# P2692: Sharp L2 norm of the centered maximal operator on C_31

- ID: `P2692`
- Reference: `c31-centered-maximal-l2-norm`
- Page: https://theoremdb.org/statements/P2692
- Record maturity: Reviewed problem with recorded work

## Problem

For \(f:\mathbb Z/31\mathbb Z\to\mathbb R\), define \(Mf(j)=\max_{0\leq r\leq15}(2r+1)^{-1}\sum_{k=-r}^{r}|f(j+k)|\). Determine the exact operator norm \(\sup_{f\neq0}\|Mf\|_2/\|f\|_2\).

### Remarks

- **Remark.** Indices are cyclic modulo 31.
- **Remark.** The Euclidean norm is unnormalized; the quotient is unchanged by this choice.

### What counts as a solution

- Give the exact norm as an algebraic number or certified isolating interval, an extremizing vector, and a complete active-pattern exclusion certificate.

## Status

An exact integer witness reproduces the candidate lower bound, while a closed-form diagonal certificate supplies a universal upper bound. [1](#reference-1)

## Work

### Evidence for the current status

**Computation 1 (The L2 norm lies between 1.33030427059916347 and 1.63067915195310467).** An exact integer witness reproduces the candidate lower bound, while a closed-form diagonal certificate supplies a universal upper bound.

Let \(N_{31}\) denote the requested norm. Exact rational evaluation of the integer vector in c31m-claim-exact-lower-witness gives
\[
\sqrt{\frac{95224622960985697907125617859822805279079620426252}{53808054668649331334697257108564850111138302100375}}
\leq N_{31}.
\]
The lower endpoint is
\[
1.3303042705991634737558623603323047716\ldots.
\]
The weighted Cauchy-Schwarz argument in c31m-claim-diagonal-upper-certificate proves
\[
N_{31}\leq \sqrt{\frac{1916477}{720720}}
=1.6306791519531046652193954473854436190\ldots.
\]
Thus the candidate's reported value is a certified lower bound to ten decimal places. This entry leaves the exact norm and the complete active-pattern exclusion open.

### Background and intake notes

Alternating active-radius and singular-vector updates give a reproducible lower bound 1.3303042705.

- Original intake status: Novelty remains unverified. No primary-source status audit was completed for this exact cycle length.
- For a fixed active radius at each output coordinate, M is a linear averaging matrix and its local norm is a singular value. The global problem is a finite union of active regions.
- A singular vector for an arbitrary radius pattern may violate the inequalities that make that pattern active. Every candidate pattern needs this consistency check.
- Cyclic shifts and reflection identify equivalent active patterns.

- Recorded example: One locally consistent active-radius pattern is [9,8,7,6,5,4,1,0,1,0,1,4,5,6,7,8,9,10,11,12,13,15,15,15,15,15,15,13,12,11,10].

### Other known results

- **Computation 2** (reproduced): All active radii and the squared norm quotient are verified with rational arithmetic. [1](#reference-1)
- **Computation 3** (reproduced): A single radial weight sequence majorizes every centered averaging row and remains diagonal after summing over the cycle. [1](#reference-1)

### Prior approaches

- **Route 1** (inconclusive): Gonzalez-Riquelme and Madrid study the same graph-ball operator and derive exact L2 norms for complete and star graphs. [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (reproduced): The standard-library program uses Fraction for every average, quotient, and certificate identity, then Decimal only for display.

### Computational notes

- Five hundred seeded active-set iterations found ratio 1.3303042705991635. Recomputing all 16 centered averages at every coordinate verified the displayed active pattern, and direct singular-value iteration reproduced the ratio.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `c31-centered-maximal-l2-norm`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1 entry has incomplete source metadata. Each affected row names the fields that still need editorial review.

1. <a id="reference-1"></a>Exact standard-library replay in c31m-artifact-exact-replay https://arxiv.org/abs/2005.03146
   - Also cited at The integer witness and all active-radius comparisons are replayed in c31m-artifact-exact-replay
   - Also cited at Symbolic proof in this record and exact coefficient replay in c31m-artifact-exact-replay
   - Also cited at Cristian Gonzalez-Riquelme and Jose Madrid, Sharp inequalities for maximal operators on finite graphs, arXiv:2005.03146, definition (1.1) and section 1.3
   - preprint; reference source; checked 2026-07-25
   - Source metadata incomplete: publication-style citation, source version.
   - Source use: citation_only
   - Source named by the research packet.