Sep 28, 2026 · Sep 28, 2026
@@ -0,0 +1,30 @@
+name: P15 overlapping-block exact controls
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/p15_bipartite_overlap_20260928/**'
+      - '.github/workflows/p15-bipartite-overlap.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  exact-overlap:
+    runs-on: ubuntu-latest
+    timeout-minutes: 10
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Verify exact sources and both-mode semantic controls
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S frontiers/p15_bipartite_overlap_20260928/run_validation.py
+          git diff --exit-code
@@ -0,0 +1,150 @@
+# Even capacities: read-two overlap without bipartiteness
+
+Object: OA-P15-EVEN-READ-TWO-20260928-v1.
+Author: OpenAI / ChatGPT, 28 September 2026.
+Disposition: AUTHOR-SIDE CANDIDATE; independent analytic review required.
+Scientific effect NONE. This is an explicit alternative to the bipartite
+hypothesis of PROOF.md, not permission to omit that hypothesis for odd capacities.
+
+## E1. Exact coloring under the alternative structural hypothesis
+
+Keep the finite original ground set X, covering blocks X_i and capacity downset
+D of (O1), but replace bipartiteness by:
+
+    every original coordinate belongs to at most two blocks;
+    every capacity a_i is a POSITIVE EVEN integer.                    (E1)
+
+**Theorem E1.** For all U subset X,
+
+    chi_D(U)=max_i ceil(|U intersect X_i|/a_i).                         (E2)
+
+*Proof.* Necessity is unchanged. Write k for the right side and assume U is
+nonempty. Represent coordinates by edges of the possibly NONBIPARTITE block
+incidence multigraph: a shared coordinate joins its two distinct block vertices;
+a private coordinate joins its block to a private auxiliary vertex.
+
+At each real vertex i, split incident edges as evenly as possible among a_i/2
+auxiliary vertices. Each new vertex has degree at most
+ceil(2|U intersect X_i|/a_i)<=2k. Private auxiliary vertices have degree one.
+Each original edge keeps its one original-coordinate label.
+
+The number of odd-degree vertices is even. Pair them arbitrarily and add one
+auxiliary edge per pair. Every odd degree was at most 2k-1, so every augmented
+degree is even and at most 2k. Each component has an Euler orientation, with
+indegree=outdegree=degree/2<=k at every vertex. Form the bipartite directed
+incidence graph with one outgoing and one incoming copy of each vertex; an
+oriented edge becomes one edge from its tail's outgoing copy to its head's
+incoming copy. Maximum degree is at most k. The matching decomposition in the
+proof of (O2) colors this bipartite graph with at most k colors.
+
+In the undirected split graph, a color has at most one outgoing and one incoming
+edge at each split vertex, thus at most two incident edges. Remove the auxiliary
+edges and combine the a_i/2 split vertices back into block i: at most a_i
+original coordinates of each color lie in X_i. This is a coloring of ORIGINAL
+coordinates, not independent copies of them. It proves (E2). QED.
+
+For |X_i|=a_i d_i+1, K=max_i d_i, the exact active-full-block obstruction (O4)
+and chi_D(X)=K+1 follow from (E2) by the same elementary argument. This holds
+even on an odd cycle of blocks. It does not extend to the odd-capacity triangle
+counterexample of PROOF.md, where capacities are one.
+
+## E2. A self-contained read-q hazard inequality
+
+The two-sided Cauchy-Schwarz proof of (O9) used a bipartition. It cannot be
+reused unchanged for an odd cycle. We prove the necessary replacement rather
+than assuming independence of the overlapping events.
+
+**Lemma E2.** Let mu be a finite product probability measure. Let A_i be events
+depending on coordinate sets S_i, each coordinate lying in at most q of those
+sets. If D is contained in every A_i, then
+
+    sum_i [-log mu(A_i)] <= q[-log mu(D)].                            (E3)
+
+The inequality uses extended hazards; mu(D)=0 is immediate.
+
+*Proof for mu(D)>0.* Set nu=mu(.|D). Denote relative entropy by
+D_KL(nu||mu), with the usual zero-mass conventions. Then
+D_KL(nu||mu)=-log mu(D). Each marginal nu_{S_i} is supported on A_i, whence
+
+    D_KL(nu_{S_i}||mu_{S_i})
+       = D_KL(nu_{S_i}||mu_{S_i}(.|A_i)) - log mu(A_i)
+       >= -log mu(A_i).                                               (E4)
+
+Order all coordinates. The chain rule and product form of mu give
+
+    D_KL(nu_S||mu_S)
+      = sum_{v in S} E_nu D_KL(nu(v | earlier coordinates in S)||mu_v).
+
+Conditioning on only the earlier coordinates in S gives a mixture of the laws
+conditioned on all earlier coordinates. Convexity of relative entropy in its
+first argument (the finite log-sum inequality) therefore bounds each summand by
+
+    c_v := E_nu D_KL(nu(v | all earlier coordinates)||mu_v) >= 0.
+
+Thus D_KL(nu_S||mu_S)<=sum_{v in S} c_v. Summing over S_i and using read-q
+incidence gives
+
+    sum_i D_KL(nu_{S_i}||mu_{S_i}) <= q sum_v c_v
+                                  = q D_KL(nu||mu).
+
+Combine with (E4). Deterministic coordinates (probability zero or one) have zero
+conditional divergence and can also be removed from the finite product support.
+No absolute-continuity problem arises because nu is a conditioning of mu. QED.
+
+This is the entropy form of a classical read-q/Finner-type inequality. Its proof
+is included; no result about arbitrary dependent measures is asserted. It
+applies to just the active blocks in (O4), as required for charging generators.
+
+## E3. Full transformed prices and a sub-2/3 factor
+
+Assume additionally |X_i|=a_i d_i+1, with d_i positive integers and K=max_i d_i>=4.
+For an active block a_i is even, hence a_i>=2. The Chernoff calculation in Lemma
+O3 gives its reference probability at p_*=1-exp(-1) at most 25/512, uniformly
+in ALL such capacities. Hence its reference hazard is at least
+
+    h_even=log(512/25)>3.                                              (E5)
+
+The last inequality has a short rational certificate:
+
+    e < 87/32,
+    512/25 - (87/32)^3 = 314641/819200 > 0.
+
+The separate-concavity transfer (O5) yields price(X_i)<=h_i/h_even. Lemma E2 with
+q=2 yields sum_active h_i<=2[-log mu_p(D)]. Since the active full-block cover is
+proved by Theorem E1, summing and applying the price-one empty-generator cap gives
+
+    covercost_c(O_K(D))
+       <= min(1, [2/log(512/25)] [-log mu_p(D)])
+       <= min(1, (2/3)[-log mu_p(D)]).                                (E6)
+
+Here 2/log(512/25)<2/3; neither factor is claimed optimal. Probabilities are the
+original independent Bernoulli probabilities in [0,1], and all prices
+0<=c_v<=min(1,-log(1-p_v)) are admitted. The zero/infinite-hazard cases are handled
+as in PROOF.md. The global palette remains K, not 2K.
+
+This supplies a second justified structural class: arbitrary read-two block
+incidence with EVEN capacities. The first class permits arbitrary positive
+capacities but requires bipartiteness. Their hypotheses should not be mixed or
+silently dropped. Inactive smaller demands, including demand one, are harmless
+to the generator budget in both classes, but (E1) still requires their capacities
+to be even for the displayed coloring proof.
+
+## E4. Verification and literature boundary
+
+The standard-library implementation follows the displayed construction:
+capacity-two splitting, pairing odd vertices, Euler orientation, a bipartite
+matching coloring, and one final color per original coordinate. It is checked
+against an independent partition oracle on small nonbipartite examples with
+heterogeneous even capacities. A triangle with four shared edges per side and
+one private coordinate per block has K=4: the twelve shared coordinates are
+4-colorable, while the full fifteen-coordinate set needs five colors.
+
+The read-two hazard inequality is tested on exact heterogeneous Bernoulli grids
+on triangle incidence. Finite grids do not prove (E3); its entropy argument does.
+
+Aboulker-Aubian-Huang, arXiv:2201.11548, studies defective edge coloring and states
+the uniform even-capacity phenomenon. This note does not claim invention of that
+classical structural mechanism. The variable-capacity construction and the
+original-coordinate price/cover assembly are written out here; no unpublished
+original checker from another packet is used. Independent mathematical review
+and any formalization are still separate obligations.