# Reading roadmap: Manuscript

The argument assumes appreciable low-level Walsh mass, applies a noise operator and a low-degree truncation, and transfers this to a correlation of ordinary exponential sums. Minor-arc bounds and major-arc estimates for nonexceptional characters are combined with a separate digit-block analysis of a possible exceptional character. Parameter choices close the contradiction; a monotone-function Fourier concentration input supplies the corollary.

Source SHA-256: `b8f3edd1342bdb9e7b54b8cd24cf3b7419a2aaf33493c4c2869ea3611e4325b2`. Indexed 34,774 characters.

Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates.

## Main statements and external context

Read the support of the Walsh spectrum and the monotone-function consequence.

Claims: theorem-1, corollary-2

## Noise operator and truncation

Track the assumed low-level mass and the decomposition into a bounded-degree part and an error.

Claims: lemma-1

## Minor arcs

Use the ordinary exponential-sum bound and the L1 estimate of the truncated Walsh polynomial.

Claims: 

## Major arcs: nonexceptional contributions

Separate the imported character estimates and sieve estimate from the reduction to the exceptional term.

Claims: lemma-2, lemma-3, lemma-4

## Major arcs: exceptional character

Trace the binary-block estimates for small and large odd conductor parts, then check the final parameter balance.

Claims: theorem-1

## Internal claim index

### Theorem 1 — `theorem-1`

Source lines 58–72.

Let $\lambda>0$ be a fixed constant and $n_0\sim (\log n)^{-1-c\lambda} n^{2/3}$. Then $$ \sum_{|A|\leq n_0} |\widehat\mu (A)|^2 <(\log n)^{-\lambda}\tag 0.1 $$ where $$ \widehat\mu (A) =\frac 1N\sum_{x\in \{0, 1\}^n} w_A(x) \mu\Big(\sum^{n-1}_{j=0} 2^jx_j\Big)\tag 0.2 $$ and $$ w_A(x) =\prod_{j\in A} (1-2x_j).\tag 0.3 $$

*Extractive statement; proof gist not independently reconstructed.*

### Corollary 2 — `corollary-2`

Source lines 84–87.

The Moebius function does not correlate with monotone Boolean functions.

*Extractive statement; proof gist not independently reconstructed.*

- `theorem-1` (proof_dependency; source_audit_annotation): Hence Theorem 1 implies
- `external:B-T` (proof_dependency; source_audit_annotation): We are invoking here an important result of Bshouty and Tamon \cite{B-T}, implying roughly that any monotone Boolean function

### Lemma 1 — `lemma-1`

Source lines 211–230.

Assume $f$ on $[0, N[$ satisfies $\Vert f\Vert_\infty\leq 1$. Given $1<K<\frac n{2n_0}$ there is a decomposition $$ T_\rho f= f_1+f_2\qquad \Big(\rho = 1-\frac 1{n_0}\Big) \tag 1.17 $$ such that $$ \widehat {f_1} (A) =0 \ \text { for } \ |A|>K n_0\tag 1.18$'$ $$ and $$ \Vert S_{f_1}\Vert_1 <(Cn)^{2Kn_0}\tag 1.18$''$ $$ and $$ \Vert f_2\Vert_2 < e^{-K} \sqrt N. \tag 1.19 $$

*Extractive statement; proof gist not independently reconstructed.*

### Lemma 2 — `lemma-2`

Source lines 330–341.

(\cite{B-P}, Theorem 3). If Case II, then $$ \int_{\Cal M_1} |S(\alpha)|^2 d\alpha\ll _A\frac N{(\log N)^A}\tag 3.6 $$ with $$ \Cal M_1 =\bigcup_{q\leq R} \bigcup_{(a, q)=1} \Big\{\alpha; \Big|\alpha-\frac aq\Big|< \frac {R(\log N)^{A+1}}{qN}\Big\}.\tag 3.7 $$

*Extractive statement; proof gist not independently reconstructed.*

- `external:B-P` (proof_dependency; source_audit_annotation): (\cite{B-P}, Theorem 3).

Candidate references: `external:B-P` (external_reference).

### Lemma 3 — `lemma-3`

Source lines 494–500.

Let $I_1\subset [1, N]$ be an interval of size $|I_1|>\frac NB$ and $\Cal X(\mod q), q\leq B$, not induced by the exceptional character. Then $$ \Big|\sum_{k\in I_1}\mu(k) \Cal X(k)\Big|\ll_A \frac {|I_1|}{(\log N)^A}.\tag 3.32 $$

*Extractive statement; proof gist not independently reconstructed.*

- `external:B-P` (proof_dependency; source_audit_annotation): We use Theorem 4 from \cite{B-P}

### Lemma 4 — `lemma-4`

Source lines 526–532.

Let $\ell\in\Bbb Z_+$. Then, with $R$ as in (3.2) $$ \sum_{q\leq R}\, \frac q{\vp(q)}\, \underset{\Cal X(\mod q)}\to{\sum\nolimits^*} \Big|\sum_{k\in I_1, (k, \ell)=1} \mu(k)\Cal X(k)\Big|\ll_\ve (\log N)^{74} (R^2 N^{\frac 12}\ell^\ve+R|I_1|^{\frac 12} N^{\frac 3{10}} \ell^\ve +|I_1|).\tag 3.36 $$

*Extractive statement; proof gist not independently reconstructed.*

- `external:B-P` (proof_dependency; source_audit_annotation): According to \cite{B-P}, Theorem 5

## External inputs and references

### [1] B1 — `external:B1`

Source lines 911–916.

\by J.~Bourgain \paper Moebius-Walsh correlation bounds and an estimate of Mauduit and Rivat \jour (arxiv 1109.2784)


Candidate references: `section:1` (external_reference); `section:2` (external_reference).

### [2] B2 — `external:B2`

Source lines 916–921.

\by J.~Bourgain \paper Prescribing the binary digits of primes \jour (arxiv 1105.3895), to appear in Israel J

The adapted digit-block method refers explicitly to section 4 of the companion paper; its source was not supplied to the panel.

Role: **proof_input**. We analyze (4.23) according to \cite{B2}, \S4.

- `section:5` (proof_dependency; source_audit_annotation): We analyze (4.23) according to \cite{B2}, \S4.

Candidate references: `section:1` (external_reference); `section:5` (external_reference).

### [3] H-K — `external:H-K`

Source lines 921–926.

\by G.~Harman, I.~Katai \paper Primes with preassigned digits II \jour Acta Arithm. 133.2 (2008), 171--184


Candidate references: `section:1` (external_reference).

### [4] Gr — `external:Gr`

Source lines 926–930.

\by B.~Green \paper On (not) computing the M\"obius function using bounded depth circuits \jour (arxiv 1103.4991)


Candidate references: `section:1` (external_reference); `section:4` (external_reference).

### [5] B-T — `external:B-T`

Source lines 930–935.

\by N.~Bshouty, C.~Tamon \paper On the Fourier spectrum of monotone functions \jour J. ACM. 43 (1996), no 4, 747-770

Fourier concentration of monotone Boolean functions, used for Corollary 2; original source not independently inspected.

Role: **proof_input**. We are invoking here an important result of Bshouty and Tamon \cite{B-T}

- `corollary-2` (proof_dependency; source_audit_annotation): We are invoking here an important result of Bshouty and Tamon \cite{B-T}, implying roughly that any monotone Boolean function

Candidate references: `section:1` (external_reference).

### [6] B-P — `external:B-P`

Source lines 935–941.

\by A.~Balog, A.~Perelli \paper On the $L^1$-mean of the exponential sum formed with the Moebius function \jour J.~London Math. Soc. (2) 57) (1998), no 2, 275--288

Specified major-arc/character estimates in Lemmas 2–4; original source not independently inspected.

Role: **proof_input**. (\cite{B-P}, Theorem 3).

- `lemma-2` (proof_dependency; source_audit_annotation): (\cite{B-P}, Theorem 3).
- `lemma-3` (proof_dependency; source_audit_annotation): We use Theorem 4 from \cite{B-P}
- `lemma-4` (proof_dependency; source_audit_annotation): According to \cite{B-P}, Theorem 5

Candidate references: `section:1` (external_reference); `section:4` (external_reference); `lemma-2` (external_reference).

## Coverage and limits

{"mode": "native_latex_index", "claims": 6, "sections": 5, "external_references": 6, "source_bibliography_entries": 6, "unresolved_bibliography": 0, "unresolved": 1}

- Reference edges are not certified proof dependencies. Edge direction is prerequisite → consumer.
- Numbers reflect simple source counters; labels and line anchors are authoritative.
- Extractive statements and proof openings are marked; they are not reconstructed proof gists.
- Macros, included files and custom counters require an expanded source or annotated fallback.