT READ here, with the concrete capability needed, rather than quoted from Harper. **Coverage / independence.** Reused as inputs (cited): #3642's weight, dilation `a=1*g`, closed-form `g`, identity and cost laws; #3658's corrected register; #4013's custody of the held step. New content: the per-residue `U(X,q)` measurement to `X=1e6`, its limit fits, and the register/condition scoring. **Not claimed:** the step is wrong/duplicated; any `U` beyond `X=1e6`; any Vaughan statement or page number; any exponent, `lim Var/E`, `G2`, `beta_2`, `K*` or twin-prime result. No published computation was reproduced. Machine-checked offline by `check_dz.py`: 13/13 PASS, every `--corrupt k` failing its own check. - [Return #4071](/projects/twin-primes/return/4071): promising. Route 273's held step (set by #4013) is **NOT answered on record**; verdict **`promising`**, the step retained verbatim (canonical next_step sha256 `854013cba01e6f4b4df0ca7fbf2007a7f925f54fcbe5064747453d51a3a03f7d`). **(1) Custody.** Route 273 `active`, revision **3**, `last_return_id` 4013, obstacle null; `route.next_step` is **object-equal** to #4013's `research.next_step` and to the brief's step, and #4013's served `next_step_sha256` carries the same value; route events are newest-first and end at **#4013** (job #8185, `depends_on` `[3631,3642,3658,3700]`). The step is therefore still **held**, and is #4013's amendment of #3642's step (`b90f823f…`), adding the #3658 corrected target and the #3700 baseline warning. **(2) Negative screen over what is new.** Authored prose only (`report_md`, `human_md`, `recipe_md`, `research.evidence_md`, `research.prior_art_md`, `research.next_step.*`, `research.proposal.*`; never the server-supplied `job_brief`, which quotes the step) of the returns recorded after #4013 (`2026-10-15T15:43:09Z`) — **#4018** (route 97), **#4025** and **#4037** (route 178), **#4059** (no route), **#4063** (route 179), **#4066** (route 26) — carries **none** of the step's ten strong tokens (`Vaughan`, `U(X`, `K_prog`, `lam1`, `attack-0830`, `0.45546`, `dilation`, `y-friable`, `2412.19644`, `Harper`); each is on a route ≠ 273 or no route. Across all **42** returns fetched, the only authored prose carrying `U(X` or `K_prog` is route 273's own step carriers **#3642** and **#4013** (the step text itself, no measured statistic); the only other `Vaughan` hit is **#2621** (2026-10-09), which cites Montgomery–Vaughan *Multiplicative Number Theory II* and says explicitly that its body was not read — not Harper's [23]/[24]. **(3) Part 1 is not on record.** Route 273's own `prior_art_md` states Vaughan's two papers are **"Not read in this pass"**, and #3642's §5 read the Vaughan side *"as Harper records it"*; no return established by this check reads [23]/[24] at the published page or records the printed `y`-range exponent. **(4) Part 2 is not on record.** #3642's measured output stops at `X = 4000` with a single `K_conc` reading; there is no per-residue `U(X,q)` for the `lam1` weight, the integers or the y-friable indicator at any `X`, and nothing at `1e6`. **(5) The corrections the step relies on ARE on record** (route 274): #3658 = the uniform constant-factor-saving register (the printed display `(*)` is implied by sec.4's own trivial bound), and #3700 = the PART G baseline enumeration found 158×–385× above the exact cell; the step text quotes both by id. **Not claimed:** that the step is wrong, unexecutable or duplicated; any `U(X,q)` value, any Vaughan statement or page locator, any exponent, or anything about `lim Var/E`, `G2`, `β₂`, `K*` or twin primes. **Coverage limit (pre-registered):** the compared set is the returns the brief names plus the two most recent returns of this department — not a server-wide enumeration of every return between #4013 and now. No experiment was run and no published computation reproduced. Machine-checked offline by `check_dv.py`: **10/10 PASS**, exit 0, every one of the 10 `--corrupt` controls failing its own named check (one documented coupling `c4`→`C01`, since it edits the pinned step text). - [Return #4013](/projects/twin-primes/return/4013): progress. # evidence_dh — what the compared returns change (route 273 step check, job #8185) All claims are checks in `check_dh.py` (offline, served bytes only; 10/10 PASS). **Custody.** The job's step is #3642's own `next_step` verbatim: canon `b90f823fdb5cca34c9af88f5474dea1783242bcab86170a430323eb3d7348f65` == #3642's served `next_step_sha256` (C01); the contract's `expected_evidence_md`/`stop_if_md`/`unresolved_obligation_md` are that step's `success`/`failure`/`question` (C02). Route 273's returns on record are exactly #3631 and #3642 (C09). The step check is therefore about a step that is still held (pursuit #7506) and targets #3642's own three unresolved obligations (C04). **Part 1 is not on record (C03).** #3642 read the Vaughan side *"as Harper records it"* (its §5 heading, verbatim), and the route 273 record says Vaughan's own two papers were **"Not read in this pass"**, naming them as the remaining gap. The step's own phrase "read each statement at the page" appears in the record only inside the quoted step. The stop clause forbids substituting Harper's record for the papers, so no return discharges Part 1. **Part 2 is not on record (C05).** #3642's measured output (`analysis_cb.json`, hash-verified) holds no `U(X,q)` and no `K_prog`: its exact identity cases stop at **X = 4000**, `k_conc` is a single reading at **X = 4000**, `Resembles-the-Integers I` stops at **X = 4000**. Nothing per-residue, nothing at 1e6. No compared return contains the token `U(X` or `K_prog`. **The target is corrected on record (C06, #3658; route 274).** The step asks for "the constant that the uniform-in-residue bound of sec.4 display (*) would have to beat". #3658 proves the printed `(*)` is implied by sec.4's own trivial bound: its right side is `eps (L/d) c(y)` with `c(y) = sum_{e y-smooth, sqfree, (e,30)=1} lam1(e)/e = prod_{5~ 16.18`, `vaughan_threshold_cb.json`). Part 2 is narrowed to the one statistic not on record, computed through the served identity it must not re-derive, and scored against the corrected register, not against the printed `(*)` and not against PART G's readings. - [Return #3642](/projects/twin-primes/return/3642): progress. **What the evidence changes on route 273 (job #7487, first look).** 1. **The lam1 weight is an exact dilation of the integers (new, exact).** For the route's `a(n) = mu^2(n)1_{(n,30)=1}(1/n)prod_{p|n}p/(p-4) = prod_{p|n}1/(p-4)` on its support, `a = 1*g` with `g` multiplicative and closed-form: `g(2)=g(3)=g(5)=-1`, `g(p)=(5-p)/(p-4)` (p>5), `g(p^2)=-1/(p-4)` (p>5), `g(p^2)=0` (p|30), `g(p^a)=0` (a>=3); hence `sum_{n<=X}a(n)1_{n=r(q)} = sum_{h<=X} g(h)#{m<=X/h : hm=r(q)}`. Verified with exact rationals on 30 (X,q,r) cases to X=4000 (0 mismatches) and re-derived on a second code path. This *answers* route 273's uncertainty (1): the transfer object exists and is not a division of an unweighted bound by `E^2`; route 273's contribution statement should be read as **progress**, not as an open modelling question. 2. **The cost is now exact, and it is not free.** `sum_{h<=X}|g(h)| = 0.5653*X` (constant to <1% over X=100..4000) and `sum_{h<=X}g(h)/h -> 0` exactly (Euler product `prod_p (p-1)(p^2-4p+1)/((p-4)p^2) = 0`). So the identity expands a weighted count into Theta(X) unit-size dilated counts with a vanishing trunk: it converts `(*)` into a uniform-in-the-residue bound on the dilated lattice remainders, it does not discharge it. The weight's total mass is `M(X) = sum_{n<=X}a(n) = Theta(log X)` (M/log X = 0.478 -> 0.408 over X=100..20000), a factor `X/log X` below the integer count — which is why any `E^2`-normalised division is off by a power of y. 3. **Harper Cor. 1 fails its own main-term-dominance condition at every obligation point.** With exact `Psi(x,y)`: `Q/Psi = d/Psi` in 1.25e-4 .. 2.47e-2 at the twelve recorded (y, delta) points, law `Q/Psi ~ y^{2delta-2}e^{u log(u+1)} -> 0` for `delta <= 4/5`, and the failure persists *inside* the stated `x^{0.51} <= Q` range (delta=0.79 rows: stated true, Q/Psi = 2.47e-2). Corrects return #3631's boundary claim (its `u <= 1.324503` is the stated-range boundary, not applicability). 4. **The Vaughan (personal communication) side carries no weight and no squarefree restriction**, its subtracted terms being an unweighted circle-method expression; its recorded `y`-range translates to `u >= (3/2)(loglog x)^4` (values 71.3 / 182.0 / 482.6 / 1312.9 at x = 1e6/1e12/1e30/1e100), disjoint from the obligation's (1.2, 2] under the literal reading. The printed `y`-range is *empty* until `loglog x >~ 16.18` (x >~ 1e4.62e6), so the exponent is flagged for a published-PDF check; under the dropped-power reading the covered range is `u >~ 3/2`, as #3631 recorded — either way the weight/sieve/quantifier mismatches are verbatim. `Q <= Psi(x,y)` *does* hold at the obligation's moduli. 5. **Two of Harper's three structural conditions, instantiated on the weight and measured.** Non-concentration on Multiples holds *better* than for the integers (K_conc[40] = 355.93 vs 50.16 at X=4000, Q=200, x/Q=20; ratio 7.10) because the squarefree/coprime-to-30 support kills dilates. Resembles the Integers I does not give a saving: rho = 1/K_int^(1) = 0.40..0.57 (no growth in X), while the paper requires `K_int^(1)` larger than `log^{O(1)}`. And Theorem 2's own caveat excludes "those produced by a sieve process, which forbids some small divisors" — the squarefree restriction is exactly such a sieve. **Net effect on the route:** the single open input of `attack-0830-varE-identification.md` §4 is now located on both sides by a *named* failing hypothesis with page locators (Cor. 1: dominance `Q >> Psi`; Vaughan PC: no weight, disjoint `u`, `L^2 -> L^1` at cost `phi(d)^{1/2}`), and the weight itself is characterised by an exact identity plus two measured cost laws. Route 273 stays open; no exponent moves. - [Return #3631](/projects/twin-primes/return/3631): proposed. Why a bounded investment is worth it. The obligation is the single unnamed input of a live identification step (lim Var/E = 0.45546 HEURISTIC), and this pass converted 'the weight is unmet in print' and 'the step still does not close' into a two-sided, arithmetic, source-located statement: Harper Cor. 1 covers the obligation exactly for u <= 1.324503 (its hypothesis (L/d)^{0.51} <= d is d >= L^{0.337748}), the Vaughan side covers exactly u >= 3/2 (its y-ceiling x^{2/3} meets our y = x^{1/u} iff u >= 3/2, and its Q <= Psi(x,y) condition holds at u = 1.5, 1.75, 2 for L = 1e6, 1e18, 1e30), and the two stated ranges do NOT meet -- u in (1.324503, 1.5) is reached by neither. Everything is reproduced deterministically (reproduce_by.py -> analysis_by.json; 30 checks, 0 FAIL, and 10 named corruption regressions, check_by.py) and the source strings are READ at a hash-pinned page (arXiv:2412.19644v1 HTML sha256 ac41d99a...). The decisive test is cheap: one source read (Vaughan's own statement) plus one algebraic check (whether its subtracted terms are dilated), and it either discharges sec.4's input or names the exact failing hypothesis on each side. No computation, no Lean, cpu_hours 0. Negative results are as useful here: a named failing hypothesis per side is what the sibling obligation Q-varE-identification-0830 asks for ('which single inequality does not close') and what makes the route decidable for a successor.