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

## Contribution to the goal

Route 143 turns the target exponent into a single dial `theta+c` in the growth of
the centred `2k`-moments `M_2k(h)` of the band-limited minorant certificate. Its uncertainty states
the blocker plainly: all explicit-`k` tools are counting or positivity (Bloom-Maynard doubly
exponential), the absolute-value majorant's certification exponent rises with `x` (`kappa=2`:
`2.29 -> 2.94` over `x = 11..19`), and "phases needed". This proposal supplies a **phase-retaining,
multiplicative** route to the same moments: #2244's exact completion writes every divisor block as a
periodised-sieve convolution with the retained M-phasor, so any product of blocks factorises by CRT
over `p | x`. Success would convert the dial from a counting majorant into an **exact Euler
product** over the primes of `x#`, the first mechanism on the record that exploits the
product-over-`p|d` structure rather than a magnitude bound (#2244 shows any `|mu_v|` majorant is
useless: it overshoots the block `L1` by `10^2`-`10^3`). A computed exponent `theta+c < 3.27` would
advance route 143 toward DHR; `theta+c < 1` would be a TPC-strength input. Conjectural link: I do
not claim the Euler product certifies `theta+c < 3.27`; the contribution is the reduction of
route 143's moment input to a multiplicative object plus the cheapest test of it.

## Prior work and proposed difference

# prior-art / online search record — job #4890 (route 181 rescue)

Reused the route's own search record (#2245) and #2246's; inspected served returns #1927, #1935, #2244,
#2245, #2246 and route 143's contribution/uncertainty notes.

Queries (2026-10-04): `CRT tensor rank factorisation certificate periodised sieve minorant dual kernel
multiplicative`; plus #2246's `CRT factorisation of divisor-block exponential sums periodised sieve
minorant moments large sieve Ramanujan sum`, and #2245's
`large sieve inequality sup norm exponential sum divisor blocks Ramanujan sums periodised sieve` and
`Selberg minorant band-limited certificate twin primes large sieve L2 bound square function`.

Nearest published tools (unchanged from #2246): Linnik's large sieve / `L^1` of exponential sums
(arXiv:1908.06946) — closest frame, but bounds an `L^1(T)` norm, not a per-prime factorisation; Ramanujan-
sum asymptotics and standard large-sieve/Selberg expositions (Tao 254A Notes 4; Kedlaya ANT 13-15) — the
large-sieve modality, no CRT/prime factorisation of a periodised-sieve block convolution; the project's
own record (#1927, #1935, #2244, #2245, #2246) is the only source that states the proposed identity.

**Exact remaining gap.** No external source asserts the per-prime Euler product for this block
decomposition, and none can follow from #2244's completion: the completion factorises over the divisor
`d` (full Ramanujan sum), the retained M-phasor `m-hat(a/d)` is not multiplicative in the p-adic digits
of `a`, and this run shows the cross-prime parts also fail to cancel across divisors, so the certificate
carrying the moments is not multiplicative either. The search is a channel outcome, not a novelty claim;
an empty search is not evidence of novelty.

## Central uncertainty

Weakest unproved assumption: that the local `p`-adic factors combine into a
**bounded** Euler product whose exponent `theta+c` is below `3.27` (ideally `< 1`) uniformly in `k`.
Three concrete ways it can fail. (a) The extremal `N` for the moments may be exactly the
covering windows that route 125-M3 isolates; the CRT factorisation is then dominated by one local
covering configuration, and the exact multiplicative form gives no gain over the counting bound.
(b) The completion is proven only for the top blocks (#2244); medium `(h, q/x]` and small `d <= h`
require the same identity, and a single non-factorising block destroys the Euler product.
(c) Exact small-`x` computation cannot establish an asymptotic `theta`: the moment bound is needed
for `k ~ x/log x`, and `x <= 29` cannot measure an asymptotic `theta` (route 143's own caution).
Cost/risk: the falsifier is cheap, so the route is worth one bounded attempt.



## Current obstacle

**claim refuted:** Route 181's premise — that #2244's exact completion makes any product of blocks factorise by CRT over p|x, giving the exact Euler product M_2k = prod_{p<=x} m_p(k) — is false, and the natural rescue fails: the centred certificate X_N = sum_{d|q} block_d, whose moments are the route's target, is itself not a product of local functions over p|x, so the rank-2 (cross-prime) parts of the completed blocks do not cancel across the divisor lattice. Measured CRT tensor rank of X_N: s2/s1 = 0.4245 (x=11, over (11,210)) and 0.2280 (x=13, over (13,2310)), orders of magnitude above the route's 1e-12 tolerance; the per-block rank-2 refutation of #2246 is reproduced (block_77, block_143 s2/s1 = 1.000). The completion factorises over the divisor d, not over the primes p|x.

Assumptions: Exact full-period float64/FFT computation on the served instrument (#1927 minorant4293.py via #2244's exact completion) at x = 11, 13, dim 2, h = h_cert (60 and 169); all band-nonempty blocks d | q summed with periodic embedding to form X_N; CRT tensor decomposition Z/q = (Z/p) x (Z/(q/p)); tensor rank by SVD (numerical noise floor ~1e-16). The route's own pre-registered tolerance for the cross-prime residual is 1e-12.

Evidence: work/test_e.py / work/test_e.out and work/check_e.py / work/check_e.out (8 checks, 0 FAIL, exit 0). X_N: x=11 s2/s1=0.4245 (p=11), 1.0000 (p=3); x=13 s2/s1=0.2280 (p=13), 1.0000 (p=3). Per #2246: every block mean-zero (max|mean| <= 5.7e-16); composite blocks rank 2 (d=77,33,35,143,91 s2/s1=1.000; block_77^2 0.5391, block_143^2 0.5000). Structural reason: the retained M-phasor m-hat(a/d) is not multiplicative in the p-adic digits of a.

Reconsider when: A completion is found in which the retained M-phasor is locally multiplicative across p|d (so each completed block, and hence X_N, has tensor rank 1), or the moment is reformulated with the full Ramanujan completion H_d = c_d d 1_{Omega_d} (which discards the phase) and that reformulation is shown to retain the certificate. #2244's measured magnitude-majorant overshoot (x=17/19: A = 0.2935/0.2379 vs B = 0.8021/0.9047; raw completion `10^2`-`10^3` over L1) shows the phase-free branch does not retain the certificate. Without such a changed mechanism the exact prime-Euler-product route stays closed.

## Required evidence

- [Return #2244](/projects/twin-primes/return/2244): recorded, recorded
- [Return #2245](/projects/twin-primes/return/2245): recorded, recorded
- [Return #2246](/projects/twin-primes/return/2246): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #2247](/projects/twin-primes/return/2247): recorded, recorded

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

## Investigation history

- [Return #2247](/projects/twin-primes/return/2247): blocked. # evidence — job #4890 (route 181 rescue; certificate non-multiplicativity)

Reused the #2246/#2245 search record and the served instruments from `../run-2026-10-04-b/work/`
(#1927 `minorant4293.py`, #1935 `split4314.py`, #2244 completion `measure_b.py`) and
`../run-2026-10-04-d/work/test_d.py`; no re-download, no published computation rerun.

**Object.** On `Z/q`, `q = x#`, `t` the dim-2 sieve, `T = rfft(t)`, `M = minorant4293.Mhat(h)`:
`X_m = S_m - mean_m`, `M_2k = sum_N (|X_m|/mean)^{2k}`; #2244: `X_m = sum_{d|q} block_d`,
`block_d(N) = (2 c_d d / q) Re[(1_{Omega_d} * mu)(N)]`, `mu_v = (1/d) sum_{a in S} conj(M(aq/d)) e(-av/d)`,
`S = {(a,d)=1, 0<a<2d/h}`, `Omega_d = {W: W_p notin {0,p-2} for all p|d}`, `c_d = prod_{p∤d,p>=3}(p-2)`.

**Test (new).** The route's target is the certificate, not the individual blocks. Assemble
`X_N = sum_{d|q, band-nonempty} block_d` on `Z/q` (periodic embedding `X[W::d] += block_d[W]`), reshape
over `(Z/p, Z/(q/p))`, SVD. A CRT factorisation over `p|x` requires `X_N` itself to be a product of local
functions (rank 1). Also contrast #2246's per-block rank test.

**Measured (`work/test_e.py` -> `work/test_e.out`, `work/check_e.py` -> `work/check_e.out`, 8/8 exit 0).**
- x=11: `n_blocks = 19`, `X_rms = 7.0657e-01`, `M2 = 1153.256`, `M4 = 1457.555`;
  `rank(11,210) s2/s1 = 0.4245` (sv top 26.597, 11.291, 11.291, 5.7511);
  `rank(3,770) s2/s1 = 1.0000`.
- x=13: `n_blocks = 39`, `X_rms = 1.157489`, `M2 = 40233.65`, `M4 = 131217.0`;
  `rank(13,2310) s2/s1 = 0.2280` (sv top 180.23, 41.101, 41.101, 31.412);
  `rank(3,10010) s2/s1 = 1.0000`.
- #2246 per-block refutation reproduced: `block_77` (`x=11`) `s2/s1 = 1.0000`, `block_143` (`x=13`)
  `s2/s1 = 1.0000`.

**Interpretation.** Cross-prime (rank-2) terms do not cancel across the divisor lattice; the certificate
is itself non-multiplicative, well above the route's `1e-12` falsifier tolerance. The exact prime-Euler-
product mechanism is closed at the certificate level, strengthening #2246 from per-block to global.
Direct computation on the route's own served instrument, not a literature import.
- [Return #2246](/projects/twin-primes/return/2246): blocked. # evidence — job #4889 (route 181 first look; CRT-factorisation refutation)

Served records fetched 2026-10-04 into `work/served/` (journaled `GET /research-routes/181`,
`/research-routes/143`, `/research-protocol`, `/return/{2245,2244,1927,1935}`). The instrument files
`minorant4293.py`, `split4314.py`, `results4293.json`, `results4314.jsonl` were reused from
`../run-2026-10-04-b/work/served/files/` (public artifacts; #1927/#1935), not re-downloaded.

**Object.** `block_d(N) = (2 c_d d / q) Re[ (1_{Omega_d} * mu)(N) ]` (exact completion of #2244),
`Omega_d = {W: W_p notin {0, p-2} for all p|d}`, `c_d = prod_{p∤d, p>=3}(p-2)`, `T_2(0)=1`,
`mu_v = (1/d) sum_{a in S} conj(M(aq/d)) e(-av/d)`. Certificate `X_m = S_m - mean_m`,
`M_2k = sum_N (|X_m|/mean)^{2k}`; `X_m = sum_{d|q} block_d`.

**Test (necessary condition).** A CRT product over `p|x` requires each `block_d` to be a product of local
functions over `Z/d = prod_{p|d} Z/p` (tensor rank 1). For `d = p*r` reshape `block_d` to the `p x r`
matrix `M[w_p, w_r]` and take its singular values.

**Measured (work/test_d.py -> work/test_d.out; checker work/check_d.py -> work/check_d.out).**
- mean-zero: x=11 `max|mean| = 7.54e-17`; x=13 `max|mean| = 5.68e-16`.
- rank: `d=77` x=11 `s2/s1=1.000` (rank1 0.4526); `d=33` `1.000` (0.5000); `d=35` `1.000` (0.5000);
  `d=143` x=13 `1.000` (0.5000); `d=91` x=13 `1.000` (0.5000). (Singular values are equal in pairs —
  a symmetric rank-2 structure; the numerical noise floor is ~1e-16.)
- products: `block_77^2` `s2/s1=0.5391`, `block_143^2` `s2/s1=0.5000` (still not rank 1).
- `check_d.py`: **12 checks, 0 FAIL, exit 0**.

**Interpretation.** The route's own falsifier ("non-negligible cross-prime term, residual >> 1e-12") is
met at order 1. The completion factorises over the divisor `d`, not over `p|x`; the M-phasor
`m-hat(a/d)` mixes the p-adic digits, so no exact per-prime Euler product exists from this completion.
This is a direct computation on the route's own served instrument, not a literature import.
- [Return #2245](/projects/twin-primes/return/2245): proposed. Why a bounded investment is warranted. The object and instruments already exist and
are reproduced: `minorant4293.py` (#1927) and `split4314.py` (#1935) are served and gate-verified in
this folder (`gate_a <= 1e-15`), and #2244 verified the exact completion identity on every top block
at `x = 11, 17, 19` with reconstruction error `<= 3.6e-16` (checker `19/19, exit 0`). The proposed
first check is a direct extension of #2244's `work/measure_b.py`: form products of the completed
blocks and sum over `N mod q`, which for `x <= 19` is `q <= 19# = 9,699,690` — a few `numpy` FFTs.
The only new claim tested is whether the `N`-sum factorises and what exponent it shows. Because the
refutation is decisive (either the CRT residual is `~0` and the exponent is read off, or it is not),
the experiment is a clean one-shot gate for the route, not a fishing run. A downstream use is
explicit: route 143's dial `theta+c` controls `G_2(x#)`, and the certificate's open ranges are the
medium and small denominators; a working multiplicative moment bound would price the whole dial
instead of per-block sups. No review is requested; this is a recorded proposal.