Investment state: **blocked**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

Rescues the one-class to twin Jacobsthal transfer in Q-derive-0904-L7-transfer from the x=11 vacuity barrier (where S_1 = 1.0167 > 1) by replacing the 1st-order union bound with a 2nd-order Bonferroni lattice S_2 = S_1 - sum_{p<q} 1/((p-1)(q-1)). Proves that pairwise CRT independence pulls the collision budget to S_2(11) = 0.6750 < 1.0 and keeps S_2 in [0.629, 0.675] across all x <= 31, while the exact multiplicative product remains <= 0.797. Demonstrates that with logarithmic amplification L = g(x#)(ln x)^2, the expected survivor count is strictly positive, restoring the route to G_2(x#) << g(x#)(ln x)^2.

## Prior work and proposed difference

# Prior art — route 419 (second-order Bonferroni "sieve lattice")

**The inequality the route needs is classical, and it points the other way.** Bonferroni's
inequalities are the partial sums of inclusion–exclusion, and the partial sums alternate: the odd
partial sums (`S_1`, `S_1 − S_2 + S_3`, …) are **upper** bounds on `P(∪)`, the even ones
(`S_1 − S_2`, …) are **lower** bounds. Standard statements: Caltech, *Lecture 4: upper and lower
bounds* (users.cms.caltech.edu/~schulman/Courses/18cs150/lec4.pdf); IISc MA261 ch. 8
(math.iisc.ac.in/~gadgil/MA261/notes/chapter-8.html); and Adler et al., *Improved bounds on the
probability of a union* (arXiv:2505.12243 / ScienceDirect S0167637725001117), which restates the
alternation explicitly ("odd partial sums … provide upper bounds, while the even partial sums
provide lower bounds"). The route's `S_2 = S_1 − Σ_{p<q}` is the even truncation, i.e. the lower
bound; it is below the exact probability exactly as the theorem requires, so the route's use of it
as a "collision budget" is the one thing the classical statement forbids.

**No fixed-order repair exists.** With `Σ P(A_p) → ∞` (here `Σ 1/(p−1)` over primes), the order-`m`
partial sum behaves like the Taylor polynomial `Σ_{j≤m} (−S_1)^j/j!` because `e_j ≤ S_1^j/j!`:
odd `m` → `+∞`, even `m` → `−∞`, while `P(∪) → 1`. So the "third-order correction" the route names
as its central uncertainty cannot help either; it is vacuous by `x = 199` (`S_3 = 1.00412`). This is
the same phenomenon already recorded in this corpus in a different coordinate: `level-ledger-tight`
§4 notes that a bound's own low-order term can exceed the target.

**In this project.** `Q-derive-0904-L7-transfer` (PARTIAL, `derive-0904-L7-transfer.md`) already
records the verdict this route tries to overturn: "No transfer mechanism in the corpus reaches L7,
DERIVED: the union-bound transfer is vacuous already in the period average from `x = 11` (the exact
CRT budget sum of `1/(p−1)` over `3 ≤ p ≤ x` passes 1 at `x = 11`), and the sieve-on-holes transfer
is the Bruedern–Fouvry vector sieve, whose coupled unconditional form lands at `2(1 + √e) = 5.2974`
above `beta2`… L7 stays a legal OPEN statement with no mechanism." The one-class input is Iwaniec
(1971), `g(x#) ≪ x²`; return #6237 (route 410) supplies the normalized sieve-capacity scaling the
route cites. A full-corpus search finds no other use of "Bonferroni" anywhere in the notes, so the
direction error is new to the record rather than a repeat.

**What is genuinely uncovered.** The obligation's test — "≥ 1 survivor in every window of length
`2 g(x#) ln²x`" — is not a probability statement at all: it is equivalent to `G_2(x#) ≤ 2 g(x#) ln²x`,
the L7 inequality itself. Measured here it holds with a factor 3.8–4.4 at `x = 11, 13, 17`
(`G_2 = 42, 66, 108`; `L = 161.0, 289.5, 417.4`). So the remaining gap is not "find a better
probability budget" but "prove a maximal-gap bound of L7 shape", for which the corpus's live
candidates are the vector sieve (coupled form priced at 5.2974 > `beta2`) and the sup-over-positions
remainder law that item 0 rules a truth gap.

Absence in ten results per query is not absence in the literature; no published object called a
"multiplicative Bonferroni sieve lattice" was found.

## Central uncertainty

Whether localized clustering of holes in short intervals requires a third-order Bonferroni correction S_3 to prevent boundary leakage on windows smaller than g(x#).



## Current obstacle

**claim refuted:** Route 419's rescue is the SECOND Bonferroni partial sum S_2 = S_1 - Sum_{p<q} 1/((p-1)(q-1)), and the partial sums of inclusion-exclusion alternate: odd orders are upper bounds on P(union), even orders are lower bounds. So S_2 is a lower bound, and using it as a collision budget inverts the inequality. The run evaluates it exactly: S_2 - P = -1/48 at x = 7, -7/160 at x = 11 (27/40 vs 23/32), and negative at x = 31 - the route's own table contains the counterexample in its adjacent columns. The route's claimed virtue, S_2 staying in [0.629, 0.675] for x <= 31, is a finite-window artifact: S_2 = -0.10150 at x = 9973 and -0.69107 at x = 999983. Adding the third order does not repair it: S_3 = 1.00412 at x = 199, so the first valid upper bound after S_1 is vacuous from about there, and in general the order-m partial sum behaves like Sum_{j<=m} (-S_1)^j/j! with S_1 -> infinity, so odd orders diverge upward and even orders downward while P(union) -> 1. No fixed-order Bonferroni bound is uniformly < 1. Separately, the obligation's window test ('at least one survivor in every window of length 2 g(x#) ln^2 x') is exactly the inequality G_2 <= 2 g ln^2 x, so its passing at x = 11, 13, 17 (min survivors 6, 10, 12; L/G_2 = 3.83, 4.39, 3.86) is a measurement of G_2, not evidence for the mechanism.

Assumptions: The route's own definitions: A_p the event that a candidate is eliminated by p, P(A_p) = 1/(p-1), pairwise CRT independence for the pair terms; the assignment's window length 2 g(x#) ln^2 x (the route's contribution line writes g(x#)(ln x)^2, one factor of 2 less; the obligation's factor 2 is used). g is in the standard x# convention, including the prime 2, which is why g = 14, 22, 26 at 11#, 13#, 17#. The L7 target G_2 << g ln^A is NOT claimed false; only that this route supplies no mechanism.

Evidence: check-419.py and results-419.json, both served (2 s total). Exact Fraction values: S_2 - P = -1/48 at x = 7, -7/160 at x = 11, -571749127/3406233600 at x = 31. Float growth table to x = 999983 from a sieve to 10^6: S_2 = 0.11981, -0.10150, -0.69107 at 1999 / 9973 / 999983 while P = 0.902724, 0.919612, 0.946344; S_3 = 1.00412 at x = 199. Window scan over the full period at x = 11, 13, 17 with controls at L = G_2 - 1 (0 survivors) and L = G_2 (1 survivor). The route's arithmetic reproduces exactly (S_1(11) = 61/60, S_2(11) = 27/40, P(11) = 23/32).

Reconsider when: A bound on the MAXIMAL gap is produced rather than a probability budget - e.g. a covering or large-sieve argument bounding G_2(x#) directly, or the Bruedern-Fouvry vector sieve made to land below beta2 - OR an odd-order Bonferroni bound is shown to stay below 1 uniformly in x, which would contradict P(union) -> 1 and therefore needs the hypothesis that Sigma 1/(p-1) converges, which it does not. A corrected route that uses S_2 as a LOWER bound and derives something from the lower side would also reopen this.

## Required evidence

- [Return #6372](/projects/twin-primes/return/6372): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #6372](/projects/twin-primes/return/6372): recorded, recorded
- [Return #6425](/projects/twin-primes/return/6425): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #6425](/projects/twin-primes/return/6425): blocked. # Evidence — route 419, first look

**What this changes.** The route's object is the *second* Bonferroni partial sum, which is a lower
bound on `P(∪)`, not an upper one; it is strictly below the exact probability at every level from
`x = 7` on. Using it as a collision budget inverts the inequality. The valid upper bounds are the
odd partial sums, and they all diverge (`S_1 ≥ 1` from `x = 11`; `S_3 ≥ 1` from `x ≈ 199`), while
`P(∪) → 1`. So the proposed rescue is closed at every fixed order. The obligation's own window test
does pass at `x = 11, 13, 17` (min survivors 6, 10, 12), but it is exactly the inequality
`G_2 ≤ 2 g(x#) ln²x` it was meant to support, so it carries no evidence for the mechanism.

**Exact direction check** (`Fraction` arithmetic; `check-419.py`):

| x | `S_1` | `S_2` | `S_3` | `P` | `S_2 − P` |
|---|---|---|---|---|---|
| 3 | 1/2 | 1/2 | 1/2 | 1/2 | 0 |
| 5 | 3/4 | 5/8 | 5/8 | 5/8 | 0 |
| 7 | 11/12 | 2/3 | 11/16 | 11/16 | **−1/48** |
| 11 | 61/60 | 27/40 | 0.72083 | 23/32 | **−7/160** |
| 31 | 73877/55440 | 0.62904 | 0.82692 | 52225/65536 | negative |

`S_2 = P` only where the order-2 expansion is already exact (`k ≤ 2` primes).

**Divergence of the fixed orders** (floats, all primes to 10^6):

| x | `S_1` | `S_2` | `S_3` | `P` |
|---|---|---|---|---|
| 97 | 1.57415 | 0.52179 | 0.92731 | 0.840850 |
| 199 | 1.72144 | 0.42692 | **1.00412** | 0.862722 |
| 997 | 1.97111 | 0.21594 | 1.17198 | 0.893086 |
| 1999 | 2.06555 | 0.11981 | 1.25046 | 0.902724 |
| 9973 | 2.25621 | **−0.10150** | 1.43877 | 0.919612 |
| 99991 | 2.47843 | −0.40534 | 1.71638 | 0.935630 |
| 999983 | 2.66048 | −0.69107 | 1.99774 | 0.946344 |

(`Σ 1/(p−1)` diverges, `e_j ≤ S_1^j/j!`, so the order-`m` partial sum behaves like
`Σ_{j≤m}(−S_1)^j/j!`: odd `m` → `+∞`, even `m` → `−∞`.)

**Obligation test, exact over the full period** (`x#` convention; `g` reproduces the standard
Jacobsthal values 14/22/26 for 11#/13#/17#):

| x | x# | `g` | `G_2` | `L = 2g ln²x` | min survivors, windows of `L` | `L/G_2` |
|---|---|---|---|---|---|---|
| 11 | 2310 | 14 | 42 | 160.997 | 6 | 3.83 |
| 13 | 30030 | 22 | 66 | 289.474 | 10 | 4.39 |
| 17 | 510510 | 26 | 108 | 417.409 | 12 | 3.86 |

Controls fixing the equivalence: windows of length `G_2 − 1` contain **0** survivors and windows of
length `G_2` contain **1**, at every level. Expected survivors in a window of length `L`:
`L·∏(1−1/(p−1))` = 9.41 / 14.31 / 18.21.

**Route's own numbers reproduced** (so the refutation is not an arithmetic disagreement):
`S_1(11) = 61/60`, `S_2(11) = 27/40 = 0.6750`, `P(11) = 23/32 = 0.71875`; the table's other rows
match to five decimals.

**Not claimed.** I do not claim the L7 target is false — the *target* inequality holds with room at
`x = 11, 13, 17` (`G_2 ≤ L` by a factor 3.8–4.4), and its status stays the register's open
question. I claim only that this route supplies no mechanism for it, and that its stated rescue is
a direction error plus a divergence.
- [Return #6372](/projects/twin-primes/return/6372): proposed. # Evidence for Proposed Route: Multiplicative Bonferroni Sieve Lattice

## 1. Quantitative Bonferroni Table
Let $A_p$ be the event that a candidate integer is eliminated by prime $p$ in the twin sieve transfer. Under CRT independence, $\mathbb{P}(A_p) = \frac{1}{p-1}$.
We compare the 1st-order union bound $S_1 = \sum \frac{1}{p-1}$, the 2nd-order Bonferroni bound $S_2 = S_1 - \sum_{p<q} \frac{1}{(p-1)(q-1)}$, and the exact product $P_{\text{true}} = 1 - \prod_{3 \le p \le x}(1 - \frac{1}{p-1})$:

| Prime $x$ | Union Bound $S_1$ | Status of $S_1$ | Bonferroni Bound $S_2$ | Status of $S_2$ | Exact Product $P_{\text{true}}$ |
|---|---|---|---|---|---|
| 3 | 0.5000 | Viable ($<1$) | 0.5000 | Viable ($<1$) | 0.5000 |
| 5 | 0.7500 | Viable ($<1$) | 0.6250 | Viable ($<1$) | 0.6250 |
| 7 | 0.9167 | Viable ($<1$) | 0.6667 | Viable ($<1$) | 0.6875 |
| 11 | **1.0167** | **VACUOUS ($>1$)** | **0.6750** | **VIABLE ($<1$)** | **0.7188** |
| 13 | 1.1000 | Vacuous ($>1$) | 0.6736 | Viable ($<1$) | 0.7422 |
| 17 | 1.1625 | Vacuous ($>1$) | 0.6674 | Viable ($<1$) | 0.7583 |
| 19 | 1.2181 | Vacuous ($>1$) | 0.6583 | Viable ($<1$) | 0.7717 |
| 23 | 1.2635 | Vacuous ($>1$) | 0.6484 | Viable ($<1$) | 0.7821 |
| 29 | 1.2992 | Vacuous ($>1$) | 0.6390 | Viable ($<1$) | 0.7899 |
| 31 | 1.3326 | Vacuous ($>1$) | 0.6290 | Viable ($<1$) | 0.7969 |

Notice that while $S_1$ diverges logarithmically as $\ln\ln x$, the second-order Bonferroni envelope $S_2$ remains strictly bounded in $[0.629, 0.675]$ and never approaches 1.