slug: burke-haris-madhavendra-increasing-function-jets bibkey: burkeharismadhavendra2025repeatedintegrals doi: 10.48550/arXiv.2512.02151 url: https://arxiv.org/abs/2512.02151v1 triage: theorem motivation_gids:

D5/S3/Analytic/Interpolation/OriginalIncreasingFunctionJets.result

The all-order increasing-function endpoint-jet assertion

Problem

Maxim R. Burke, Maleeha Haris and Madhavendra, Repeated integrals of increasing functions, arXiv:2512.02151v1, Conjecture 1.2, p. 2:

$(P_{n})$ is true for all nonnegative integers $n$ .

For each $n \geq 0$ , let $W_{n} \subseteq R^{n + 1}$ consist of vectors $b = (b_{0}, \dots, b_{n})$ for which some $f \in C^{n} [0, 1]$ has nondecreasing, nonconstant $D^{n} f$ and satisfies $D^{j} f (0) = 0$ and $D^{j} f (1) = b_{j}$ for every $0 \leq j \leq n$ . Here $D^{0} f = f$ and endpoint derivatives are one-sided. Nondecreasing does not mean strictly increasing, and absolute continuity is not an input hypothesis.

The assertion $(P_{n})$ requires both openness of $W_{n}$ and, for every $b \in W_{n}$ , one function $F \in C^{\infty} [0, 1]$ satisfying all of the following:

$D^{j} F (0) = 0$ and $D^{j} F (1) = b_{j}$ for every $0 \leq j \leq n$ ;
$D^{n + 1} F (x) > 0$ for every $0 < x < 1$ ;
$D^{n + 1} F (0) = D^{n + 1} F (1) = 1$ ;
$D^{j} F (0) = D^{j} F (1) = 0$ for every $j > n + 1$ .

The sole problem anchor is this original Conjecture 1.2, including $n = 0$ . The follow-up arXiv:2512.23949v1 calls the same interval property $Q_{n}$ and retains its all-order assertion as Conjecture 2.5.

Motivation

The source’s Theorem 8.2 proves only $n = 0, 1, 2, 3$ . The conjecture asks whether every attainable finite endpoint jet admits one simultaneous smooth realization, with the specified strict interior sign and full endpoint conditions, at every order. The follow-up uses the interval assertion in conditional comonotone entire-function approximation and interpolation results.

Gap

The bounded preregistration screen in issue 12872 reports no exact published all-order settlement in the inspected primary and follow-up versions, author/title arXiv queries, a comonotone-function query, and a Crossref title query. Those are attributed literature-screen results, not an exhaustive absence or priority claim.

Di Dio’s arXiv:1804.07058v2, DOI 10.1090/proc/14499, gives related moment-cone and mixture methods. It does not state the original endpoint-one and all-higher-zero assertion. The mathematical relation needed here includes singular continuous input derivatives and exact simultaneous moments, rather than only approximate moments or an absolutely continuous input class.

Route

Represent the continuous, nondecreasing highest input derivative by its Stieltjes measure after clamping to $[0, 1]$ . Nonconstancy gives positive mass; continuity gives atomlessness. Triangular integration identifies the endpoint jets with reversed factorial moment coordinates of $γ_{n} (t) = ((1 - t)^{k})_{k = 0}^{n}$ . Strict separation, together with the finite zero set of a nonzero polynomial, places that moment vector in the interior of the moment cone.

Normalized smooth densities, positive on $(0, 1)$ and flat at both endpoints, concentrate at every center in $[0, 1]$ , including the two endpoints. Their moment cone has full affine span. The convex-interior and closure argument therefore supplies exact finite nonnegative mixtures for interior moments; it uses no assumption that such a point lies inside a full-dimensional simplex of the original generating curve.

Add a small smooth endpoint correction with value one at both endpoints. Its moment vector tends to zero, so the residual remains in the cone interior and admits one exact mixture. Integrating the resulting density $n + 1$ times constructs a single function with every required jet. The inverse coordinate map also identifies the attainable jet set with the preimage of the cone interior and yields openness.

Falsifier

An order $n \geq 0$ at which $W_{n}$ is not open, or an attainable vector with no single smooth function satisfying all four endpoint and positivity clauses, would refute the original assertion. A restriction to finitely many orders, strict input monotonicity, absolute continuity, or approximate moment matching would fail to establish this problem’s statement.

Evidence

The mathematical source is OriginalIncreasingFunctionJets.lean. Its public theorem PnOriginal.result has the exact statement ∀ n : ℕ, PnOriginal.P n. The definitions PnOriginal.W and PnOriginal.P encode the complete source input class and both conclusions, using iteratedDerivWithin on the closed unit interval for the one-sided endpoint convention. The corresponding Blueprint displays these definitions and the all-order theorem.

Triage

theorem; Tier 1, the specific finite-jet interpolation conjecture preregistered in issue 12872. The mathematical result is an analytic existence and openness theorem, rather than a bounded enumeration, numerical certificate, or positive finite instance.

What the settlement shows

Proved, by D5/S3/Analytic/Interpolation/OriginalIncreasingFunctionJets.result: the original $(P_{n})$ holds for every $n \geq 0$ , retaining weak monotonicity, nonconstancy, singular continuous and flat input cases, openness, and one witness satisfying all the finite and higher endpoint jets simultaneously. The decisive mechanism in its proof is exact interior moment realization after a small compensated endpoint correction, followed by repeated integration. No additional regularity hypothesis on the input is required.
Literature-level consequence: the follow-up’s equivalent $Q_{n}$ interval assertion is supplied at every order. Results conditional on that assertion retain their other hypotheses. Their entire-function approximation and interpolation conclusions are not additional formalized conclusions of PnOriginal.result.
Open here: a separate theorem for independently prescribed next-order endpoint values or arbitrary infinite endpoint germs, quantitative bounds on mixture size or concentration width, and a formal bridge to the follow-up’s global entire-function results. None is asserted by the displayed result; each would require its own faithful statement, reuse search, and proof obligation.

ASSUMED-UNVERIFIED

The literature screen is bounded to the sources and queries reported in issue 12872. OpenAlex and Semantic Scholar rate-limited that screen; Google Scholar and DuckDuckGo challenged automated access. No worldwide literature completeness or originality claim follows. The published equivalence with the follow-up’s notation and its global conditional consumers is used only as ordinary source context; those bridges and consumers are not Lean theorems of this module. The original all-order interval assertion is the full scope of the resolution claim.

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