Deferred readouts on both stopped networks: conditional κ = 8.58478e-5

Makefile

@@ -30,6 +30,7 @@ verify-community: community-audit-check community-followup-check copied-reversed
 	$(MAKE) climbed-48-verify
 
 verify-producers:
+	$(MAKE) stopped-product-verify
 	$(MAKE) copied-centers-verify
 	$(MAKE) structured-bulk-verify
 	$(MAKE) endpoint-gauge-producer endpoint-gauge-certificate
@@ -428,3 +429,12 @@ pair-assembly-verify: pair-assembly-producer pair-assembly-check
 .PHONY: verify-pair
 verify-pair: pair-assembly-verify
 	python3 scripts/audit_pair_candidate.py --check docs/research/community-pair-arithmetic.json
+
+.PHONY: stopped-product-producer stopped-product-certificate stopped-product-verify
+stopped-product-producer:
+	python3 scripts/stopped_product_producer.py
+
+stopped-product-certificate:
+	python3 scripts/stopped_product_network.py
+
+stopped-product-verify: stopped-product-producer stopped-product-certificate

NOTICE

@@ -302,3 +302,20 @@ inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
 uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
 are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
 verify_skip_frame.py. Original notices are retained.
+
+Stopped product-ring extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The opposite-bank generic projector factorization, componentwise atom-ring
+adapters, stopped recurrence and outer ordinary-interchange conversion,
+aligned pair-star producer with rational centers, exact denominator-21
+grid and selected (23,23)/(24,24) composition are contributed by icekylinx
+with substantial OpenAI GPT-6 Astra and Codex assistance. This extension
+builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is
+integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37.
+
+The retained ordinary leaf uses the preceding projector, corner and
+copied-center proofs. The selected two-stage topology retains Aurel Prosz
+(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly
+retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching,
+original source notices and each imported license remain unchanged.

PATCHING.md

@@ -1,6 +1,6 @@
 # Applying the inherited manuscript patch
 
-This patch is the PR #10 baseline. The current copied-center extension is
+This patch is the PR #10 baseline. The PR #36 copied-center extension is
 presented in `notes/copied-centers-note.tex`; it is not included
 in this historical combined manuscript patch.
 
@@ -30,3 +30,6 @@ alternatives for the same base. PR7 is pinned at
 `6725c6a17b17871a35353fd29157f4ed851bc114` for the retained producer sources.
 The old uniform recurrences keep their original exponents. The active consumers
 use `lem:batched-chunk-swap` and `prop:batched-simultaneous-layer`.
+
+The stopped product-ring extension is supplied separately in
+`notes/stopped-product-note.tex`; it does not regenerate this historical patch.

README.md

@@ -3,6 +3,36 @@
 **Community research maintained by Douglas Colkitt — conditional on the original
 OpenAI #109 framework.**
 
+## Stopped product-ring extension
+
+The new construction contributed by **icekylinx**, extending merged
+[PR #36](https://github.com/CrocSwap/integer-mult-bounds/pull/36), gives
+
+$$
+T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+\kappa=\frac{194869}{2500000000}=7.79476\times10^{-5}.
+$$
+
+It combines a stopped product-ring bit interchange on `(23,23)` with an
+all-disjoint rational-center complex network on `(24,24)`. The new generic
+opposite-bank factorization pays one reversed child per projector rank;
+atom adapters, ordinary leaves, endpoint copies and the exact denominator-21
+grid are included in the proof. The bound retains the original analytic
+and fixed-tape hypotheses.
+
+[Proof source](notes/stopped-product-note.tex) ·
+[Exact certificate](certificates/stopped-product-network.json) ·
+[Incremental reproduction](docs/stopped-product.md)
+
+```sh
+make stopped-product-verify
+```
+
+The maintainer-reviewed community checkpoint below remains its own result
+and validation record.
+
+## Reviewed community checkpoint
+
 The reviewed community witness gives
 
 $$

SOURCES.json

@@ -234,5 +234,47 @@
       }
     ],
     "continuing_analytic_dependencies": "See structured_bulk_extension.adopted and references/semantic-bulk/MANIFEST.json."
+  },
+  "stopped_product_extension": {
+    "contributor": "icekylinx",
+    "base_commit": "56b66d58297deca1d7dd130247d720e960f77a37",
+    "proof_predecessor_PR": "https://github.com/CrocSwap/integer-mult-bounds/pull/36",
+    "proof_predecessor_commit": "11817ccacb564bb7f98789c20dc11d3fece207e3",
+    "predecessor_merged": true,
+    "archive": "integer_mult_round5_kappa_7_79476e-5_complete.zip",
+    "archive_sha256": "714cbffa0cccba3c18f046a63e62d6981683cb386b5d5cd09761be7b70f0b0be",
+    "kappa": "194869/2500000000",
+    "proof": "notes/stopped-product-note.tex",
+    "certificate": "certificates/stopped-product-network.json",
+    "producer": "scripts/stopped_product_producer.py",
+    "assistance": "Substantial OpenAI GPT-6 Astra and Codex assistance.",
+    "scope": "Conditional on the retained analytic and fixed-tape framework; new product-ring and exact rational-grid arguments supplied as written proofs.",
+    "continuing_dependencies": [
+      "copied_centers_extension",
+      "structured_bulk_extension"
+    ],
+    "icekylinx_lineage": [
+      {
+        "pr": 10,
+        "commit": "62691e395a0458ce089a1c7b5d89e74291e95e29"
+      },
+      {
+        "pr": 18,
+        "commit": "f2ab41aebad47861caf6316282c1793e5513845e"
+      },
+      {
+        "pr": 24,
+        "commit": "ed90fd940279c336ebc45968631bdebfb087b505"
+      },
+      {
+        "pr": 32,
+        "commit": "0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52"
+      },
+      {
+        "pr": 36,
+        "commit": "11817ccacb564bb7f98789c20dc11d3fece207e3"
+      }
+    ],
+    "curation": "Selected h23/h24 inputs, generator, exact certificate and proofs only; excludes raw graphs, research alternatives and recursive predecessor copies."
   }
 }

certificates/stopped-product-bit-axis.json

@@ -0,0 +1,38 @@
+{
+  "h": 23,
+  "v": 1771,
+  "c": 34764,
+  "q": 5336,
+  "baseline_R": 40100,
+  "matching": 1324,
+  "R": 38776,
+  "loss": 506,
+  "histogram": [
+    9840,
+    38409,
+    32471,
+    8898,
+    9754,
+    2116,
+    5737,
+    1725,
+    4145,
+    1547,
+    3128,
+    1551,
+    2893,
+    1925,
+    2841,
+    2450,
+    2327,
+    3096,
+    2344,
+    3628,
+    9435,
+    2110,
+    23,
+    23
+  ],
+  "label_source": "positive",
+  "description": "Retained-total aligned triple producer, base-two recursion, original maximum carrier matching, and compatible positive labels."
+}

certificates/stopped-product-complex-input.json

@@ -0,0 +1,41 @@
+{
+  "h": 24,
+  "v": 2024,
+  "c": 45764,
+  "q": 8120,
+  "baseline_R": 53884,
+  "matched": 8966,
+  "R": 44918,
+  "orientation_changes": 7338,
+  "rank_sum": 1079136,
+  "loss": 552,
+  "histogram": [
+    8930,
+    49042,
+    30844,
+    15972,
+    11883,
+    8272,
+    6399,
+    4356,
+    6682,
+    2263,
+    5109,
+    1808,
+    5317,
+    1733,
+    5391,
+    1757,
+    4908,
+    3290,
+    6233,
+    4363,
+    4056,
+    264,
+    564,
+    24,
+    24
+  ],
+  "central_disjoint": 24,
+  "center_denominator": 21
+}