Offline, standard-library-only verification software accompanying Seth Christopher's preprint “A Split-Graph Counterexample to a Roman–Double Roman Candidate Inequality”. Version 1.0.0; source-only publication revision figshare.1. This item contains Roman_Double_Roman_Verification_Software_v1.0.0.zip, including the byte-preserved mathematical modules and original standalone verifier, a new strict-JSON wrapper, a new public process adapter, a bounded sequential replay instrument, and a separate LaTeX build driver. The software and its software-use documentation are offered under MIT to the extent applicable rights are held. The original full archives did not have a blanket MIT grant and are not redistributed here. Required companion inputsThe manuscript, exact inputs and research evidence are in the companion Preprint “A Split-Graph Counterexample to a Roman–Double Roman Candidate Inequality”, file Roman_Double_Roman_Manuscript_Source_and_Evidence_v1.0.0.zip, under CC BY 4.0. Merge the two extracted archives into their common Roman_Double_Roman_v1.0.0 directory. Their file paths do not overlap. ReproductionFrom the assembled directory run: python3 -I -S -B verification/check.py inputs/certificate.jsonFor the full bounded replay run: python3 -I -S -B reproduce.py --out ../roman_double_roman_replayUse a new external output directory. Expected replay status: SOURCE_ONLY_ROMAN_REPLAY_PASS. Full replay requires Linux and Python 3.10+; each serial child defaults to 60 wall seconds and 4 GiB address space. No network or model call is required. ScopeFresh public-source replay completed 15 bounded serial processes. The standalone check accepted the exact 15-vertex certificate, verified its upper witnesses and chordality, and exhausted all 42,129 Roman assignments of weight below seven and all 1,076,624 double Roman assignments of weight below ten. The original differential suite passed 1,323,108 assertions across 1,098 labeled graphs on two through five vertices, with 24 relabelings and nine deliberately invalid or unsupported certificates rejected by both production checkers. Additional protocol and strict-JSON rejections passed. These checks share authorship and are not independent peer review. No global priority, minimum-order counterexample, or proof-assistant-kernel certification is claimed. The public source generation is distinct from the original private-runtime generation. The host Python interpreter is not included in the protocol-generation hash. This source-only publication does not reproduce the private sandbox or claim confidential-read isolation. AI assistance: ChatGPT provided substantial assistance with question selection, the graph construction and proofs, source research, verification software, test execution, and manuscript preparation. Aurelia performed bounded candidate search; Selene checked adjacency-set and reduced-label obligations; Mica checked the full original label alphabets and freshly rechecked use. These are computational roles, not coauthors or independent peer reviewers. Seth Christopher initiated and directed the research programme and is the sole author responsible for the submitted content.
This preprint combines three investigations of independent sets on finite trees. It proves arbitrarily long consecutive blocks of strict log-concavity failures in a fixed family of unweighted subcubic trees. For every fixed positive epsilon less than one, a suitable fixed terminal path gives at least a constant multiple of n to the power one minus epsilon consecutive failures along that family. The coefficients in the resulting blocks are nevertheless decreasing. The paper also develops a message-based covariance bound and then proves its exact activity threshold: (500 + 225 sqrt(5))/64, approximately 15.6736764834. The threshold is the same for uniform activity and for positive vertex activities bounded by a common ceiling. At the critical value, the worst normalized covariance norm on trees with at most n vertices grows on the order of the square of log n. Full proofs, counterexamples to specific proof shortcuts, and exact finite, rational-interval and symbolic checks are included. These results do not prove or disprove the general tree independence-sequence unimodality conjecture. The universal assertions rest on the written arguments, not the finite tests. Prior constructions and external results are credited; global priority, independent peer review and proof-assistant certification are not claimed. Author: Seth Christopher, Independent Researcher. Version 1.0.0. Manuscript source and evidence accompany the paper; verification software is supplied separately. CC BY 4.0 applies only to author-held rights; third-party rights remain unchanged. AI assistance: ChatGPT substantially assisted with mathematical development, proof drafting, literature work, verification software, testing and manuscript preparation. Seth Christopher directed the research and is the sole listed author responsible for the content.
Offline, standard-library-only verification software accompanying Seth Christopher's preprint “A Split-Graph Counterexample to a Roman–Double Roman Candidate Inequality”. Version 1.0.0; source-only publication revision figshare.1. This item contains Roman_Double_Roman_Verification_Software_v1.0.0.zip, including the byte-preserved mathematical modules and original standalone verifier, a new strict-JSON wrapper, a new public process adapter, a bounded sequential replay instrument, and a separate LaTeX build driver. The software and its software-use documentation are offered under MIT to the extent applicable rights are held. The original full archives did not have a blanket MIT grant and are not redistributed here. Required companion inputsThe manuscript, exact inputs and research evidence are in the companion Preprint “A Split-Graph Counterexample to a Roman–Double Roman Candidate Inequality”, file Roman_Double_Roman_Manuscript_Source_and_Evidence_v1.0.0.zip, under CC BY 4.0. Merge the two extracted archives into their common Roman_Double_Roman_v1.0.0 directory. Their file paths do not overlap. ReproductionFrom the assembled directory run: python3 -I -S -B verification/check.py inputs/certificate.jsonFor the full bounded replay run: python3 -I -S -B reproduce.py --out ../roman_double_roman_replayUse a new external output directory. Expected replay status: SOURCE_ONLY_ROMAN_REPLAY_PASS. Full replay requires Linux and Python 3.10+; each serial child defaults to 60 wall seconds and 4 GiB address space. No network or model call is required. ScopeFresh public-source replay completed 15 bounded serial processes. The standalone check accepted the exact 15-vertex certificate, verified its upper witnesses and chordality, and exhausted all 42,129 Roman assignments of weight below seven and all 1,076,624 double Roman assignments of weight below ten. The original differential suite passed 1,323,108 assertions across 1,098 labeled graphs on two through five vertices, with 24 relabelings and nine deliberately invalid or unsupported certificates rejected by both production checkers. Additional protocol and strict-JSON rejections passed. These checks share authorship and are not independent peer review. No global priority, minimum-order counterexample, or proof-assistant-kernel certification is claimed. The public source generation is distinct from the original private-runtime generation. The host Python interpreter is not included in the protocol-generation hash. This source-only publication does not reproduce the private sandbox or claim confidential-read isolation. AI assistance: ChatGPT provided substantial assistance with question selection, the graph construction and proofs, source research, verification software, test execution, and manuscript preparation. Aurelia performed bounded candidate search; Selene checked adjacency-set and reduced-label obligations; Mica checked the full original label alphabets and freshly rechecked use. These are computational roles, not coauthors or independent peer reviewers. Seth Christopher initiated and directed the research programme and is the sole author responsible for the submitted content.
Offline verification software accompanying Seth Christopher's preprint “Roman Labels Yield One-Round Power Domination”. Scientific version 1.0.0; packaging revision figshare.2. All 24 Python files are byte-identical to figshare.1. The README, release metadata and checksum manifest have been updated; historical versions of changed software metadata are retained. Roman_Power_Verification_Software_v1.0.0_figshare2.zip contains the verification implementation, the original-definition standalone checker, a strict-JSON wrapper, a bounded public-source replay runner, and a two-pass LaTeX build driver. This authored software subset is offered under MIT to the extent the author holds the relevant rights. It does not relicense the original private archive or third-party material. Required inputs are in Roman_Power_Manuscript_Source_and_Evidence_v1.0.0_figshare2.zip in the companion Preprint item. Extract both archives into one NEW parent directory, merging Roman_Power_v1.0.0. Their paths do not overlap. The optional full historical audit archive is not required for the complete replay. From the merged directory run:python3 -I -S -B verification/check.py inputs/primary_task.json inputs/historical_primary_admitted.json --censuspython3 -I -S -B reproduce.py --out ../roman_power_replay Full replay uses Linux, Python 3.10+ and the standard library. The output directory must be new and outside the release. Expected status: SOURCE_ONLY_ROMAN_POWER_REPLAY_PASS. Fresh testing from the final rebuilt ZIPs completed 83 bounded serial processes; each child defaults to 60 wall seconds and 4 GiB address space. No model call, network service or embedded runtime is needed. A resource stop is unresolved execution, not a counterexample or theorem. The public runner does not recreate the original private sandbox or claim confidential-read isolation. Asserted computational roles are not authenticated agent identities or independent reviewers. The universal theorem rests on the written proof, not these finite tests. AI assistance: ChatGPT provided substantial assistance with target selection, the constructive proof, source research, verification software, test execution, and manuscript preparation. Aurelia generated augmentations; Selene reconstructed neighborhood, payment, and simultaneous-force obligations; Mica reconstructed byte adjacency, checked admission, and freshly rechecked use. These are computational roles, not coauthors or independent peer reviewers. Seth Christopher initiated and directed the research programme and is the sole author responsible for the submitted content.
We give a connected chordal graph on 15 vertices and 30 edges with Roman domination number 7 and double Roman domination number 10. It refutes the candidate inequality (3/2) gamma_R(G) <= gamma_dR(G) recorded at rank 67 of AutoGraphForge's public audit, together with its equivalent rank-68 restatement. The graph consists of a clique K5 and one independent vertex adjacent exactly to each pair of clique vertices. More generally, for the analogous split graph G_k, the manuscript proves gamma_R(G_k) = 2k - 3 for k >= 3 and gamma_dR(G_k) = 2k for k >= 4. Their ratio tends to one, so no universal multiplier greater than one lower-bounds gamma_dR/gamma_R on connected chordal graphs. The finite witness is checked on the original graph and original label alphabets. Author: Seth Christopher — Independent Researcher. Version 1.0.0. Files in this itemRoman_Double_Roman_Counterexample.pdf is the complete mathematical preprint. Roman_Double_Roman_Manuscript_Source_and_Evidence_v1.0.0.zip contains its LaTeX source, exact finite inputs, selected original research evidence, fresh public-source replay receipts, and a preservation map. This authored manuscript/evidence subset is offered under CC BY 4.0 to the extent applicable rights are held. Mathematical facts are not claimed as newly copyrighted; cited works retain their own rights. Companion verification softwareThe executable source is published separately under MIT as “Verification software for A Split-Graph Counterexample to a Roman–Double Roman Candidate Inequality”, file Roman_Double_Roman_Verification_Software_v1.0.0.zip. Extract both ZIPs into one parent directory, merging their shared Roman_Double_Roman_v1.0.0 folder. The source-only replay requires Linux, Python 3.10+ and no third-party packages. This public release excludes private knowledge, embedded runtimes and unrelated investigations; it is not a redistribution of the full private agent archive. Verification and limitsFresh public-source replay completed 15 bounded serial processes. The standalone check accepted the exact 15-vertex certificate, verified its upper witnesses and chordality, and exhausted all 42,129 Roman assignments of weight below seven and all 1,076,624 double Roman assignments of weight below ten. The original differential suite passed 1,323,108 assertions across 1,098 labeled graphs on two through five vertices, with 24 relabelings and nine deliberately invalid or unsupported certificates rejected by both production checkers. Additional protocol and strict-JSON rejections passed. These checks share authorship and are not independent peer review. No global priority, minimum-order counterexample, or proof-assistant-kernel certification is claimed. AI assistance: ChatGPT provided substantial assistance with question selection, the graph construction and proofs, source research, verification software, test execution, and manuscript preparation. Aurelia performed bounded candidate search; Selene checked adjacency-set and reduced-label obligations; Mica checked the full original label alphabets and freshly rechecked use. These are computational roles, not coauthors or independent peer reviewers. Seth Christopher initiated and directed the research programme and is the sole author responsible for the submitted content.
We give a connected chordal graph on 15 vertices and 30 edges with Roman domination number 7 and double Roman domination number 10. It refutes the candidate inequality (3/2) gamma_R(G) <= gamma_dR(G) recorded at rank 67 of AutoGraphForge's public audit, together with its equivalent rank-68 restatement. The graph consists of a clique K5 and one independent vertex adjacent exactly to each pair of clique vertices. More generally, for the analogous split graph G_k, the manuscript proves gamma_R(G_k) = 2k - 3 for k >= 3 and gamma_dR(G_k) = 2k for k >= 4. Their ratio tends to one, so no universal multiplier greater than one lower-bounds gamma_dR/gamma_R on connected chordal graphs. The finite witness is checked on the original graph and original label alphabets. Author: Seth Christopher — Independent Researcher. Version 1.0.0. Files in this itemRoman_Double_Roman_Counterexample.pdf is the complete mathematical preprint. Roman_Double_Roman_Manuscript_Source_and_Evidence_v1.0.0.zip contains its LaTeX source, exact finite inputs, selected original research evidence, fresh public-source replay receipts, and a preservation map. This authored manuscript/evidence subset is offered under CC BY 4.0 to the extent applicable rights are held. Mathematical facts are not claimed as newly copyrighted; cited works retain their own rights. Companion verification softwareThe executable source is published separately under MIT as “Verification software for A Split-Graph Counterexample to a Roman–Double Roman Candidate Inequality”, file Roman_Double_Roman_Verification_Software_v1.0.0.zip. Extract both ZIPs into one parent directory, merging their shared Roman_Double_Roman_v1.0.0 folder. The source-only replay requires Linux, Python 3.10+ and no third-party packages. This public release excludes private knowledge, embedded runtimes and unrelated investigations; it is not a redistribution of the full private agent archive. Verification and limitsFresh public-source replay completed 15 bounded serial processes. The standalone check accepted the exact 15-vertex certificate, verified its upper witnesses and chordality, and exhausted all 42,129 Roman assignments of weight below seven and all 1,076,624 double Roman assignments of weight below ten. The original differential suite passed 1,323,108 assertions across 1,098 labeled graphs on two through five vertices, with 24 relabelings and nine deliberately invalid or unsupported certificates rejected by both production checkers. Additional protocol and strict-JSON rejections passed. These checks share authorship and are not independent peer review. No global priority, minimum-order counterexample, or proof-assistant-kernel certification is claimed. AI assistance: ChatGPT provided substantial assistance with question selection, the graph construction and proofs, source research, verification software, test execution, and manuscript preparation. Aurelia performed bounded candidate search; Selene checked adjacency-set and reduced-label obligations; Mica checked the full original label alphabets and freshly rechecked use. These are computational roles, not coauthors or independent peer reviewers. Seth Christopher initiated and directed the research programme and is the sole author responsible for the submitted content.
We construct a connected graph on 15 vertices with burning number 4 and Slater number 2, refuting the candidate inequality b(G) <= s(G) + 1 recorded in AutoGraphForge's public conjecture audit. The graph consists of a clique K7 joined by one edge to a path on eight new vertices. Its Slater number is determined by its two largest degrees. A ten-vertex geodesic excludes three burning rounds, while an explicit four-round schedule burns the whole graph. More generally, joining a path on t new vertices to K_(t-1), for t >= 8, gives s(G) = 2 and b(G) >= ceil(sqrt(t + 2)). Burning number is therefore unbounded on connected graphs with Slater number two. The proof uses the established diameter lower bound for graph burning. Complete finite certificates and reproducible verification accompany the self-contained arguments. Author: Seth Christopher — Independent Researcher. Scientific version 1.0.0. Files in this itemUnbounded_Burning_Number_at_Slater_Number_Two.pdf is the complete seven-page mathematical preprint. Burning_Slater_Manuscript_Source_and_Evidence_v1.0.0.zip contains the LaTeX source, finite certificate, tables, original research evidence, historical receipts, and fresh source-replay results. These materials are offered under CC BY 4.0, to the extent the licensor holds applicable rights. Mathematical facts and raw numbers are not claimed as newly copyrighted, and cited works retain their own rights. Companion verification softwareThe executable Python code is supplied separately under the MIT License in the companion Software item titled “Verification software for Unbounded Burning Number at Slater Number Two”, file Burning_Slater_Verification_Software_v1.0.0.zip. This separation preserves the paper/evidence and software licensing scopes. Extract both ZIPs into the same parent directory and merge their shared Burning_Slater_v1.0.0 directory. The software is offline and standard-library-only; the full source replay is supported on Linux with Python 3.10+. Verification and limitsThe finite checker evaluates all 3375 ordered three-center sequences for the 15-vertex graph, checks the capacity obstruction, and verifies the four-round witness. The full replay includes bounded search, checking, admission, regression tests and three finite-family instances. The infinite-family statement is established by the manuscript's mathematical argument, not by testing finitely many examples. These computational checks share authorship; they do not constitute independent peer review or a proof-assistant formalization. This work targets the burning–Slater candidate inequality, not the distinct square-root graph-burning conjecture. No claim of global priority or of a smallest counterexample is made. AI assistance: ChatGPT (OpenAI, used in this project as TPM45) provided substantial assistance with problem selection, mathematical development and proof checking, literature searches, verification-software preparation and execution, and manuscript drafting. Aurelia performed bounded candidate search; Selene checked the exhaustive three-round obstruction; Mica checked the capacity certificate and four-round witness. These are computational tools, not coauthors or independent reviewers. Seth Christopher initiated and directed the project and is the sole author responsible for the submitted content.
This preprint proves that, for every finite simple graph without isolated vertices, the power domination number is at most half the Roman domination number, rounded down. More strongly, any Roman dominating function of weight w yields a sensor set of size at most floor(w/2) that observes the graph after initial domination and at most one parallel forcing round.The paper gives a constructive proof, examples showing that the coefficient is sharp, a residual-completion bound, and an adjusted inequality for graphs with isolated vertices. The main statement addresses a candidate in the AutoGraphForge audit. Manuscript source and computational inputs accompany the paper; verification software is supplied separately. Finite computational checks supplement the proof and are not a substitute for it.Author: Seth Christopher, Independent Researcher. Scientific version 1.0.0. Preprint; not independently peer reviewed. No claim of global priority or proof-assistant certification is made.The manuscript and authored supporting evidence are offered under CC BY 4.0 to the extent the author holds applicable rights. Third-party works retain their own rights.AI assistance: ChatGPT substantially assisted with target selection, mathematical development, source research, verification software, testing, and manuscript preparation. Aurelia, Selene, and Mica performed computational roles; they are not coauthors or independent reviewers. Seth Christopher is the sole author responsible for the submitted content.
Seth Christopher· Figshare· 0 citations
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