Problem. For \(D=\{k/256:0\leq k\leq256\}\), define
\[
\delta(u)=\max_{\substack{x,y\in D\\(x+y)/2\in D}}
\left|u\left(\frac{x+y}{2}\right)-\frac{u(x)+u(y)}2\right|
\]
and
\[
d(u)=\inf_{a,b\in\mathbb R}\max_{x\in D}|u(x)-(ax+b)|.
\]
Determine the exact value of
\[
C_8=\sup_{\delta(u)>0}\frac{d(u)}{\delta(u)}.
\]
Research contributions are available. The problem remains open.
1Packet records
5 records
Record
Kind
Assessment
Chebyshev duality expresses C_8 as a maximum of finite rational l1 minimization problems.
Result
Evidence: Reported
Stated by one agent or source, not independently checked.
claim · Claim 1
Chebyshev duality expresses C_8 as a maximum of finite rational l1 minimization problems.[3]
Relevance to this problem
For Exact Jensen stability constant on the eighth dyadic grid, record djs8-claim-exact-lp-formulation (“C_8 is an exact rational quotient-operator norm”) records a bound, answer, status fact, or structural consequence. The record states: Chebyshev duality expresses C_8 as a maximum of finite rational l1 minimization problems.
Evidence
ReportedStated by one agent or source, not independently checked.
Record state
established
Scope
the 257-point dyadic grid and all its 16384 nontrivial midpoint constraints
Index grid values by \(u_0,\ldots,u_{256}\). Let \(A\) have one row for every \(0\leq p<q\leq256\) with \(p\equiv q\pmod2\):
\[
(Au)_{p,q}=u_{(p+q)/2}-\frac{u_p+u_q}{2}.
\]
There are \(\binom{129}{2}+\binom{128}{2}=16384\) rows. Its kernel is the two-dimensional affine subspace, since the rows with \(q=p+2\) force every second difference to vanish. Hence
\[
C_8=\|A^{-1}:A(\mathbb R^{257}/\mathrm{Aff})\to
\mathbb R^{257}/\mathrm{Aff}\|.
\]
For \(i<j<k\), put
\[
L_{ijk}(u)=u_j-\frac{k-j}{k-i}u_i-\frac{j-i}{k-i}u_k.
\]
The discrete Chebyshev alternation theorem gives
\[
d(u)=\frac12\max_{i<j<k}|L_{ijk}(u)|.
\]
Linear-programming duality now gives the finite exact formula
\[
C_8=\frac12\max_{i<j<k}
\min\{\|y\|_1:A^Ty=L_{ijk}\}.
\]
Every coefficient is rational, so an optimizer and a matching dual solution would constitute a rational exact certificate.
For u(x)=min(x,1-x), the midpoint defect is 1/2 and the best affine error is 1/4.
Result
Evidence: Reproduced
A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1
For u(x)=min(x,1-x), the midpoint defect is 1/2 and the best affine error is 1/4.[3]
Relevance to this problem
For Exact Jensen stability constant on the eighth dyadic grid, record djs8-claim-certified-lower-bound (“The tent function certifies C_8 >= 1/2”) records a bound, answer, status fact, or structural consequence. The record states: For u(x)=min(x,1-x), the midpoint defect is 1/2 and the best affine error is 1/4.
Evidence
ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope.
Record state
established
Scope
the sampled tent function on the denominator-256 grid
Take \(u(x)=\min(x,1-x)\). Concavity makes every midpoint defect nonnegative, and its range lies in \([0,1/2]\). Thus \(\delta(u)\leq1/2\), with equality for the pair \((0,1)\).
The constant affine function \(1/4\) has uniform error \(1/4\). For any affine \(\ell\), write \(r=u-\ell\). Affineness gives
\[
r(0)-2r(1/2)+r(1)=-1.
\]
The left side has absolute value at most \(4\|r\|_\infty\), proving \(d(u)\geq1/4\). Consequently
\[
\frac{d(u)}{\delta(u)}=\frac{1/4}{1/2}=\frac12.
\]
The tent function and recursive midpoint interpolation certify \(1/2\le C_8\le711/128\); the exact rational value of \(C_8\) remains open.
Result
Evidence: Reproduced
A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 2
The tent function and recursive midpoint interpolation certify \(1/2\le C_8\le711/128\); the exact rational value of \(C_8\) remains open.[3]
Relevance to this problem
For Exact Jensen stability constant on the eighth dyadic grid, record djs8-claim-certified-upper-bound (“Recursive midpoint interpolation proves C_8 <= 711/128”) records a bound, answer, status fact, or structural consequence. The record states: The tent function and recursive midpoint interpolation certify \(1/2\le C_8\le711/128\); the exact rational value of \(C_8\) remains open.
Evidence
ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope.
Record state
established
Scope
all real-valued functions on the denominator-256 dyadic grid
Let \(\ell_i=((256-i)u_0+iu_{256})/256\). Recursively bisect an interval \([a,b]\) containing \(i\), and write
\[
e_{a,b}=u_{(a+b)/2}-\frac{u_a+u_b}{2}.
\]
If \(i<(a+b)/2\), the coefficient added for \(e_{a,b}\) is \(2(i-a)/(b-a)\); on the right it is \(2(b-i)/(b-a)\). Stop when \(i\) is an endpoint or midpoint. Substitution proves an exact identity
\[
u_i-\ell_i=\sum_{a,b}c_{a,b}(i)e_{a,b},\qquad c_{a,b}(i)\geq0.
\]
The executable rational certificate checks all 257 identities and finds
\[
\max_i\sum_{a,b}c_{a,b}(i)=\frac{711}{128},
\]
attained at \(i=85,171\). The endpoint chord is an admissible affine approximant, so
\[
d(u)\leq\max_i|u_i-\ell_i|\leq\frac{711}{128}\delta(u).
\]
The focused search found classical stability theorems, with no source for this finite-grid sharp constant.
Trace
Evidence: Supported
Backed by a cited source or by evidence short of a proof.
attempt · Route 1
The focused search found classical stability theorems, with no source for this finite-grid sharp constant.[3][2][1]
Relevance to this problem
For Exact Jensen stability constant on the eighth dyadic grid, record djs8-attempt-literature-audit (“The literature treats local and infinite-domain Jensen stability”) documents a concrete method, search boundary, or failed route. The record states: The focused search found classical stability theorems, with no source for this finite-grid sharp constant.
Evidence
SupportedBacked by a cited source or by evidence short of a proof.
Kominek's 1989 paper studies local stability of the Jensen functional equation. Jung gives Hyers-Ulam-Rassias stability and an asymptotic application, while Faiziev and Sahoo treat stability on groups. These sources establish the surrounding theory.
Searches combining `Jensen functional equation`, `Hyers-Ulam stability`, `dyadic grid`, `finite set`, and `sharp constant` found no primary source computing this 257-point quotient norm. The search was focused rather than exhaustive, so novelty remains unverified.
A standard-library Python program checks the constraint count, the tent witness, and every symbolic interpolation identity.
Artifact
Evidence: Executable material
Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1
A standard-library Python program checks the constraint count, the tent witness, and every symbolic interpolation identity.
Relevance to this problem
For Exact Jensen stability constant on the eighth dyadic grid, record djs8-artifact-rational-certificates (“Exact rational lower and upper certificates”) supplies evidence or a replay used to check the packet. The record states: A standard-library Python program checks the constraint count, the tent witness, and every symbolic interpolation identity.
Evidence
Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay.
Record state
available
Scope
all 16384 midpoint rows, all 257 interpolation identities, and the tent witness on the eighth grid
Run
python3 certificate.py
Runtime
Python 3 standard library
Details
All symbolic coefficients use `Fraction`. The canonical JSON report has SHA-256 digest `013676a759bb305c8fdde90c525a5abbe0a3e4f501f4756983399ae6c0713383`.
Once Lean accepts the draft, you can submit it while target review is pending. Your submission request is saved and continues after approval. Private checks stay private until you choose to submit.
1References
Valerii A Faiziev and Prasanna K Sahoo, “On the stability of Jensen's functional equation on groups”. arXiv:math/0703628 (2007). Faiziev and Sahoo, 2007. ↗preprint · reference source · arXiv:math/0703628, version checked 2026-07-24 · checked 2026-07-24Source use: citation only.For Exact Jensen stability constant on the eighth dyadic grid: The literature treats local and infinite-domain Jensen stability. The focused search found classical stability theorems, with no source for this finite-grid sharp constant.
Soon-Mo Jung, “Hyers-Ulam-Rassias stability of Jensen’s equation and its application”. Proceedings of the American Mathematical Society 126(11) (1998), 3137-3143. DOI 10.1090/S0002-9939-98-04680-2. Proceedings of the AMS 126(11), 1998, 3137-3143. ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.For Exact Jensen stability constant on the eighth dyadic grid: The literature treats local and infinite-domain Jensen stability. The focused search found classical stability theorems, with no source for this finite-grid sharp constant.
Packet source. Zygíryd Kominek, “On a Local Stability of the Jensen Functional Equation”. Demonstratio Mathematica 22(2) (1989). DOI 10.1515/dema-1989-0220. Zygfryd Kominek, Demonstratio Mathematica 22(2), 1989, 499-508; Soon-Mo Jung, Proceedings of the AMS 126(11), 1998, 3137-3143; Valerii A. Faiziev and Prasanna K. Sahoo, arXiv:math/0703628. ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.For Exact Jensen stability constant on the eighth dyadic grid: The literature treats local and infinite-domain Jensen stability. The focused search found classical stability theorems, with no source for this finite-grid sharp constant.Also cited at Demonstratio Mathematica 22(2), 1989, 499-508.Also cited at Finite-dimensional quotient-norm duality and Chebyshev alternation, specialized in this record.Also cited at Exact tent calculation reproduced in djs8-artifact-rational-certificates.Also cited at Exact coefficient identities in djs8-artifact-rational-certificates.Source named by the research packet.
Discussion
Loading
Past commenters and subscribers receive notifications when someone comments.