Mister-Meeseeks · Oct 8, 2026 · Oct 8, 2026
diff --git a/CITATION.cff b/CITATION.cff
@@ -6,18 +6,21 @@ title: "A sharper exponent for integer multiplication"
 authors:
   - family-names: Colkitt
     given-names: Douglas
+  - family-names: Chen
+    given-names: Zhihao
+  - name: eumemic
+  - name: icekylinx
 license: Apache-2.0
 type: software
 repository-code: "https://github.com/CrocSwap/integer-mult-bounds"
 url: "https://github.com/CrocSwap/integer-mult-bounds"
 abstract: >-
-  Exact rational certificates, source patches, and research notes for conditional
-  improvements to OpenAI's integer-multiplication manuscript. Compact dirty-control
-  fields remove the simultaneous layer's spacing penalty. Together with complete
-  temporary reservations, local exceptional repair, a separately sized complex
-  network, and retained refinements, the supplied written proofs support
-  kappa = 83/10^12 > 2^-34. The complete upstream algorithmic theorem is assumed;
-  the new arguments are not independently reviewed or formally verified.
+  Conditional bulk-recursion improvement of the retained integer-multiplication
+  framework, with kappa = 6149999/50000000000000 > 2^-23. Controlled rational
+  projector blocks and whole complex residuals give mixed-width recurrences;
+  a dependency-path guard controls exact intermediate precision. The retained
+  Gaussian scaling enclosure is corrected without changing P=34p. Finite
+  checks and exact certificates do not formally verify the full theorem.
 keywords:
   - integer multiplication
   - computational complexity

diff --git a/CONTRIBUTING.md b/CONTRIBUTING.md
@@ -7,7 +7,7 @@ For a mathematical issue, identify the exact proposition, source label, or
 inequality; give a counterexample or the missing implication when possible;
 and distinguish numerical validation from an algorithmic proof obligation.
 For the current result, start with the
-[compact-control review guide](docs/research/compact-control-review.md).
+[batched-recursion review guide](docs/research/batched-review.md).
 The [original audit](docs/audit.md) describes the retained upstream assumptions.
 
 For a parameter improvement, supply exact rational choices, the full dependency
@@ -21,7 +21,7 @@ make verify
 ```
 
 Include regenerated certificates and patches in the same change. If the note
-changes, also run its corresponding PDF target (`make compact-note` for the
+changes, also run its corresponding PDF target (`make batched-note` for the
 current result) and include the updated PDF. Review changes to
 claims in the README and note together. Finite tests should address a mathematical
 identity or a failure mode, rather than simply restating implementation details.

diff --git a/Makefile b/Makefile
@@ -1,6 +1,10 @@
 .PHONY: verify note audit-note tuned-note reuse-note incidence-note dag-note shared-point-note paired-note compact-note fetch
 
 verify:
+	python3 scripts/prime_field_network.py
+	python3 scripts/complex_network.py
+	python3 scripts/fast_gaussian.py
+	$(MAKE) batched-certificate batched-patch
 	python3 scripts/certify.py
 	python3 scripts/search_network.py
 	python3 scripts/make_patch.py
@@ -54,6 +58,7 @@ verify:
 	git apply --check --directory=upstream patches/h46-shared-point.patch
 	git apply --check --directory=upstream patches/h50-paired-59.patch
 	git apply --check --directory=upstream patches/compact-control-34.patch
+	git apply --check --directory=upstream patches/batched-23.patch
 
 note:
 	mkdir -p artifacts
@@ -93,3 +98,25 @@ compact-note:
 
 fetch:
 	python3 scripts/fetch_upstream.py
+
+.PHONY: batched-certificate batched-patch batched-note
+PDFLATEX ?= pdflatex
+TEX_ENGINE ?= pdflatex
+TECTONIC ?= tectonic
+
+batched-certificate:
+	python3 scripts/controlled_bit_rank_moment.py --output certificates/controlled-bit-rank-moment.json > /dev/null
+	python3 scripts/batched_network.py --output certificates/batched-network.json --summary
+
+batched-patch:
+	python3 scripts/make_batched_patch.py
+
+batched-note:
+	mkdir -p artifacts
+ifeq ($(TEX_ENGINE),tectonic)
+	$(TECTONIC) -Z search-path=$(CURDIR) --outdir artifacts notes/batched-23-note.tex
+else
+	$(PDFLATEX) -interaction=nonstopmode -halt-on-error -output-directory=artifacts notes/batched-23-note.tex
+	$(PDFLATEX) -interaction=nonstopmode -halt-on-error -output-directory=artifacts notes/batched-23-note.tex
+	$(PDFLATEX) -interaction=nonstopmode -halt-on-error -output-directory=artifacts notes/batched-23-note.tex
+endif
diff --git a/NOTICE b/NOTICE
@@ -14,3 +14,44 @@ as separate files under patches/; upstream source files remain unchanged.
 The new research note, scripts, certificates, patches, and documentation
 were prepared by Douglas Colkitt with assistance from OpenAI Codex.
 This project is not an official OpenAI release or an OpenAI endorsement.
+
+Ternary five-subset construction and common-four-point resynthesis
+Copyright 2026 Zhihao Chen (jacklightChen)
+
+This is Zhihao Chen's first submission of this specific construction to this
+repository. Research, proof development, code and verification were prepared
+with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by
+the contributor. This is not independent review or an OpenAI endorsement.
+Future research building on this contribution should explicitly acknowledge
+Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and
+retain the preceding contributors' attribution. This notice records research
+provenance; the repository's Apache-2.0 license remains unchanged.
+
+Dependencies retained from eumemic's Claude-assisted PRs #3 and #5:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+https://github.com/CrocSwap/integer-mult-bounds/pull/5
+PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893
+These include complex_circuit.py, complex_network.py, fast_gaussian.py,
+their generators, construction notes and certificates. The Gaussian input
+scaling proof is corrected in this derivative; see the modification notice below.
+Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank
+sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6.
+Their results and ideas are credited separately from this new construction.
+
+Bulk recursion and controlled tensor basis, 8 October 2026
+Copyright 2026 icekylinx
+
+The new projector batching, controlled simultaneous basis, mixed-width tape
+recursions, whole-residual complex recursion, and dependency-path precision
+guard in the batched-23 proof package are contributed by icekylinx, developed
+with substantial OpenAI GPT-6 Astra assistance. Review corrections and
+repository integration were prepared with OpenAI Codex assistance. They build on the retained
+PR #7 finite producers and the preceding contributions credited above.
+The new package retains the repository's Apache-2.0 license and the original
+authors' attribution. It is not an official OpenAI release or endorsement.
+
+Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian
+proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2).
+The P=34p precision budget and asymptotic exponent are unchanged. New batching
+checks are integrated into make verify. Existing attribution and licenses
+are retained.
diff --git a/PATCHING.md b/PATCHING.md
@@ -0,0 +1,28 @@
+# Applying the manuscript patch
+
+`patches/batched-23.patch` applies to the original manuscript snapshot at
+`adc7f1241b42e322a6451854ab7e4b4c146bf78a`, retained under `upstream/`.
+It includes the PR7 dependencies, batched construction, and corrected Gaussian
+input error enclosure.
+
+To generate a complete copy for review, choose a new destination:
+
+```sh
+python3 scripts/make_batched_patch.py --materialize /tmp/integer-mult-batched-review
+```
+
+The entry point is `/tmp/integer-mult-batched-review/build/main.tex`.
+The bundled `upstream/` stays unchanged. With pdfLaTeX installed, compile from
+that copy's `build/` directory, including BibTeX and the usual reference reruns.
+
+Check applicability without modifying the source:
+
+```sh
+git apply --check --directory=upstream patches/batched-23.patch
+```
+
+The combined patch starts from the original snapshot; historical patches are
+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`.
diff --git a/README.md b/README.md
@@ -1,151 +1,55 @@
-# A sharper exponent for integer multiplication
+# Integer multiplication with a conditional saving beyond 2^-23
 
-**Research draft by Douglas Colkitt — conditional on the underlying manuscript
-and the written extensions supplied here.**
-
-This draft improves OpenAI's
-[*Integer multiplication below n log n*](https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integer-multiplication-below-n-log-n-September-23-2026)
-(result family #109). In its fixed finite-alphabet Turing-machine model with a
-fixed number of one-dimensional tapes, the strongest supplied witness is
-
-$$
+\[
 T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
-\boxed{\kappa=\frac{83}{10^{12}}=8.3\times10^{-11}>2^{-34}}.
-$$
-
-The simpler **`kappa = 2^-34`** is a corollary. The witness remains below
-`2^-33`. It increases the exponent saving by approximately **47.85 million
-fold** over our preceding published `2^-59` witness. The original manuscript
-uses `2^-182`. These compare asymptotic exponents, not practical runtimes.
-
-**[Read the compact-control proof note (PDF)](artifacts/compact-control-note.pdf)** ·
-[Review the combined source patch](patches/compact-control-34.patch) ·
-[Inspect the exact certificate](certificates/compact-control-layer.json) ·
-[Review guide and dependencies](docs/research/compact-control-review.md)
-
-This is a research claim supported by written proofs and reproducible checks.
-The complete upstream theorem is assumed; the new arguments have not received
-independent mathematical review or formal verification.
-
-## What changed
-
-The new construction moves **compact control fields instead of entire spaced
-windows**. For `f` selected axes, it replaces the layer's movement cost
-`O(V*((f*K)^tau+1))` by
-
-$$
-O\!\left(V\bigl((f\log p)^\tau+1\bigr)\right).
-$$
-
-The proof reserves temporary fields from existing address coordinates,
-allows arbitrary initial temporary values, restores them exactly, and charges
-exceptional-address repair at every recursion node. The temporary ranges
-remain complete through padding and recursive row splitting.
-
-Removing `K^tau` removes the restriction responsible for the preceding
-quadratic dependence on the finite-network saving. The bit network stays at
-`h=50`. The original complex network is separately instantiated at `h=25`,
-and a generalized stopping-depth guard completes the new parameter witness.
-This is a change to the movement construction and its proof, beyond parameter
-tuning of the preceding algorithm.
-
-The exact minimum assembly margin is
-
-$$
-G_* = \frac{333833}{4\cdot10^{15}}
-    = 8.345825\times10^{-11} > \kappa.
-$$
+\kappa=\frac{6149999}{50000000000000}>2^{-23}.
+\]
 
-The remaining bottleneck is the complex layer's saving. With the **fixed
-`h=25` complex motif and retained Gaussian/leaf inequalities**, the scoped
-ceiling is below `8.369598075e-11`, hence below `2^-33`. This is not a ceiling
-for other networks or integer multiplication in general.
+The construction batches large rational projector blocks and whole complex
+residuals in the finite networks of [PR #7](https://github.com/CrocSwap/integer-mult-bounds/pull/7).
+A controlled common basis, mixed-width recursion, and an arithmetic dependency
+path bound supply the stated exponent. The conclusion is conditional on the
+retained upstream multiplication framework and the written interface proofs.
 
-## Evidence and scope
+## Proof and reproduction
 
-| Component | Evidence |
-| --- | --- |
-| Parameters, logarithm enclosures, final margins | Exact rational certificate |
-| Dirty-control identities, inverses and repair | Finite exhaustive cases and seeded tests |
-| Wider-control tape bound, reservations and recursion | Written general proofs |
-| Separate complex arity and precision guard | Written proofs and exact accounting |
-| Source integration | Combined patch, reference checks and manuscript build |
-| Full upstream multiplication theorem | Assumed |
-| Independent review / full formalization | Not supplied |
-
-The [review guide](docs/research/compact-control-review.md) identifies the new
-proof obligations and their tests. [Current research status](docs/research/current-status.md)
-is authoritative when older notes describe superseded barriers or hypothetical
-witnesses. The earlier artifacts remain available and unchanged.
-
-## Reproduce
-
-With Python 3.11 or newer, Git and Make, run from the repository root:
+- [Proof note](artifacts/batched-23-note.pdf) and [LaTeX source](notes/batched-23-note.tex).
+- [Exact certificate](certificates/batched-network.json): rank moments, guard,
+  29 strict constraints, and seven assembly margins.
+- [Combined manuscript patch](patches/batched-23.patch) and [application instructions](PATCHING.md).
+- [Review guide](docs/research/batched-review.md) and [source provenance](SOURCES.json).
 
 ```sh
 make verify
-git diff --exit-code -- certificates patches
+make batched-note
 ```
 
-No third-party Python packages or network access are needed for these checks.
-They regenerate the certificates and patches, run the tests, verify upstream
-hashes, and check each patch against the pinned manuscript. The second command
-checks exact regeneration on a clean checkout.
-
-With Tectonic installed, rebuild the latest note using:
+Verification requires Python 3.11+, a C++17 compiler, Git, and Make. It includes
+retained producer checks, the new certificates, regression tests, and patch
+application checks. The PDF target requires pdfLaTeX.
+For just the new arithmetic and manuscript patch:
 
 ```sh
-make compact-note
+make batched-certificate batched-patch
 ```
 
-The output is `artifacts/compact-control-note.pdf`. The first PDF build may
-download TeX resources. See [reproducibility instructions](docs/reproducibility.md)
-for applying the combined patch in a disposable copy and building older notes.
-[GitHub Actions](.github/workflows/verify.yml) runs the arithmetic and patch checks.
-Passing tests does not establish the complete multiplication theorem; this
-repository contains no full multiplication-machine implementation.
-
-## Earlier witnesses and independent patches
-
-Each patch applies independently to the **unmodified** pinned source; they are
-alternatives, not a sequence to apply together. The
-[result history](docs/research/result-history.md) records the earlier mechanisms
-and scoped ceilings.
-
-| Patch | Conditional saving | Scope |
-| --- | --- | --- |
-| [frozen-154](patches/frozen-154.patch) | `2^-154` | Original network and recurrence exponents |
-| [balanced-153](patches/balanced-153.patch) | `2^-153` | Balanced assembly parameters |
-| [same-network-129](patches/same-network-129.patch) | `2^-129` | Original network, sharper recurrence comparison |
-| [h46-111](patches/h46-111.patch) | `2^-111` | Smaller network, dyadic parameters |
-| [h46-109](patches/h46-109.patch) | `2^-109` | Rational recurrence saving, strict final margin |
-| [h46-108](patches/h46-108.patch) | `2^-108` | Variable stopping exponent |
-| [h46-rational](patches/h46-rational.patch) | `5.8e-33` | Strongest supplied parameter-only witness |
-| [nonadjacent-layout](patches/nonadjacent-layout.patch) | Original parameters retained | Routing proof and revised layout cost only |
-| [frozen-nonadjacent-107](patches/frozen-nonadjacent-107.patch) | `2^-107` | Direct routing, original network and recurrence exponents |
-| [h46-nonadjacent-78](patches/h46-nonadjacent-78.patch) | `2^-78` | Direct routing with the h = 46 network |
-| [h46-nonadjacent-76](patches/h46-nonadjacent-76.patch) | `2^-76` | Direct routing with tuned dimension and stopping parameters |
-| [h46-shared-side-75](patches/h46-shared-side-75.patch) | `2^-75` | Stage-1/stage-3 side-role sharing, routing, and parameter tuning |
-| [h46-incidence-67](patches/h46-incidence-67.patch) | `2^-67` | Rectangle incidence circuits, full auxiliary sharing, routing, and parameter tuning |
-| [h46-dag-63](patches/h46-dag-63.patch) | `2^-63` | Shared intermediate sums and reversible role allocation |
-| [h46-shared-point](patches/h46-shared-point.patch) | `13*2^-66` | Cross-group sharing |
-| [h50-paired-59](patches/h50-paired-59.patch) | `2^-59` | Paired sums, stopped guard and tighter Gaussian setup |
-| **[compact-control-34](patches/compact-control-34.patch)** | **`83/10^12 > 2^-34`** | **Compact controls, complete reservations, local repair and separate complex arity** |
-
-## Attribution, citation, and license
-
-Author: **Douglas Colkitt**. Research, implementation and drafting were performed
-with assistance from OpenAI Codex. The compact-control proposal originated
-with a separate research agent; the supplied note develops its tape, layout,
-repair and assembly arguments. AI assistance is not independent review or
-endorsement by OpenAI. No priority or unrestricted optimality claim is made.
-
-The original manuscript is by OpenAI, pinned at commit
-`adc7f1241b42e322a6451854ab7e4b4c146bf78a`. Source URLs and SHA-256 hashes are in
-[upstream/manifest.json](upstream/manifest.json). Files under `upstream/` remain
-unchanged; modifications are supplied as separate patches.
-
-Use [CITATION.cff](CITATION.cff) and also cite the
-[upstream manuscript](upstream/README.md). Until a release is archived, include
-the repository commit used. Licensed under [Apache-2.0](LICENSE); see
-[NOTICE](NOTICE) and [CONTRIBUTING.md](CONTRIBUTING.md).
+The bit and complex savings are `246/10^9` and `7/10^7`; the precision guard
+exponent is `11999/10000`. The final absorption gap is
+`362000001/80000000000000000000000 > 0`.
+The retained Gaussian input scaling proof uses the corrected enclosure
+`F = 2^ceil(1.14 alpha^2)`, with the existing precision `P = 34p`.
+
+## Attribution
+
+Bulk recursion and this integration are contributed by **icekylinx**, with
+substantial OpenAI GPT-6 Astra and Codex assistance. The finite ternary network
+and paired producers are due to **Zhihao Chen (jacklightChen)**. The construction
+builds on **Douglas Colkitt**'s compact-control framework and contributions by
+**Bortlesboat**, **eumemic**, and **dleen**; see [NOTICE](NOTICE).
+
+OpenAI's manuscript is pinned at
+`adc7f1241b42e322a6451854ab7e4b4c146bf78a`; its files under `upstream/` are
+unchanged. The repository and retained source use Apache-2.0; see
+[LICENSE](LICENSE) and [upstream/LICENSE](upstream/LICENSE).
+Exact certificates and finite tests do not constitute formal verification
+of the full multiplication theorem.