--- title: "The Picard group of a triangulated product of graphs" date: 2026-09-08 version: "0.1.0-candidate" doi: 10.5281/zenodo.22656789 pdf: https://github.com/ipitchford/triangulated-graph-product-picard/releases/download/v0.1.0-candidate/SOLUTION.pdf repository: https://github.com/ipitchford/triangulated-graph-product-picard archive: https://zenodo.org/records/22656789 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # The Picard group of a triangulated product of graphs ## Summary Take two networks, form a square for each pair of edges, and divide every square into two triangles. The resulting surface-like object carries a tropical version of a divisor class group: a way to classify integer configurations after identifying those related by allowed piecewise-linear changes. This unrefereed candidate gives a uniform description of that group. The two original networks contribute their own classes; pairs of independent cycles contribute an additional free part. The answer works for every choice of square diagonals. Parallel edges require an explicit local convention, and the paper carefully distinguishes the discrete ridge group from the larger group with continuous Jacobian factors. ## Summary for specialists Let $G,H$ be finite connected loopless multigraphs, each with at least one edge, with circuit ranks $g,h$. Give edges unit length and use the Cartwright–Lazar structure constants on any independently diagonally triangulated product $\Gamma$. Under the branch-local Cartier convention, the written proof gives $$\operatorname{Pic}_r(\Gamma)\cong\mathbb Z^{2+gh}\oplus K(G)\oplus K(H),$$ and $$\operatorname{Pic}(\Gamma)\cong J(G)\times J(H)\times\mathbb Z^{2+gh}.$$ Here $K$ denotes finite graph critical groups and $J$ denotes metric-graph Jacobian tori. The splittings are noncanonical. For simple factors, this proves Lazar's product formula; the multigraph extension uses branch germs, not restriction of global vertex potentials to stars. ## Technical account Signed diagonal coefficients turn compatibility into two incidence equations. Decomposing each rational edge space into cuts and cycles eliminates the mixed cut–cycle blocks, leaving exactly the cut–cut and cycle–cycle blocks. Rooted path integration supplies integer lifts, so the argument does not confuse rational rank with integral solvability. The principal diagonal lattice is primitive. Its quotient therefore contributes no new torsion. A separate integer argument identifies the factor kernel. Explicit diagonal flips preserve the signed data, intertwine principal divisors and commute even when squares share a boundary. For two cycles, the all-ones tensor can be a nonprimitive class: it is the greatest common divisor of the cycle lengths times a primitive generator. The full Picard group is obtained through Cartwright's sheaf comparison. Its Jacobian torus is not the finite ridge Chern kernel. The general results rest on these written arguments, not extrapolation from the software. ## Evidence, assurance and limitations The package contains the complete proof, exact-arithmetic diagnostic code, manifests, source attribution and a response to the supplied review. Producer checks exercise 27 product/triangulation cases, 682 local germ patterns, all 512 diagonals for the three-cycle product, 36 primitive cycle lifts, 12 individual flips and 66 commuting pairs. These overlapping test categories are not a count of distinct product complexes. The publication workflow adds internal AI editorial review, fresh-extraction replay and fail-closed controls. These are not unaffiliated specialist acceptance, independent reimplementation or formal proof verification. Historical priority remains uncertain. The theorem excludes loops, disconnected factors and arbitrary edge lengths; point factors have a separate lower-dimensional answer. No practical performance or workflow-acceleration effect has been measured. ## Relationship to earlier work Lazar's 2017 paper supplies the conjectural formula and the earlier factor-map and tree results. Cartwright supplies the tropical divisor framework, the ridge/full comparison and an earlier torus example. The torus group shape itself is not claimed as new. An earlier UnsolvedMath catalogue record supplied the diagonal exact-sequence and incidence-primitivity reduction. The release archives that record with attribution and identifies its version limits and corrected claims. The present contribution is the uniform integral closure, explicit compatibility lattice and constructive refinements, not discovery of the conjectural formula or of cut–cycle linear algebra. ## Who should care, and why | Audience | Potential use | Required caution | | --- | --- | --- | | Tropical geometers and chip-firing researchers | Inspect a uniform integral product argument and ridge/full comparison. | The parallel-edge convention and cited sheaf inputs are essential. | | Computational algebra researchers | Implement integral class normal forms and test triangulation transport. | The supplied code is diagnostic, not a general-purpose normal-form library. | | Interested mathematical readers | See why rational dimensions do not determine an integer class group. | Finite checks support inspection, not universal proof certification. | ## Why the problem matters The product question asks how divisor theory changes when graph factors form a two-dimensional complex. Recovering the entire integer group requires controlling torsion and local integrality, not just counting dimensions. The explicit lattice makes that distinction inspectable and offers a concrete starting point for further tropical product calculations. ## How to inspect or reproduce the recorded checks Download the evidence ZIP or clone the research repository. With Python 3.10 or later, install the pinned SymPy dependency using `python3 -m pip install -r requirements.txt`, then run `python3 release_check.py` from the extracted directory. It checks file hashes, exhaustive diagnostic parity and optimization-mode rejection. Expected output ends in PASS. Read Theorems 1–2 and Sections 2–6 for the universal proof. The verifier's success is not a substitute for inspecting those arguments. The environment is specified but not hermetically pinned. ## The most valuable next projects 1. Unaffiliated scrutiny of branch-local Cartier lifting, the integer factor kernel and the full-Picard comparison. 2. An independently written normal-form implementation with documented basis conventions. 3. A specialist antecedent search for equivalent product formulas and integral lattice constructions. 4. Carefully formulated extensions to arbitrary metric lengths or other excluded graph classes. ## Who might contribute Specialists in tropical divisor theory can assess the sheaf conventions; integer-lattice software authors can independently implement the quotient construction. Neither task should inherit correctness from producer replay alone. ## What is in the evidence package The archive provides the PDF and Markdown proof, verifier and release checker, pinned dependency, exact receipt, claim index, audit, response matrix, source identity records and component licences. Original prose is CC0-1.0 and original code is MIT; the attributed catalogue record retains CC-BY-4.0. GitHub and Zenodo preserve the same release assets. Communication art and audio explain the result but are not additional mathematical evidence. ## Open directions for follow-up research - Unaffiliated mathematical review and independent implementation. - Specialist historical-priority audit. - General-purpose integral normal-form software. - Extensions to arbitrary lengths and excluded graph classes. ## Research process, metrics and reusable methods This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:triangulated-graph-product-picard - Attempt and metric receipts: ep-attempt:triangulated-graph-product-picard-assurance-publication: published / positive; scope assurance-through-publication; target Review repairs, deterministic baseline, five-role internal acceptance, immutable GitHub and Zenodo assets, and first canonical Evidence Press publication.; active forecast 150 minutes (120-210); Fermi components Review repairs and sources: 1 x 25/30/40 minutes low/central/high (Minor attribution and metadata revisions.); Package baseline and internal review: 1 x 35/45/65 minutes low/central/high (Compact proof with fast exact replay.); Public identity and media: 1 x 30/35/50 minutes low/central/high (Existing publishing tools.); Site seals and canonical readback: 1 x 30/40/55 minutes low/central/high (Mandatory new-slug two-deployment cycle.); positive-signal/closure probabilities 0.95/0.85 within 240 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 2/unknown/unknown/3/0/0; cycles positive/negative/inconclusive 0/0/0; falsification gates 1; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -148 minutes; ratio 0.013333333333333334; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.0225; missing telemetry activeHumanMinutes: No contemporaneous human active-time instrument.; computeMinutes: No complete scoped substantive-computation wall clock; ordinary CI/build time is not research compute.; appended measurement corrections measurement.agentRuns -> metrics.outcome.agentRuns: Terminal receipt records six agent runs; original intake snapshot retained. (reason: Prospective snapshot predates five completed internal reviewers.); measurement.reworkMinutes -> metrics.outcome.reworkMinutes: Actual repairs are disclosed in outcome varianceReason; zero does not mean no repair. (reason: No complete repair-specific active-time instrument; terminal zero is an explicit lower bound.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json - Intended aims: science - Artifact roles: research-output, evidence-assessment, communication - Decision object: reusable-method — Integral compatibility lattice and ridge/full Picard classification. Scope: Connected loopless unit-edge graph products, branch-local Cartier convention, all diagonal choices. - Reusable methods: Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: assurance, publication - Semantic bridge: explicit — Local branch lifts and endpoint incidence equations identify the integer Cartier lattice; primitive principal diagonals control torsion; the cited sheaf comparison supplies the full-group bridge. Remaining risks: Uniform written proof awaits unaffiliated scrutiny.; Finite diagnostics cannot establish universal correctness.; The global-star convention on parallel edges is a different object.; Equivalent earlier results are not excluded.. - Human judgement gates: Audit integral local-to-global sufficiency and sheaf comparison.; Assess antecedents and historical priority.; Preserve branch conventions, attribution and rights. - Next assurance action: Unaffiliated mathematical audit, independent implementation and primary antecedent comparison. - Claim ceiling: Unrefereed uniform written proof candidate under explicit branch-local conventions. Producer exact diagnostics and internal AI review do not establish unaffiliated whole-proof acceptance, formal verification, historical priority or impact. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Inspectable integral product classification candidate in Producer-coordinated publication; design none; comparator No matched comparison; estimand No acceleration or downstream effect estimated; no real-world effect evidence asserted - Parent handoffs: depends-on-claim https://doi.org/10.37236/5533; inherited claim: Product construction, conjectural formula and factor/tree antecedents.; inherited ceiling: Earlier theorem does not validate this general closure or altered multigraph convention.; depends-on-claim https://arxiv.org/abs/1506.02023; inherited claim: Ridge/full sheaf comparison and exponential sequences.; inherited ceiling: Classical framework, not external review of the present specialization.; depends-on-claim https://huggingface.co/datasets/ulamai/UnsolvedMath; inherited claim: AIM-COMBINATORICS-0121 attempt 1 diagonal exact-sequence and primitivity reduction; hash-bound snapshot in package.; inherited ceiling: Unreviewed catalogue input with explicit upstream version limits and corrected claims. ## Verification status Unrefereed uniform written proof candidate under explicit branch-local conventions. Producer exact diagnostics and internal AI review do not establish unaffiliated whole-proof acceptance, formal verification, historical priority or impact. ## References 1. Lazar (2017), A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group, Conjecture 1.2 and Proposition 4.5. 2. Cartwright (2015), Combinatorial tropical surfaces, Section 3 and Example 3.2. 3. Cartwright (2020; online 2019), Tropical complexes. 4. AIM chip-firing workshop (2013), Problem 31. 5. UnsolvedMath, catalogue record AIM-COMBINATORICS-0121, imported attempt 1; snapshot archived with explicit provenance limits.