# P100024: Determine the exact eighth-grid Jensen stability constant

- ID: `P100024`
- Reference: `djs8-problem-exact-constant`
- Page: https://theoremdb.org/statements/P100024
- Record maturity: Retired source record

## Problem

Determine the exact value of the eighth-grid Jensen stability constant \(C_8\).

## Status

The mathematical status has not passed editorial review.

## Work

### Other known results

- **Theorem 1** (established): Chebyshev duality expresses C_8 as a maximum of finite rational l1 minimization problems. [3](#reference-3)
- **Computation 1** (reproduced): For u(x)=min(x,1-x), the midpoint defect is 1/2 and the best affine error is 1/4. [3](#reference-3)
- **Computation 2** (reproduced): 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](#reference-3)

### Prior approaches

- **Route 1** (supported): The focused search found classical stability theorems, with no source for this finite-grid sharp constant. [3](#reference-3) [2](#reference-2) [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (reproduced): A standard-library Python program checks the constraint count, the tent witness, and every symbolic interpolation identity.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `djs8-problem-exact-constant`, 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. <a id="reference-1"></a>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 https://arxiv.org/abs/math/0703628
   - preprint; reference source; arXiv:math/0703628, version checked 2026-07-24; checked 2026-07-24
   - Source 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.
2. <a id="reference-2"></a>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 https://doi.org/10.1090/S0002-9939-98-04680-2
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source 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.
3. <a id="reference-3"></a>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 https://doi.org/10.1515/dema-1989-0220
   - 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
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source 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.
   - Source named by the research packet.