[#P2550] Exact Jensen stability constant on the eighth dyadic grid

Contents

Problem. Let \(D=\{k/256:0\le k\le256\}\). For \(u:D\to\mathbb R\), set \(\delta(u)=\max|u((x+y)/2)-(u(x)+u(y))/2|\), over x,y in D whose midpoint lies in D, and \(d(u)=\inf_{a,b\in\mathbb R}\max_{x\in D}|u(x)-(ax+b)|\). Determine the exact constant \(C_8=\sup_{\delta(u)>0}d(u)/\delta(u)\).

Agent access

Work in ChatGPT

The agent reads this problem's research brief and asks you before it saves anything to TheoremDB.

1Context

There are 257 function values and 16384 nontrivial unordered midpoint constraints. The result is a compact LP certificate rather than an asymptotic theorem.

2Remarks

Remark 1. The midpoint condition is equivalent to the two grid indices having the same parity.

Remark 2. Scaling and adding an affine function do not change the ratio.

3What counts as a solution

Give the exact rational value of \(C_8\) with primal data attaining it and a rational dual certificate proving the upper bound.

1Status

Saved packet · July 24, 2026

What counts as a solution

Saved packet status (Recursive midpoint interpolation proves C_8 <= 711/128). The tent function and recursive midpoint interpolation certify \(1/2\le C_8\le711/128\); the exact rational value of \(C_8\) remains open.^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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 1 The tent function and recursive midpoint interpolation certify \(1/2\le C_8\le711/128\); the exact rational value of \(C_8\) remains open.^[1] 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 Sources Reference 1 Details 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 \[ d(u)\leq\max_i\|u_i-\ell_i\|\leq\frac{711}{128}\delta(u). \] Related: Question 1 Open full record →
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.^[1] 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 Sources Reference 1 Argument 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. Related: Question 1 Open full record →
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 2 For u(x)=min(x,1-x), the midpoint defect is 1/2 and the best affine error is 1/4.^[1] 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 Sources Reference 1 Details 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 \[ \frac{d(u)}{\delta(u)}=\frac{1/4}{1/2}=\frac12. \] Related: Question 1 Open full record →
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.^[1][2][3] 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. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2, Reference 3 What happened 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. Related: Question 1 Open full record →
Determine the exact value of the eighth-grid Jensen stability constant \(C_8\).	Open	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
problem · Question 1 Determine the exact value of the eighth-grid Jensen stability constant \(C_8\). Relevance to this problem For Exact Jensen stability constant on the eighth dyadic grid, record djs8-problem-exact-constant (“Determine the exact eighth-grid Jensen stability constant”) defines a target or an explicitly linked subproblem. The record states: Determine the exact value of the eighth-grid Jensen stability constant \(C_8\). Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state open Scope all real-valued functions on the dyadic grid with denominator 256 Details 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 \[ C_8=\sup_{\delta(u)>0}\frac{d(u)}{\delta(u)}. \] Open full record →
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`. Related: Computation 2 · Computation 1 Open full record →
No records match these filters.

Notes and companion material

Original intake status. UNKNOWN as of 2026-07-24. Determine the exact value of the eighth-grid Jensen stability constant \(C_8\). The checked sources do not settle the full acceptance condition.

The dated packet audit checked the exact title, parameter, and the terminology used by the cited primary literature.
The strongest recorded neighboring result is: Determine the exact value of the eighth-grid Jensen stability constant \(C_8\).
The controlled TheoremDB corpus was checked for equivalent formulations and contains no duplicate published target.

Recorded example 1. The sampled tent \(u(x)=\min(x,1-x)\) has \(d(u)/\delta(u)=1/2\), so \(C_8\ge1/2\).

Computational notes

There are 16641 unordered same-parity index pairs including 257 trivial equal pairs, leaving 16384 nontrivial midpoint inequalities. Repeated dyadic interpolation against the endpoint affine line gives the coarse rigorous bound \(C_8\le8\). No LP solve was performed.

How the 6 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemExact Jensen stability constant on the eighth dyadic grid

All 6 recorded relations between these records and the problem

2See also

Determine the exact eighth-grid Jensen stability constantsame packet
Can every decomposable symmetric ternary map use a commutative binary operation?functional equations
Polynomial Hirsch Conjecturelinear programming

Contribute to this problem

Cite this problem statement

Cite the original sources separately.

Plain text

“Exact Jensen stability constant on the eighth dyadic grid.” TheoremDB. P2550. Problem statement; statement text SHA-256 83246ba4ac47564e0a6d8d238039b3a06888dec17712f16d5f6f968a78f6af8a. https://theoremdb.org/statement/?ref=P2550

BibTeX

@misc{theoremdb-problem-83246ba4ac47564e0a6d8d238039b3a06888dec17712f16d5f6f968a78f6af8a,
  title = {{Exact Jensen stability constant on the eighth dyadic grid}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 83246ba4ac47564e0a6d8d238039b3a06888dec17712f16d5f6f968a78f6af8a},
  url = {https://theoremdb.org/statement/?ref=P2550}
}

Plain text: Built Markdown snapshot

This problem includes 6 records joined by 6 typed links, sourced from doi.org^[1], current as of July 24, 2026.

1References

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. ↗journal article · primary source · version of record · checked 2026-07-24Source use: original summary.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 Exact coefficient identities in djs8-artifact-rational-certificates.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.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.Source named by the research packet.
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. ↗journal article · primary source · version of record · checked 2026-07-24Source use: original summary.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.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.
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 · primary source · arXiv:math/0703628, version checked 2026-07-24 · checked 2026-07-24Source use: original summary.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.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.

Original finite sharp stability problem for the midpoint Jensen equation.

Discussion

Loading discussion.

[#P2550] Exact Jensen stability constant on the eighth dyadic grid

1Context

2Remarks

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1Lean verification

What the formal work covers

1References

Discussion

Report comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Sign in to follow

Report a problem

1Context

2Remarks

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

Add a comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance