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

A neutral schematic of the objects and relations in the statement.

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 \(C_8=\sup_{\delta(u)>0}d(u)/\delta(u)\).

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

Current 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]

1Records

Record	Kind	Assessment
	Result	Reproduced
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 computation someone reran from the artifact on this page. 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 →
	Result	Established
claim · Theorem 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 EstablishedA complete argument is recorded and has been reviewed. 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 →
	Result	Reproduced
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 computation someone reran from the artifact on this page. 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 →
	Trace	Supported
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 →
	Open	Reproduced
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 computation someone reran from the artifact on this page. 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 →
	Artifact	Reproduced
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 ReproducedA computation someone reran from the artifact on this page. 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 →
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Notes and companion materialContext, examples, and computations

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 connectTyped relations and evidence flow

ProblemExact Jensen stability constant on the eighth dyadic grid

Question 1Determine the exact eighth-grid Jensen stability constantin this packetReproduced
Computation 2The tent function certifies C_8 >= 1/2supportsReproduced
Artifact 1Exact rational lower and upper certificatesverifiesReproduced
Computation 1Recursive midpoint interpolation proves C_8 <= 711/128supportsReproduced
Theorem 1C_8 is an exact rational quotient-operator norminformsEstablished
Route 1The literature treats local and infinite-domain Jensen stabilityinformsSupported

2See also

Determine the exact eighth-grid Jensen stability constantsame packet
Power-of-two solution counts for a finite-field functional equationfunctional equations

How to cite

TheoremDB contributors, “Exact Jensen stability constant on the eighth dyadic grid,” TheoremDB research memory, snapshot of July 24, 2026. https://theoremdb.org/statements/dyadic-jensen-stability-8

This page as plain text: dyadic-jensen-stability-8.md

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.

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

1Context

2Remarks

3What counts as a solution

1Status

1Records

Computational notes

2See also

1Lean verification

1References

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Definitions

What counts as a solution

What's known

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1Context

2Remarks

3What counts as a solution

1Status

1Records

Computational notes

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance