Name: Frontier Score for "Complexity functions of infinite sequences"
Rating: 69
Author: Frontier

What it is

We describe an asymptotic solution to the inverse problem, realizing every function satisfying two necessary conditions as the complexity function of an infinite word up to asymptotic equivalence. We then turn to phenomena beyond asymptotic equivalence, presenting constructions of infinite words with seemingly contradictory behavior of the discrete derivatives of their complexity functions.

Every metric behind this entry is listed, with its source, under Sources and data below.

Filed undersemigroups and automata theory, Mathematical Dynamics and Fractals, Algorithms and Data Compression

Scored 2026-11-04How scoring works

The Frontier Score, in full

How this entry's score is built, term by term. Open any pillar to see each sub-metric's raw value, how it maps to 0 to 100, and a link to its source (OpenAlex for most sub-metrics that combine several sources, NIH iCite where its ratio is part of field-normalized impact).

The three pillars as one shape: a result strong on every axis fills the triangle.

How the Frontier Score is calculated for this breakthrough
Term	Score	Weight	Points
EvidenceHow established and verifiable the result is.	86	× 0.25	21.6
ImpactHow much the result matters, judged field-relative and by current momentum.	48	× 0.25	12.1
NoveltyHow genuinely new the result is, and how fast-moving.	71	× 0.50	35.6
Weighted pillars			69.3
Frontier recency0 months since publication		× 0.99
IntegrityNot retracted		× 1
Frontier Score			69

Confidence: Early signalEarly signal: little citation history yet, so impact is an estimate and may shift as the work is taken up.

Every sub-metric, traced to its source

EvidenceHow established and verifiable the result is.86

Evidence sub-metrics
Sub-metric	Raw value	0 to 100	Weight	Points	Source
Peer review & venueWhether the work has cleared peer review, and how selective its venue is. Peer review is necessary but not sufficient: before citations accrue, a fresh result in one of the five general flagships (Nature, Science, Cell, NEJM, the Lancet) is a stronger evidence signal than one in a selective specialist or society journal, which in turn outranks one in a legitimate but very high-volume mega-journal, and any of them outranks a preprint that has not been reviewed at all.Maps to 0 to 100: Peer-reviewed article graded by venue tier: flagship journal (Nature, Science, Cell, NEJM, the Lancet) 100; selective journal (every other journal the index admits: the specialist journals of those families, JAMA, PNAS, Physical Review Letters and X) 90; high-volume mega-journal (Nature Communications, Science Advances) 78; venue not recorded 88. Other records: review 90, book 70, dataset 60, preprint 45, an article that has not been peer reviewed 45, other 55.	Peer-reviewed, selective journal	90	0.30	27.0	OpenAlex type / primary_location / source
CorroborationIndependent corroboration that the result is on the record and reaching beyond a single source, read as the BREADTH of independent records rather than the size of the citation pile. Counts how many of the five independent citation indices (OpenAlex, Crossref, Semantic Scholar, OpenCitations, Europe PMC) report a citation count for the work at all, zero included, plus orthogonal, non-citation lines of corroboration: citation in Wikipedia, registered clinical trials or clinical citations, released datasets or software or a repository copy, public funding on record, and technical-community discussion. Citation MAGNITUDE is deliberately judged under Impact, not here, so Evidence and Impact measure genuinely different things instead of both rewarding the same citation pile twice.Maps to 0 to 100: Starts at 28 and rises by 9 for each of the five independent citation indices that returns a citation count for the work (zero counts; OpenCitations returns zero even for a DOI it does not hold) and by 6 for each orthogonal non-citation line (Wikipedia, clinical trials or clinical citations, open datasets, software or a repository copy, public funding, community discussion), up to the top of the 0-100 scale.	4 of 5 citation indices report a count	64	0.30	19.2	OpenAlex, Crossref, Semantic Scholar, OpenCitations, Europe PMC, DataCite, NIH RePORTER, Wikipedia, Hacker News count of indices that report a count for the work + orthogonal reach
VerifiabilityHow openly the result can be read, reused, and reproduced. Rewards open access, a Creative Commons BY or CC0 license, a green repository copy anyone can archive, openly minable full text, and released datasets or software a reader can actually run. Cross-checked across OpenAlex, Unpaywall, Europe PMC, and DataCite.Maps to 0 to 100: Closed access 45; open access 88; any Creative Commons BY license (including the NC, ND and SA variants) or CC0 100. A repository copy raises the floor to 85, openly minable full text to 90, and released data or software to 92.	Open access (hybrid), CC-BY-NC-ND	100	0.20	20.0	OpenAlex, Unpaywall, Europe PMC, DataCite is_oa / license / has_repository_copy / open datasets
IntegrityRetraction and post-publication correction status. A retracted result is not evidence of anything and collapses the whole Frontier Score; a correction or expression of concern is a softer flag. A correction the publisher labels as its own error (a Publisher Correction) is not a flag on the work. Read across OpenAlex and Crossref (both update-to and updated-by notices).Maps to 0 to 100: Clean 100 (also when the only notice is a Publisher Correction); a correction or expression of concern 75; retracted 0. Flagged if OpenAlex OR Crossref records the notice.	No retraction or concern on record	100	0.20	20.0	OpenAlex, Crossref is_retracted / update-to / updated-by

ImpactHow much the result matters, judged field-relative and by current momentum.48

Impact sub-metrics
Sub-metric	Raw value	0 to 100	Weight	Points	Source
Field-normalized impactImpact relative to the world average for the same field, year, and type, so a small field and a large field are judged fairly. Triangulates three independent field-normalized measures: OpenAlex's Field-Weighted Citation Impact, its citation percentile within the exact field-and-year cohort, and the NIH iCite Relative Citation Ratio. 1.0x is average; the percentile is the share of same-field, same-year work it out-cites.Maps to 0 to 100: Each ratio (1.0 = field average) maps 1.0 to 50, 3.0 to 75, 9.0 to 90; the field+year percentile maps directly (top 1% -> ~99). The available measures are averaged. When none is available yet (very recent work), this sub-metric is left out and Impact is computed from the others.	top 14% of field-year vs field	86	0.45	38.7	OpenAlex, NIH iCite fwci / citation_normalized_percentile / relative_citation_ratio
Influential citationsCitations that genuinely build on the work rather than mention it in passing (Semantic Scholar's influential-citation measure). A sharper signal of real impact than a raw count.Maps to 0 to 100: Saturating: 15 influential citations maps to 50, and none maps to 0. When Semantic Scholar has no measure for the work, this sub-metric is left out and Impact is computed from the others.	Not yet available	n/a	redistributed	0.0	Semantic Scholar influentialCitationCount
Citation velocityCitations received in the trailing twelve months (estimated from yearly citation counts: this year's so far plus the share of last year's that falls inside the window), judged as a rate rather than a lifetime total. Momentum is what a frontier index cares about: a result being taken up fast right now is landing, whether its lifetime pile is large or still small. Where an expected field citation rate is known, the momentum is also read relative to it, so a fast-moving result in a quiet field is not overlooked.Maps to 0 to 100: Saturating: 30 citations in the last twelve months maps to 50. When NIH iCite provides the field's expected rate, this is blended evenly with the ratio of observed to expected momentum.	0 in last 12 months	0	0.35	0.0	OpenAlex, NIH iCite counts_by_year / field_citation_rate

NoveltyHow genuinely new the result is, and how fast-moving.71

Novelty sub-metrics
Sub-metric	Raw value	0 to 100	Weight	Points	Source
Conceptual noveltyHow genuinely new the contribution is, measured at publication from the work's OWN content, not its date: how atypical its combination of research areas is versus all prior work, and whether it builds on a broad, deep base rather than only the newest papers in a fast-moving area. This is what separates a real advance from a recent-but-commodity result, and it needs no forward citations, so it is stable for brand-new work. An honestly noisy proxy (even state-of-the-art novelty measures agree only moderately with expert judgment), so it carries bounded weight and never drives the ranking alone.Maps to 0 to 100: Two publication-time signals, each 0-100. The primary one is atypical combination: how unusual the work's median pairing of research topics is versus the prior literature, by pointwise mutual information (a pairing made far more often than chance scores low). The second, foundational reach (the median age of the work's references and the share older than five years), can only lift it: when reach is higher, the score rises by 35% of the gap, and it is never lowered. When only one signal is measurable it is used alone; when neither is (a work too thin in topics or references), this sub-metric is left out and Novelty is computed from the other three. A review is damped to 60%.	Median topic pairing semigroups and automata theory x Mathematical Dynamics and Fractals (5,428 prior works made this pairing); builds on refs a median 18.4y deep	55	0.45	24.5	OpenAlex topics + referenced_works (publication-time)
RecencyHow recently the work appeared. The frontier is now, so newer scores higher.Maps to 0 to 100: Exponential decay with a 12-month half-life from the publication date.	This month	99	0.30	29.7	OpenAlex publication_date
Attention accelerationWhether attention is rising: citations in the latest full year versus the year before. Accelerating interest signals an active, opening frontier.Maps to 0 to 100: Latest/prior-year ratio: 1x maps to 50, 2x to 75, 4x to 100. The prior year counts as at least 3 citations, so a jump from one citation to a few is not read as explosive growth. With no citations in the latest full year it reads as unknown, and unknown is neutral (50).	Not enough history	50	0.10	5.0	OpenAlex counts_by_year
Edge of reviewPreprints sit ahead of the peer-review process and brand-new papers have only just cleared it. Both earn novelty here (while a preprint is discounted under Evidence), because that is where the frontier forms first.Maps to 0 to 100: Preprint 95; any other record 80 when under 6 months old, 65 under 18 months, 45 after that.	Peer-reviewed record	80	0.15	12.0	OpenAlex type / publication_date

ProvenanceAll sources

Sources and data

The source figures behind this entry, grouped by the source that holds them: up to five citation indices cross-checked, field-normalized impact, then real-world reach (Wikipedia, community discussion, clinical trials, released data and public funding) where it exists. Each value links to its record.

OpenAlex

Primary record

Field-weighted citation impact: 0.0x field average
Citation percentile: Top 14% of its field-year
Publication date: 2026-10-29
Access: Open access (hybrid)

Crossref

Citations: 0
Works it builds on: 52
Funders on record: 3

OpenCitations

Citations: 0

Europe PMC

Citations: 0

Unpaywall

Reuse license: CC-BY-NC-ND

OpenAlex record Read it in Proceedings of the National Academy of Sciences

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