A portable tail-certificate principle behind interlacing families

Definitions and mathematical statement

This item is an aspirational workshop prompt about Kadison--Singer (KS), attributed to Adam Marcus by the attached aid, rather than a mathematical proposition with fixed hypotheses. It asks for a tool expressing the useful content of a KS-type proof in another mathematical area, with enough portability to support further applications. The source's statement that polynomial tools are not portable is motivation and opinion, not a quantified impossibility theorem. No target area, admissible objects, conclusion, constant, or computational requirement is specified. Consequently there is no uniquely determined theorem to prove or disprove from this record alone.

The attached aid proposes one scoped abstraction: selection along a finite rooted tree of recursive outcomes using tail certificates. Its objects include a finite tree, a notion of badness or a certified threshold at each node, certificate functions, and local compatibility, tail-soundness, growth, guard, and oracle conditions. With compatibility error $a\geq0$, absolute oracle error $\varepsilon\geq0$, and positive normalized growth $\kappa>0$, it reports a deterministic descent whose allowed badness bound increases by at most \[ \frac{a+2\varepsilon}{\kappa} \] per level. For a root-to-leaf path of depth $h$, the corresponding accumulated allowance is at most $h(a+2\varepsilon)/\kappa$ when these parameters are uniform along that path. Exact compatibility and an exact oracle correspond to $a=\varepsilon=0$.

This numerical conclusion is conditional on the aid's axioms. Their definitions, the sign and domain of the certificates, the exact local guard, and the relation of certified thresholds to outcome badness are not supplied in the record. Merely naming affine compatibility and tail soundness does not provide a complete theorem statement. The aid also mentions a polynomial corollary and two exact examples claiming that the needed tail condition is necessary for this particular mechanism yet weaker than common interlacing; neither the corollary nor the examples are reproduced. These claims concern a selection module once suitable certificates are available. They do not construct certificates in every area, solve KS anew, or fulfill the source's broad portability request for all mathematical domains.

Short English statement

This is a call to develop portable KS-inspired tools, with no fixed theorem specified. The attached aid offers a conditional finite-tree selection principle and an explicit per-level error allowance, but omits the axioms needed for a complete theorem and does not provide certificates for arbitrary applications.

Sources

• AIM workshop Beyond Kadison--Singer: paving and consequences, item b28: https://docs.google.com/document/d/1XRw0WyG5PshqboCp99tovCRD1GCK0jXZiNLH_nw618c/edit. The attached aid attributes this aspirational prompt to Adam Marcus; the original extracted item supplies no individual attribution.
• Local registry data/unsolvedmath/problems.json, permanent ID $20000569$, exact source code catalogue entry; canonical aim-analysis-notes.json, notes[100].
• No authoritative external citation for the proposed tail-certificate theorem or examples is supplied. The attached attempt is explicitly machine-generated, not peer reviewed, and requires independent verification before citation.