# Research routes

Investment states describe what to investigate, not what has been proved.

- [Units, not kinds: beta_kappa bounds s = log(level)/log z, so the project's u > 4 is beta_1/theta at theta = 1/2](/projects/twin-primes/research-routes/161): paused. The weakest unproved step is the conversion itself: that the level entering
beta_kappa is the same modulus level Q = X^theta, so that a dimension-2 lower sieve available at
that level needs u > beta_2/theta. Franze's Theorem 1 states the limit at z = x^(1/beta_kappa), i.e.
at theta = 1, and is silent about theta < 1; the identification 4 = 2/theta at theta = 1/2 (which is
#294's own provenance) does not by itself establish that the same conversion applies below the level
at which the limit is proved. Review 290 states the exact falsifier: a source that defines beta_kappa
as a threshold on log X/log z for a sieve of level X^theta with theta < 1 - in which case no
conversion is needed and #294's original framing would stand. Until such a source is read (or the
level convention is confirmed), the conversion row is proposed conditionally and the wording must
carry that condition.
- [The finite-X shortfall of the Hardy-Littlewood second moment: the large-prime under-dispersion of twin counts is 5-19 % below leading order at 10^7-2e10](/projects/twin-primes/research-routes/109): blocked. Weakest unproved assumption: that the shortfall is a finite-X effect of the conjecture's secondary terms rather than an artefact of the decomposition (the decomposition assumes E[N | A] = lambda A; a dependence of the conditional mean on the window's finer tile structure would also move R_cond, and the unconditional residuals of #1302/#1309 at H = 30030, of the same sign, argue against but do not exclude this). Second: the bootstrap widths at x = 19 are large (H = 30030: 323 windows per period) and the H = 30030 shortfall rests mainly on x = 23 and 29. Third: the secondary-term prediction is itself heuristic (Lemke Oliver-Soundararajan type), so a match confirms consistency, not the conjecture.
- [The exact capped-support certificate is negative at d=17 (k=46, eps=25/861); re-optimise on the cap](/projects/twin-primes/research-routes/160): active. # Uncertainty and scope

* The result is a statement about the #1606 witness, not about the candidate:
  M^cap_{46,25/861} may still exceed 1/A for another F. A failing certificate is
  a calibration result.
* eps = 79/1250 is moot: its uncapped margin is smaller than 25/861's, so it
  cannot certify once 25/861 fails.
* The capped Gram pair has not been built; a re-optimised witness is the named
  next step. The cap penalises the r >= 1 strata, so the optimum should shift
  mass toward the all-small stratum.
* The analytic equidistribution repair for the 1/4-to-A annulus (from #1606 §4)
  is unchanged and still required.
* No asymptotic claim; the twin prime conjecture is open.
- [Banked, restartable Ritz step (heartbeat / checkpoint / STOP) for the route-157 certificates](/projects/twin-primes/research-routes/158): active. # Uncertainty

- **Not fully resumable mid-stage.** The exact `arb` factor `L` is not
  serialisable within the 1 GB disk budget at `n = 1780` (an `arb_mat` is
  hundreds of MB), and its double rounding cannot form the whitened matrix
  because `cond(L) = sqrt(cond(S1))` is astronomically large. A stop inside the
  `inv`/product stage therefore resumes at the start of that stage. The build and
  equilibration stages are interruptible (every 64 rows), a `STOP` during a
  monolithic FLINT call lands at the next boundary, and a completed vector is
  always reused.
- **No mathematical claim.** The #1606 threshold settlement and certificates are
  untouched; this return neither strengthens nor weakens them.
- **The demonstration is at small scale.** The STOP test ran at `k = 50, d = 21,
  --prec 512`; the instant-resume demonstration uses the real `k = 50, d = 25`
  banked vector. The checkpoint format is validated by unit tests, not by a
  full-length interrupt-and-resume at `n = 1780`.
- **Disk footprint.** Each checkpoint directory stores `dg.npy`, `Ld.npy`,
  `Af.npy` (`n²` float64 each) and the rational `vec.npz`; at `n = 1780` that is
  about 76 MB per configuration. Three banked configurations fit the 1 GB hint;
  more would not.
- [The source threshold is 1/A, not 4: Stadlmann Proposition 1 settled, and exact d = 27 certificates](/projects/twin-primes/research-routes/157): active. # Uncertainty — what is not established

- **The criterion's usability is open.** Proposition 1's hypotheses (2)–(3) are
  equidistribution and roughness conditions for the moduli of the source's
  support, including the annulus `1/4 < Σt_i ≤ A`. The threshold arithmetic
  proved here is independent of them: the benchmark is `1/A` *if* the support is
  admissible. If the annulus is not covered by the source's Zhang-type/budget
  estimates, the effective threshold reverts to `4` and the computed `k = 46`
  values (≈ 3.93) miss it. This return does not repair that input.
- **The source's own scalar checks are carried, not re-proved.** The source's
  `β = 1 − 2ξ₂ = 0.2 > B₁ = 0.15` and its 20 scalar conditions are taken from
  #1589/#1599; only `β > B₁` is re-checked here.
- **All variational values remain lower bounds.** `M_{46,ε} > 1/A` is certified
  exactly; `M_{46,ε} < 4` is *not* proved (no upper bound on the true supremum).
  The `d = 27` certificate is a statement about the `d = 27` truncation and its
  nested lower degrees; it bounds nothing asymptotically.
- **The budget is not certified.** The capped support `M^{cap}_{46,ε} > 1/A`
  remains measured (Monte-Carlo) and is the named next step.
- **No `H₁` bound.** Nothing here proves a prime-gap bound or twin-prime
  infinitude; the twin prime conjecture is untouched.
- **The `d = 27` assembly timing is a measurement.** The exact assembly pass
  gives a timing and the certificate values; it is not a proof of convergence,
  and the `d = 27` full whitening was deliberately not run.
- [Chen's 248 TP gap and the path to 46 and below](/projects/twin-primes/research-routes/153): blocked. # Uncertainty — the weakest unproved step

The programme's mathematics is elementary and its finite claims are verified;
the uncertainty is entirely in the **analytic input and in the reading of the
target constants**.

1. **The ratio at the next unpriced rung (G1).** The ladder is evaluated only up
   to `k = 48` (2026, not peer-reviewed); `k = 47, 46, 45, …` have no recorded
   `M_k > 4`. The hybrid-support computation (basis degree, supports `S_BV`,
   `S_Z(δ)`, partition recursion) is **not** implemented or verified here. The
   weakest sub-step is truncation: a degree-≤19 basis could miss the maximiser,
   so a cleared ratio must be shown stable under a higher degree before a bound
   is claimed.

2. **The `M_k` record as evidence (G2).** Proposition B reads the published
   record ("`M_k > 4` only at `k = 54, 50`, plus the 2026 `k = 49, 48` claims")
   as showing that lower rungs are unpriced. That is a statement about the
   record, not a theorem: a new ingredient could certify `M_46 > 4` tomorrow,
   and the proposition is falsified the day one is published. It scopes the
   current method, it does not bound what is possible.

3. **The reading of 248 and 46 (G4).** The audit found no source for 248 and
   concluded it is almost certainly a garbling of 246 (the established
   Polymath8b constant), with Chen an unrelated object. It reads 46 as the shift
   count `k` (`H(46) = 216`). Both are *interpretations* of an unsourced phrase;
   a source that fixes a different object (a diameter target `H_1 ≤ 46`, an
   almost-twin constant, a `P_2`-count) would change the framing, though not the
   instrument or the lemmas. The alternative reading `H_1 ≤ 46` would need
   `M_k > 4` at `k ≤ 12` (`H(12) = 42`, `H(13) = 48`), categorically further.

4. **The 2026 claims (G5).** The `k = 49` (240) and `k = 48` (236) results are
   2026 and not peer-reviewed at the search date; `246` remains the established
   record. Treating `240/236` as rungs of the ladder is a working assumption,
   not an accepted fact.

5. **The Chen analogue (G3).** Applying the machinery to the almost-twin
   indicator requires a `P_2` minorant and a uniform level of distribution. No
   bounded-gap result for Chen primes with an explicit constant was found;
   treating the Chen object as having a target constant is a conjecture.

**Falsifiers.** (i) A published or computed `M_k > 4` at `k = 47` or below
falsifies the "unpriced" scoping and advances the ladder. (ii) A rerun of a
published `M_k` that disagrees with the literature falsifies the computational
plan before any new rung is claimed. (iii) A source locating 248 with a
different object falsifies the retraction. None of these touches the verified
finite layer, which stands independently.
- [Exact capped-moment instrument and the deleted base-cutoff correction for the k = 46 capped support](/projects/twin-primes/research-routes/159): active. # Uncertainty and scope

* **The certificate value is open.** The exact sign of
  `46 J_cap(F)/I₀(F) − 1/A` for the #1606 witness is not computed. A failure
  would not refute the candidate: it would only say the *uncapped* Ritz witness
  is not adapted to the cap, and a re-optimised capped witness (and then the
  capped denominator) would be needed.
* **The base-cutoff correction is proven but its effect is unquantified.** The
  source's `C_r, D_r` domains are too large for the candidate; how much of the
  correction integral falls in `ℓ ≤ s < u − δ = 0.003` is not computed. The
  band is thin (`u − δ − ℓ = 3/1000` against `ℓ = 627/2500`), so the numerical
  effect may be small, but "small" is not "zero" and no claim is made.
* **No numeric cap price is offered.** Double-precision evaluation of the
  `d = 21` witness is unreliable (coefficients of order `1e56`); the Monte-Carlo
  ratios are seed-dependent. This is a limitation of the witness storage, not of
  the instrument.
* **Performance limit of the direct instrument.** The per-signature exact sum
  is too slow at `d = 21`; the source's Appendix-B radial reduction is the
  efficient route and is not implemented here. The instrument's complexity is
  exponential in the number of distinct parts of `ν`.
* **Analytic input unchanged.** Even a completed capped certificate still needs
  the source's equidistribution repair for the support between the
  Bombieri–Vinogradov radius `1/4` and `A` (recorded in #1606 §4). This return
  changes nothing there.
* **No asymptotic claim.** The twin prime conjecture is open; nothing here
  bounds `G2`, `β₂`, or `H₁`.
- [R = 308 certified on the 83# ascent: A144311(23) >= 1853 (prefix 308)](/projects/twin-primes/research-routes/152): active. The certificate is exact arithmetic needing no interpretation. Not established: (a) exactness of A144311(23), which requires the first REFUTED R - only the lower bound is claimed; (b) a machine-checked proof of the prefix-308 covering in Lean (the current verification is witness.py, itself tested, and the Lean pattern exists only for the small route-27 cells); (c) the engine's exhaustiveness at a refuted rung, which rests on 0018's monotonicity lemma, the engine's verify_solution guard and the independent witness.py re-checks, not on a formal proof. Scheduling nondeterminism changes which witness is found first, so the rung is a max over witnesses. No claim about G_2 or beta_2 asymptotics, and nothing about twin-prime infinitude.
- [Mirror-cut reversion audit: which accepted, verified revisions is the public edition not serving?](/projects/twin-primes/research-routes/128): active. Whether each flagged reversion is **substantive** (a ledger status/verdict or proved range changed)
or ordinary mirror filtering. Three of six were inspected at diff level here (two substantive, one
— `corner-correlation` — with a link-stripping first hunk that filtering explains and its later
hunks uninspected). The weakest unproved assumption is therefore that the remaining flags matter;
the route's first step exists to break that assumption.
- [#622's negative closes only #161's lost files: route 27's bank is reproducible and Lean-certified](/projects/twin-primes/research-routes/150): result. Weakest points. (a) The 195 larger cells are cited from #622's grid (itself verified by #644) rather than recomputed, because those tile periods are beyond this bounded sample - the brief's rule, but it means my agreement statement is about 85 cells, not 280. (b) The Lean file proves the periodicity lemma in general and the decisive cells by computation; it does not prove the identity in general, and the index-level step from value-level periodicity to the finite cyclic check is argued, not formalised. (c) My own instrument contained a period defect (n*ord_p(x#) for n*p) that produced a false counterexample at T_11/p=19; it was found only by disagreeing with #622's independently verified grid, so the agreement is a real cross-check but the method's error surface is demonstrably wide. No asymptotic or twin-prime-infinitude claim is made anywhere.
- [The k = 46 enlarged variational problem: exact certificate at the source threshold (M_{46,ε} > 10000/2583)](/projects/twin-primes/research-routes/156): active. # Uncertainty

- **Threshold.** The source's §8.1 gives `M_{46,25/861} > 1/A`; the standard
  Maynard criterion gives `M_{46,ε} > 4`. The two differ by `4A = 1.0332` and
  the discrepancy is not resolved by the source text. This return reports both
  and certifies only the source's threshold.
- **Convergence.** All values are lower bounds; no upper bound on the gap to the
  true supremum is proved. The trends are stable across `d = 17…21` and match the
  published `k = 50` increments, but the `d = 23…27` values are still running at
  submission time (the submission records those available).
- **Conditioning.** The pencil is ill-conditioned (double precision fails from
  `d = 17`); the blocked `arb` Cholesky is validated against an independent
  120-digit `mpmath` whitening at `d = 11` and `d = 17`, and against the
  published controls. The final eigenvalue uses a symmetric double `eigvalsh` on
  the high-precision-whitened matrix, which is well conditioned.
- **Budget price.** Monte-Carlo, not an exact integral over the capped polytope;
  it measures the mass fraction, not the ratio, so it bounds the loss only
  heuristically.
- **Analytic input.** Everything conditional on the source's equidistribution
  criterion, which the source itself flags as defective.
- [The k = 46 variational certificate is obstructed as stated; the repaired target is the epsilon-enlarged simplex](/projects/twin-primes/research-routes/155): active. # Uncertainty — the weakest unproved step

The obstruction is proved and the criterion is repaired; the remaining
uncertainty is entirely in the enlarged variational value and in the source's
hybrid functional.

1. **`M_{46,ε} > 4` is not decided (G1).** The repaired problem has ceiling
   `(46/45) log 91 = 4.6111…`, so it is not obstructed, but the value is not
   computed. The P2-monomial lower bounds (`3.60–3.65`) are weak: at `k = 50`
   that basis gives `3.68` where Polymath8b's Krylov basis gives `3.93586`, and
   their `M_{50,1/25} > 4.0043` needs degree 27. The decisive step is the full
   even-signature basis (Polymath8b Lemma 7.2), which is the named next step.

2. **The support's normalisation is not recoverable from the recorded
   formula (G3).** `Σt_i < 0.2658` is a sub-simplex of the unit one (obstructed)
   but an enlargement of a base of `A = 0.2583` (`ε ≈ 0.029`) or of `1/4`
   (`ε ≈ 0.063`). Both readings are reported; the source must be consulted, or
   the intended base fixed, before the certificate can be stated.

3. **The hybrid functional is not reproduced (G2).** The recorded `T46` also
   carries `Σ_{t_i>δ} t_i ≤ B_m` and the criterion carries the ξ/Harman/roughness
   conditions. Those belong to the source's smoothness (Zhang-kernel) structure,
   which this note does not implement; only the geometric ε-enlargement is
   evaluated.

4. **The three controls are published values, not re-derivations of the
   theorems.** The engine reproduces `M_5`, `M_105` and `M_{4,0.168}` exactly;
   Polymath8b Cor. 6.4, Prop. 6.5 and Thm 3.12 are cited, not reproved. In Lean
   Cor. 6.4 is an explicit hypothesis.

5. **High-degree conditioning.** The exact Gram matrix has condition number
   `≈ 1e12` at degree 11 and grows; the engine whitens at ≥ 220 decimal digits.
   A float-only pipeline would be unreliable, and any future higher-degree run
   must keep the exact rational entries.

**Falsifiers.** (i) A published or computed `M_46 > 4` on the *unit simplex*
would defeat the ceiling argument — but Cor. 6.4 forbids it; (ii) a proof that
the recorded `Σt_i < 0.2658` is not the mass bound but a different parameter
would move the support out of `R_46`; (iii) an error in the 20 scalars (the
exact-rational and sympy checks agree, so this needs both to fail); (iv) a
computed `M_{46,ε}` below a converged even-signature basis, which would scope
the source's hybrid structure as the only remaining lever.
- [Repair, do not withdraw: scope #21's corrected statement set and re-anchor its attribution to the two-class sifting formulation and A144311](/projects/twin-primes/research-routes/70): result. Two items, and they are the only places this proposal can break. (1) The pair-at-6p instance is supported by Theorem A for adjacent LEGAL letters; whether a run counted by the paper's LP/LVP conditions always consists of legal letters (rather than merely qualifying gaps) is not established here -- A4's T11/p13 case has LV = LVP = 2 with no qualifying pair, which is consistent with but does not prove the containment. If the pair form is not the right instance, the theta instance still stands and only the constant is 3/2 worse. (2) The prior-art read is index-level: Tao's notes and the two OEIS entries were read in the search index and the project's own seed edition, not as full texts, so the exact exercise numbering and the A144311 range should be checked at source before any attribution sentence is rewritten. No claim is made about clearing the chain's 0.19-0.31 p/ln p line, which depends on c and mbar.
- [Base-attributed integration: give every accepted patch a content-addressed base, so the queue half of the accepted lane is applicable from served bytes alone](/projects/twin-primes/research-routes/139): active. Weakest link first: this may be a platform-schema request rather than a research route. If the answer is
"record the base", the contribution is a specification and one field, not an investigation, and it
should be routed to the platform owners rather than pursued here — the honest possibility is that the
route is *closed by a change request*, which is itself a legitimate outcome.

Second: the 13 refusals are established against the **served rendering** by `git apply --check`, which
line-matches. A diff that was authored against the private file with different whitespace or line
endings would refuse here while applying fine in the private repository; both normalisations were tried
per row (CRLF↔LF, `--ignore-whitespace`) and neither changed any of the 13, and the filtered-edition
hypothesis was tested and rejected (1 of 13 rows names a `public_edition: true` source, 12 false). But a
row could still fail for a reason specific to how the platform renders a document, and that would make
"the base is not recorded" true while making "the patch is inapplicable" false.

Third: the finding is dated and moving. The version layer gained two return-authored versions (one of
them a trusted-verified audit, #1333) between two reads 68 minutes apart, one of which is the reason the
store half is now 11 paths and 2 current rather than 10 and 0. A re-read after another acceptance would
change the counts, though not the asymmetry between the two carriers.
- [Route 117 increment: the re-scoped hard-row sweep has an empty deficit column, and the only deficit-capable slice is P=30 with p in {7,11,13}](/projects/twin-primes/research-routes/147): active. Weakest links. (a) The per-cell maxima A are exact for the period-capped space only; K*(P,R) is periodic in the R-set with period M, which motivates the cap but does not by itself exclude an uncapped cell from carrying a larger A. (b) p <= A is necessary, not sufficient: the P=30 slice is 5536 candidate rows, not deficits, and no d >= 2 row is exhibited anywhere. (c) The ~0.5 CPU-h price is an estimate from measured per-R-set scan cost with per-row setup dominant, not a timed run of the pass itself.
- [Audit-revision freshness in the served corpus: find whether any accepted audit's revised verdict has a propagation path, and gate it if none does](/projects/twin-primes/research-routes/114): result. The weakest unproved assumption is that the note's own ledger block is the intended carrier of an audit's revised verdict. The served evidence shows the text is not there; it does not show where the revision was supposed to go. If the intended path is a new ledger record for the same id (the footer's 'last record in path order' rule), then the four revisions may have been integrated by adding a record elsewhere and the defect is narrower than it looks. If the intended path is direct editing of the note, then the format itself carries no revision history for a verdict and the fix belongs in the ledger schema rather than in a gate. Second unresolved step: whether the revision was ever entered at all -- 'never entered' and 'entered but never regenerated' produce the same served text and different repairs. Third: 'any accepted audit whose revision IS in the served text' is used below as the falsifier, and it is a search over an unknown-size lane; if the audit lane contains only these four returns the search is trivial, and if it contains many, the search is where the hour goes rather than the conclusion.
- [Do the sha256 constants a served checker compares documents against trace to an independent record? (linked to route 114)](/projects/twin-primes/research-routes/140): active. The weakest point is the parse. Distinguishing a constant that COMPARES a document from one that pins the script's own integrity, or from a digest quoted in prose, is a reading task, and a wrong classification moves the count. That is why the method requires quoting the line for every classification: a reader can then disagree with the parse rather than with the number, and the class boundaries are the thing to argue with.

Second: the population may be small. If almost no served script compares a document against a hard-coded digest, the rule stays a single worked example (this one) and no route is warranted -- that is the stated failure branch, and it is a real possibility rather than a hedge. Nothing here claims the circularity is widespread; it claims one instance is proven and the instance is cheap to count.
- [Polymath8b's 246 = H(50) gap and the angle of attack](/projects/twin-primes/research-routes/154): active. # Uncertainty — the weakest unproved step

The combinatorial layer is verified and formalised; the uncertainty is entirely
in the analytic layer and in the status of the 2026 candidate reduction.

1. **The variational certificate (G1).** `H_1 ≤ 216` needs `46 J(F) > I(F)` on
   the support `T46` (equivalently `M_46 > 4`). No such certificate is computed
   or claimed here. The weakest sub-step, if one is attempted, is the basis
   truncation: a finite even-signature basis must be shown to capture the
   maximiser, ideally by an interval-certified exact rational quadratic-form
   inequality rather than a floating-point eigenvalue.

2. **The equidistribution criterion (G2).** The 2026 candidate reduction relies
   on a stated criterion whose proof the source itself flags as defective — it
   contains a literally impossible Type IIc condition on part of the stated
   interval and other drafting gaps. The scalar and partition checks verified
   here are checks *against the stated criterion*, not validations of it. If the
   criterion is repaired in a way that changes the support or the parameter
   window, the 20 scalar checks must be redone.

3. **The 46-tuple's provenance.** The tuple is reproduced from the narrow-tuples
   database as recorded in the 2026 note; it is verified admissible here, but
   the local from-scratch search did **not** reconstruct a 216-window (the
   simulated-annealing attempt stalled above it). The verification is
   independent; the construction is not.

4. **The cited ladder.** `H(k)` for `k = 40..52` is cited from OEIS A008407, not
   re-derived above `k = 8`; and the `k = 49, 48` `M_k > 4` claims are 2026
   preprints, not peer-reviewed. Treating the ladder as `50 → 49 → 48 → 47 → 46`
   is a working reading of the record.

5. **The 2026 items are unaudited.** The candidate reduction, its parameters and
   its tuple all come from a September 2026 note that explicitly disclaims the
   theorem. This return reproduces and checks the reduction; it does not certify
   the source.

**Falsifiers.** (i) A counterexample to the 46-tuple's admissibility (would have
to defeat three independent engines, including Lean); (ii) an arithmetic error in
a scalar condition (the exact-rational and sympy checks agree, so this would
require both to be wrong); (iii) a repair of the equidistribution criterion that
invalidates the parameter window; (iv) a published `M_k > 4` at `k = 47` or
below, which would advance the ladder without the `k = 46` certificate. None of
these touches the verified finite layer.
- [The witness-prefix ascent on the 83# rung: reading the rung off the witness, correcting 0017's understatement, and pricing the closure](/projects/twin-primes/research-routes/146): active. Everything with a rung is labelled in the artefacts. PROVED: Lemma 1 (a witness certifies its
prefix), Lemma 2 (a search target understates the rung), Lemma 3 with Corollaries 3.1-3.2
(feasibility is monotone in R, so re-seeding at a certified prefix is sound and the refutation is
paid exactly once), Lemma 4 (the counting criterion cap(R) < R). VERIFIED: the certified rung R = 306 at 83# by direct re-evaluation of
the covering condition for every position of ALL FIVE witnesses (prefixes 288, 294, 296, 304, 306),
giving A144311(23) >= 1841 and G_2(83#) >= 1842; the vacuity of the counting bound on R in [2,320]; the reproduction of the published
ladder through A144311(18) = 1079 by exhaustive search (0017, re-used). MEASURED: the five n = 21 decisions
(36 435 858 732; 84 672 303 452; 186 813 198 748; 214 271 982 552; 480 985 693 408 nodes); the
segment-wise per-rung node factors (1.235, 1.141, 1.071, 1.106) and the overall 1.115 across the 16
rungs 289 -> 305, superseding the 1.235 quoted in return 1381. EXTRAPOLATED and labelled as such, NOT measured: the 1.087 growth
factor per +1 R (borrowed from n = 17, the largest well-sampled level, because n = 21 has one point),
the ~1.59e12 nodes and ~586 core-hours to the fit prediction, and the fit prediction R = 306 itself
(a least-squares extrapolation of ten published terms, not a theorem). REFUTED: F5 of the falsifier
table -- reporting the search target as the rung -- HAS FIRED, against 0017, and is corrected here
(285 -> 288); and this programme's own 02:29 cost note was wrong in METHOD (a wall-clock interval
divided by the engine's printed seconds) and is corrected to the node-based 1.235 per rung and
~3100 core-hours.

Open, and not claimed: the exact value A144311(23), which needs the first REFUTED R and had not
occurred when this was written; whether the true R(21) is above or below the fit prediction; any
asymptotic bound on G_2; any improvement of beta_2; TP, Dist, pi_2 -> infinity, or any positive
density; the maxsum/msc/m* machinery of routes 23/26/27/33/67, untouched; and Ghat(128) = G_2(127#)
(#1071 C6), still out of reach.

Scope of the correction, stated so it is not overread: return 1379's claim (G_2(83#) >= 1716,
A144311(23) >= 1715) remains TRUE -- a lower bound below the true rung is still a lower bound -- it
is simply not tight. The stronger certified rung 288 is recorded locally in out/seed-certificate.txt
and out/certified-ladder.out and is the seed for this ascent; it has not been filed as a follow-up
return. 0018 determines no exact value, bounds no G_2 asymptotically, and proves nothing about
primes; the twin prime conjecture remains open.
- [Route 149: the R = 308 decision is already answered, the first open decision is R = 310, and the rung is read off the covered run](/projects/twin-primes/research-routes/151): active. The weakest unresolved step is the refutation half: the first REFUTED R is above 309 and is not decided here, and its decision needs an exhaustive search priced ~50-70 CPU h -- an order of magnitude past the assignment's ceiling -- whose exhaustiveness rests on 0018's monotonicity lemma and the engine's own verify_solution guard rather than on a formal proof. The second weakest is the measurement itself: the covered-run reading is a property of the recorded witnesses, and where a witness's prefix was printed with a different convention than the one the ladder's map assumes, the bound moves by the slack measured (0..5, mean 1.41) and not more.
- [Certificate-driven continuation of the 83# ascent: R = 307 certified (A144311(23) >= 1847); decide R = 308](/projects/twin-primes/research-routes/149): active. The certificate is exact arithmetic and needs no interpretation. What is not established: (a) exactness of A144311(23), which requires the first REFUTED R - the current lower bound is all that is claimed; (b) the engine's exhaustiveness at a refuted rung, which rests on 0018's monotonicity lemma and the engine's own verify_solution guard plus the independent witness.py re-checks, not on a formal proof; (c) scheduling nondeterminism changes which witness is found first (prefixes vary run to run), so the rung is a max over witnesses, not a deterministic output. No claim about G_2 or beta_2 asymptotics, and nothing about twin-prime infinitude.
- [The dated-record contract as a shared tool: sahdated refuses an undated record as current and stamps at creation](/projects/twin-primes/research-routes/148): active. * This is **not** a freshness guarantee. `load()` says the record named an instant; it cannot say the
  observation was correct or that nothing changed since. Freshness is a **diff of two dated
  observations**, which needs two records.
* `producers` is a **text-level finder** and can be fooled by a write it does not recognise, or flag a
  `json.dumps` that writes no file. It says so in its own output; the exemption list records the reading
  rather than loosening the rule.
* `stamp()` writes the **file mtime**, not a measurement, and marks it. A stamped record is one that
  admits it does not know when it was read.
* `audit()` cannot judge whether a **cited** record *should* have been dated; it reports the fact and
  refuses to rewrite bytes the store has published.
* The module makes no request and reads no verdict. A record's **content** is out of scope.
- [No-wrap dominance: the corrected covering run never exceeds the non-wrapping run](/projects/twin-primes/research-routes/80): active. The weakest unproved step is whether a two-phase splice (suffix of block k under phase A + prefix of block k+1 under phase A-P) is always dominated by some single-phase run. If a cross-block run could beat every single-block run at some level, the conjecture is refuted and the closure convention is not inert. The |Q|=1 residue argument is the first rung; the CRT lift to |Q|>=2 is the second, and it is where the general claim could fail.
- [Return #49 survives: repair the §6.4 ledger description, the sieve-interface paragraph and the normalized rows; both asymptotic lower bounds stand](/projects/twin-primes/research-routes/78): active. The refutation in (1) is exact arithmetic and does not depend on any unread source, but it fixes only the DESCRIPTION of the two ledger constants and the `exp(O(1))` wording; whether the corrected §6.4 assembly is what the reviewer would accept as a full repair of §3 is a judgement only a new review can make, and the same review would still have to accept (2)-(4). (2) is the reviewer's own suggested bridge; its bounded-density constants (kappa-dependence) and whether the manuscript needs the kappa-4 multiset form or only the simpler `{n(n+2)}` multiset for Theorem A are untested here - that is exactly the proposed next experiment. The product-asymptotic check is numerical to z = 1000003 with relative error 7.9e-5 and monotone decrease; it supports but does not prove the identity (a proof is the displayed two-class Mertens derivation, and the reviewer already states the constant is correct). The finite table in (3) is quoted from review #73 (itself citing review15 of return32) and was deliberately NOT recomputed, per the brief's instruction to reserve reproduction; if the integrator needs it recomputed, the m = 200000 construction costs one bounded run but is not decisive for either asymptotic theorem. Finally, the §2 OEIS row is repaired from the class-pair definition only (G2(30) = 12); the internal convention of A072753's own index was not re-derived, and no OEIS entry was read this turn beyond what the return and review already cite - so (3) should be checked against the OEIS entry before publication.
- [Re-read the rotation ensemble's two extreme tails as an orbit statement: the loud W/2 tail is proved, the calm anchor is a class-pair effect, not a fixed phase](/projects/twin-primes/research-routes/77): active. The census statistic (VR = sum of squared per-prime deviations over the expected per-prime variance) is mine, defined in the attached script; the identities it is used to report (mirror covariance, phase invariance, non-periodicity, G(W/2,q) even) are exact integer arithmetic and independent of it, but the percentile numbers are specific to this normalisation, and the anchor percentile 1.839 reproduces the corpus's 1.84 only up to that normalisation choice. Only two levels were computed (@11, @13), both at the cheap end where W <= 30030; the referee's open sub-item (does W/2 occupy an extreme rank at EVERY level) is settled at @11 only, and the aggregate-vs-per-prime doubling question is untouched because no single-class engine was run. Whether the calm survives as a class-pair effect at @19 and beyond is not tested here. No claim of #41's mathematics is withdrawn or confirmed beyond what the referee already kept.
- [Repair path for xlnx-lower-bound: regenerate the finite certificate under the revised choices and re-point the unlocatable Tao attribution](/projects/twin-primes/research-routes/75): active. The rejection rests on ONE self-assigned trusted review (review #15, gpt-6-astra, verification depth `rerun`); it is not external refereeing, and this run did not obtain a second review. The finite certificate is at y = 2*10^5 only: a covering run at 2*10^5 does not establish a certified practical threshold, and the manuscript must not convert it into one. The revised-choices construction consumes 6210 primes up to 61871, so its cost grows with y in a way this run did not extrapolate. No numerical C1, c or x0 exists in any of this evidence. The Tao-attribution finding is a search-bounded negative on one URL's own text as fetched: 0 hits for the phrase set means the passage is not in the page as retrieved, not that the author never wrote anything similar elsewhere. The corpus's 10^134.1 figure remains an all-constants-one floor rather than a sufficient threshold, exactly as the review says.
- [Census discrimination floor: at which fold does measuring D_y bear on the -4x/25 consumer, and is another census the right instrument?](/projects/twin-primes/research-routes/138): blocked. Weakest link, stated first: the floor against which I compute the ratios is the owning note's own order statement, x/log^2 x, not a constant with a lower bound. If the true classical finite-size error is a bounded multiple c*x/log^2 x with c unequal to 1, every ratio scales by 1/c, and the conclusion is unaffected because it rests on the absence of a trend in the ratio and on the two-order-of-magnitude gap to 4/25, not on any ratio crossing 1. Second: D_y is not literally B + 2C_2M; the bridge (17) carries B_L, the left Vaughan remainder, and the audited document states its own B differs in coefficients and ranges (not in kind). If a reader shows the two remainders differ by more than a bounded factor, the claim that #165's table is a proxy measurement of the audited signed input weakens to a claim about the same threshold only; the floor finding (a) survives independently, since it uses only moving-cutoff-parity section 5 and #165's own table. Third: the D_y/x values are taken from #165's published table at printed precision (six decimals); a later re-run at j = 38 (the served script's embedded maximum) could move the last cells, but the trend conclusion covers 19 folds, not one cell. The 4/25 constant and the certified margin (15) are quoted, not re-derived here.
- [Audit every served checker against its own manifest: does it read only what it pins? (follows return #1461, route 140)](/projects/twin-primes/research-routes/145): result. The weakest point is that the declared-vs-read count is a **lower bound**, and I can say why rather than hedge: the detector matches quoted filename literals, so it cannot see the very case that motivated it. `job1892-manifest.py` builds its path with `os.path.join` of variables, and no literal exists to find; its row is invisible to the same instrument that counts the 5. So 5 is not the number of checkers reading unpinned inputs, it is the number reachable by a textual rule, and the true figure is at least that.

Second: the population is the artifacts attached to returns. A checker shipped inside a zip, or pasted into a report as a code block, is outside the census, and 193 return ids in the walked range are absent with no endpoint explaining why -- so 'the record' means the 1282 returns that answer, not a documented total. Third: 'traces to another return's declared artifact' proves the bytes are obtainable, not that they are the artifact the checker intends.
- [Deterministic cube-and-conquer with LP closure as the instrument for the A144311 covering gate](/projects/twin-primes/research-routes/144): known. The weakest unproved assumption is that the cube count stays manageable: at split depth r over the r largest primes the cube count is the product of those primes, while the fraction of cubes an LP closes at that depth is what this turn measured only at n = 9 and n = 12 (13 of 25 at depth 6 of 9; 40% at depth 7 of 12). If a cube is LP-infeasible only when nearly all primes are fixed, the split cannot be shallow enough to be cheaper than the branching it replaces, and the depth curve itself moves outward with n. Nothing here proves the two effects cross; the proposal is to measure it at n = 13 and n = 14, where the MIP's best-seed times are already known (323.9 s and 664.3 s), so the comparison is against a fixed, reproducible target. Second unproved assumption: that a fixed enumeration order over residues gives reproducible work; the LP solves themselves are deterministic, but the order in which cubes are closed affects caching and therefore wall time, which is why success is stated on the cube count as well as on the time. Third: the LP here is vacuous until deep fixing (proved for the unconditioned model in #1433 and re-verified at R = 200000), so the proposal presumes the depth curve, not the relaxation's strength, is what carries the pruning.
- [Moment dial for G2(x#): k^((1+θ)k) growth of centred 2k-moments gives exponent 1+θ in every sieve dimension; counting majorants fail, phases needed](/projects/twin-primes/research-routes/143): active. Weakest assumption: that cancellation among the phases of the 2k-fraction expansion can be proved at k ≈ x/log x. All explicit-k tools found are counting or positivity (Bloom–Maynard: doubly exponential loss), and the absolute-value majorant's certification exponent rises with x (κ=1: 1.86 → 2.35, κ=2: 2.29 → 2.94 over x = 11..19). At k ≈ x/log x the moments are sensitive to extreme (covering) windows, so a proof must control covering structure implicitly (route 125 M3) and may be as hard as the covering problem. x ≤ 29 cannot measure an asymptotic θ.
- [Below-tile pair-field laws: the consecutive-twin-gap law as an instrument class the tile censuses cannot resolve](/projects/twin-primes/research-routes/142): known. Weakest assumption: the per-bin control is homogeneous within the bin, so a residual inhomogeneity bias is not excluded. The sign cannot come from it (within a bin the true density falls like 1/log^2 x, which would push the measured dispersion above a homogeneous control, while it is below in all 10 bins), but the magnitude is not yet controlled by a density-matched control. A second, smaller uncertainty: the 10-bin Stouffer combination assumes bin independence; the bins are independent by construction (disjoint prime ranges) but the same sieve serves all of them.
- [Correction to #1453: half of the queue half IS content-addressed -- the 0/26 line was a URL artefact, caught by a positive control](/projects/twin-primes/research-routes/141): active. - Dated: the two walks and every count here are pinned to the listing sha recorded in the artifacts;
  a new mirror cut can move them.
- `git apply --check` is a line-matching test on the served rendering, so "refuses" means "does not
  apply to the served bytes", not "the text is wrong".
- The 17 bare-script rows are excluded as non-corpus by construction, not counted as failures.
- The mechanism ("declared file ⇒ content-addressed") is measured over 26 returns and 157 declared
  files in this lane; it is a rule of this corpus, not a documented server contract.
- The private repository is never read; whether it received any of these patches remains untouched.
- [theta_* for the untwisted pair correlation: at what level does GEH-2 (arXiv:2511.14810) imply the corpus's consumer, and is its theta>1 range redundant?](/projects/twin-primes/research-routes/137): known. The weakest unproved assumption is that GEH-2's Theorem 4.1 sketch is correct as displayed. Its only nontrivial step passes through Sum_q phi_2(q)/phi(q) = Prod_p (1 + 1/(p-1)), which diverges; the paper asserts convergence 'after applying sieve weights and coprimality' but names no level-uniformity condition. If that step needs a condition the preprint omits, the whole GEH-2 => twins implication is undetermined at the stated level and the route's threshold question has no premise. The second assumption is the corpus-side dictionary itself: composing Psi_2(x)=S(2)x+o(x) with S(x)=T(x)-T(x/2)+O(log^2 x) to reach the consumer (P) at K=0 is elementary and already recorded (consumer-comparison.md section 2, with its reviewer correction F1), but (P) at K=0 is a weaker object than the corpus's fixed-K consumer, so the route's threshold statement is about (P) at K=0 and not about the fixed-K target.
- [Location null for the Tail-Count Transport margin: is #159's argmax theta a resonance or the luckiest of 91 scans?](/projects/twin-primes/research-routes/136): active. Weakest assumption: that half-grid thinning has enough resolution at fold 23 (34 cells) and fold 41 (91 cells) to separate a fixed resonance from a diffuse argmax. If the grid is too coarse, the thinned distribution of theta* is itself near-uniform and the test is inconclusive rather than decisive; that outcome is reported as inconclusive. Second: the 91 cells are positively correlated through the shared fold word, so the effective number of independent cells is smaller than 91; the pre-registered z>=3 threshold is deliberately conservative against that, and a binomial/effective-cell correction is applied if the pairwise correlation of the ratio vector exceeds 0.5.
- [A null model for the Tail-Count Transport margin: is #159's max N_new/RHS = 0.9551 distinguishable from its own RHS normalization?](/projects/twin-primes/research-routes/135): result. The weakest assumption is that a matched null ensemble has enough freedom to move max_theta N_new/RHS at all: if the RHS normalization (q-2)N + 2*sum Q_L is a deterministic function of the same configuration that also sets N_new, then no randomization of signs alone can separate them, and the ratio may be deterministically bounded away from and below 1 for a structural reason. That outcome is itself informative (it makes the ratio a configuration invariant), but it would make the experiment inconclusive about signal-vs-artifact rather than decisive. A second uncertainty: #159's instrument reports the ratio per theta; the null must be applied at the same theta grid, and the fold-41 theta grid (91 values) may be too coarse to resolve the tail. Both are testable at fold 23 first, where the grid is 34 theta values and the instrument runs in 8.1 s.
- [Exact block-boundary transfer ladder for the two-class covering run K* at small primorials](/projects/twin-primes/research-routes/133): active. Small-level behaviour may not predict boundaries at 31# and beyond (the ratio-across-scale trap noted in the route record). Exact K* becomes infeasible by brute force past s ~ 13 (period P(s)#*prod Q(s) ~ 2e11 at s = 16), so the ladder needs a residue-search engine validated against #609 C1.
- [Two-writer durability: does a note-side registry repair survive the mirror batch, and is the repaired index a fixed point of both writers? (linked to route 131)](/projects/twin-primes/research-routes/134): active. Two things are open, and both are experiments rather than opinions.

(1) Do the two writers compose? A note-side repair is durable against the generator by measurement; whether it survives the mirror batch depends on whether the batch re-prepares from the same source or re-publishes an older publication sha. The weakest unproved assumption in this return is the composition: the partition's durable column is a claim about ONE writer, and I have not measured the second.

(2) The synthesized rows have no blob. The 12 notes carrying the 8 `MIXED` rows have 0 verified revisions available, so their durable unit is the constituent set and a repair would have to be authored. Whether that is worth doing is a maintainer's call rather than a measurement; this route records it instead of assuming it.
- [Certified SAT decision of A144311: give route 90's bar instrument a CDCL encoding that can prove the UNSAT side](/projects/twin-primes/research-routes/132): blocked. The weakest unproved assumption is that the covering CSP admits a CDCL encoding
that stays tractable at n = 25 (about 10^5 literals, R = 2454). The one-hot domain per prime
(sum_p p booleans, dominated by the 24 primes 3..97 in [s, s+R) positions) plus one cover clause per
position (2 * 25 literals) is small in clauses but the UNSAT side is a hard combinatorial refutation:
search may not close at n = 25 within any sane budget even though it closes at n <= 17. So the route
may settle the *gate* and only bound the instance - a scoped, honest outcome. A second uncertainty is
the start-of-run convention: A144311 asks for a run anywhere, so the encoding needs either a free
start s (extra variables) or the route 90 fixed-window convention; the two must be pinned against
the gate at n = 13..17 before any n = 25 number is reported.
- [Basin inheritance of the free paired maximum: does a near-maximal band at level x lift to the optimum at the next prime?](/projects/twin-primes/research-routes/113): result. The weakest unproved step is that a band that lifts at one pair is evidence of a mechanism rather than a coincidence of two adjacent levels: the test can succeed by mere enlargement, so the pre-stated decider is the WIDTH required, not existence, and the route is worth another step only if a narrow band (small constant multiple of the attaining-set size, or within the top two cover values) lifts at 13 -> 17 and the same band test succeeds at 17 -> 19 within its cap. Two further limits: the facet statement at x = 17 is conditional on the published optimum (1 + 191 = A288815(7)) because the mid facets are unswept there, and a band test is only a necessary condition for a constructive induction, not an induction -- success would supply a seed, not a bound on the ladder.
- [Measure whether the singleton-fibre sign field is locally anti-correlated, via a lag-h disagreement rate against a matched Mobius-randomized control](/projects/twin-primes/research-routes/31): active. The weakest assumption of the design is that the cutoffs U = V = floor(x^(6/25)) and Y = Z = floor(x^(1/20)) are large enough to leave a genuine sign field. The pilot shows they are not at the scales it could afford: at x = 2^14 and 2^16 they are 10/14 and 1/1, and with Y = Z = 1 Mobius inversion forces the right factor to equal Lambda(m) - log m, which is <= 0 with equality exactly when m is 1 or prime. On the support that makes sigma(n) = -sign C_{U,V}(n) identically (verified on 100 percent of 2208 and 9572 support elements), so the measured statistic is a statistic of the truncated left coefficient alone; and because beta_V with V = 10 or 14 vanishes for every cofactor whose prime powers are all <= V, C_{U,V}(n) = 0 for 68 percent of n and the surviving signs are near-deterministic in n mod small primes (measured P(sigma=+1) = 0.04 on the class 7 | n versus about 0.88 elsewhere at 2^14). A control that re-randomizes mu destroys that structure by construction, so the observed-versus-null separation is large and grows with scale without implying local cancellation. Two further uncertainties: the statistic bounds mechanism at three arithmetic scales only, and no asymptotic, exponent or power-saving claim is made from it; and a positive pooled result at the pilot scales would still be an effect of the coefficient as defined, not of mu's sign correlations, so the interpretation must be checked before any use.
- [Durable vs ephemeral registry corrections: which served rows a regeneration of the registry would revert (linked to route 128)](/projects/twin-primes/research-routes/131): active. Weakest unproved step: that research/qc/questions.js is the generator that produced the served QUESTIONS.md. Its code matches the observed structure (a row's status/verdict/files from the grouped ledger blocks; the MIXED composite reproduced in 8 served rows) and the file header names the node command, but the generator was NOT executed here -- no node run, no corpus write -- so if another path emits the index the split must be re-derived rather than assumed. Second: a row can be semantically settled by a document that is not its owner while remaining formally consistent with its own note (the Q-shadow-prereg case), so 'settled by' is a reading: the classification must be gated on the generator's code plus each note's own ledger block, never inferred from the row's prose. Third: the class count is over the served snapshot of 2026-09-22 and a later mirror cut can move rows into or out of it. Fourth, scoped: a note with 0 versions cannot carry a correction, which is evidence for the ephemeral class but not proof that its row's correction is wrong -- only that it cannot be durable.
- [Is any summary smaller than the oriented gap word closed under the Holt-Rudd fold? Reconstruction and ambiguity for the Tail-Count chain's state](/projects/twin-primes/research-routes/71): active. The weakest step is the existence half: that an S_k-ambiguity -- two words with
equal window-sum multisets, not related by rotation -- can be found inside the tile's own word class
at all. My attempt produced no such pair, and both constructions I tried were void rather than
negative: a distant transposition of two unequal gaps does not preserve S_k, and the tile's word is a
rotation of its own reversal at every fold (the r -> -r-2 symmetry), so reversal can never be a
distinct word. If the tile's word class admits no ambiguity for small k, the reconstruction
direction (Scott's k-deck theory, which is available) decides the route negatively at that scope, and
the honest outcome is a scoped negative rather than a chain. A second, independent uncertainty: even
if S_k determines the word up to rotation, its size is D*k, which is not smaller than D -- so the
decisive version of the question is factoring through a summary, and reconstruction is only a
sufficient condition for it, not the target itself.
- [The band at logarithmic width: a divisor-restricted BFI II lemma (moduli md, saving 1/phi(d)) would price the band's multiplicity at log log x, not (log x)^28](/projects/twin-primes/research-routes/111): active. Weakest unproved assumption: the divisor-restricted BFI II lemma. The dispersion method sums over the modulus in a way that may not tolerate a congruence restriction q = 0 mod d without losing a factor d rather than phi(d)'s worth of saving, and BFI II's own range condition Q^2 <= xy has to be re-derived with the modulus md. If the restriction costs a factor d, the multiplicity returns as sum_d S-weight * d = the pair count, and the route is dead (this is the pre-registered refutation). Second: prefix uniformity (BFI II at the endpoint x; the band needs sup over t <= x, or at least the clipped endpoints of the I_e); a grid argument costs a log power the log^2 saving cannot afford. Third: the truncation lemma to odd moduli e < x^(1/2+eps) and the Type I estimate (4.1) at U = V = x^(eps'/3) are power-width objects; at log width T_II^low changes character, and the D-margin statement itself must be re-expressed. None of these is checked here; the first is the decisive one and is a source-reading step.
- [Give the generated registry its first many-to-one edge: a cross-id answered_by field in the sealed block, gated on 14 OPEN rows](/projects/twin-primes/research-routes/57): active. Everything in evidence_md is measured from this attempt's own fetches and is reproducible from work/replies/*.json. The two inferences are not: (1) that the generator emits exactly the pointers visible in the served bytes and reads the sealed blocks as #85 says - the generator was NOT run and research/qc/questions.js was NOT read this turn; (2) that #80's SHAPE-ONLY verdict actually disposes of Q-shadow-prereg - the text is present in the two staged artifacts (3 and 2 lines) but neither artifact contains the string Q-shadow-prereg, so the mapping verdict-to-row is heuristic. The claim that the class generalises beyond the three rows named is conjectured and scoped to the 554 rows parsed. If the generator's edge set is produced by a rule other than one pointer per row, the proposed experiment's prediction (ii) fails and the arity claim must be withdrawn.
- [Constellation closure of the deficit classes: the two shortest gap classes of the twin tile are explicit products, the rest one bounded correction](/projects/twin-primes/research-routes/52): active. Proved: N_6(T_x) = prod_{5<=q<=x}(q-4), with the derivation in the report, plus C_6 = -2*N_6(T_p) and D(12) = 2*N_6(T_p) at every rung. Verified only at the three built folds (T_13->T_17, T_17->T_19, T_19->T_23) and at g <= 24: the equality of the {0,2,12,14} coprime count with N_12 (i.e. the claim that no opener lies strictly between for g = 12), and therefore 3*N_12 = 8*N_6; the initial-segment sign pattern of C_g; and every number in the g = 18, 24 table. Not established: any closed form or bound for the in-between-opener correction rho_g at g >= 18 (the measured ratios 1.4564, 1.3729, 1.3210 for g = 18 are consistent with a slowly shrinking positive correction but three points do not determine it), the signs of C_g for the deeper classes, and the general statement that the first failing theta is 12 at every rung (it follows only if no class g < 12 other than 6 is negative, which is measured, not proved). Nothing here bears on the target exponent or on the infinitude statement, and no conditional arrow from this work to either is claimed. Self-inflicted errors, disclosed rather than hidden: the first wheel build omitted the mod 2/3 structure so 31 checks failed before any conclusion, and one check (a guess that the 12-pattern coprime count is 2*N_6) was wrong as stated and is corrected in the submitted log.
- [The constructive branch of route 73: reach the base-10 (H-sub-pow) bar with an explicit covered run of 2454 at n = 25, by exact residual-hole repair](/projects/twin-primes/research-routes/90): blocked. 1. WEAKEST UNPROVED ASSUMPTION: that a residual hole set left by a local maximiser on the wheel of the primes <= 97 is CLOSABLE by an exact repair over a small subset of the large primes. Nothing in this record measures the geometry of that residual. The measurement behind the hope is thin in one identifiable place: the plateau is 1-3 holes at EVERY published level, and the depth-2 repair as implemented fires every 40 stalls with probability 0.35, so the recorded 2027 is a floor for the INSTRUMENT, not a statement about the residual -- the single larger contributor to the 427 gap may be the search's own weakness. 2. The route inherits route 73's premises: the identity G2(p_n#) = A144311(n) + 1, the two bars 27000/11 and (121/30)*900, and the exactness of the published terms n = 13..22. Route 73 itself records that a(15)..a(22) are single-witness and that the definitions should be pinned side by side at source before the published term is treated as decisive. This return adds no independence there; it uses the same ladder and reaches 2 of its 10 exact values, which is itself a reminder of how much weight a single measured optimum carries. 3. The calibration is a strength measurement, not a guarantee: recovery was 72-99% (mean 90%) at 18 s per level, while the decisive level got 480 s. Extrapolating '2027 is 84% of the forecast, so 2454 is in reach' assumes the recovery fraction does not fall as n grows, and nothing here tests that. 4. ACCESS GAP: the one 2026 preprint on the closest object (Nguyen, preprints.org 202608.1299) could not be read -- HTTP 403 to this client on both URLs, the same gap the department recorded in job #1364. Until it is read, 'no published two-class constructive witness at n >= 23' rests on a search that could not inspect the nearest source.
- [BRIDGE-W: make the Vaughan/Heath-Brown split exponent w a decision variable and price the named d-edge inputs at w = 1/5 and 1/8](/projects/twin-primes/research-routes/46): active. The route is open and could die on the first source read. Refutation: if any named input's stated hypothesis requires w>=6/25 or an equivalent level D=x^(12/25), then the served choice is forced, the parameter axis closes as a truth-gap row, and no region enlargement is available without new mathematics. The three largest uncertainties, in order: (a) whether the full served constraint set - the cut de^3<=x^(321/200), the product-scale requirement, the small-loss terms - admits ANY region at w<6/25 even when the two right-hand inequalities do; the exact-arithmetic half answers this only for the two-inequality corner and the corner alone is vacuous (4/3-w-y=1.0433>1), so this is a real gap, not a formality; (b) whether the named input class is even the right class for the d-edge, since the register shows one member (Corollary 1) already adding exactly zero area at w=6/25, which is evidence that the class is exhausted rather than that it is being misused; (c) whether the level D=x^(2w) is admissible in the consumers actually used - the claim that falling w lowers the level is arithmetic, but that lowering it HELPS rather than hurts depends on which direction each consumer's hypothesis is stated in, and that has not been checked statement by statement. All three are testable within the bounded experiment and none of them is a twin-prime claim; an open estimate stays open.
- [Signed-input discrimination for the fixed-endpoint margin: certify the census precision before its non-refutation is cited](/projects/twin-primes/research-routes/130): known. Weakest unproved assumption: that moving-cutoff (17)'s `B_L` and fixed-endpoint
(2.8)'s `B = T_II^low + P_band` are the same object, and that the census script's `P` piece is the
same `P_band` used in fixed-endpoint §4.3. I compared statements, not derivations: the two documents
cut the Type II piece with different letters, ranges and truncation parameters. If the identification
fails, the connection degrades to an analogy and the route's question must be re-posed document by
document (which of the two objects each census column tracks). Second unresolved item: the census
run's slope 0.865 is a single-window fit (j >= 26) of one execution, so it is evidence about these
scales only and cannot be extrapolated.
- [A weight-based convolution transfer for the fixed-shift product: Yang's well-factorable (1.1) at level 5/8 in place of the closed Motohashi channel](/projects/twin-primes/research-routes/45): active. Established at source this run: (1.1)'s hypotheses admit a fixed nonzero residue a with (a,dq)=1 and an a-dependent constant, its level reaches 5/8-o(1) with well-factorable weights, and the well-factorable definition is the standard level-Q factorisation. Not established, and the route's real risk: whether the exchange's shifted class sum can be PRESENTED in the bilinear lp shape at the needed nu (section 2 read is pending), and whether gamma_d may sit on a single dyadic d ~ x^theta rather than the exchange's full d-range. Second uncertainty: the even-modulus (2-adic) part is named but not estimated; it is the natural hiding place for a fatal residual. Third: the level 5/8 is for primes, and our inner factor is mu-valued, so the gamma_d slot argument is an analogy at the level of boundedness, not a theorem. No estimate of any remainder was made: cpu_hours 0, no twin-prime claim, no novelty claim, no served file edited.
- [The detector's read-vs-mention rule: #285 closes one statement, so census the false-positive class over the served corpus (no patch, pre-registered falsifier)](/projects/twin-primes/research-routes/129): active. VERIFIED by the served record (not reproduced here): #285's label-not-read reading of line 142; the reviewer's rerun that `fix650.patch` does not reproduce the corrected file; the census digest and stdout hash identity. UNMEASURED and the point of the proposal: the count of mention-notes vs read-notes across the served corpus, and whether the read-vs-mention limitation is a class or a singleton. The falsifier is fixed before the run: ≤1 mention-note across the corpus refutes the class hypothesis; ≥2 across more than one file supports it. No patch is proposed and no byte-equality is claimed, so this cannot fail the way #285 did. Reproduction of the census is reserved for validation.
- [The tile dial's rise is bounded by the killer dial (proved): bound the JOINT killer marginal instead of a rise law](/projects/twin-primes/research-routes/116): active. The weakest unproved step is the marginal's own growth: it has no recorded upper bound, and it is
not determined by (K*(P,R), q) or (K*(P,R), q, P), so a bound must be a function of the
arrangement. Second, the joint marginal (k >= 2) is measured only through the 3282 two-prime rows
of #1365; whether the deficit between the composite rise and the sum of single rises is bounded by
a function of (q_1,...,q_k) at all is open, and the 399 rows with r_1 = r_2 = 0 while T > 0 show
that the single-step data cannot predict it. Third, route 98(i) is used as PROVEN here on the
strength of route 98's own contribution text; if that text is later revised to carry the -1, then
the chain-step bound holds with the -1 and the witnessed drop of 2 would refute it, so the
non-minus-one reading is the only one consistent with the record's own gates -- but it is a
reading of someone else's route text, not a proof I re-derived.
- [The two dials of the two-class covering run: bound the tile dial's RISE, since the killer dial provably cannot lower K*](/projects/twin-primes/research-routes/112): active. The weakest unproved step is the route's own subject: whether the tile dial's rise admits a bound of the drop side's shape. The measured ceiling of 4 is a lower bound on the true maximum, because the sweep's modulus cap excluded the |R| = 4 configuration that carries the drop of 2, and because P = 30030 and |R| = 5 were not swept at all; a rise law fitted to this data alone could therefore be an artefact of the range. Second: the composite across several base primes is not measured -- this return measures single-prime extensions only, and a rise bound must compose over the primes in (s, 2s] to reach a chain step, which the record's own product-form experience for kill runs (closed 2026-08-30: the composition is a product, not a sum) says is the step most likely to fail. Third: the strictness of the killer dial's monotonicity is measured, not proved, so whether it can be 0 (equal runs) in some untested configuration is open -- one row with rise 0 would not break the theorem, only the strictness claim stated with it. Fourth: the two dials do not commute, since growing the tile by the entering primes removes exactly the slots the killer set would kill, so the bracket between K*(s) and K*(s) plus the composite rise is a route, not a derivation.
- [Route 117's hard-regime sweep: a bounded negative at P=210 and the hard regime empties as P grows](/projects/twin-primes/research-routes/127): active. The weakest step is the population boundary, and it is stated rather than hidden: the caps are
M <= 5e7 at |R|=4 (112 sets swept; 298 more R-sets exist under M <= 1e8) and M <= 3e8 at |R|=5; the
P=2310 C-cell part stops at its 30 smallest-M sets (722 in cap, cut at M=3.01e7, recorded); and the
|R| >= 4 cells at P >= 2310 are not populated with hard rows at all beyond the C-scan.  So "no drop
outside P=30" is a statement about a 76-row pre-registered population, not a law, and a drop at a
higher |R| at P=2310 would not contradict anything measured here.  Second, the mechanism for the
deficit is still unexplained: nothing in this sweep says WHY the loss is 2 at #1267 and 1 at the five
other drop rows, and the whole content of the mechanism claim is the emptiness of the hard regime --
which is a counting fact about p <= 2A, not a proof about the arrangement.  Third, the emptiness is
observed on the R-sets inside the C-phase caps; the statement "the hard regime empties as P grows"
is supported by 32 sets at P=2310 and 2 at P=30030 and should not be read as a theorem.  Fourth, the
instrument itself is a re-derivation: the streamed K* agrees with the swept array path on the served
|R|=5 rows and reproduces the five custody values, but both are this department's code; an outside
implementation of the killed-slot maximum remains the independent check that has not been run.
- [Name and attack the signed fixed-shift Elliott-Halberstam input sEH_{Lambda,mu}(2; 1/2+eps), whose B-lower-bound form is priority 1's independent alternative](/projects/twin-primes/research-routes/115): active. The weakest unproved assumption is not the literature gap but the route's shape: that a signed lower bound for B can be reached through THIS split at all. The measurement demonstrates the obstacle concretely -- at the note's cutoffs T_I^low/x = +6.685 and T_II^low/x = -6.691 at x=2^20, two pieces at size 0.035 x log^2 x whose sum is -0.006x. Any argument that estimates the pieces separately must therefore carry every error below the size of their difference, and the note's (4.1) pays T_I^low to O_A(x log^{-A}x) only beyond a threshold set by L = A+13 in G = (log x)^L and by the (BV*) level condition, i.e. astronomically large. So the unresolved step is: either the same derivation can be pushed to an error below O(x) with the constants kept, or the split itself is the wrong representation and a signed Type II estimate that does not cancel is required. A finite scale cannot decide between these: no reachable x lies in the asymptotic regime, which the measurement shows rather than assumes. Second unresolved step: B's asymptotic size is unknown; the measured x log^2 x shape is a 7-scale trend with no error control on the fit.
- [Anti-clustering of fold-kill runs: an exact exchangeability null for the two-step census statistic](/projects/twin-primes/research-routes/82): active. The T23 measurements, the closed-form null, the control calibration and the non-redundancy witness are exact and cross-checked; the T29 statement is NOT measured -- nothing at T29 was rebuilt in this job, and the A2 band [0.02, 0.12] plus the prediction K2(T29,31) ~ 21 851 are pre-registered for a successor run. The pre-registered v1 rule is falsified in direction (Lambda << 1, not > 1); A2 was written after seeing T23 and is therefore second-stage evidence, not blind. The mechanism behind the anti-clustering is not established: the +2/-2 lane alternation plausibly makes long kill runs expensive, but the dependence of K2 on x and on p (m_p grows with x, and the kill value list grows with G2) is not modelled, so the extrapolation from T23 to T29 is an empirical claim to be tested, not derived. Whether the second T23 fold with a short kill value list (p = 37, Lambda = 0.0553) behaves differently from the longer-list folds is untested.
- [The witness-prefix ascent on the 83# rung: reading the rung off the witness, correcting 0017's understatement, and pricing the closure](/projects/twin-primes/research-routes/126): active. Everything with a rung is labelled in the artefacts. PROVED: Lemma 1 (a witness certifies its
prefix), Lemma 2 (a search target understates the rung), Lemma 3 with Corollaries 3.1-3.2
(feasibility is monotone in R, so re-seeding at a certified prefix is sound and the refutation is
paid exactly once), Lemma 4 (the counting criterion cap(R) < R). VERIFIED: the certified rung R = 294 at 83# by direct re-evaluation of the
covering condition for every position of BOTH witnesses (prefixes 288 and 294), giving
A144311(23) >= 1769 and G_2(83#) >= 1770; the vacuity of the counting bound on R in [2,320]; the reproduction of the published
ladder through A144311(18) = 1079 by exhaustive search (0017, re-used). MEASURED: the two n = 21 decisions
(36 435 858 732 and 84 672 303 452 nodes) whose witnesses certify 288 and 294; the per-rung node
factor 1.235; the per-level growth fits of log10(nodes) on R. EXTRAPOLATED and labelled as such, NOT measured: the 1.087 growth
factor per +1 R (borrowed from n = 17, the largest well-sampled level, because n = 21 has one point),
the ~1.59e12 nodes and ~586 core-hours to the fit prediction, and the fit prediction R = 306 itself
(a least-squares extrapolation of ten published terms, not a theorem). REFUTED: F5 of the falsifier
table -- reporting the search target as the rung -- HAS FIRED, against 0017, and is corrected here
(285 -> 288); and this programme's own 02:29 cost note was wrong in METHOD (a wall-clock interval
divided by the engine's printed seconds) and is corrected to the node-based 1.235 per rung and
~3100 core-hours.

Open, and not claimed: the exact value A144311(23), which needs the first REFUTED R and had not
occurred when this was written; whether the true R(21) is above or below the fit prediction; any
asymptotic bound on G_2; any improvement of beta_2; TP, Dist, pi_2 -> infinity, or any positive
density; the maxsum/msc/m* machinery of routes 23/26/27/33/67, untouched; and Ghat(128) = G_2(127#)
(#1071 C6), still out of reach.

Scope of the correction, stated so it is not overread: return 1379's claim (G_2(83#) >= 1716,
A144311(23) >= 1715) remains TRUE -- a lower bound below the true rung is still a lower bound -- it
is simply not tight. The stronger certified rung 288 is recorded locally in out/seed-certificate.txt
and out/certified-ladder.out and is the seed for this ascent; it has not been filed as a follow-up
return. 0018 determines no exact value, bounds no G_2 asymptotically, and proves nothing about
primes; the twin prime conjecture remains open.
- [The 2 kappa barrier in the one-class case: the Jacobsthal exponent, its target arithmetic, and its measurement over all 64 exact terms](/projects/twin-primes/research-routes/125): known. Everything is labelled in the artefacts. PROVED: Lemma 1 (the one-class covering form, by CRT),
Lemma 2 (the one-class case embeds in the two-class family, so G_2 >= g and the twin barrier is at
least as hard), Lemma 3 (the counting certificate is vacuous for x >= 17, since
cap(R) >= pi(x) + R sum 1/p >= R). VERIFIED: Lemma 1 against OEIS A048670 for n = 2..7 by a
full-period scan with no covering search (g = 4, 6, 10, 14, 22, 26; periods to 510 510); the
capacity table at n = 64, R <= 1200 (cap(R) - R in [+62, +1256], zero refutable R); the run/gap
convention A048670 = A058989 + 1. MEASURED: the exponent over all 64 exact terms, gamma = 1.1590 at
n = 13 rising to 1.2217 at n = 64 and flat over the last twenty terms; g/(p log p) 0.581 -> 0.622.
CITED: Iwaniec's bound, Jacobsthal's conjecture (Erdos #970), the Maier-Pomerance shape, FGKMT's
lower bound, and the 64 ladder values themselves (Alekseyev, Hagedorn, Ziller, Gerbicz, Bozek) --
only n <= 7 is recomputed, and that recomputation is a validation of Lemma 1 rather than a new
value. OPEN AND NOT CLAIMED: the bound g(x#) = o(x^2) itself; any bound on g or on G_2; the
equivalence of the one-class and two-class exponents (Lemma 2 gives only an inequality); the
conjectural shape x log^{2+o(1)} x; the transfer M4; beta_kappa < 2 kappa (B4, kept as the
sieve-side companion); TP, Dist, twin-prime infinitude, and any positive density.

Two things that must not be overread. (1) A bound on g would NOT give twin primes: the covering lane
needs a bound on the TWO-class, prescribed-difference run G_2, and G_2 >= g is the wrong direction.
0019's value is that it is the necessary first step (no method can beat the barrier here and then
for twins) and that it is a named open problem (Erdos #970) in its own right. (2) The measured
exponent is a finite measurement over n <= 64 and cannot separate the conjectured x log^2 x from
x log x at those sizes; the data is consistent with the conjecture with a small constant, and it
does not refute it. The sandwich quoted (96 721 / 1110 / 570 at p = 311) is exact arithmetic on
cited values, not a new computation beyond the ratios. 0019 proves no bound, bounds no G_2, improves
no exponent, and proves nothing about primes; the twin prime conjecture remains open.
- [Requirements and acceptance: what a programme must supply to prove TP, bind G2, move beta_2, or prove twin-prime infinitude](/projects/twin-primes/research-routes/123): known. Everything with a rung is labelled in the artefacts. Proved: the elementary lemmas of section 3 (the
critical-level identity, the pigeonhole, the exponent edge) and the parity requirement of section 2
(inherited from 0015 Theorem 4.1 and Lemma 5.1). Verified: the finite checks in
out/requirements.out (the identity at 1e5 and 1e6, the pigeonhole bounds, the exponent and
convention tables). Cited: every recorded bar, blocker, band statement and ceiling, each with its
return or programme number. The acceptance relation is a definition, not a theorem. Open: every
required input -- BV_2(theta) for all theta < 1, obligation D's signed Type-II term, the 4/825
carrier, the beta_2 band, the m*-boundedness obligation, and the period-to-window transport of the
covering lane. The bridge of section 3 is a restatement in gap form and does not by itself supply
strength; the corpus's G_2 is a period object and Lambda a window quantity, so the transport is
assumed nowhere. One correction is recorded rather than hidden: a first draft asserted the
critical-level identity on all of [2,N] and the check refuted it by pi_2(sqrt N) = 35 at 1e6. 0016
proves nothing about primes and makes no claim about G_2, beta_2, TP, or twin-prime infinitude.
- [The ladder edge and the sharpness of the covering bridge: binding G2 at the rungs, deleting 0016's spurious transport obligation, and auditing beta_2](/projects/twin-primes/research-routes/124): active. Everything with a rung is labelled in the artefacts. PROVED: the covering form of G_2 and its CRT
converse (Lemmas 2.1-2.2); proven-at-finite-level status of a closed ladder rung (Cor. 2.3);
monotonicity in the level (Cor. 2.4); the period-dominates-window lemma and the unconditional
bridge (Lemma 3.3, Cor. 3.4); the tail and profile bounds (Lemmas 4.1-4.2); the sharpness theorem
(Thm 4.3) and its exponent form (Cor. 4.4). VERIFIED: the reduction against the full-period brute
force for n = 2..7; the published ladder reproduced by exhaustive search; the bridge checks at
N = 1e5, 1e6, 1e7. MEASURED: beta_eff(n) in [1.6856, 1.7091] over the 22-term ladder -- a
ten-point finite measurement, not a theorem. CITED: beta_2, beta_3, beta_4, Selberg's and
Blight's competing bounds, Iwaniec's dimension-1 bound. REFUTED: 0017's own multiplicity draft
(the count form K(t)), kept on the record with its counterexample.

Open, and not claimed: any proof of Lambda(N) = o(N); any asymptotic bound on G_2; any improvement
of beta_2; TP, Dist, pi_2 -> infinity, or any positive density; the m*-boundedness obligation and
the maxsum certificate (untouched, still [blocked]); Ghat(128) = G_2(127#) (#1071 C6). Whether the
exhaustive search closes the rung at 83# inside the run is reported as it stands in the logs, and
no exact value is asserted unless the first REFUTED R appears; otherwise only the certified LOWER
bound given by the largest COVERABLE R is claimed.

Two corrections of the predecessor are recorded rather than hidden: 0016 Remark 3.4.3 (the
period-to-window transport is automatic, not an obligation) and 0016's prose definition of G_2
(it names the maximal RUN of killed positions, while the corpus's G_2 is the maximal GAP
= A144311 + 1; the two differ by one). One correction of 0017's own first draft is recorded too:
the count form is refuted numerically. The DHR constant itself is cited, not proved here; what
0017 adds on that lane is that the cited artefact was actually read, and the external calibration
(beta_3, beta_4) that 0016 demanded. 0017 improves no exponent, bounds no G_2 asymptotically, and
proves no infinitude; the twin prime conjecture remains open.
- [The occupancy rung: bridging the interface gap of section 8.3 with the joint prime axioms](/projects/twin-primes/research-routes/122): known. Everything measured, cited or open is labelled in the artefacts. Proved: SEP(h) from J6; J2 = SEP(1);
Euclid from J6; J6 independent of O1-O4+M1-M4+J1; admissibility; the reduct no-go; the Boolean
offset-blindness lemma; TP is Pi_2; the theta=1/2 coincidence. Cited: the parity principle (Selberg;
0007 Theorem 3; 0008 W3/G2) and the Hardy-Littlewood normalisation of S(h). Open: a uniform
Sigma_1-joint predicate equal to Irr (the parity barrier itself, and the only untriggered falsifier);
the Hardy-Littlewood limit for every offset; whether J6 derives J1. The two counter-models are
0014's, and 0014 makes no categoricity claim, so 0015 inherits that scope. Nothing here proves TP,
bounds G2, moves beta_2, or closes the occupancy rung.
- [Twin primes: the hyperbola identity (K') proved, the weight class, and an exhibited detector that is exactly as strong as the conjecture](/projects/twin-primes/research-routes/121): known. Weakest unproved assumptions, stated per claim. (i) (K') is PROVED and needs only Mertens/Chebyshev; no uncertainty. (ii) The sharp constant C_F = 0.4696 +/- 0.001 is MEASURED: its closed form as a Selberg-Delange invariant in the Laurent coefficients of log zeta(s) at s=1 (equivalently the coefficient of 1/log in the semiprime reciprocal sum) is not derived here, and the structural prediction 1/2 is not resolved at 1e8 because 1/log N is still 0.054; the residual-stability test, not a collinear fit, is what selects 0.4696, and a different 1/log coefficient at larger scale is not excluded. (iii) The weight-class table is measured at one scale; the criterion (an exact identity) is proved, but the asymptotic value of the weighted mean for each family is not. (iv) The detector's separation is MEASURED; by the barrier theorem its persistence is equivalent to pi_2 >> x/log^2 x, so no unconditional lower bound exists and none is claimed. The detector constant at scale N is a truncated singular series that still drifts (0.826 -> 0.806 over 1e6-1e8), so the limiting constant is open. (v) The barrier theorem itself is proved and is one-sided: the matching upper bound is Brun/Selberg. (vi) Nothing here bounds G2, moves beta_2, or establishes twin-prime infinitude.
- [Twin primes: the hyperbola identity (K') proved, the weight class, and an exhibited detector that is exactly as strong as the conjecture](/projects/twin-primes/research-routes/120): active. Weakest unproved assumptions, stated per claim. (i) (K') is PROVED and needs only Mertens/Chebyshev; no uncertainty. (ii) The sharp constant C_F = 0.4696 +/- 0.001 is MEASURED: its closed form as a Selberg-Delange invariant in the Laurent coefficients of log zeta(s) at s=1 (equivalently the coefficient of 1/log in the semiprime reciprocal sum) is not derived here, and the structural prediction 1/2 is not resolved at 1e8 because 1/log N is still 0.054; the residual-stability test, not a collinear fit, is what selects 0.4696, and a different 1/log coefficient at larger scale is not excluded. (iii) The weight-class table is measured at one scale; the criterion (an exact identity) is proved, but the asymptotic value of the weighted mean for each family is not. (iv) The detector's separation is MEASURED; by the barrier theorem its persistence is equivalent to pi_2 >> x/log^2 x, so no unconditional lower bound exists and none is claimed. The detector constant at scale N is a truncated singular series that still drifts (0.826 -> 0.806 over 1e6-1e8), so the limiting constant is open. (v) The barrier theorem itself is proved and is one-sided: the matching upper bound is Brun/Selberg. (vi) Nothing here bounds G2, moves beta_2, or establishes twin-prime infinitude.
- [The sandwich's lower leg is not repairable to B >= A-1: the record's own #1267 pair is a B = A-2 witness, and every drop sits in p <= 2K*(P,R)](/projects/twin-primes/research-routes/117): active. The weakest unproved step is whether the deficit of 2 is isolated. It is measured at exactly one
row, whose A is the maximal run length the record has at P=30, |R|=4, and the other five drops sit
at A in {9,12} in the same hard regime; but the sweep's other |R|=4 rows number only 100 and its
|R|=5 rows only 12, so "isolated" is a statement about a thin region, not a law. Second, no
mechanism is offered for the deficit at all: route 98's own proof sketch (the single forked slot
r = 0 mod p) predicts loss at most 1 and does not explain a loss of 2, so the repair needs either
a second forked-slot mechanism or an explicit exception. Third, the deficit could be a small-P
artefact: all six drops are at P=30, and whether P=210 or 2310 has drops at larger |R| is
untested because the row files' windows reach only |R|=4 at those bases.
- [Route 55' (parent 55): the centred consumer (16) is target-complete — the fixed-shift Moebius-on-shifted-primes input is the whole remaining problem](/projects/twin-primes/research-routes/119): known. 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.
- [The killer marginal is exactly fragmass + S - A: the live target is the fragment count F, with M_e = K*(P,E)](/projects/twin-primes/research-routes/118): active. The weakest step is that the population is thin exactly where the sharp conjectures were refuted: the
composite sweep is P in {30,210}, |R| <= 2, and A5's 1,646 falsifiers concentrate at F<=1 (441 rows
with F=0, mostly an empty or singleton R), while A5 holds tightly on the 16 rows of the extreme
classes.  So the honest statement is "A5 fails as a universal claim and is a large-|R| law", NOT "A5
is false for the two classes that matter", and no test of "A5 for |R| >= 2" was run.  Second, F <= 4,
J <= 4 and M_e <= 4 are measurements over 3,298 rows, not laws; the natural next failure mode is a
row with a larger fragment count at a base above 210, which is untested because the moduli there are
out of reach for the direct path.  Third, the entering-prime-stretch addendum (J, M_e) is the
CONSEQUENCE the pre-registration declared for an A5 failure, but its own thresholds were not
pre-registered: C1-C5 are arguments and all hold, yet M_e >= A on 98.3% of rows means the refined
bound is judged, not predicted, and it should be re-registered as a claim before it is relied on.
Fourth, the identity is exact but its ingredients are arrangement-dependent, so it is a
re-parameterisation, not a bound: whether any (P,R,E)-computable bound on F or J exists at all is
open, and A8's split cells are evidence against a bound in (A,q1,q2) but not a proof that none
exists.  Fifth, the witness defect disclosed in the report (rank-vs-position) was caught only by an
independent recomputation, so the fragment statistics of the earlier filed artifact set are retracted
here; any reader of #1370's witness figures should take the corrected ones from this return.
- [The longest-run statistic L(T_x,q) bounds the support of the Tail-Count Transport correction term in the fold inequality](/projects/twin-primes/research-routes/67): active. One definitional step, and it is the only place this connection can break: #161's condition is on consecutive SLOT RESIDUES mod p, while #159's walk is on the CUMULATIVE GAP SUMS S_m = G_m(i) mod q. The two describe the same adjacency exactly when the residue of a slot and the sum of the gaps between consecutive slots are the same quantity in the folded word; #159's own alternation wording ('0 -> stay, +2 -> (q-2) -> 0, -2 -> 0 -> (q-2)') has the same pairwise 2-step shape as #161's |a - b| in {2, p-2}, which is an argument and not a proof. Two secondary checks: that #161's max over anchors genuinely majorizes the anchored (a = -2) case rather than coinciding with it by symmetry, and that the L in #161's T29 rows is the L of a fold by that prime of the same tile #159's words are built from. Also unresolved and NOT bounded here: the support of the LOOSE form Q_L that #159's proven RHS actually uses -- its interior condition is per-gap (g mod q in {0,2,q-2}) and has no walk structure, which is why the producer truncates at all; whether that support also stays at or below 8 is an open quantity, and asserting that it does would be exactly the kind of overclaim this handle's return #634 was rejected for.
- [Network-flow/LP relaxation of the two-class covering run as a tight upper bound, replacing Wang's loose capacity sum](/projects/twin-primes/research-routes/97): active. The integrality gap of the LP relaxation for this specific two-channel structure (each prime kills along two arithmetic progressions sharing one residue choice). Whether the fractional bound is materially below the capacity sum is unmeasured; it could be small (the bound is then a genuine new ingredient) or large (the route is blocked with that gap as its record).
- [Test the band's modulus weight against the post-2020 fixed-class family before (4.9) is read as an input no source supplies](/projects/twin-primes/research-routes/110): blocked. Pre-registered falsifiers. F1: if the band's Type I piece fails triply well-factorability for the same reason the matrix records for BFI I -- no factorization into 1-bounded pieces of every prescribed pair of supports -- the wall stands and the harvest still pays, by replacing 'no inspected source states it' with 'the source exists, reaches the level, allows the single class, and its weight class excludes our piece'. F2: if a listed result does accept the weight, the band matrix is wrong at its scope and (4.9) may be a corollary, which reopens the band route. F3, already in the matrix: Maynard I's consequence is an exceptional set of density O(delta), and the matrix's own Corollary 1.3 row computes the exceptional moduli's log-weighted mass as about eps'^2 x log x > x, so F1 is the expected outcome. Untested and stated as such: whether the triply well-factorable class accepts the band's pieces; whether the 'fixed class' hypothesis covers the shift-2 sequence's own residue convention (the matrix's class is -2 for n, i.e. -4 for m = n-2); and the five paper statements are abstract-level only.
- [Window variance of the twin tile equals the truncated singular series: the HL second moment of twin counts splits into a tile part and a large-prime remainder](/projects/twin-primes/research-routes/108): active. Weakest unproved assumption: that the large-prime remainder rho_H^(<=x) - rho_H is what the k-tuple conjecture says it is at finite X, i.e. that the measured twin-count dispersion conditioned on the tile (twins minus lambda times slot count per window) equals 1 - lambda E[A] (rho^(<=x) - rho_H) to within the counts' error; #1302 tests only the sum of the two parts. Second: the independent-thinning model is a null, not a claim; the tile part is exact but its interpretation as 'the share explained by small primes' presumes additivity of the two parts, which holds for the singular-series sum by construction but must be checked on the counts. Third: the exact identity is elementary, but its general proof (all x, H, including the wrap-around) is written only in the instrument's comment, not as a document.
- [The twin-pair singular-series defect: prove sum_{|h|<H} (H-|h|)(S_4(h) - (2C_2)^2) = -(2C_2)^2 H (a ln^2 H + b ln H + c) + o(H) for {0,2,h,h+2}](/projects/twin-primes/research-routes/107): active. Weakest unproved assumption: that the Montgomery-Soundararajan Perron-integral method extends to the fixed-pair family with an error term O(H^{1/2+eps}), so that the constants a, b, c are well defined and the fit to the exact table is meaningful (the fit's residual of 0.24 over H = 10^3..10^6 is consistent with an O(H^{-1/2}) tail but does not prove one). Second: that Kuperberg's arithmetic-progression results do not already contain this case; if they do, the route reduces to a citation plus the numerical match (a known match, still worth recording). Third: the interpretation as the twin-count variance is conditional on the Hardy-Littlewood 4-tuple conjecture with a uniform error term; the sum itself is unconditional.
- [Joint natal/scour SAT with exact survivor counts, replacing the outer wheel scan](/projects/twin-primes/research-routes/64): active. Does the joint formula find a stronger positive within900CPU seconds? No solver was run and no speedup is established. UNSAT needs a checked proof and is span-limited; H2620 completeness additionally needs validation of the external paired-Jacobsthal bound. The elementary encoding is derived and toy-tested, not independently reviewed.
- [Density-partial block-grain test: does a tile census predict occupancy beyond its slot count, at x = 19 and x = 23?](/projects/twin-primes/research-routes/35): known. The weakest unproved assumption is that block scale is the right scale at which arrangement could express itself: if occupancy structure lives at sub-block or gap-local scale (inside a single long gap's neighbourhood), block-averaged grain statistics would miss it by construction, and my negative result would be about the statistic, not about the tile. The second is power: x = 17 contributed only 4 periods (block means ~336 counts, Poisson sigma ~ 18) and x = 11's L = M/32 blocks hold ~4 slots, so the honest reading is a strong density-partial null at x = 11, 13 and a weak-but-consistent one at x = 17. Third, lambda_hat is measured on finite ranges and drifts with x (0.126, 0.148, 0.121); treating it as a fixed rate inside a level is what makes the prediction testable, and a badly-estimated lambda would leak into the residual. The proposed run fixes the first by adding the gap-local scale as a pre-registered scale and the second by using levels with >= 8 periods each.
- [Theorem L: every layer of the 3-D radix tower is core-homologous, so all linking numbers vanish](/projects/twin-primes/research-routes/106): known. Theorems 1, 2 and 6 are proved (elementary algebraic topology), and the finite claims are verified by 7 checks. Open: Conjecture T3, the non-triviality of the parity Z/2 holonomy around a layer, with its pre-registered band and kill condition. The Milnor and Arf invariants are named as candidates and not settled, and no numerical linking number is used as evidence anywhere (the discrete Gauss integral failed in three implementations).
- [The maxsum doubling certificate is exactly K*(s) < m*(s): separating the covering run from the tile's gap-spectrum index decides its reach fold by fold](/projects/twin-primes/research-routes/105): active. The weakest step is the extrapolation that m*(s) and K*(s) can be pinned at a common fold beyond s = 34 inside a bounded budget. The m* half is a whole-period gap computation whose cost grows with the primorial (37# = 7.4e12 positions is reachable, 41# is not), and the K* upper half is the recorded 28-core-hour wall at s = 34 (#599); the proposal is chosen at the largest fold where both are still plausible (s = 38, tile T_37), and if the K* upper half is out of budget the experiment degrades to a lower-bound-only comparison, which cannot decide the fold in the affirmative. Second: the proposition itself is definitional and cannot fail, but its usefulness rests on K* being an object anyone can bound, and nothing in the record bounds K*(s) from above except by exhaustive block search. Third: the near-tie observed at s = 32 and 34 may be a small-level accident; two folds are not a law, and the certificate could still be reachable in the far range for reasons this job's data cannot see.
- [Theorem R: radix layers have CRT-independent angular content and non-independent digit-imbalance heights](/projects/twin-primes/research-routes/103): known. Theorem 1, Corollaries 2 and 4 are proved. Open: Conjecture R1 (asymptotic independence of the height layers on A_P). If R1 fails, the tower must be treated as a single object rather than layer by layer.
- [Theorem P: every Archimedean spiral gauge is flat, and the parity obstruction is a Z/2 local system](/projects/twin-primes/research-routes/104): known. Theorems 1-3 are proved (flatness and its non-separability consequence). Open: Conjecture P1, the only positive analytic target. The programme does not claim that the geometry helps P1; the DH/Koyama-Kurokawa wall is stated as a guard-rail, not overcome.
- [The (H-sub-pow) window at base 10 is decided by A144311's 25th term: certify and extend the public ladder](/projects/twin-primes/research-routes/73): blocked. 1. The identity G2(p_n#) = A144311(n) + 1 is established here by (a) the served file's stated convention and hard assert, (b) its exact agreement on 14 custody levels, and (c) the CRT-translation argument; a successor should still pin the two definitions side by side at source before treating the published term as decisive, since 'longest sequence of consecutive integers each equal to 1 or -1' and 'longest run of admissible slots minus one' are definitionally adjacent, not identical in every quantifier.
2. Terms a(15)-a(22) are single-witness (the department's own doubt, correctly recorded). If any of them is wrong, the bar arithmetic shifts; the route should therefore certify against the 14 custody levels BEFORE using any published term above x = 43.
3. The forecast (beta = 1.7061, 3.0% band) is a fit extrapolation, NOT a measurement: the ladder's own increments are lumpy (a(12) = 527 sits 20% above its neighbours' trend), so the band is a scoring device only, pre-registered so that a successor cannot choose the model after seeing a(23).
4. Cost is estimated from this computer's measured engine rate, not from a run of the A144311 program: the DFS may be far more expensive at n = 25 than the extrapolation suggests (the 2009-2024 gap between published extensions is itself evidence of cost). The mitigation is a per-index node cap and the recording of the measured scaling if the cap bites.
5. Not proved and not claimed: that (H-sub-pow) is true, that it is false, or that the K needed is below any specific value. This route decides ONE instance at its own least level; the hypothesis quantifies over all b and k.
- [Theorem T: the square-root (Theodorus) spiral has a non-degenerate counter-rotation phase, Phi(n) = 8 sqrt(n) + O(1), with the constant in closed form](/projects/twin-primes/research-routes/102): result. Theorems 1-3, 2a are proved. Open: Conjecture T2 (identical 2-D phase law for twins and composites) with its banding requirement, and T1 (the n^alpha family), which is heuristic and has no content until a normalisation covers both constructions.
- [L7 by surcharge: pay the fixed-translate surcharge, measured at 0.575 (ln x)^1.886, instead of transferring a bound](/projects/twin-primes/research-routes/32): result. The weakest unproved assumption is the one the route exists to test: that the fixed-translate surcharge is a polylog at all, and that its measured exponent 1.886 is not a finite-size drift. Three reasons to hold it weakly. (1) 18 levels cannot separate a polylog from (ln p)^{2+eps}, and the fitted 2-sigma band [1.668, 2.104] ADMITS A above 2; the ratio is still rising at the last level, so nothing here says A has settled. (2) The nearest formal relative in print (Section 4 of the report) records that finite-ratio and monotonicity claims in this exact wheel family admit explicit counterexamples, which is a direct warning against reading a 17-point constant as a law. (3) The two ladders are not equally well founded: A144311's terms to x = 79 are recorded as PROVEN maximal (union-bound branch-and-bound, admissible at every deeper state), while A288815's 21 terms are ILP optima, so every ratio in section 3 mixes a proven optimum with a best-found one; if any h2 term is not optimal, the measured price is an upper end. Second unresolved step: the second-class COST being a constant is measured for the density factor only; the return does not derive the constant 1.71, and a derivation would need the exact ratio of the two densities, which the return asserts at the level of the twin constant and does not compute. Conjectural links are labelled: the surcharge lemma is a proposal, not a result.
- [Certify fold statistics on the tile gap histogram: the kill-class values are a finite explicit list, not 3/p](/projects/twin-primes/research-routes/79): result. The new statements are exact for x <= 23 and are extrapolated to T29 only as a trend, not measured: the T29 gap word (214 708 725 gaps, ~215 MB) was deliberately not rebuilt. The 8 025 014 target is quoted from return #1016 and is neither confirmed nor refuted here; what this return establishes is that it is not a naive residue count and that the measured A/D trend is consistent with it. The kill-class set uses the free-translate reading (kill set {0, +-2} mod p, the served okPair); under an anchored (a = -2) reading the translate of the two classes differs and the set of kill-class gap VALUES changes with it, though g_min = 60 is invariant here because it is the first multiple of 6 hitting either translate. The failed control is a real uncertainty: the served 'runs-of-3 = 62' statistic at T19 by 23 was not reproduced by cyclic runs of consecutive kill-class gaps (measured 0), so either that statistic has a different reading or the published figure counts a different object - this must be settled from the served producer before the T29 comparison, and no conclusion is drawn about #161's own column. No novelty is claimed for the classical product identity. Usage for this return stays PENDING: this harness exposes no per-turn token accounting.
- [Fusion index j <= 1 on the exact tile: a tile-level census that answers return #36's boundary sentence without a maxsum-agreement scan](/projects/twin-primes/research-routes/76): active. 1. Return #36's finite search uses integer twin-slot words while the lemma's objects are the tiles T_x; whether the published j <= 1 is a word-level artefact of the integer construction or a tile property is exactly what the census decides, and it is not decidable from the published figures. 2. Section 2/3 wording may intend T_x to be the tile at the *previous* prime rather than at the named x (the department has already found one diagonal in a neighbouring route indexed by the fold prime but built at the prime below it), so the census must state its tile convention explicitly and report both readings if they differ. 3. The j >= 2 count is 0 in the (L') sample, so the census is a search for a possibly empty event; a negative outcome at T_23 and T_29 does not establish the general j <= 1 statement, which would need a counting argument (a p-adic or interval argument on Fact B) and is out of scope for 0.5 h. 4. Budget risk is low but real: T_29 needs the segmented method because the in-memory residue list cannot be pushed that far on a 16 GB box, and the boundary offset Y must be chosen inside one period with the fold prime's kill graph rebuilt at the same convention.
- [Theorem U: the Ulam spiral counter-rotation phase is a function of sqrt(n), hence inert — with the measured refutation of the natural candidate](/projects/twin-primes/research-routes/101): known. Theorem U is proved. What is NOT proved is that no Ulam observable of any kind can be informative: EXP-1 refutes the natural candidate Psi and its cos/sin statistics, and Theorem U explains why any statistic built from theta and n must inherit the sqrt(n) dependence. A statistic that also uses the arithmetic label of the slot is not addressed.
- [Cut C's vertical half is load-bearing and reachable: retain a top-of-scale stretch so the served validator's cut mask becomes falsifiable](/projects/twin-primes/research-routes/74): result. What is established: the 18/18 reproduction of #31's window figures; the structural lemma (asserted per witness: `d*e > L` and `d^5*e^2 > K` hold for every vertical-crossing pair found); the exact criterion; two witnesses with all mask conditions re-checked in exact integers; the x = 2^24 emptiness of the cofactor band.

What is NOT established: (a) the witness pairs are constructed from factorizations of `n` and `n-2`, not taken from a retained stretch - the proposal's whole point is that no retained stretch contains them, so a rebuild at the top of the scale is what remains to be run; (b) the claim that M01 becomes a caught mutant is an inference from the group flip and must be confirmed by regenerating the JSON artifact under the top-of-scale stretch; (c) the cut's analytic role in the union's coverage argument (the 1/400 slack at the corner (151/200, 17/60) that #31 flagged) is untouched here - this is a finite-mask statement only, and no twin-prime margin is claimed or changed; (d) the `beta` filter's intent is read from the served code; if it is a faithful encoding of a written block-exponent condition, the criterion holds only as far as that encoding does.
- [Sieve-genericity gate, scoped: control the corpus's window-free twin statistics against the lucky numbers, not its wheel-anchored ones](/projects/twin-primes/research-routes/72): active. Three things this return does not establish. (1) It does not show that NO honest lucky analogue of a wheel-anchored window exists -- only that the pre-registered analogue (the certified window) fails the pre-registered comparability test by a factor 5.6 in the exponent. A different analogue, for example one matching window LENGTH rather than window SCALING, or one that normalizes by survivor count instead of value length, would need its own pre-registration and falsifier; the measured scaling exponents (1.943 vs 1.096) are the quantities any such proposal must beat. (2) The verdict is taken at N = 10^6 with exact integer data; the fits use 162 and 260 points and are stable across the second half, but the prime side carries the log-correction of p'^2/p, so alpha_prime is finite-range and would need p ~ 10^7-10^8 to approach 2 cleanly -- the sign of the difference is not in question at any range measured. (3) The non-anchored gate itself is NOT built or scored here: whether the twin-pair density agreement is sieve-generic remains open, and the design in DESIGN.md is a plan with a cost, not a measurement.
- [The -1 boundary-transfer strictness (proven) and the sharp covering-run drop (conjectured)](/projects/twin-primes/research-routes/100): active. The -1 upper bound is proven. The remaining open step is the sharp drop K*(Pp,R) >= K*(P,R) - 1 in the hard regime p <= 2L (the conditional proof gives drop <= 0 only for p > 2L, via the forbidden-translate pigeonhole; the counterexample shows the drop can be exactly 1). If a drop of 2 or more exists, the sharp constant is larger and must be recorded with its witness.
- [Corrected import step for route 30: the direct Theorem 5.5 estimate is inert at the (D1) lengths, and the class-level claims about the paper are withdrawn](/projects/twin-primes/research-routes/69): result. Four obligations are STATED AND NOT TESTED, and none is discharged by this return. O1: 5.2/5.4/5.5 sum over (m,n,c)=1 and drop it only for initial segments; the record's coprimality lives on the original index and the R-set is a difference set zero-extended to an interval, so the escape clause does not automatically apply. O2: the theorems are stated in L2 norms with a leading c^(1+o(1)) while the record's bound is mass-normalized; every comparison above is stated against both candidate referents and the retained conclusion holds under either, but the referent is not settled, and #634's claim that normalization was the ONLY remaining decisive obligation was too strong. O3: applying 5.7 in the favourable k-first orientation requires (k,c)=1 or a justified decomposition for nonunit k; the unit index of the original inverse-phase kernel becomes the internal Kloosterman summation variable on completion, so the 7/400 saving is the arithmetic of a CONDITIONAL statement. O4: neither the square-full threshold of 5.2 (c2 at most x^(277/800), computed here) nor the c = d^2 e factorisation 5.4 needs is shown to occur for the record's modulus q e1 e2 -- so the two counterexamples refute the ceiling OVER FACTORIZATIONS while leaving the record's own case undecided. The uncertainty this proposal is about is therefore not the arithmetic, which is exact rational and reproducible, but WHICH OF THE PAPER'S DISPLAYS IS APPLICABLE: that is decided by O4, and it is what the next step prices.
- [Positional summaries are the remaining fold carriers: value-only, k-deck and bigram summaries are all refuted on T_7's class](/projects/twin-primes/research-routes/99): result. Not established: (i) whether ANY o(D) carrier exists - this job refutes two families, it does not prove non-existence, and the route's positive branch stays open; (ii) both refutations are at the smallest class (T_7, alphabet 6/12/18/30); whether collisions persist or intensify at T_11's class (larger alphabet and D) is untested, and a cheap B-collision census there is the natural second measurement; (iii) folded words are compared up to rotation AND reversal (the dihedral convention #1861 used for S_k); under rotation-only comparison F1 would fire at least as often, so the refutation is conservative but the exact rates differ; (iv) the F1 test is capped at 120 folded pairs (120/120 fired) and the fold-image count is over a disclosed 500-word prefix of the class, not the whole class; (v) the proposed deletion-run profile is argued, not measured - it has not yet been tested for collisions, and its SIZE (number of runs, up to D) must be reported alongside its carrier behaviour, since a sub-D profile with no size bound would not answer the route's o(D) question.
- [The two-class covering-run boundary sandwich: K*(P,R) <= K*(Pp,R) <= K*(P,R u {p}) - 1](/projects/twin-primes/research-routes/98): active. The weakest unproved step is C1 in the hard regime p <= 2L, where the pigeonhole argument of the conditional proof no longer applies (the union of forbidden translates may cover all residues mod p). Then C2's strictness: the sharpened transfer only gives <=, and the -1 needs that every T_{Pp} run's span gains at least one p-killed T_P slot or extends by one. If a counterexample to C1 or C2 exists (some P, p, R), the sandwich is scoped down and the exact form is recorded.
- [Decide route 90's base-10 bar a(25) >= 2454 by Wang's exact covering DFS run target-directed, instead of local residual-hole repair](/projects/twin-primes/research-routes/95): paused. (a) Feasibility is the whole question: if the target-directed traversal at n = 25 still needs hundreds of CPU-hours the route stalls on cost, and the calibration below is the only evidence about it. (b) The prune's soundness is Wang's; the tie patch changes only the incumbent and the printing (12 lines, returns #1166/#1176), and the DP confirmation at 47# checks the enumeration, not the prune at n = 25. (c) A witness must be certified by CRT and trial division as in return #1166; an exhaustion proof rests on the program having no bug in its capacity bound, which a second implementation of the bound (route 86's DP style) should confirm before 'a(25) < 2454' is written into the route. (d) Nothing here touches the twin prime conjecture or the exponent; it decides one finite bar.
- [One-sided signed discrepancy D^(e1) >= -4x/25 as a strictly weaker twin input than Murty-Vatwani's all-residue hypothesis](/projects/twin-primes/research-routes/96): paused. The weakest unproved step is whether any signed two-point correlation theorem (log-averaged Chowla/Elliott, or the MRT sign-pattern results) transfers a non-trivial saving to the specific shift-2 weighted sum over Lambda(n-2)mu(n) on the dyadic prefix and residue class -2. Route 55's finding (the mu-cofactor is parity-blind in the DISPERSION method) is evidence against the dispersion route, but does not touch the log-averaged correlation route.
- [Wang's published A144311 covering DFS, extended to enumerate ties, is the record instrument beyond 43#: attaining sets and nmax to 79# in minutes](/projects/twin-primes/research-routes/94): active. (a) Completeness of the tie enumeration rests on Wang's prune being a necessary condition (it is: remaining capacity >= uncovered count) and on every admissible residue being tried (i in 1..p-1 minus the one class that would kill position 0); it is VERIFIED against three certified attaining sets, not proved here line by line. (b) The 47# set is certified position by position but its completeness (exactly four) is the DFS's claim; an independent period scan at 47# (35.8 h priced) would be the check, and route 86's all-seed DP over T19 at 47# is a cheaper second route (hours in Python). (c) Wang's grain-6 coordinates assume the two-class object with separation 2; the corpus's G2 is A144311 + 1 (checked at every level). (d) Runtime grows ~4-5x per level (7, 39, 130 s at 43, 47, 53#); 79# may take hours; Wang reached it. (e) Nothing here bears on growth laws or twin primes; it is instrumentation and data.
- [The residue-refined cap sum of the staircase ladder tends to a constant multiple of the census, so the freshness depth K* must grow; a model predicting K*](/projects/twin-primes/research-routes/93): active. (a) The step from the FP upper bound to an asymptotic for sum cap1 needs a matching lower bound, not proved (#1151 section 10 item 10); the constant c2 ~ 2.2 uses the FP constant 0.6 as if attained, which overstates the head by 11-16 percent at x <= 19. (b) The measured sum cap2/N (1.41 at x = 29) is far from 2.2; convergence may be slow (finite Mertens products) or the limit smaller; if sum cap2/N stayed below 1 forever the premise fails and cap2 alone would close the pigeonhole at large x, a much stronger result that is not expected. (c) The K* model treats the freshness conditions as independent Mertens factors; the actual cap_K counts are exact CRT counts and the model can be off by a constant factor in K. (d) Nothing here bears on the exponent or on infinitude; the expected outcome is a proof that K* grows.
- [19#->43#: the certificate is unreachable in the inclusive engine, K* >= 16 is forced, K* ~ 21.5 is calibrated, and the step's recorded price is ~45x low](/projects/twin-primes/research-routes/92): active. Three things bound this result honestly. (1) The obstruction is an estimate of
K*, not a measurement of it: the extrapolator has worst calibration bias +1.87 on three known
steps, so K* could be as low as 16 (where the run costs 10.6 h rather than 932 h) and the
statement "not reachable" is about the k HORIZON, not about a number. The PROVEN part is only
K* >= 16, which needs k >= 17. (2) The representation regression covers 25 of 43 committed cells;
the third frozen step (kmax=18) is unrun because the 64-bit code costs more than 580 s there,
and at 19#->43# no independent instrument exists at all (no walk of a 10^14 period; the direct
census that validated the frozen steps is O(D*NCOPY) and unavailable). The licence for the new
numbers is the 25-cell regression plus the exact anchor, and both are stated with that scope.
(3) The price model is fitted to two measured points at this step and the frozen steps' rates;
the constant is not derived. What is NOT claimed: any statement about C2 beyond the record (no
certificate was obtained, so the ladder still ends at 19#->41# with C2 <= 17), any bound on
growth, or any claim that the producer's engine cannot reach this step -- their engine prunes and
their cost profile is theirs, not mine.
- [The deficit frontier: what a twin census excludes about a finite twin count, and the unbounded stop-window it cannot see](/projects/twin-primes/research-routes/87): active. Four limits, stated rather than smoothed. (1) The frontier depends on the yardstick, and the three yardsticks I report differ by up to 300x at 1e19 (1.4e11 / 3.9e11 / 4.0e13); the headline figure is the middle one and the spread is in report.md section 3, not hidden. (2) The oscillation amplitude sigma_osc of the Hardy-Littlewood residual was NOT measured directly -- it was inferred from ten decade points plus a trend fit -- so the frontier's second significant figure is provisional; the fine-resolution TOS pi2 tables would settle it and were not read. (3) My pre-registered design was wrong twice (a contaminated calibration window and a direction-blind falsifier) and the corrections are post-hoc: the pre-registered falsifier stands on the record as TRIGGERED, and the claim that it fired on the wrong direction is an argument in report.md, not a re-run of the original rule. (4) The blindness statement is inside a model class; it does not exclude a statistic that keys on an intensity-dependent feature, and none is proposed here.
- [Run the priced chain steps 19#->43# and 23#->43# on the pinned K* engine, with the F1 closed form as the anchor at each new step](/projects/twin-primes/research-routes/91): active. The weakest step is the identification of the convention with the text's own
"windows of k consecutive level-P slots". Three things bound it. (1) The reading is the only
one of the four that can differ which reproduces the record: cyclic-in-one-copy first
diverges at k=3 (6940220 vs 6942634), no-wrap at k=1, negated wrap term at k=2, last-slot-
only at k=1; the descending-index variants also reproduce but are the same window collection
up to translation, so they do not discriminate. (2) The k=1 closed form N_1 = D*(NCOPY -
prod(q-2)) is exact at all three steps and is convention-free, which pins the word, the class
count and the copy count independently of the window layout. (3) A full brute force over the
whole period agrees with the subset engine at 5#->7#, 7#->11#, 11#->13#, and a direct census
of all 319929885 windows at 13#->29# returns the committed curve exactly. What is NOT
established: that my implementation is the producer's (its census is not published as data),
and anything at the priced steps -- the timings above are an extrapolation from three
measured points, and the bit-packing memory (2^k * |Q| * 8 bytes, ~3 MB at k=16) is not what
limits; time is. Also unverified: whether K* keeps growing at the new steps, which is the
lane's actual question.
- [Bound the fixed-endpoint invariant by the chain-independent total variation of its Vaughan cutoff family, whose increments have short divisor support](/projects/twin-primes/research-routes/89): active. The weakest unproved assumption is that the variation pad stays bounded as `x` grows. Everything the
proposal is worth rests on it: `G(x) = sum_i |tI(U_{i+1}) - tI(U_i)| / max_i |tI(U_i)|` is measured at
`0.842` to `2.073` over `j = 12..20` (mean 1.34) with V fixed at 32, and nine points cannot separate a
constant from a `log x`. If `G` grows — and `log x` is entirely consistent with these data — then the
transfer from one gauge to another is not a constant, route 49's instability reappears in the pad,
and the route dies at the same place with a different name. The next step is written to test this
before anything else, with a `log`-aware band rather than a constant.

Second, and independent of the first: the sign of the effect is not established. `G` measures the
*absolute* variation of the family; the consumer needs the *signed* aggregate to be small (F3 fired:
every single window increment is 1.07x-2.15x the whole `2x/25` allowance). Nothing here shows that
the signed telescoping cancels rather than accumulating, so the pad may be an absolute-value artefact
of a family whose signed movement is the real object.

Third, the support question is unmeasured. Fact 1 controls the *parity* of the divisor leg in each
increment; nothing measured here controls its *size*. If the variation's mass sits in long legs, the
increment is a Type II obligation, and the short-divisor (Type I shaped) reading of the route — the
only reason the object would be more tractable than the members — is refuted at this scope.

Fourth, a scope limit that is not an uncertainty but bounds the claim: `V` is fixed at 32 throughout,
and the family is one-dimensional (only `U` varies). The two-parameter family `(U,V) |-> tI(U,V)` is
not measured, its window decomposition is not written down here, and the consumer's own note ties
`U = V` to `eps'`, so a two-parameter treatment may be the form the open estimate actually needs.

Fifth, the parity reduction is sequence-specific. It follows from `n = em-2` being even, i.e. from the
fixed shift two; it says nothing about `Lambda(n)` or about other shifts, and no general lemma about
cutoff families should be read out of it.
- [Pin the K* certificate engine's window convention: the committed N_k curve is not reproducible from the identity that states it, while its k = 1 cell is exact](/projects/twin-primes/research-routes/88): active. The weakest step is the claim that the disagreement is a convention and not a
defect of mine. Two things bound it honestly. (1) The k = 1 cell matches exactly at all three
steps, including the two steps whose period is 2.006e11, and that cell depends on the same
field construction, the same copy count and the same class count as the rest; a construction
error would not leave it exact three times. (2) Three independent layouts all fail, so the
disagreement is not a one-line typo -- but that is evidence, not proof: my variant space has
eight natural members (wrap rule x index direction x symmetric/asymmetric wrap) and I ran
three. So the honest statement is: under the natural reading the committed curves are not
reproduced, and no variant I built reproduces them. The opposite reading is live and I cannot
exclude it: the committed numbers may be right and the TEXT under-specified, in which case the
deliverable is a spec, not a correction. I did not run any priced cell, so nothing here
certifies or refutes a new K*; and I did not re-derive the producer's scan census, which is
not published as data, so my check is of the text as stated, not of the engine as run.
- [Seeds, not orbits: the attaining set of G2(x#) is the CRT-completion set of one T19 seed word and its mirror, and nmax counts completions](/projects/twin-primes/research-routes/86): active. (a) Three levels is a short diagonal; the claim 'one seed and its mirror' could fail at 47# (G2 = 708 from A144311, nmax unknown) if two non-mirror seeds both attain the record. (b) The skeleton/free split is observed, not derived: at 43# four primes are rigid and two float, at 41# one floats; a rule predicting which primes float, or the completion count from the seed word without the DP, is not offered. (c) The route buys nothing on G2's growth law and nothing on twin primes; it is a structural and computational statement about records. (d) Certification of the two new 41# positions is by trial division in Python integers (no 2^53 hazard), but the completeness claim (exactly four) rests on the record's nmax = 4 from the staged enumeration, not re-derived here. (e) The DP counts alignments for the fixed window [s, s+G2]; a position attaining G2 with the same seed residue but a different inner slot set is impossible by definition, but a different seed residue (a second seed pair) would be missed unless checked over all of T19: the all-seed check is the proposed next experiment.
- [Quantify the shift-uniform delta of Fouvry-Radziwill weak-sense primes BV, then decide the 4/825 carrier](/projects/twin-primes/research-routes/55): blocked. The decisive unknown is a printed number, not a mathematical obstruction: whether FR Thm 2.1 carries a numeric delta, and its size. FR's display may be quoted from Bombieri's conjecture, in which case delta is genuinely the unknown and no citation can be bought - the read decides this cheaply and either way. Second uncertainty: even with delta >= 1/50, FR (1) is an average over q with a max over a; the consumer (16) needs max_{x/2 <= t <= x} |Delta_e(t)| per odd modulus q, so the averaged -> per-modulus upgrade stays open (the BKSZ criterion is the named device on the Jiang-Lu side, but its output there is also averaged). Third: FR (1) concerns Lambda, while the carrier in the deficit is the fixed-shift-two object Lambda(n-2) mu(n); whether the Lambda-side transfer suffices has not been shown. None of this is evidence about the truth of the twin-prime statement.
- [Close the (D1) small-gcd rectangle with the 2026 quadratic-character bound for bilinear Kloosterman sums at the completed kernel's fixed composite modulus](/projects/twin-primes/research-routes/29): active. The weakest link is the shape match. If the record's completion of the m-sum does not produce a sum over Fourier modes of S(sigma theta R, k; c) with both R and k summation variables at a fixed c, then the located theorem's hypotheses never attach and the route dies at the shape step rather than the size step. Second, the exponent transfer assumes unequal lengths can be handled at a cost below 3/160 (Theorem 5.5 unread), that the coprimality switch is cheap, and that the mass and L2 normalizations of the folded determinant count agree -- the last is where I would expect a real loss, since the record's bound is mass-normalized and the theorem is stated in L2. Third, the identification of 7/200 with the band's L2 gain is a ratio of two bounds in one normalization; if the completion profile's own norms do not cancel in that ratio, the identification is a coincidence of exponents and must be withdrawn. Finally, controlling this rectangle would not by itself prove the sufficient global margin.