PureOne/Ilyrun-Qhora-II-Gapped-Consecutive-Ones

Ilyrūn Qhōra II

(2,1)-Gapped Consecutive-Ones Property: Exact Laminar Counting, Uniform Sampling, Minimal Interfaces and Counting Complexity

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Research version: 2.0 · 31 October 2026
Original manuscript title: Ilyrūn Qhōra II: Exact enumeration, minimal interfaces, and universality for the (2,1)-gapped consecutive-ones property

This research package studies how binary-matrix columns can be ordered when each row permits at most one internal zero. Its central contribution is an exact three-profile composition calculus for laminar row families: supports that are nested or disjoint. It counts all valid labeled-column permutations in polynomial time, computes the complete total-defect distribution, and supports exact uniform generation through bijective integer unranking. A transfer theorem permits bounded arbitrary nonlaminar cores inside a validated enclosing hierarchy.

The release includes the 45-page manuscript, editable Markdown and LaTeX, Python implementations, input instances, independent checkers, mathematical audits, certificates, and recorded measurements. It retains the original NP-completeness reduction and subsequent universality, structural-recognition and counting-complexity results. Supplementary inventory theory gives exact repeated-column counting laws and a sharp abundance threshold for ordinary consecutive ones.

This repository is a mathematical research and reproducibility archive. Its heterogeneous JSON files are solver inputs, results and certificates; they are not a single machine-learning training dataset. The automatic dataset viewer is disabled.

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Material	Link
Complete manuscript	Ilyrun_Qhora_II_Manuscript.pdf
Original research entry point	source/README.md
Detailed research guide	source/research/README.md
Solver formats and reproduction commands	Software guide
Editable manuscript	Markdown · LaTeX
Claim scope and evidence	Theorem manifest
Untouched original 354-file archive	Original research ZIP
Citation	BibTeX
Packaging provenance and compatibility fix	PACKAGING_NOTES.md

Mathematical model

For a nonempty support $S$ and a permutation $p$, the row defect is

$$h_p(S)=\max p(S)-\min p(S)+1-\mathrm{\mid }S\mathrm{\mid },h_p(\mathrm{\varnothing })=0.h\_p(S)=\backslash max\; p(S)-\backslash min\; p(S)+1-|S|,\; \backslash qquad\; h\_p(\backslash varnothing)=0.$$

The defect counts the total number of internal zeros between the row's first and last selected columns. With budget zero or one this is the one-hole, $(2,1)$-gapped consecutive-ones setting. Larger total-zero budgets must not be conflated with the general $(k,\delta)$-C1P model, which separately limits the number of one-blocks and individual gap lengths.

Columns are distinct and labeled, even when their incidence patterns coincide. Reversed orders are distinct when their labeled sequences differ. Duplicate rows contribute separately to the total-defect statistic.

Principal structural results

For laminar supports with row budgets $\beta_i\in{0,1}$, the manuscript gives an algorithm for

$$Z_M(z)=\sum _{\begin{array}{c}p\text{ a labeled permutation}\\ h_p(S_i)\le {\beta }_i\mathrm{\forall }i\end{array}}{z}^{\sum _i{h}_p(S_i)}\mathrm{.}Z\_M(z)=\backslash sum\_\{\backslash substack\{p\backslash text\{\; a\; labeled\; permutation\}\backslash \backslash h\_p(S\_i)\backslash le\backslash beta\_i\backslash \; \backslash forall\; i\}\}\; z^\{\backslash sum\_i\; h\_p(S\_i)\}.$$

It computes the exact count $Z_M(1)$, full defect histogram and uniform row-hole marginals. A polynomial-time unranking bijection maps tickets $0,\ldots,Z_M(1)-1$ to every valid labeled order exactly once. A uniform ticket therefore gives an exactly uniform order.

The recurrence retains three boundary responses per cluster:

Profile	Meaning
$C$	Valid internal orders.
$D$	Valid orders containing one distinguished foreign marker strictly inside the cluster hull.
$R$	The subset of those orders with the marker immediately before the cluster's final column.

The crossing lemma shows that overlapping hulls of disjoint nontrivial children form isolated pairs. Each crossing pair contributes $2R_A R_B$. This yields a complete composition rule without discarding legal interleavings. A rank-three context matrix proves that three coordinates are necessary within exact linear boundary representations; it is not an unrestricted lower bound on algorithms or nonlinear encodings.

After polynomial preprocessing, scalar counting and each laminar unranking operation use $O(N^3)$ integer arithmetic operations. A conservative full-histogram bound is $O(N^3(R+1)^2)$. The explicit-input model also has polynomial bit complexity. Row count, row size, hierarchy depth, column frequency and incidence width can grow with the input. Feasibility alone is already easy for laminar families: the main advance is enumeration of the entire permitted gapped solution space.

The bounded-core transfer theorem allows arbitrary internal rows in supplied disjoint cores, each nontrivial core itself an actual support with budget at most one, inside a checked laminar enclosing hierarchy. It gives fixed-parameter tractability in maximum core size. It does not find an optimal decomposition, and the provided CLI does not implement arbitrary-core uniform sampling.

Inventory laws and retained hardness results

The supplementary methods preserve separated copies of identical incidence types, which row-specific budgets can require. Finite run skeletons and a marked-automaton compiler provide exact counts and defect spectra. Once each type multiplicity strictly exceeds the sum of budgets of rows omitting that type, feasibility becomes equivalent to ordinary C1P and the count, divided by the product of type factorials, obeys an ordinary multivariate polynomial law. The strict inequality is essential; a supplied counterexample fails at equality.

Retained results include the audited NP-completeness reduction, compact guarded-slot forcing, actual-row representation of upward-closed verifier predicates, a $\Pi_2^P$ barrier for a specified nested-extension recognition problem, and #P-completeness of counting labeled valid orders under polynomial-time Turing reductions. The latter is not a parsimonious-reduction claim. The second-level recognition barrier concerns that structural certification problem, not ordinary matrix recognition.

Run the examples

Use Python 3.12, the version used for the recorded research checks. The laminar enumerator, sampler, core enumerator, run-skeleton method and independent checker use the Python standard library. From the repository root:

cd source
python research/software/laminar_enumerator.py research/examples/laminar_hierarchy_eight.json --marginals > laminar_result.json
python research/software/checker.py research/examples/laminar_hierarchy_eight.json laminar_result.json
python research/software/laminar_sampler.py research/examples/laminar_hierarchy_eight.json --samples 3 --seed 20261031
python research/software/hierarchical_enumerator.py research/examples/two_path_cores.json --time-limit 30
python research/software/hierarchical_enumerator.py research/examples/two_nonlaminar_cores.json --time-limit 30

Expected results: the eight-column hierarchy has 1,312 valid orders and coefficients 288, 384, 256, 256, 128; the two path cores have 320 valid orders; the nonlaminar-core example has 32 orders, all requiring eight total additions.

Optional general CP-SAT recognition uses the dependency pinned in source/research/software/requirements.txt. See the software guide before running the full test or benchmark scripts: some deliberately expensive cases take much longer and some scripts write fresh result files. Work in a copy to preserve the historical measurements.

Portable completion and infeasibility checkers cover their documented certificate classes. A completed exact search may return UNSAT; resource-limit exhaustion is UNKNOWN. General CP-SAT infeasibility has no exported portable proof in this release.

Evidence and measured limits

The original research records the following executed validation. These are retained historical results, distinct from the fresh packaging smoke checks in validation/packaging_validation.json.

Component	Recorded validation
Laminar enumeration	172 instances; full histograms compared against 602,120 direct permutations.
Uniform sampler	65 global instances; all 5,730 global tickets and 1,524 marker tickets checked.
Bounded-core transfer	30 instances; 813,600 direct permutations.
Run skeleton	768 decision, count, optimization and spectrum comparisons over 192 instances.
Inventory compiler	128 direct histogram comparisons; 94,624 permutations; accepted certificates and rejected corruptions.

The recorded full-histogram runs include a 512-column balanced hierarchy at 0.0876 seconds and a 512-column nested chain at 1.2450 seconds. The generic recurrence took 97.1747 seconds on 256 disjoint pairs; a separate disjoint-family formula reproduced that histogram in 0.013952 seconds under its own measurement protocol. These are exploratory shared-machine measurements, not universal performance guarantees. Raw inputs, outputs and environment records are included.

The manuscript supplies full proofs and documents internal independent audits. Its literature logs state the inspected prior-work boundary. External peer review, proof-assistant formalization, absolute priority and empirical biological or device validation are not claimed. The exact sampling ensemble is a declared combinatorial null model, not an inferred probability distribution of real biological histories or physical layouts.

Potential uses

The methods support exact uncertainty calculations for hierarchical marker-order constraints, nested one-dimensional placement models, and reproducible study of bounded-gap matrix ordering. These interpretations require the stated support, labeling, pitch and observation assumptions. Missing observations, unequal device widths, electrical or thermal effects need additional models.

Citation, provenance and rights

Use CITATION.bib. No DOI, journal acceptance or published repository URL is invented. The package credits its research author exactly as supplied. Its original materials, prior-work references, audits and superseded research stages remain available under source/ and in the unchanged archive.

The supplied research contained no explicit reuse license. This prepared release applies no new blanket license and deliberately omits a license identifier from the card metadata. See RIGHTS_AND_PROVENANCE.md.

Search terms: (2,1)-C1P; (2,1)-GCOP; gapped consecutive-ones property; one-hole binary matrix; laminar support family; exact labeled permutation counting; uniform random generation; bijective unranking; minimal boundary interface; total internal zeros; matrix completion; fixed-parameter enumeration; repeated column inventory; #P-completeness; consecutive-ones NP-completeness.