Add exact digital Alexander rank oracle

analysis/digital_alexander_rank_manifest.json

@@ -0,0 +1,23 @@
+{
+  "counterexample_search": {
+    "entry_bound": 9,
+    "matrix_count": 160,
+    "max_order": 31,
+    "min_order": 5,
+    "samples_per_matrix": 640,
+    "seed": 2690829
+  },
+  "exhaustive_geometries": [
+    {"id": "axis-L2", "kind": "axis", "L": 2},
+    {"id": "axis-L3", "kind": "axis", "L": 3},
+    {"id": "gaussian-2-1", "kind": "gaussian", "a": 2, "b": 1},
+    {"id": "diamond-L2", "kind": "diamond", "L": 2},
+    {"id": "c4-self-matching-3-1", "kind": "c4_self_matching", "a": 3, "b": 1}
+  ],
+  "scientific_boundary": [
+    "The exhaustive checks establish finite configuration identities only on the declared quotients.",
+    "The fixed-seed general-period scan is a counterexample search, not statistical evidence or a proof.",
+    "A general result still requires a periodic digital Alexander/relative-homology argument with the 4/8 adjacency pair.",
+    "No CFT field, universal amplitude, or closed form for square-site p_c is inferred."
+  ]
+}

notes/digital-alexander-rank-oracle.md

@@ -0,0 +1,83 @@
+# Digital Alexander rank identity: finite exact oracle
+
+Status: Phase A finite theorem search for Issue 269.  The oracle is exact on
+the declared quotients and deliberately stops short of a general Alexander-
+duality proof.
+
+## Algebraic reduction
+
+Let `r_b,r_w` be the ambient torus-homology ranks of the black-primal and
+white-matching complements.  Since each rank lies in `{0,1,2}`, define
+
+```text
+q_either = 1[r_b>0] - 1[r_w>0],
+q_cross  = 1[r_b=2] - 1[r_w=2].
+```
+
+Exhausting the nine abstract rank pairs proves the elementary implication
+
+```text
+q_either = q_cross  =>  2 q_either = r_b-r_w.
+```
+
+The only rank pairs satisfying the premise are
+
+```text
+(0,0), (0,2), (1,1), (2,0), (2,2).
+```
+
+Thus the genuinely geometric step is the already observed equality of the
+`either` and `cross` matching differences.  Once that premise holds, the
+factor-two identity is an exhaustive rank lemma rather than a new conjectural
+fit.
+
+## Finite checks
+
+The additive oracle reuses the canonical general-period homology engine and
+does not depend on the open configuration-Betti PR.  It enumerates every mask
+on axis L=2/3, Gaussian `(2,1)`, diamond L=2, and the self-matching C4 N=10
+control.  For every mask it archives
+
+```text
+(r_black, r_white, q_either, q_cross),
+q_either-q_cross,
+2q_either-(r_black-r_white),
+r_black+r_white-2.
+```
+
+The last residual tests the stronger rank-sum proposal separately; it is not
+silently folded into the weak identity.
+
+The exact joint tables contain only the three rank pairs
+
+```text
+(0,2), (1,1), (2,0).
+```
+
+Across the five geometries this gives zero failures for both
+`2q=r_black-r_white` and the stronger `r_black+r_white=2`.  It also resolves
+the older raw-rank-equality counts: `q=r_black-r_white` occurs exactly on the
+`q=0`, `(1,1)` configurations, never for `q=+/-1`.
+
+## Deterministic general-period search
+
+A fixed-seed scan samples configurations on 160 nonsingular integer period
+matrices with orders 5 through 31.  Empty and full configurations are always
+included.  This is a reproducible counterexample search, not a confidence
+statement: a null scan cannot replace a proof on arbitrary periodic digital
+complexes.
+
+The frozen run evaluated 85,152 configurations and again saw only `(0,2)`,
+`(1,1)`, `(2,0)`, with zero weak-identity, common-channel, or strong-rank-sum
+counterexamples.
+
+## Boundary and next proof obligation
+
+If the finite oracle remains null, Phase B must show why the 4-connected
+foreground and 8-connected matching complement have equal rank-one indicators
+under periodic digital Alexander/relative homology.  The proof must handle
+nonseparating cycles and distinguish the weak identity from the stronger
+rank-sum statement.
+
+No continuum field, amplitude, or value of the square-site threshold follows
+from this finite oracle alone.