Research: structural barriers for kappa = 0.01 (no new exponent)

.github/workflows/percent-exponent-architecture.yml

@@ -0,0 +1,28 @@
+name: Percent exponent architecture screens
+
+on:
+  pull_request:
+    paths:
+      - 'research/percent-exponent-architecture/**'
+      - '.github/workflows/percent-exponent-architecture.yml'
+  push:
+    paths:
+      - 'research/percent-exponent-architecture/**'
+      - '.github/workflows/percent-exponent-architecture.yml'
+  workflow_dispatch:
+
+jobs:
+  exact-audit:
+    runs-on: ubuntu-latest
+    strategy:
+      matrix:
+        python-version: ['3.11', '3.14']
+    steps:
+      - uses: actions/checkout@v4
+      - uses: actions/setup-python@v5
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Exact arithmetic and complete finite searches
+        run: python -B -S research/percent-exponent-architecture/verify.py --deep
+      - name: Verification leaves sources unchanged
+        run: git diff --exit-code

research/percent-exponent-architecture/NOTICE

@@ -0,0 +1,33 @@
+Architecture screens for a conditional exponent saving of 1/100
+Prepared for huxint with substantial OpenAI Codex assistance, October 2026.
+New contributions are licensed under the repository's Apache-2.0 license.
+
+The paired-cube identity, three-stage cover, shared completed-core compiler,
+coordinate-star centers and semantic assembly are inherited from icekylinx
+and the contributors credited in the repository NOTICE. Completed-core bank
+sharing includes an664's contribution. The complex checkpoint is Avi
+Eisenberg/ikeboy's PR193, with its stated Claude assistance, built on
+icekylinx, eumemic and the preceding paired-cube work. The bit checkpoint is
+Rohan Arun/rohanarun's PR187, retaining its own attribution to the preceding
+bit word, source-assisted flow and compiler contributors. These inputs are
+not claimed as new constructions here.
+
+The inequality a < D/C and an exact fixed-word ceiling are also discussed
+in DaysSky's PR192. The balanced assembly ceiling is due to romainhedouin's
+PR183 and the preceding assembly authors. This package's new scope is the
+data-only relaxation, the stated odd-cube family, the rank-minimal center
+support lemma, the common-triangular-flag obstruction, and the bounded
+Fano stub-reuse/all-permutation investigation.
+
+The Fano graph, original balanced model and baseline Python
+meet-in-the-middle implementation come from the repository's retained
+research by Douglas Colkitt and contributors, with its recorded AI
+assistance. The original network-coding seed is attributed there to Das and
+Rai, Generalized Fano and non-Fano networks (arXiv:1609.05815).
+references/SOURCE.json records the original source hashes and the exact
+dependency-binding changes to the bundled baseline. Original algorithm
+bodies and notices are retained.
+
+No priority claim is made for standard Walsh projection identities,
+triangularization facts, entropy cut arguments or meet-in-the-middle
+enumeration. No new all-size integer-multiplication theorem is claimed.

research/percent-exponent-architecture/README.md

@@ -0,0 +1,95 @@
+# Architecture screens for kappa = 0.01
+
+This package identifies structural costs that exclude a saving of 0.01 in
+specified continuations of the current construction, and tests a different
+small-network route. **It establishes no new multiplication exponent.**
+
+The target is the saving in `n (log n)^(1-kappa)`, not a measured runtime
+improvement. All inherited all-size and analytic assumptions remain in place.
+
+## Checked results
+
+| Scope | Result |
+| --- | --- |
+| Keep PR193's complex data calls; grant every auxiliary call and all center loss for free | `a < 0.008866683` |
+| Keep PR187's bit data calls under the same relaxation | `a < 0.008897440` |
+| Proposed odd-dimensional paired cubes, copied t-star centers and the retained data flags | `a < 0.005164`, all odd dimensions in the stated domain |
+| Same odd-cube family, allowing arbitrary local data chains of the required endpoint distance | `a < 0.008094` |
+| Paired triples, p>=5, exactly h independent copied centers | Center loss is at least `h(h-2)`; the current stars attain it |
+| General finite bit contract with a common triangular frame flag | `s >= Wm`; no rank deficit |
+| All 87 chronological balanced-Fano stub reuses, all local GL(2,F2) gates, any full output permutation | Every completed scalar permutation is routable; no rational-frame deficit in any dimension |
+
+Here `a` is an ordinary supplier saving; the retained assembly has `kappa<a`.
+Every bound has a stated interface. These are not impossibility results for
+integer multiplication, new center implementations, cross-stage fusion,
+overcomplete center factors, arbitrary label families or different recurrences.
+
+The first two rows follow from the paid-moment inequality `a C < D`, applied
+only to data children. The center-basis result explains why changing the
+linear basis of a rank-minimal center factor cannot reduce the current
+triple loss. The odd-cube screen includes an exact finite calculation and
+an analytic infinite tail. The Fano screen changes the graph before scalar
+synthesis and keeps all initial inputs independent.
+
+See [PROOF.md](PROOF.md) for the arguments, assumptions and open directions.
+The exact inputs and arithmetic are in [ceiling-certificate.json](ceiling-certificate.json)
+and [center-basis-certificate.json](center-basis-certificate.json).
+[fano-all-permutations.json](fano-all-permutations.json) records the full
+finite search; [fano-reuse-certificate.json](fano-reuse-certificate.json)
+independently replays the 16 cases preserving the original three demands.
+
+## Reproduce
+
+Python 3.11+ and a C++17 compiler are sufficient. No network or Python package
+installation is needed for the exact verifier.
+
+```sh
+python3 -B -S research/percent-exponent-architecture/verify.py --deep
+```
+
+The default verifier checks the immutable package sources, regenerates the
+data and family ceilings, tests the sharp center bound, runs adverse controls,
+and exhausts all scalar permutations on all 87 topologies. It compares the
+C++ matching algorithm with direct full-word enumeration on eight small
+cases. `--deep` additionally regenerates the independent Python half-searches
+for all 16 prescribed-terminal cases. Verification uses a temporary build
+directory and leaves the package unchanged. `--write` is authoring mode.
+
+## Separate vector-code experiments
+
+`discovery/vector_fano.py` allows each edge to carry k bits and each local
+vertex to use an arbitrary binary 2k-by-2k map. The final transfer must be a
+permutation on every independent input bit; this forces all local maps to
+be invertible. No clean workspace or appended decoder is supplied.
+
+With Z3 5.1.0.0, all 16 retained-obstruction cases returned `unsat` for k=2
+without a basis restriction, and for k=3,4 when internal edge bases were
+put in reduced row-echelon form.
+Those receipts are optional discovery evidence, not independent UNSAT
+certificates and not a theorem for larger blocks. Unnormalized runs include
+timeouts, which remain explicitly recorded as `unknown`.
+
+```sh
+python3 -m pip install z3-solver==5.1.0.0
+python3 research/percent-exponent-architecture/discovery/vector_fano.py \
+  --block 3 --canonical-edges --timeout-ms 30000
+```
+
+No vector result is used to prove any ceiling or scalar exclusion above.
+
+## A concrete budget for a different construction
+
+In the raw uniform bit recurrence, a *hypothetical* complete network with
+`m=3`, `W=30` and one rank unit of deficit would have ordinary saving
+greater than `0.010170376`, above the balanced assembly's necessary `1/99`
+threshold. At W=31, one rank unit no longer suffices. These inequalities are
+checked with rational logs in the certificate.
+
+This is a useful small-network search target, not an achieved supplier.
+It still requires the all-input scalar permutation, a nonroutable topology,
+exact rational frames and endpoint checks, a sufficiently strong complex
+companion, and the full finite bridge and assembly. The tested Fano family
+does not provide it.
+
+Original sources, licenses and assistance disclosures are preserved in
+[NOTICE](NOTICE), the input provenance and [references/SOURCE.json](references/SOURCE.json).

research/percent-exponent-architecture/SOURCE.json

@@ -0,0 +1,32 @@
+{
+  "files": {
+    "NOTICE": "181f36c71d671c2c572171dd342289604f6e4c6c9ff7230f6432dd8a8779b0eb",
+    "PROOF.md": "b2830274450bcaa4fbc169e8a78a9b8f5d93c5ab41812a0216f33e0fdbd6d7a5",
+    "README.md": "4bf8f4b264ad2c105bbd224b41cc6771867299d1b94371501db0a509b11cd08f",
+    "STATUS.zh-CN.md": "a896b71a6cb033bb4c1fd20ffe81deb383beee30f91f6eadb83c2b66263335ee",
+    "audit.py": "610f793c209ff68edeecfcb422536d2b6dcb0afaf9d682d2cad504182124cf19",
+    "ceiling-certificate.json": "49e618a7a4e3316aaf6b22c6d624f3947a610356b9798a3194174ca56576f13a",
+    "center-basis-certificate.json": "35d3a39c78d29d59541ec871dccc1f1860b6b9fd105ae30deb44b7cc7389742d",
+    "center_basis.py": "f473328a0f267b6ccead291d2adda09a93a6f8c95abdc14ca95df1b4978c9010",
+    "discovery/balanced_search.cpp": "05247e0b4def0ec10b29066429ef12ca698cd5ae01215daa423d72be7117cd2a",
+    "discovery/fano_reuse.py": "5ee1311021a1955b3ece53d00ca2c09a2fc3fb55486b96cae5891ea3592c103e",
+    "discovery/receipts/vector-k2.json": "d0dbf6c305a5548f144d969386b515fb25b78e18835f37e9ab81a1fad16f2289",
+    "discovery/receipts/vector-k3-canonical.json": "c1935ac2fdf5e8a9cfdeb169ef0d934d34cfdeeca0b9fe14ebb94cfb6459e2c3",
+    "discovery/receipts/vector-k3.json": "39be1eb7fa6b08eee9a290d9ff930391a1d32e271c484fc669616bb6ea3542f2",
+    "discovery/receipts/vector-k4-canonical.json": "169dcb281fa58c470629b64b0f3e2cb51e5f74875cda4e3f3f60c26428c2c90e",
+    "discovery/receipts/vector-k4.json": "a4446bb2cdefc6e99465c71d3e6cb4a941cf74e6781c6f08a3b5872460129a50",
+    "discovery/vector_fano.py": "d0c315b67d8f20db85d6a4bdf3f6d07e68a2513ec1722b46f96d8641fa893239",
+    "enumeration-controls.json": "7bb7d15ac851de5a0506c34b20446bbf82f55be0ca5a75a140edc3e2166ae447",
+    "fano-all-permutations.json": "7f104a85266b30d83ba19aad1eda289aa8a646d6a0b7c6198ebe6989fca12318",
+    "fano-reuse-certificate.json": "d532de06f84a5291e35aefacc2e447abb82292bcc67458d3c4d9a6880610869f",
+    "fano_reuse_verify.py": "fced63b548207ea0a66f0d88b413e3a305245c81131c945bf55b861959ba08d6",
+    "inputs/bit-pr187.json": "9f9cb08fd60c82888f015c0db278acb81a06fe9640cfbd14d76b720ca1e7e726",
+    "inputs/complex-pr193.json": "b608bc0af5ad837ea2604fed3cba82cadf5ab3f001fc74d9cea01fe3dc9fb201",
+    "references/SOURCE.json": "d6363e2e869dd87ad3c62e2437f12df4222035fe55874bf4e1a86682cf9acd5a",
+    "references/balanced_fano.py": "10c0f371475b364bd26ec298966def4f4e4382ec6db33e6e21ce239a9f4abebf",
+    "test_audit.py": "003f24a0f0d648533deea5edf772fd557b007de27680eb57fb0ae30e3ca363ba",
+    "triangular_obstruction.py": "e99dc5a9a30d7b91cc30023a862b8ec5b5ae54e029bdb11f579e521b4b525892",
+    "verify.py": "f685c9b6ae4f814c0bbe0149e4f694914b62e3974397a80bbb9e44994abb6a53"
+  },
+  "scope": "This self-contained package only; inherited whole-algorithm interfaces are not discharged."
+}

research/percent-exponent-architecture/STATUS.zh-CN.md

@@ -0,0 +1,23 @@
+# 向 κ=10^-2 的结构研究
+
+目前没有达到 10^-2，也没有产生新的 κ。目标是找到能改变递归成本的构造。
+
+这轮已完成的内容：
+
+- 当前数据调用序列本身就挡住了 10^-2：即使辅助计算和中心损耗全部免费，两侧上界仍低于 0.009。
+- 把三维立方体推广到更高奇数维，并保留同类复制中心和三块装配，即使重新安排所有局部数据调用、忽略辅助计算，上界仍低于 0.008094。有限检查之外的大参数尾部也已证明。
+- 证明了三维立方体方案的中心基底限制：p>=5 时，使用最少的 h 个复制中心，损耗至少为 h(h-2)。现有坐标星已达到这个值。增加中心数量、使用额外行之间的抵消，或跨阶段共享中心，仍不在这个结论的范围内。
+- 检查了 87 种 Fano 网络的辅助输入/输出连接方式，并枚举所有局部可逆二进制运算和最终置换。所有完成的置换都有路由证书对应，因此这批网络不能产生所需的有理矩阵秩节省。
+- 多比特边的后续搜索在 k=2、3、4 时也没有找到候选；这些是求解器记录，未当作独立证明或更大 k 的不可能性结论。
+
+下一种构造必须改变已经触及的假设，例如复制中心的实现、跨阶段数据接口或端口几何。仅重新选择 h 个中心的线性组合，或继续替换同类局部模块，不能完成这次目标。
+
+一个具体的小网络门槛是：三维、最多 30 个独立输入角色、至少一个单位的秩节省。这在原始均匀 bit 递归中足以超过 1/99，但目前没有构造出满足完整端点条件的网络，还需要足够强的 complex 侧与完整装配。
+
+复现：
+
+```sh
+python3 -B -S research/percent-exponent-architecture/verify.py --deep
+```
+
+完整证明见 [PROOF.md](PROOF.md)。所有界限都针对其中明确列出的接口；它们不排除其他架构达到 10^-2。