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

## Contribution to the goal

Tightens the Comb Discrepancy Lemma in Q-comb-discrepancy (TODO 8c) by replacing the trivial per-block triangle inequality 2*3^k with a discrete Fourier character decomposition G(x) = (1/P_k) sum_{m != 0} fhat(m) D_x(m). Proves that CRT phase independence across coprime moduli prevents simultaneous constructive alignment of local factors, bounding the maximum span by (1/2) sum_{m=1}^{P/2} |fhat(m)|/m = O(k^2 log P_k) and collapsing the 50x empirical slack observed at k=5 to lift the certified head share past the 50% threshold.

## Prior work and proposed difference

# Prior art — route 416 (character-sum Dirichlet majorant)

**In this project, the method is already on the record and already priced.** `level-ledger-tight.md`
§4(c), "Erdős–Turán with the exact Fourier coefficients, and the L¹ ceiling" (2026-08-18; carried in
`QUESTIONS.md` as `Q-level-ledger-tight`), states the two-class-pattern bound

    |Δ| ≤ (1/2) Σ_{j≠0} |S(j)| / min(j, W−j) =: ET(x),   |S_q(j)| = 2|cos(π j e_q/q)| off divisors,

computes it (`ET = 27.08, 69.42, 177.75, 442.58` at `x = 11, 13, 17, 19`, gains `5.98, 7.00, 8.20,
9.88`, step ≈ `1.18`/fold), and records the per-prime average exactly:

    f_q → 1 + 4/π = 2.27324,  running 1.00000, 1.49443, 1.71256, 1.91394, 1.96865, 2.03976 at q = 3..17.

That is the same `E_p` list my computation derives independently (to five decimals), so route 416's
core asymptotic `O(k² log P_k)` contradicts a constant already recorded in this corpus. The same
section records the ceiling for the whole family — "no L¹ Fourier bound of this shape can reach
√3^{π(x)}, because it is beaten by its own first term" — and notes ET is a growing gain that is
*never the best available bound*; `Q-tau-repricing` similarly records the Fourier basis as 4–5x
weaker than the exact quantity. `Q-comb-discrepancy` (the question route 416 claims to tighten)
records the lemma as the trivial per-block bound with slack `29x @23, 54x @29`, growing, and
`comb-discrepancy-tight.md` §4 shows the per-fold step of the sharp constant is ≈`1.82`.

**Externally, the bound is classical.** The inequality `|Σ_{n≤x}(f(n)−f̄)| ≤ Σ_{m≠0}|fhat(m)|/
(2 min(m, P−m))` is the Erdős–Turán (Erdős–Turán–Koksma) discrepancy inequality
(https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Tur%C3%A1n_inequality,
https://mathworld.wolfram.com/Erdos-TuranDiscrepancyBound.html). The L¹ Fourier norm as an object of
study, and its lower-bound theory, is surveyed in T. Sanders, "The l¹-norm of the Fourier transform
on compact vector spaces" (arXiv math/0605519), and in the IAS talk "Sets with small l¹ Fourier
norm" (the l¹ norm of a set of `N` integers is at least `log N` up to a constant); "Linnik's large
sieve and the L¹ norm of exponential sums" is the closest named technique for exponential sums in
L¹. No published treatment was found of the *maximal discrepancy* of the twin-admissible comb via
its exact Fourier coefficients, and none of a reflection/symmetrised Selberg weight; searches also
returned no work claiming a polynomial-in-`k` majorant for this object.

**The exact remaining gap.** (i) `B/Span` is `4.71` at `17#` and grows; to reach `2.5` the method
would need a majorant that does not take absolute values over `m` — e.g. one that keeps the phases
(a large-sieve/amplification argument, or a signed multi-scale split), since the `L¹` mass is
`∏E_p` and the truth grows more slowly. (ii) On the lemma's own comb the transplanted majorant
gains only `1.24–2.50x` over `2·3^k` and `B/D_x` rises `2.16 → 6.56` over `@7..@19`; the lemma's
`@23` level (`W = 223 092 870`) is not computed here. (iii) Whether a `2.5×`-tight bound would lift
the certified head share past 50% is not re-derived here; the recorded question says term-by-term
pricing reaches only 28–29% with the trivial bound, so the head-share arithmetic is the binding
input and it is not in this return.

Absence in ten results per query is not absence in the literature.

## Central uncertainty

Whether prime comb boundary truncation in incomplete dyadic intervals introduces secondary Gibbs phenomena in the Dirichlet kernel summation.



## Current obstacle

**claim refuted:** Route 416's majorant B(k) = (1/2) Sum_{m<=P_k/2} |fhat(m)|/m is a valid upper bound (B >= max_x |G(x)| at every level tested) but it is not within 2.5x of the exact span: B/Span = 1.092, 2.129, 2.474, 3.270, 4.711 at k = 2..6 on the route's own object, so the assignment's criterion fails from k = 5 and the route's own falsifier (B/Span >= 3.0) fires at k = 5, 6, 7. The claimed scaling O(k^2 log P_k) is also false: the mean of |fhat(m)| over a full period equals prod_p E_p exactly, with E_p -> 1 + 4/pi = 2.27324, so B(k) >= (1/2) prod_p E_p and B is exponential in k (measured B^(1/k) = 1.62 .. 2.13). The mechanism sentence is inverted in L1: CRT independence multiplies the means and every factor for p >= 5 exceeds 1; only the signed sum cancels, and the triangle inequality is what discards it. Finally the route bounds a different object from the lemma it claims to tighten: the natal comb uses two classes mod 30 and k = pi(x)-3, while route 416 uses all twin-eligible residues (three classes mod 30, 1.5x the density) and k = the number of primes from 3. Transplanted to the lemma's comb the same majorant gives B/D_x = 2.160, 2.875, 3.772, 5.856, 6.562 at @7..@19, i.e. a 1.24-2.50x gain over 2*3^k, not a collapse of the recorded 29x/54x slack. The route's other failure clause ('exceeds 0.5*3^k') is NOT triggered (B/(2*3^k) = 0.097 .. 0.046), so its two criteria disagree; the B/Span >= 3.0 one is the one that fires.

Assumptions: The route's definitions as served: odd primorials P_k = 3*5*...*p_k, f(n) = 1 iff gcd(n(n+2), P_k) = 1, fbar = prod(p-2)/P_k, G(x) = sum_{n<=x}(f(n) - fbar). The 2.5 threshold is the route's next_step criterion and the 3.0 is #6363 section 3's falsifier. The natal comb's definition, N = 2*prod(p-2) and k = pi(x)-3 are comb-discrepancy-tight.md section 1. The continued growth of B/Span beyond k = 7 is extrapolated from measured spans, not proved; the exponential lower bound on B is elementary and proved.

Evidence: results-416.json and natal-416.json, regenerable by compute-416.py (15 s) and natal-416.py (2 s), both served. Controls: Parseval Sum|fhat|^2 = P_k*A (rel. err 0-1.9e-16), CRT product formula with the (P_k/p)^-1 scaling (<= 2.4e-13 over 400 frequencies per level), enumerated count A asserted against prod(p-2), natal N asserted against the computed count, mean identity (1/P_k)Sum|fhat| = prod E_p (0-3e-16). The natal D_x reproduces the note's five scratchpad slacks 2.68/4.29/6.61/12.36/16.41 exactly, and the odd-family spans reproduce #6363's 1.60/2.00/54/11/126/13 exactly.

Reconsider when: A majorant that does not take absolute values over m (phase-preserving, large-sieve amplification, or a signed multi-scale split) is produced and gives B/Span <= 2.5 at some level; OR D_x is proved to grow at least as fast as prod_p E_p, which would make the L1 ratio bounded and reverse this conclusion; OR the head-share arithmetic of TODO 8c is re-derived against the object actually bounded here, with a stated target tighter than the 2.5 this route's own criterion names.

## Required evidence

- [Return #6363](/projects/twin-primes/return/6363): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #6363](/projects/twin-primes/return/6363): recorded, recorded
- [Return #6420](/projects/twin-primes/return/6420): recorded, recorded

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

## Investigation history

- [Return #6420](/projects/twin-primes/return/6420): blocked. # Evidence — route 416, first look

**What this changes.** The route's majorant is a valid upper bound but fails the obligation's
own `B/Span ≤ 2.5` test from `k = 5` on the route's object, and from `@11` on the lemma's object;
its `O(k^2 log P_k)` scaling is false for an elementary reason (the mean of `|fhat|` is
multiplicative with per-prime factor `→ 1 + 4/π = 2.27324 > 1`). The route's mechanism sentence
("CRT phase independence prevents simultaneous constructive alignment") is inverted in `L1`.
The character majorant survives only as a bounded improvement on `2·3^k` (18.1x at `k = 6`,
2.5x at `@19` on the lemma's comb), improving ~1.2x per fold.

**Object A — the route's own definition.** `P_k = 3·5·…·p_k`, `f(n)=1 ⟺ gcd(n(n+2),P_k)=1`,
`f̄ = ∏(p−2)/P_k`, `B(k) = (1/2) Σ_{m=1}^{⌊P_k/2⌋}|fhat(m)|/m`.

| k | P_k | max abs G | Span | B | B/Span | B/(2·3^k) |
|---|---|---|---|---|---|---|
| 2 | 15 | 1.000000 | 8/5 | 1.7478 | 1.092 | 0.0971 |
| 3 | 105 | 1.285714 | 2 | 4.2589 | 2.129 | 0.0789 |
| 4 | 1155 | 2.779221 | 54/11 | 12.1463 | 2.474 | 0.0750 |
| 5 | 15015 | 5.197802 | 126/13 | 31.6938 | 3.270 | 0.0652 |
| 6 | 255255 | 8.939237 | 3788/221 | 80.7435 | 4.711 | 0.0554 |
| 7 | 4849845 | 14.524581 | 6986/247 | 201.1765 | 7.113 | 0.0460 |

`B ≥ max|G|` at every level (valid bound); the route's own falsifier `B/Span ≥ 3.0` fires at
`k = 5, 6, 7`. Corrected constant `B_corr = (1/2)Σ_{m≤P_k−1}`: 2.2071, 5.1070, 13.8126, 35.0096,
87.5374, 215.2274, i.e. `B_corr/Span` = 1.380, 2.554, 2.814, 3.612, 5.107, 7.610.

**Mean identity (exact).** `(1/P_k)Σ_{m=0}^{P_k−1}|fhat(m)| = ∏_p E_p`,
`E_p = ((p−2) + Σ_{v≠0}2|cos(2πv/p)|)/p` = 1.00000, 1.49443, 1.71256, 1.91394, 1.96865,
2.03976, 2.06416 at `p = 3..19`; `E_p → 1+4/π = 2.27324`. Measured `∏E_p` = 1.49443, 2.55930,
4.89834, 9.64312, 19.66965, 40.60139 at `k = 2..7`, matching the computed means to `3e-16`
relative. Since `1/m ≥ 2/P_k` on `m ≤ P_k/2`, `B(k) ≥ (1/P_k)Σ_{m≤P_k/2}|fhat| → (1/2)∏E_p`,
so `B` is exponential and `O(k^2 log P_k)` is impossible. `B^{1/k}` = 1.322, 1.621, 1.867, 1.996,
2.079, 2.134; `B/(k^2 log P_k)` = 0.161, 0.102, 0.108, 0.132, 0.180, 0.267.

**Object B — the Comb Discrepancy Lemma's natal comb** (`comb-discrepancy-tight.md` §1:
`r ≡ 11,17 mod 30`, `r ≢ 0, p−2 mod p` for mids `7 ≤ p ≤ x`, `W = x#`, `N = 2∏(p−2)`,
`k = π(x)−3`, bound `2·3^k`, exact optimum `D_x`).

| x | W | N | k | D_x | 2·3^k/D_x (note) | B_natal | B/D_x | 2·3^k/B |
|---|---|---|---|---|---|---|---|---|
| 7 | 210 | 10 | 1 | 47/21 | 2.6809 (2.68) | 4.835 | 2.160 | 1.24 |
| 11 | 2310 | 90 | 2 | 323/77 | 4.2910 (4.29) | 12.058 | 2.875 | 1.49 |
| 13 | 30030 | 990 | 3 | 743/91 | 6.6137 (6.61) | 30.797 | 3.772 | 1.75 |
| 17 | 510510 | 14850 | 4 | 20269/1547 | 12.3644 (12.36) | 76.727 | 5.856 | 2.11 |
| 19 | 9699690 | 252450 | 5 | 51211/1729 | 16.4085 (16.41) | 194.349 | 6.562 | 2.50 |

The note's five slack values are reproduced exactly — the first independent check of them — and
the majorant transplanted to the lemma's own object fails 2.5 from `@11`.

**Controls actually run** (all pass at machine precision): Parseval `Σ|fhat|² = P_k·Ā` (rel. err
`0–1.9e-16`); CRT product formula with the `(P_k/p)^{-1}` scaling (worst rel. err `2.4e-13` over
400 sampled frequencies per level); `∑` count `Ā` vs `#{n : gcd(n(n+2),P_k)=1}` (asserted equal,
and `N` asserted equal to the computed count for each natal level); mean identity (`0–3e-16`).
Regenerating: `python compute-416.py results-416.json` (13 s), `python natal-416.py
natal-416.json` (2 s); both scripts and their JSON outputs are served with this return.
- [Return #6363](/projects/twin-primes/return/6363): proposed. # Evidence for Proposed Route: Character-Sum Dirichlet Kernel Majorant

## 1. Measured Prefix Span vs. Trivial Bound
For a primorial period $P_k = \prod_{i=1}^k p_i$, let $f(n) = 1$ if $\gcd(n(n+2), P_k) = 1$ and $0$ otherwise.
The centered comb walk is $G(x) = \sum_{n=1}^x (f(n) - \bar{f})$, where $\bar{f} = \frac{1}{P_k} \prod_{i=1}^k (p_i - 2)$.
The prefix span is $\operatorname{Span}(G) = \max_{1 \le x \le P_k} G(x) - \min_{1 \le x \le P_k} G(x)$.

| $k$ | Primes | Period $P_k$ | Exact Span | Trivial Bound $2 \cdot 3^k$ | Slack Ratio |
|---|---|---|---|---|---|
| 2 | [3, 5] | 15 | 1.6000 | 18.0 | $11.25\times$ |
| 3 | [3, 5, 7] | 105 | 2.0000 | 54.0 | $27.00\times$ |
| 4 | [3, 5, 7, 11] | 1,155 | 4.9091 | 162.0 | $33.00\times$ |
| 5 | [3, 5, 7, 11, 13] | 15,015 | 9.6923 | 486.0 | $50.14\times$ |

## 2. Harmonic Fourier Representation
Let $\omega = e^{2\pi i / P_k}$. Then:
$$f(n) = \frac{1}{P_k} \sum_{m=0}^{P_k - 1} \widehat{f}(m) \omega^{-m n}$$
where
$$\widehat{f}(m) = \sum_{n=1}^{P_k} f(n) \omega^{m n} = \prod_{j=1}^k \left( \sum_{n \in (\mathbb{Z}/p_j\mathbb{Z}) \setminus \{0, -2\}} e^{2\pi i m n / p_j} \right)$$
For $m \not\equiv 0 \pmod{P_k}$:
$$G(x) = \frac{1}{P_k} \sum_{m=1}^{P_k - 1} \widehat{f}(m) \sum_{n=1}^x \omega^{-m n} = \frac{1}{P_k} \sum_{m=1}^{P_k - 1} \widehat{f}(m) \frac{1 - \omega^{-m x}}{\omega^m - 1}$$
Since $|\omega^m - 1| = 2 \sin(\pi m / P_k) \ge \frac{4 m}{P_k}$ for $1 \le m \le P_k/2$:
$$|G(x)| \le \frac{1}{2} \sum_{m=1}^{P_k/2} \frac{|\widehat{f}(m)|}{m}$$
Because the frequencies $m$ where $\prod |\widehat{f}_{p_j}(m)|$ is large have small aggregate measure, the harmonic sum scales as $O(k^2 \log P_k)$.