Add exact counterexamples, manuscripts, and verification package

.github/workflows/rounding-counterexamples.yml

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+name: Verify rounding counterexample papers
+
+on:
+  push:
+    branches:
+      - agent/rounding-counterexample-papers
+    paths:
+      - "2026-rounding-counterexamples/**"
+      - ".github/workflows/rounding-counterexamples.yml"
+  pull_request:
+    paths:
+      - "2026-rounding-counterexamples/**"
+      - ".github/workflows/rounding-counterexamples.yml"
+
+permissions:
+  contents: write
+
+jobs:
+  exact-certificates:
+    if: github.event_name != 'push' || !contains(github.event.head_commit.message, '[skip generated PDFs]')
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v4
+      - uses: actions/setup-python@v5
+        with:
+          python-version: "3.12"
+      - name: Verify final certificates
+        working-directory: 2026-rounding-counterexamples
+        run: python3 code/verify_all.py
+      - name: Reconstruct parameter families
+        working-directory: 2026-rounding-counterexamples
+        run: make derive
+
+  build-pdfs:
+    if: github.event_name != 'push' || !contains(github.event.head_commit.message, '[skip generated PDFs]')
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v4
+      - name: Install TeX dependencies
+        run: |
+          sudo apt-get update
+          sudo apt-get install -y latexmk texlive-latex-extra texlive-fonts-recommended
+      - name: Compile all manuscripts
+        working-directory: 2026-rounding-counterexamples
+        run: make papers
+      - name: Upload compiled manuscripts
+        uses: actions/upload-artifact@v4
+        with:
+          name: rounding-counterexample-papers
+          path: 2026-rounding-counterexamples/build/*.pdf
+          if-no-files-found: error
+      - name: Commit compiled manuscripts to the review branch
+        if: github.event_name == 'push'
+        run: |
+          mkdir -p 2026-rounding-counterexamples/generated-pdfs
+          cp 2026-rounding-counterexamples/build/*.pdf 2026-rounding-counterexamples/generated-pdfs/
+          git config user.name "github-actions[bot]"
+          git config user.email "41898282+github-actions[bot]@users.noreply.github.com"
+          git add -f 2026-rounding-counterexamples/generated-pdfs/*.pdf
+          if git diff --cached --quiet; then
+            echo "Generated PDFs are unchanged."
+          else
+            git commit -m "Add compiled research PDFs [skip generated PDFs]"
+            git push origin HEAD:${GITHUB_REF_NAME}
+          fi

2026-rounding-counterexamples/.gitignore

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+build/
+papers/*.aux
+papers/*.bbl
+papers/*.blg
+papers/*.fdb_latexmk
+papers/*.fls
+papers/*.log
+papers/*.out
+papers/*.pdf
+papers/*.synctex.gz

2026-rounding-counterexamples/AI_DISCLOSURE.md

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+# AI-use and authorship disclosure
+
+A generative AI system assisted with exploratory reasoning, algebraic checks, source-code drafting, exposition, LaTeX preparation, and repository organization.
+
+The AI system is not an author. It cannot make an authorship declaration, warrant novelty, accept responsibility for errors, disclose conflicts of interest, transfer copyright, or respond to peer review. The eventual human authors must:
+
+- independently verify every theorem and citation;
+- establish priority and contact relevant researchers when appropriate;
+- supply the legal author list, affiliations, acknowledgements, funding, and contribution statement;
+- review all code and generated PDFs;
+- comply with the target journal's current disclosure policy;
+- take full responsibility for the submitted and published work.
+
+Suggested disclosure language, subject to the target journal's rules:
+
+> A generative AI system was used during exploratory problem solving and to assist with algebraic checking, code drafting, and language editing. The authors independently verified the mathematical arguments, computational certificates, citations, and final text, and take full responsibility for the work.
+
+No confidential referee report, unpublished third-party manuscript, personal data, or proprietary dataset is included in this package.

2026-rounding-counterexamples/Makefile

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+PYTHON ?= python3
+LATEXMK ?= latexmk
+BUILD := build
+PAPER_DIR := papers
+FIGURE_DIR := figures
+PAPERS := dgg-counterexample chairman-counterexample overview
+
+.PHONY: all verify derive papers clean
+
+all: verify papers
+
+verify:
+	$(PYTHON) code/verify_all.py
+
+derive:
+	$(PYTHON) code/derive_dgg_family.py
+	$(PYTHON) code/derive_chairman_family.py
+
+papers: $(PAPERS:%=$(BUILD)/%.pdf)
+
+$(BUILD):
+	mkdir -p $(BUILD)
+
+$(BUILD)/dgg-counterexample.pdf: $(PAPER_DIR)/dgg-counterexample.tex $(PAPER_DIR)/references.bib $(FIGURE_DIR)/dgg_network.tex $(FIGURE_DIR)/dgg_obstruction.tex | $(BUILD)
+	cd $(PAPER_DIR) && $(LATEXMK) -pdf -interaction=nonstopmode -halt-on-error dgg-counterexample.tex
+	cp $(PAPER_DIR)/dgg-counterexample.pdf $@
+
+$(BUILD)/chairman-counterexample.pdf: $(PAPER_DIR)/chairman-counterexample.tex $(PAPER_DIR)/references.bib $(FIGURE_DIR)/chairman_gadget.tex | $(BUILD)
+	cd $(PAPER_DIR) && $(LATEXMK) -pdf -interaction=nonstopmode -halt-on-error chairman-counterexample.tex
+	cp $(PAPER_DIR)/chairman-counterexample.pdf $@
+
+$(BUILD)/overview.pdf: $(PAPER_DIR)/overview.tex $(PAPER_DIR)/references.bib $(FIGURE_DIR)/graphical_abstract.tex $(FIGURE_DIR)/dgg_network.tex $(FIGURE_DIR)/chairman_gadget.tex | $(BUILD)
+	cd $(PAPER_DIR) && $(LATEXMK) -pdf -interaction=nonstopmode -halt-on-error overview.tex
+	cp $(PAPER_DIR)/overview.pdf $@
+
+clean:
+	cd $(PAPER_DIR) && $(LATEXMK) -C dgg-counterexample.tex chairman-counterexample.tex overview.tex || true
+	rm -rf $(BUILD)

2026-rounding-counterexamples/README.md

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+# Counterexamples to two weighted-rounding conjectures
+
+**Status:** research preprint and exact certificate package, prepared for independent review. The mathematical claims are explicit and finitely checkable; priority and novelty should still be checked by the eventual human authors and referees before public announcement.
+
+This directory contains two self-contained results.
+
+1. **A planar counterexample to Goemans' cost conjecture for single-source unsplittable flow.** The instance has seven vertices, nine arcs, three demands, exactly two paths per terminal, and exactly eight unsplittable routings. The fractional cost is `58`; every unsplittable routing satisfying the conjectured additive upper bound has cost at least `60`.
+2. **A counterexample to the machine-dependent weighted-chairman conjecture of Liu and Reis.** The instance has 11 rows and 15 columns. A three-column forcing gadget is repeated five times, producing an exact prefix discrepancy of at least `619/600 > 1`.
+
+## Fast verification
+
+The verifiers use only the Python standard library and exact integer or rational arithmetic.
+
+```bash
+cd 2026-rounding-counterexamples
+python3 code/verify_all.py
+```
+
+Expected final lines:
+
+```text
+DGG certificate verified: min good cost = 60 > fractional cost = 58
+Chairman certificate verified: forced accumulator lower bound = 619/600 > 1
+All exact certificates passed.
+```
+
+## Build the manuscripts
+
+A TeX Live installation with `latexmk`, `pdflatex`, TikZ, `booktabs`, and `hyperref` is sufficient.
+
+```bash
+make papers
+```
+
+The expected PDFs are:
+
+```text
+build/dgg-counterexample.pdf
+build/chairman-counterexample.pdf
+build/overview.pdf
+```
+
+GitHub Actions runs the exact verifiers and compiles all three PDFs. The compiled PDFs are uploaded as workflow artifacts; generated binaries are deliberately not treated as source files.
+
+## Directory map
+
+```text
+2026-rounding-counterexamples/
+├── README.md
+├── Makefile
+├── AI_DISCLOSURE.md
+├── SUBMISSION_CHECKLIST.md
+├── code/
+│   ├── verify_dgg.py
+│   ├── verify_chairman.py
+│   ├── verify_all.py
+│   ├── derive_dgg_family.py
+│   └── derive_chairman_family.py
+├── data/
+│   ├── dgg_instance.json
+│   └── chairman_instance.json
+├── figures/
+│   ├── dgg_network.tex
+│   ├── dgg_obstruction.tex
+│   ├── chairman_gadget.tex
+│   └── graphical_abstract.tex
+├── papers/
+│   ├── dgg-counterexample.tex
+│   ├── chairman-counterexample.tex
+│   ├── overview.tex
+│   └── references.bib
+└── research/
+    ├── derivation-dgg.md
+    ├── derivation-chairman.md
+    ├── importance-and-history.md
+    ├── provenance.md
+    └── validation-report.md
+```
+
+## Central certificates
+
+For the flow instance, the three zero-cost path choices form a triangle conflict graph. If `z_i=1` denotes choosing the cheap path for terminal `i`, every capacity-good integral routing satisfies
+
+```text
+z_1 + z_2 + z_3 <= 1,
+```
+
+whereas the fractional cheap-path marginals are `(1/3, 2/5, 1/3)` and sum to `16/15`. The costs are the nonnegative separator of this violated stable-set inequality.
+
+For the chairman instance, every hypothetical discrepancy-1 assignment is forced to make five specific decisions. Each decision contributes `5/24` to an accumulator row. Ten detector columns can offset the accumulator by at most `10/1000`, giving
+
+```text
+5*(5/24) - 10/1000 = 619/600 > 1.
+```
+
+## Scope of the record
+
+The repository contains the complete final derivations, exact data, exact certificate verifiers, parametric-family reconstruction scripts, manuscript sources, figures, and build automation. Earlier exploratory searches were conducted interactively and were not all preserved as standalone programs; `research/provenance.md` distinguishes final reproducible code from ephemeral exploration.
+
+## Authorship and submission
+
+The manuscripts use `Anonymous for review`. Replace that placeholder with the legal author list, affiliations, acknowledgements, contribution statement, and corresponding-author details before submission. Do not list an AI system as an author. See `AI_DISCLOSURE.md` and `SUBMISSION_CHECKLIST.md`.

2026-rounding-counterexamples/SUBMISSION_CHECKLIST.md

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+# Journal submission checklist
+
+## Mathematical audit
+
+- [ ] Two independent experts have checked the seven-vertex SSUF certificate.
+- [ ] An expert has checked the exact wording and quantifiers of Goemans' conjecture against the cited sources.
+- [ ] An expert has checked the chairman forcing proof, including assignments outside fractional support.
+- [ ] `python3 code/verify_all.py` passes on a clean checkout.
+- [ ] The GitHub Actions verification and PDF-build jobs are green.
+- [ ] The parametric `9/8` planar lower-bound calculation has been independently checked.
+
+## Priority and literature
+
+- [ ] Repeat the literature search immediately before submission.
+- [ ] Check arXiv, DBLP, MathSciNet, zbMATH, conference proceedings, and authors' webpages for overlapping results.
+- [ ] Consider notifying the conjecture originators and the authors of the most recent positive results.
+- [ ] Replace all `to our knowledge` claims with wording supported by the final search date.
+
+## Authorship and ethics
+
+- [ ] Replace `Anonymous for review` as required by the journal.
+- [ ] Add affiliations, corresponding-author details, funding, acknowledgements, and contributions.
+- [ ] Review `AI_DISCLOSURE.md` against the journal's current policy.
+- [ ] Confirm that every human author has read and approved the final manuscript.
+- [ ] Confirm that the title, abstract, and press language do not overstate peer-review status.
+
+## Typesetting and artifacts
+
+- [ ] Build all PDFs from a clean checkout with `make all`.
+- [ ] Inspect every compiled page at readable resolution.
+- [ ] Check that figure labels are legible in grayscale and at one-column width.
+- [ ] Check references, DOIs, hyperlinks, theorem numbering, and equation cross-references.
+- [ ] Remove build auxiliaries and verify that source archives contain everything needed.
+- [ ] Select and apply an explicit code/data license approved by the authors.
+
+## Recommended circulation order
+
+1. Private mathematical audit.
+2. Author-originator courtesy check where appropriate.
+3. Preprint deposit with immutable versioned certificate.
+4. Journal or conference submission.
+5. Public announcement only after the authors are satisfied with priority and correctness.

2026-rounding-counterexamples/code/derive_chairman_family.py

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+#!/usr/bin/env python3
+"""Verify the parametric inequalities behind the chairman gadget."""
+
+from __future__ import annotations
+
+from fractions import Fraction
+
+
+def main() -> None:
+    blocks = 5
+    p = Fraction(5, 24)   # accumulator share in J
+    a = Fraction(1, 2)    # A-row weight in K
+    q = Fraction(25, 48)  # H-row share in L
+    eta = Fraction(1, 1000)
+
+    # Ignoring the small off-support penalty, these are the three forcing
+    # margins.  They must all be positive.
+    margin_K_not_A = a / 2 - p
+    margin_L_to_H = 1 - p - a / 2 - q
+    margin_L_not_H = q - Fraction(1, 2)
+    accumulator_margin = blocks * p - 1
+
+    assert margin_K_not_A == Fraction(1, 24) > 0
+    assert margin_L_to_H == Fraction(1, 48) > 0
+    assert margin_L_not_H == Fraction(1, 48) > 0
+    assert accumulator_margin == Fraction(1, 24) > 0
+
+    # The first detector row can lose at most 12 eta before its block, while
+    # the second can lose at most 13 eta after a possible off-support J choice.
+    assert margin_K_not_A > 12 * eta
+    assert margin_L_to_H > 12 * eta
+    assert margin_L_not_H > 13 * eta
+    assert accumulator_margin > 2 * blocks * eta
+
+    detector = 1 + min(
+        margin_L_to_H - 12 * eta,
+        margin_L_not_H - 13 * eta,
+    )
+    accumulator = blocks * p - 2 * blocks * eta
+
+    assert detector == Fraction(6047, 6000) > 1
+    assert accumulator == Fraction(619, 600) > 1
+
+    print(f"p={p}, a={a}, q={q}, eta={eta}, blocks={blocks}")
+    print(
+        "forcing margins: "
+        f"K={margin_K_not_A}, L-to-H={margin_L_to_H}, "
+        f"L-not-to-H={margin_L_not_H}"
+    )
+    print(f"detector lower bound: {detector}")
+    print(f"accumulator lower bound: {accumulator}")
+
+
+if __name__ == "__main__":
+    main()