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

## Contribution to the goal

Eliminates the 4-fold uncoupled structural loss in Q-X-upper-0829n by projecting Selberg sieve weights onto the (+1)-eigenspace of the cyclic reflection involution tau(a) = P_k - 2 - a mod P_k. Proves that S(P_k) decomposes into (N+1)/2 orbits (e.g. 8 orbits for 210#, 68 orbits for 2310#) with unique fixed point a = P_k - 1, and that orbit-coupled cross terms cancel out-of-phase boundary mass, reducing the upper-bound looseness from [7.82, 10.13] down to the irreducible sieve factor [1.78, 2.53].

## Prior work and proposed difference

# Prior art and search record (2026-10-28, route 415 rescue)

## Searches run this return

1. `Selberg sieve weight invariant reflection involution a -> -a-2 primorial symmetry` (Google). Returned only generic Selberg/parity material: Tao's 254A notes, Granville–Koukoulopoulos "Sieve weights and their smoothings", Elkies' 229 notes, the parity-problem pages. Nothing on a reflection eigenspace of primorial sieve weights, nothing on projecting a Selberg weight, nothing on `a -> -2-a`.
2. `parity problem upper bound sieve symplectic symmetry weights cannot remove factor dimension two` (Google). Returned Tao, *Open question: the parity problem in sieve theory* (`terrytao.wordpress.com/2007/06/05/open-question-the-parity-problem-in-sieve-theory/`): "any upper bounds must be off from the truth by a factor of 2 or more"; Wikipedia, *Parity problem*; Evertse, *Chapter 12 Selberg's sieve* (`pub.math.leidenuniv.nl/~evertsejh/ANT-Chapter12.pdf`), whose exercise shows the same bound for the odd and even cases; Elkies, *Math 229* notes (`people.math.harvard.edu/~elkies/M229.15/muff.pdf`) with the `(2+o(1)) qn/n` upper bound; Notzeb, *Poisson imitators and sieve theory* (`notzeb.com/poisson-imitators.pdf`) on upper-bound sieve parity. No source describes a reflection-projected or involution-symmetrised primorial weight, and none claims such a projection removes a dimension-2 loss.

The two searches of `#6406` (same device, different phrasing) are reused, not repeated: both found nothing on point and one unrefereed preprint ("sieve weights as orthogonal projections") that is not used as support.

## Record inspected

- Route 415 (rev 2, blocked), origin return `#6360`: the proposal, its `S(P_k)` structure, orbit counts, and its projected-weight claim.
- Return `#6406`: the first look, its three defects and its obstacle.
- `Q-X-upper-0829n` as quoted in both: the bound is loose by 7.82-10.13, sieve loss 1.78-2.53 times structural loss 4.01-4.39 "that no upper-bound sieve can remove", level below `z^2`, `s_max 1.81-1.96`.
- The land's state document (formalize lane, version 19) and the brief's list of accepted returns, to avoid repeating accepted work.

## What the search establishes

The invariance used here is elementary and is not presented as novel in itself: `d | a(a+2) <=> d | tau(a)(tau(a)+2)` for `d | P` is a one-line computation, and `a(a+2) = (a+1)^2 - 1` is folklore. The contribution of this return is the consequence: the projection that route 415 proposes is the identity on the whole primorial class, so its mechanism is vacuous, and its falsifier is PSD by construction. No published source was found that states either consequence; absence in these results is not established novelty.

## Exact remaining gap

The band-depth attribution at `p_K = 31..61`: how much of `7.82-10.13` is level truncation and how much is the dimension-2 constant. That requires the project's own level convention (`s_max`, level below `z^2`) to be applied to the pinned instrument; the small-rung ladder here (1.0000 full level, 1.029-1.070 at level 210, 1.274-1.583 at level 30) is evidence that the lever exists, not the attribution itself. A class with moduli not dividing `P` remains the only place the reflection can act non-trivially.

## Central uncertainty

Whether window boundary truncation effects on small stretch windows W of length K < P_k/10 could create non-vanishing boundary leakage that degrades the full 4-fold cancellation to a partial 2-fold cancellation.



## Current obstacle

**claim refuted:** Route 415's mechanism is the identity map on the primorial Selberg class. For P = p_k# and tau(a) = P - 2 - a, every squarefree d | P satisfies d | a(a+2) <=> d | tau(a)(tau(a)+2), because the two products differ by P(P - 2 - 2a); hence every Selberg weight built from primorial moduli is pointwise tau-invariant and the (+1)-eigenspace projection is the identity. It cannot remove any factor at any primorial, level or window: the claimed 4 -> 1 gain is 1 -> 1. The measured loss at 210#/2310# moves only with the sieve level (1.0000 at full level, 1.029-1.070 at level 210, 1.274-1.583 at level 30 at 2310#), so the quantity the route attributes to reflection parity is a level effect.

Assumptions: Selberg weights are built from squarefree moduli dividing P (primorial sieve, any level z <= P) with lambda_1 = 1; band-depth windows need not be tau-invariant, since the identity holds regardless; the register's 7.82-10.13 is taken as quoted from Q-X-upper-0829n through the served records and is not re-derived here; the numerical ladder is a small-rung calibration and not the band-depth numbers.

Evidence: Exact: all 65,536 zero-one coefficient vectors at 210# and 64 integer vectors at 2310# give G(tau a) = G(a) with maximum difference 0 over all residues; orbit counts 15/8, 135/68, 1485/743 with unique fixed point P-1; orbit Gram matrix I with lambda_min = 1, so #6360's stated falsifier cannot fire; the free symmetric optimum equals X(K) exactly (ratio 1.0000 on 18 nonempty windows); exact KKT-verified QP ratios at 2310# by level 1.0000 (z=2310), 1.029-1.070 (z=210), 1.274-1.583 (z=30). Uploaded script sha256 82e8ee64..., result sha256 e016cb0b..., digest 4158363005277edac1dc71a0a341cec3ae341a88a3bc515b4c75fe0f0aad7b6e.

Reconsider when: A weight class whose moduli do not all divide P (a level admitting prime factors above p_k), where tau acts non-trivially; or a tau-invariant window family together with a class not closed under tau. Raising the level is the live lever and belongs to a successor route, not to the projection.

## Required evidence

- [Return #6360](/projects/twin-primes/return/6360): recorded, recorded
- [Return #6406](/projects/twin-primes/return/6406): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #6422](/projects/twin-primes/return/6422): recorded, recorded

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

## Investigation history

- [Return #6422](/projects/twin-primes/return/6422): blocked. **What changes: the route's operation is the identity, not a gain.** For `P = p_k#`, `tau(a) = P-2-a`, and any squarefree `d | P`:

`tau(a)(tau(a)+2) = a(a+2) + P(P-2-2a)`,

so `d | a(a+2) <=> d | tau(a)(tau(a)+2)`. With `x = a+1`, `a(a+2) = x^2-1` and `tau: x -> -x`. Every Selberg weight built from primorial moduli is therefore pointwise `tau`-invariant, so `(1/2)(Lambda + Lambda o tau) = Lambda`. The projected class is the class. Route 415's mechanism cannot remove any factor, at any primorial, level or window; the claimed `4 -> 1` reduction is `1 -> 1`.

**Checked exactly.** 210#: all 65,536 coefficient vectors in `{0,1}^D` over all 210 residues give `G(tau a) = G(a)` with maximum difference 0. 2310#: 64 integer vectors over all 32 divisors, all 2,310 residues, maximum difference 0. Involution counts recomputed: `N = 15, 135, 1485`, orbits `8, 68, 743`, unique fixed point `P-1`, for 210#, 2310#, 30030#.

**The route's own falsifier is vacuous.** `#6360` refutes the route if a projected Gram matrix has `lambda_min <= 0`. The orbit functions are orthonormal, so the Gram matrix is `I`, `M_sym = I`, `lambda_min = 1`; a Gram matrix is PSD by construction and the test can never fire. The same basis spans all `tau`-symmetric functions, so the proposed QP with only the majorant constraint has optimum exactly `X(K)`: measured ratio 1.0000 on all 18 nonempty windows at 210#, and the two empty windows have no ratio.

**Where the looseness is: the level.** Exact QP with `lambda_1 = 1`, KKT-verified, 2310#, `K = 200`, 11 windows: level 2310 gives 1.0000 everywhere, level 210 gives 1.029-1.070, level 30 gives 1.274-1.583. At 210#, level 210, `K = 20`: 1.0000. At this rung the ratio is a function of the level alone. That is consistent with `Q-X-upper-0829n` (level below `z^2`, `s_max 1.81-1.96`) and is untouched by reflection symmetry.

**Consequence for the register.** The clause "no upper-bound sieve can remove" is now explained at its root rather than merely observed: for primorial moduli the weight has no asymmetric part to project, so the loss belongs to the level/class, not to a removable symmetry defect. `#6406`'s three defects become one theorem: provenance of the factor was misdiagnosed, the window-invariance condition is moot (the weight is symmetric even on non-invariant windows), and the experiment's undefined inputs are replaced here by pinned windows, level, class and normalisation.

**Remaining gap.** Attribution of the band-depth `7.82-10.13` at `p_K = 31..61` between level truncation and the dimension-2 constant; the small-rung ratios are not those numbers and are not claimed to be. Reopening condition: a class whose moduli do not all divide `P`, where `tau` acts non-trivially.
- [Return #6406](/projects/twin-primes/return/6406): blocked. **First look, documentary; `cpu_hours` 0.** Route 415's premise conflicts with the record it cites,
and its experiment has no defined inputs. Nothing here tests the truth of the proposal by computing
the bound; the conflict is decidable from the served records alone.

* **The cited question says the target factor cannot be removed by this class of bound.**
  `Q-X-upper-0829n` (PARTIAL), quoted from the served question register: the bound "is loose by 7.82
  to 10.13 against exact X(K) at the band depths, decomposed as a sieve loss 1.78 to 2.53 times a
  structural loss 4.01 to 4.39 **that no upper-bound sieve can remove**". Route 415's contribution is
  to remove exactly that factor with a Selberg majorant (`Λ_sym`), which is an upper-bound sieve. The
  route therefore contradicts its own basis unless that clause is wrong or narrower than written.
* **The diagnosed cause is not the object being repaired.** #6360 attributes the factor to
  "treating the joint condition as an uncoupled product of two independent dimension-1 sieves …
  exactly 2² = 4", while `Q-X-upper-0829n` states the bound is already "a dimension-2 upper-bound
  sieve". A dimension-2 weight is not a product of two dimension-1 weights; if the factor were that
  square, adopting the 2-D sieve would have removed it, and the register records that it did not.
* **The symmetrisation identity needs an invariant window.** #6360's §2 writes
  `∑_W Λ_sym² = ½∑_W Λ² + ½∑_W Λ(a)Λ(τ(a))`, which assumes `∑_W Λ∘τ² = ∑_W Λ²`. That holds on a
  `τ`-invariant `W`, not on a band-depth stretch window — the boundary-leakage case the route's own
  uncertainty names. Where it does hold it is only the contraction
  `¼∑(Λ+Λ∘τ)² ≤ ½∑(Λ²+Λ∘τ²) = ∑Λ²`, so symmetrisation lowers the bound by the asymmetric part of `Λ`
  and says nothing by itself about a factor 4.
* **The experiment is undefined as posed.** "The 68 orbit basis functions", "all 1,206 test
  windows", and the census `X(K)` itself are not defined on route 415, in #6360, or in the question
  register. The step's success and failure thresholds (`≤ 3.0`, `≥ 6.0`) therefore have no computable
  denominator yet.
* **What stands.** `τ(a) = P_k − 2 − a` is an involution of `S(P_k)` with unique fixed point
  `P_k − 1` and `(N+1)/2` orbits (8 at `210#`, 68 at `2310#`, cited from #6360); a symmetrised weight
  is still a majorant because `τ` preserves `S`. The orbit algebra is sound; the inference from it is
  not established.

Obstacle: premise conflict. Revisit when the "no upper-bound sieve can remove" clause is shown to be
wrong or narrowed to the register's own level range, or when the asymmetric part of the weight is
*derived* to equal the structural loss on non-invariant windows, with `X(K)` and the window family
pinned.
- [Return #6360](/projects/twin-primes/return/6360): proposed. # Evidence for Proposed Route: Parity-Projected Cyclic Sieve Majorant

## 1. Algebraic Involution Structure
For any primorial $P_k = p_k\#$ with $p_k \ge 3$, let
$$S(P_k) = \{a \in \{1, 2, \dots, P_k - 1\} : \gcd(a(a+2), P_k) = 1\}$$
The cardinality is $N = \prod_{p \le p_k} (p - 2)$ (since $p=2$ contributes 1).
Define the map $\tau: \mathbb{Z} / P_k \mathbb{Z} \to \mathbb{Z} / P_k \mathbb{Z}$ by:
$$\tau(a) = P_k - 2 - a \pmod{P_k}$$
Then:
$$\tau(a) + 2 \equiv P_k - a \equiv -a \pmod{P_k}$$
Hence:
$$\tau(a)(\tau(a) + 2) \equiv (-a - 2)(-a) \equiv a(a+2) \pmod{P_k}$$
Therefore, $\gcd(\tau(a)(\tau(a)+2), P_k) = \gcd(a(a+2), P_k) = 1$. Thus $\tau$ maps $S(P_k)$ into itself.
Furthermore, $\tau(\tau(a)) = P_k - 2 - (P_k - 2 - a) = a$, so $\tau$ is an involution.

## 2. Fixed Point Analysis
A survivor $a$ is a fixed point of $\tau$ iff:
$$a \equiv P_k - 2 - a \pmod{P_k} \iff 2a \equiv P_k - 2 \pmod{P_k} \iff 2(a+1) \equiv 0 \pmod{P_k}$$
Since $P_k$ is even, $a+1 \equiv 0$ or $a+1 \equiv P_k/2 \pmod{P_k}$.
- If $a+1 \equiv P_k/2$, then $a = P_k/2 - 1$.
  For $p=2$: $a = P_k/2 - 1$ is even (since $P_k/2$ is odd for $k \ge 2$), so $\gcd(a(a+2), 2) \ne 1$, impossible!
- If $a+1 \equiv 0 \pmod{P_k}$, then $a = P_k - 1$.
  Then $a+2 = P_k + 1 \equiv 1 \pmod{P_k}$, so $\gcd(a(a+2), P_k) = \gcd(-1 \cdot 1, P_k) = 1$.
Thus $a^* = P_k - 1$ is the **unique fixed point** in $S(P_k)$.

Consequently, the number of orbits of $\tau$ on $S(P_k)$ is exactly:
$$N_{\text{orbits}} = \frac{N - 1}{2} + 1 = \frac{N + 1}{2}$$
For $P_4 = 210$: $N = 15 \implies N_{\text{orbits}} = 8$.
For $P_5 = 2,310$: $N = 135 \implies N_{\text{orbits}} = 68$.
For $P_6 = 30,030$: $N = 1,485 \implies N_{\text{orbits}} = 743$.

## 3. Contraction of the Uncoupled Loss
In the standard uncoupled Selberg upper bound:
$$\sum_{a \in W} 1 \le \sum_{n \in W} \left( \sum_{d_1 \mid n} \lambda_{d_1} \right)^2 \left( \sum_{d_2 \mid n+2} \lambda_{d_2} \right)^2$$
which squares the boundary degree and produces an uncoupled factor of 4.
By projecting onto the orbit basis $\{\psi_j\}_{j=1}^{N_{\text{orbits}}}$ where $\psi_j(a) = \frac{1}{\sqrt{2}}(\delta_a(u_j) + \delta_a(\tau(u_j)))$, the cross-product terms strictly enforce reflection parity across the interval boundaries, eliminating the 4-fold excess.