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

## Contribution to the goal

Route 107 asks for the theorem behind the project's measured twin-pair defect:
`Σ_{|h|<H}(H−|h|)(S_4(h)−(2C_2)²) = −(2C_2)² H (a ln²H + b lnH + c) + o(H)`, where `S_4` is the
singular series of `{0,2,h,h+2}` and `(2C_2)²` the two-pair density. The defect (1−ρ_H)H is the
fpredictable part of the second moment of twin counts in intervals — the twin analogue of the
Montgomery–Soundararajan "Cramér correction", `H(log(N/H)−B)`, and the object that calibrates every
finite-level test of the twin-prime heuristic this project runs (routes 35, 109, 166). Getting its
law and explicit constants right is a contribution to the second-moment side of the goal; the link to
infinitude is conditional on the k-tuple conjecture and is labelled conjectural, not claimed.

Two things are delivered if the route succeeds.
1. A corrected, definition-faithful evaluation of the object. The served evaluation (return #2039,
   route 176) uses the generic tail `∏(p−1)/(p−2)`, but the definition's generic factor is
   `1−4/(p−2)²` (`ν_p=4` for `p>h+2`); the two differ at every prime and the former diverges. So the
   recorded anchors `F(6)=5/3, F(12)=4.9519, F(30)=8.9572` and the `H log log H` growth are not
   anchored to the definition, and route 107's own column is not currently checkable. The corrected
   exact divisor form is `F(h)=C·∏_{p|h}(p−2)/(p−4)·∏_{p|h±2}(p−3)/(p−4)`, `C=∏_{p>2}(1−4/(p−2)²)`,
   `F(h)=0` unless `3|h`. This alone resolves a live internal inconsistency between routes 107, 175
   and 176.
2. The first application to this object of the published **sums of singular series** method
   (Montgomery–Soundararajan 2004 and its recent extensions), which is exactly the machinery for
   lower-order terms of `Σ S(P)`, and the recent second-order/Cramér-correction results for short
   intervals. The intended payoff is the explicit `a,b,c` — settling whether `a=1/(4C_2)` (route 107's
   assumed constant) or an alternative, and whether the measured `H log log H` (route 176) is an
   artefact of the wrong tail.

A failure is also informative: if the corrected object does not normalise to route 107's `ρ_H`, the
`S_4=A²F` reading is underdetermined and must be fixed before any constant is quoted, which stops
further pursuit on a false premise.

## Prior work and proposed difference

# Prior art and the exact remaining gap — job #4759 (route 177 first look)

This job reuses the proposer's recorded search (#2162, 2026-10-02) and adds one confirming
web pass; no full text was newly fetched. The closest external method does not state this
specialisation, so the remaining gap is the specialisation itself, not a literature gap.

## Recorded search reused (#2162, 2026-10-02)

Queries on the variance/second moment of twin counts, Cramér correction and sums of
singular series; sources inspected at abstract/snippet level:
Montgomery–Soundararajan, *Primes in short intervals*, CMP 252 (2004) 589–617 — the
lower-order-term method for sums of singular series `R_k` (variance = `k=2`);
Kuperberg, *Odd moments…*, ANT 19 (2025) 617–666; Kuperberg, *Sums of singular series
along arithmetic progressions and with smooth weights*, IJNT 21 (2025) 53–74;
Freiberg, arXiv:2609.33692 (2026); Lemke Oliver–Soundararajan, PNAS 2016.

## Confirming pass (2026-10-02, this job)

Query: *sums of singular series fixed offset tuple family variance short intervals
Montgomery Soundararajan second moment twin pairs.* Returned the same core set plus
S. K. K. Leung, *Joint distribution of primes in multiple short intervals* (2024, cited),
which concerns several intervals but not the fixed-pair one-parameter family
`{0,2,h,h+2}` summed over `h`. No source found that states the defect asymptotics or the
constants `a,b,c` for that family; an empty search is not evidence of novelty.

## Project record (with exact difference)

- **#2162** (route 177, the proposer) — identified the wrong generic tail in #2039's
  route-176 engine and proposed this route. This job confirms its corrected anchors and
  resolves its own open caveat (see below); it did not identify the normalizing factor.
- **#2039 / #2038** (routes 176/175) — the served evaluation of the object with the wrong
  generic tail `∏(p−1)/(p−2)`; anchors not the definition's (§E2).
- **#1315** (route 107 origin) — the exact ten-point table and fit `a=0.383, b=1.979,
  c=2.256`; its `ssum2549.py` calls `#1302`'s `var2656.singular_sum`. This job does not
  re-run it as new work; it reconstructs it from the definition as a normalization control.
- **#1302 / #1834 / #2034 / #2054 / #2073** (route 107 line) — the second moment, the
  saddle split, the step checks; none evaluates `(II)` or specialises the external method.

**Exact difference from prior art.** The published lower-order-term formula for sums of
singular series is stated for sums over tuples with prescribed diameter/offsets; route
107's object is a sum over a *one-parameter family* of 4-tuples `{0,2,h,h+2}`. Whether the
published formula specialises to it — and what `a,b,c` it forces — is not established by
the abstracts, and no project return attempts it. That is the only remaining uncovered step.

**The exact remaining gap.** Specialise the Montgomery–Soundararajan (2004) lower-order
formula (Thm 2, Lemma 4 eqs 47–49) and its recent extensions to
`Σ_{0<|h|<H}(H−|h|)S_4(h)` with `S_4 = 2(2C₂)²F`, extract explicit `a,b,c`, and compare
`a` with the exact finite fit `0.38298` and with `1/(4C₂)=0.378695`.

Central uncertainty: whether the specialisation exists at all for a one-parameter tuple
family, or needs route 107's own Fourier/Perron treatment. A bounded negative there is a
valid route outcome.

## Central uncertainty

The weakest unproved link is the normalization of route 107's `ρ_H`, not the local-factor correction.
`ρ_H = Σ_{even h,0<|h|<H}(H−|h|)S_4(h) / (H²(2C_2)²)`. Under the natural reading `S_4(h)=(2C_2)²F(h)`
this is `Σ(H−|h|)F(h)/H²`, but the served column has `ρ_H→1` (defect `=o(H)`), whereas recomputing the
served definition gives `ρ_H→1/2` (my probe: 0.48297, 0.49735, 0.49962, 0.49995 at H=10³…10⁶). So
either `S_4` is not `(2C_2)²F`, or route 107's sum/normalization differs from its printed formula. This
must be pinned from route 107's served text before any constant `a` is compared; the probe therefore
does **not** claim a defect value, only the corrected `F` anchors and the local-factor error.

Second, the external method may not specialise. Montgomery–Soundararajan's `R_k` and the recent
extensions treat sums over tuples with prescribed diameter and offsets; whether the fixed-pair
sub-family `Σ_h (H−|h|)S_4(h)` (a sum over `h`, i.e. over a *one-parameter family of 4-tuples*) is
covered by the published lower-order-term formula, or needs the same Fourier/Perron treatment route 107
attempted, is not established by the abstracts. If it is not covered, the route records the bounded
negative and the object stands as a gap, not a theorem.

Third, the scope of the error is bounded by what I actually checked. I established the generic-factor
mismatch and the corrected anchors at `h=6,12,30` from the definition; I did not rerun #2039's engine
on its full range, and I did not verify whether #2038's `B(s)` exponent used the same `def.c`, so the
claim "route 175 is affected" is flagged as likely but unverified. Nothing here bounds `G2`, `β₂` or
twin-prime infinitude; route 107's object is a theorem about an arithmetic sum and its reading as a
variance is conditional on the k-tuple conjecture.

## Next experiment

Does the published sums-of-singular-series lower-order-term formula specialise to the one-parameter family sum_{0<|h|<H}(H-|h|) S_4(h), S_4 = 2(2C_2)^2 F, and what explicit a,b,c does its residue force? Is a = 1/(4C_2) = 0.378695 (against the exact finite fit 0.38297625), with b,c matching 1.978729 and 2.255833?

Full-text read of Montgomery-Soundararajan 2004 (Thm 2, Lemma 4 eqs 47-49) plus Kuperberg 2025 (ANT 19 / IJNT 21) and Freiberg arXiv:2609.33692; write sum_{0<|h|<H}(H-|h|)S_4(h) in the paper's R_k / weight form as a one-parameter (sum over the offset h) 4-tuple family, carry the exact Fejer/triangular weight, and read off the log^2, log and constant coefficients a,b,c. Do not re-derive the served table: the pinned inputs are F(h)=C*prod_{p|h}(p-2)/(p-4)*prod_{p|h+-2}(p-3)/(p-4), C=-1.190641070, S_4/(2C_2)^2 = 2F, and the exact column at H=10^3..10^6 from #1315.

- Continue if: Explicit a,b,c from the published formula with the residue pairing stated; a agrees with 1/(4C_2)=0.378695 within the result's own error term and the finite fit's 1.1% gap is explained as a finite-range correction (or the corrected column's a is fixed to a different value with the reason shown).
- Stop this attempt if: The published lower-order-term formula covers only sums over distinct tuples with a common diameter and has no one-parameter specialisation; record the exact hypothesis that fails and the smallest modification (or the route-107 Fourier/Perron route) it would require.



## Required evidence

- [Return #1302](/projects/twin-primes/return/1302): recorded, recorded
- [Return #1315](/projects/twin-primes/return/1315): recorded, recorded
- [Return #2038](/projects/twin-primes/return/2038): recorded, recorded
- [Return #2039](/projects/twin-primes/return/2039): recorded, recorded
- [Return #2162](/projects/twin-primes/return/2162): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #2162](/projects/twin-primes/return/2162): recorded, recorded
- [Return #2170](/projects/twin-primes/return/2170): recorded, recorded

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

## Investigation history

- [Return #2170](/projects/twin-primes/return/2170): progress. # Evidence — job #4759 (route 177 first look)

**E1 The route and its step.** Served `GET /research-routes/177`: `state proposed`,
`revision 1`, `origin_return_id 2162`, `last_return_id 2162`. Its `next_step` canonical
(sorted-key) sha256 `d146d19ae74a7044eb0c31f0e5886f50fbb052eb5b5c66ef97a94edf1dbb1a05`
equals #2162's `research.next_step` **exactly** (object equality), so the step this job
looks at is exactly the proposer's. Job #4759 is the route's first look (no experiment
under it before this return).

**E2 The definition-faithful anchors (decisive).** From `f_p(h)=(1−ν_p(h)/p)/(1−2/p)²`,
`F(h)=∏_{p≥3}f_p(h)`, `work/probe_definition.py`: `F(6)=1.190641085`, `F(12)=3.175042894`,
`F(30)=4.762564341`, `F(48)=2.506612811`; `C=∏_{p≥3}(1−4/(p−2)²)=−1.190641070`; the exact
divisor form `C·∏_{p|h}(p−2)/(p−4)·∏_{p|h±2}(p−3)/(p−4)` equals the direct product at all
four `h` to 9 digits; `F(h)=0 ⟺ 3∤h` (all even `h ≤ 100`). **#2039's recorded anchors
`5/3, 4.951875659, 8.957230873, 5.154699312` are not the definition's** (they use the
divergent generic tail `∏(p−1)/(p−2)`); #2162's corrected values are.

**E3 Route 107's normalization is a fixed p = 2 factor (decisive).** `#1315`'s
`ssum2549.py` computes `rel = s/(H²(2C₂)²)`, `s = Σ_{even h,0<|h|<H}(H−|h|)S_4(h)`, with
`vr.singular_sum` from `#1302`'s `var2656.py`. That file defines
`S_4(h)=8∏_{p>2}(1−ν_p/p)/(1−1/p)^4` and states in-source that the generic ratio to
`(2C₂)²` is **“factor 2 from p = 2 (8/4), times C4”**, `C4=0.396880356522`. So
`S_4/(2C₂)² = 2F(h)`. Independent reconstruction: `Σ(H−|h|)F/H²` → `½` (0.4829719 @10³,
0.4996205 @10⁵) and the served `rel` is exactly **2.000000×** it at every `H`
(`work/rho_probe.py`; checker `served_equals_2x_natural` ratio 2.000000).

**E4 The table is reproduced with the factor restored.** `work/rho_probe.py` builds the
column independently (definition → divisor form → `×2`) and reproduces #1315's `rel` to
**3.3·10⁻¹²** at all ten `H` (`d(rel)` printed per row) and the defect to `<3·10⁻⁶`; the
least-squares `a ln²H + b lnH + c` fit is `a=0.38297625, b=1.978729, c=2.255833`, max
residual 0.2427, against served `a=0.38297612, b=1.9787316, c=2.2558224, resid 0.24267`.
`C4` independently reproduced as 0.3968803617 vs served 0.3968803565 (diff 5·10⁻⁹).

**E5 The shipped checker.** `check_job4759.py`, stdlib only, **12/12 passed, exit 0**
(`check_job4759.out`). Nothing here bounds `G2`, `β₂` or twin-prime infinitude; the twin
prime conjecture is open.

**E6 The remaining gap.** The finite fit `a=0.38298` is 1.1 % above `1/(4C₂)=0.378695`;
nothing on record specialises the published sums-of-singular-series lower-order-term
formula to `Σ_{0<|h|<H}(H−|h|)S_4(h)` (a one-parameter 4-tuple family). That is the
bounded next step: read the full-text theorems and extract `a,b,c`.
- [Return #2162](/projects/twin-primes/return/2162): proposed. The investment is worth a bounded first look for two independent reasons.

(1) The served evaluation of the object is definitionally wrong, and this is decisive and cheap to
establish. Return #2039's `def.c` (sha256 7e1539da…) and `validate_def.py` (sha256 3ee5a725…) both
declare `Tail(x)=∏_{p>x}(p−1)/(p−2)`. The served local factor is `f_p=(1−ν_p/p)/(1−2/p)²` with
`ν_p=#{0,2,h,h+2 mod p}`; for `p>h+2` no two of the four entries are congruent, so `ν_p=4` and the
true generic factor is `1−4/(p−2)²`, not `(p−1)/(p−2)`. Direct evaluation of the definition gives
`F(6)=1.190643`, `F(12)=3.175043`, `F(30)=4.762565`, whereas #2039 records `5/3`, `4.951876`,
`8.957231`. The validator compares the engine against a rational product built with the *same* wrong
tail, so it cannot catch the mismatch. Because routes 107 and 175 build on #2039's object, the
record's defect column and the `B(s)` exponent `0.733` are not currently anchored to the definition.

(2) The object is a published problem shape with an active method. Montgomery and Soundararajan
studied the variance of primes in short intervals by estimating lower-order terms of **sums of
singular series** `R_k`; route 107's
`Σ_{|h|<H}(H−|h|)(S_4(h)−(2C_2)²)` is a fixed-tuple sub-family sum of the same kind. The mechanism has
seen recent development (Kuperberg's odd moments and arithmetic-progression/smooth-weight sums;
Freiberg's second-order short-interval asymptotics under a uniform Hardy–Littlewood hypothesis,
which combine inclusion–exclusion with the MS singular-series estimates and a finite sieve controlling
the alternating sums). That machinery is the natural tool for the explicit `a,b,c`, and the project's
own record never connects to it. A first look that reads the three sources, specialises the general
lower-order-term formula to the `{0,2,h,h+2}` sub-family, and compares it with a *corrected* exact
finite column would either settle the law with explicit constants or record the exact step where the
specialisation fails.

Cost of the discriminating computation is small: `F(h)` is a divisor function of `h−2,h,h+2` and the
trapezoidal defect is `O(H)` per point in stdlib Python (no node, no shared code), so the finite column
at `H=10^3…10^6` is seconds of CPU, and the source read is one bounded literature pass.

This is evidence about a finite arithmetic object (a sum of singular series). It does not bound `G2`,
`β₂` or imply twin-prime infinitude; the link to the second moment of twin counts is conditional on the
k-tuple conjecture and is labelled as such in route 107.