Scoped negative result: cost-six terms force 67-copy E8 decoders at 25 punctures

research/complex-general-normal-boundary/README.md

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+# A cost-six term at 25 punctures forces a 67-copy completion
+
+**Scoped negative result: written proof and exact finite certificates. No new
+kappa or physical supplier.**
+
+The exact eight-term complex E8 decoder floor extends from `p<=24` to
+`p<=25`. At 25 punctures the minimum, over puncture choices and decoders, is
+**56 copies on 95 ports**. Parallel root normals can already give an individual
+term of cost six, but every exact eight-term completion containing it costs
+**at least 67**, and 67 is attained. The cheaper term does not lower the total.
+
+For **nonparallel** normals with nonzero ordinary bilinear product, including
+arbitrary complex normals, the first cost-six puncture size is **26**.
+PR1020 already supplies the attaining root-pair witness at 26. This result
+extends its lower bound beyond root normals and classifies the parallel
+exception; it does not contradict PR1020's distinct-root statement.
+
+## Reproduce
+
+From the repository root, Python 3.12, standard library only:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/complex-general-normal-boundary
+```
+
+The registered command runs the primary replay, separate Gram-minor checker
+and scoped mutation tests. Expected final wrapper line:
+
+```text
+PASS_REGISTERED_RESEARCH_PACKAGE research/complex-general-normal-boundary outcome=negative-result
+```
+
+Both mathematical checkers also support direct Python 3.10+ execution:
+
+```sh
+python3 -I -O -B research/complex-general-normal-boundary/verify.py
+python3 -I -O -B research/complex-general-normal-boundary/check_gram.py
+```
+
+Expected markers: `PASS_COMPLEX_GENERAL_NORMAL_BOUNDARY` and
+`PASS_GENERAL_NORMAL_GRAM_CHECK`. Ordinary replay recomputes and compares
+[CERTIFICATE.json](CERTIFICATE.json); `verify.py --write-certificate` deliberately
+regenerates only that file.
+
+Coverage is **7,560** distinct boundary punctures, **63** for each of 120
+distinguished roots, with explicit symmetry transport. All **32,193** rational
+matrices (`63*511`) in the representative slice are checked. Two full decoder
+examples are checked on **18,050** ordered live pairs. The primary controls
+include an incorrect dual scale, the omitted unit label, the parallel-root
+counterexample and all 25 single-line restorations.
+
+## Model and theorem
+
+Use PR717/PR968's 120 real E8 root lines, with norm-squared-two representatives
+`r_i`, and label pairing `N_ij=(r_i dot r_j+1) mod 2`. Delete a common set `P`
+of `p` lines; `R` denotes the live-root matrix. Complex matrices `D,T`, each
+with eight columns, must satisfy **every entry** of
+
+`D T^transpose = 2 R R^transpose`.
+
+Transpose is ordinary transpose. Each term costs the F2 rank of `N` on the
+two column supports. Masked identities, arbitrary outside-code overcomplete
+factorizations and different label pairings are excluded.
+
+**Theorem, with the inherited real root-flat classification F6.**
+
+1. At `p<=25`, normals with nonzero bilinear product can have term cost at most
+   six only at `p=25`, with both normals proportional to one real root `alpha`
+   and `P` one of its 63 boundary cuts. The cost is exactly six. Each of the
+   7,560 cuts has a unique distinguished root.
+2. For every such puncture, every exact eight-term decoder containing the
+   cost-six term costs at least 67. The certificate supplies attaining frames
+   for all representative cuts and verified transport to every root.
+3. The exact minimum at `p=25`, over punctures and decoders, is 56. Combined
+   with PR968/PR1077, `cost>=max(72-p,56)` holds through `p=25`, and the best
+   copy/port ratio remains `7/13` at `p=16`.
+4. Nonparallel normals with nonzero product first attain cost six at `p=26`,
+   using PR1020's witness.
+
+Statement 3 does not say every puncture admits 56. Statement 2 concerns
+decoders containing a cost-six term, not all decoders on that puncture.
+
+## Proof
+
+### Reduce an arbitrary complex support to the first boundary
+
+The label space has nondegenerate dimension nine. For support spans `U,V`,
+
+`rank(N|U x V) >= dim(U)+dim(V)-9`.
+
+A term of cost at most six has one side of dimension at most seven. A generic
+real combination of a complex normal's real and imaginary parts preserves
+its exact root-zero pattern, as in PR1077. This is a support statement; it
+does not independently replace two dual normals by real representatives.
+
+Extend a root basis of the real zero flat to seven independent roots. The
+resulting root-spanned hyperplane has a normal `omega` with
+`C(omega) subset C(a)`. Its surviving support also has dimension at most seven.
+
+The inherited **F6 classification of real root-spanned E8 hyperplanes** says
+that every such hyperplane is Weyl-conjugate to one of eight fundamental
+types. The primary checker recomputes their complete two-functional census:
+
+| Hyperplane root lines | Minimum deletions for label dimension at most seven |
+|---:|---:|
+| 63, root-normal E7 | 25 |
+| 42 | 42 |
+| 37 | 51 |
+| 28 | 56 |
+| 23 | 61 |
+| 22 | 62 |
+| 16 | 68 |
+| 14 | 70 |
+
+Thus `p<=25` forces `p=25` and `omega` proportional to a root `alpha`.
+Let `V_lambda` be the root set where a nonzero label functional evaluates to
+one. Two independent functionals annihilating the surviving support require
+
+`C(alpha) intersect (V_lambda union V_mu) subset P`.
+
+All 130,305 independent functional pairs are enumerated. Minimum size 25
+occurs for 189 pairs, producing 63 distinct sets. Since `|P|=25`, containment
+is equality. All 120 root censuses and their transport by eight explicit root
+reflections are checked. Their 7,560 punctures are distinct; transitivity is
+verified rather than assumed.
+
+### Recover every possible reduced normal, including complex ones
+
+For each cut and each of the 511 nonzero functionals, form `Z_lambda` with
+the live roots in `V_lambda` as rows. A normal `b` has support label dimension
+at most eight exactly when it belongs to the complex kernel of some
+`Z_lambda`.
+
+For each of the 63 representative cuts, **483** matrices have rank eight and
+**28** have rank seven. Every rank-seven kernel is a single real root
+direction, extended to complex scalars. The 28 kernels represent **26**
+distinct directions: only `alpha` has support dimension seven; the other
+25 have dimension eight. Exactly one reduced direction, `beta`, lies in
+`alpha^perp`.
+
+These are exhaustive linear constraints on arbitrary complex coefficients.
+A rational matrix has the same rank over Q and C. The separate checker uses
+exact integer Gram determinants: positive 8-by-8 determinants certify rank
+eight; a positive 7-by-7 principal minor and an explicitly annihilated root
+vector certify rank seven. These Gram matrices come from the **real rational
+rows** of `Z_lambda`; no complex positive-definiteness assumption is made.
+
+The cheap term's smaller side must therefore be proportional to `alpha`.
+If the other side has dimension nine, its cost is at least seven. Otherwise
+it is one of the 26 directions. Exact binary ranks show that among directions
+with nonzero product with `alpha`, only `alpha` has cost at most six, and its
+cost is six. This proves statement 1 and excludes all nonparallel normals
+through 25 punctures.
+
+### Charge the seven remaining terms
+
+At least 95 roots survive. F6 also gives a maximum of 63 lines in a real root
+hyperplane, so the live-root matrix has rank eight. The all-pairs identity
+implies `D=RA`, `T=RB` and
+
+`A^transpose B = A B^transpose = 2I`.
+
+The two normal lists are bases with paired products two. If one term uses
+parallel `alpha` normals, the seven others on each side lie in `alpha^perp`
+and are independent. At most one on either side can be reduced: it must be
+proportional to the unique `beta`. Each side's support-dimension sum is at
+least `8+6*9=62`. The remaining seven term costs sum to at least
+
+`62+62-7*9 = 61`.
+
+Including the cost-six term yields **67**. No blanket prohibition on complex
+parallel cross-terms is used; duality and same-side independence suffice.
+
+Every representative cut has an explicit orthogonal root frame beginning with
+`alpha,beta`. Using that frame on both sides gives
+`(6,7,9,9,9,9,9,9)`. All frames and costs are checked, and the displayed
+seed-selected example is checked entrywise on all 9,025 live pairs. Verified
+symmetries transport these constructions, proving sharpness.
+
+If no term costs six, all eight cost at least seven. Otherwise 67 applies.
+Hence the total is always at least 56. A second example extends PR968's
+frame-pair construction by nine explicit punctures, with costs
+`(7,7,7,7,7,7,7,7)` and all 9,025 entries checked. PR1020's nonparallel
+26-puncture witness is independently reranked. Statements 2--4 follow.
+
+## Budget and limits
+
+The hypothesis was that the first seven-dimensional support could beat the
+copy/port objective `7/13`. Even the favorable copy-only price fails:
+`67/95 > 56/95 > 7/13`. Physical synthesis stopped at this budget gate.
+This is the full **eight-term copy invoice**, not a recursive supplier
+invoice. Source access, helper work, frame changes, restoration, recursive
+children, hosting, precision/tape and assembly costs have not been priced or
+removed. A different contract could change the comparison.
+
+F6 is cited, not proved here; the primary replay recomputes its eight-type
+table. The written argument quantifies over arbitrary complex normals, while
+the finite checks establish its stated premises and examples. Neither is a
+formal proof of multiplication. The Gram checker imports no producer or
+predecessor code, but both implementations were prepared in the same
+AI-assisted session; no independent human review is claimed.
+
+No physical word, native admission, full selected-supplier cold replay,
+all-size compiler proof, bit H1--H5 proof or analytic transfer is supplied.
+The selected record remains unchanged at the base's conditional kappa
+`836330098538459/500000000000000000`.
+
+## Provenance, overlap and credit
+
+Base: `ef5f39403610d231a0c27e8a6b4f08c10209bcd6`. OS assignment seed **31**
+selected the complex-supplier track, candidate `31 mod 3 = 1`, root line 31
+and witness index `31 mod 63`. Randomization is not evidence of novelty.
+The declared census is complete and has no randomized optimizer.
+[TASK.json](TASK.json) records the three candidates, overlap and scope change
+from the catalogue's masked-decoder card. No pinned sprint issue existed;
+the donor's local-claim fallback was used.
+
+[SOURCES.json](SOURCES.json) binds base Git objects and exact byte hashes.
+The selected record and catalogue are historical provenance. Proof sources
+and the retained witness are explicit inputs in [VERIFY.json](VERIFY.json).
+
+- **DreamingOfClouds, PR717:** B14 label pairing and decoder cost model.
+- **themilkman216, PR968:** eight-type census, F6 premise, floor through 24
+  punctures and the frame-pair attaining construction.
+- **woklerx, PR1077:** arbitrary-complex support bridge and isotropic caution.
+- **woklerx, PR1020**, head `bf4ee95df4ee6e3101a88f5d015c8aa85b688b6d`:
+  distinct odd-root threshold 26 and its attaining witness. This package
+  extends its lower-bound scope and credits the existing witness.
+- **PR1180:** overlapping factor-linear overcomplete work through `p<=24`;
+  its exact pending head is recorded for overlap, not used as a premise.
+- **PR1187/PR1455:** thresholds and aggregate drops for a fixed P104 decoder,
+  with a different prescribed decoder and puncture range; no proof dependency.
+- **Rohan Arun, Jeremy Kothe, Hao Yan and the PR1094/1087/k8/T16,
+  PR589/PR555 lineage:** reviewed supplier context.
+
+Prepared by **Hanson Sun (@bbMisk)** with substantial **OpenAI Codex assistance**
+in task selection, research, proof, both implementations, verification and
+writing. New material is **Apache-2.0**; predecessor sources remain unchanged.
+
+## Open next questions
+
+1. **What is the least complete eight-term cost at 26 punctures?** The known
+   nonparallel term costs six, but its retained completion costs 69. Start
+   with its annihilator-kernel inventory and an optimistic full copy bill.
+   Dual-basis compatibility beyond one cheap term is the blocker; this
+   package does not enumerate 26-line punctures.
+2. **Can F6 be replaced by a self-contained root-flat certificate?** That
+   would remove the cited classification premise. Start with a canonical
+   exact enumeration of maximal root-spanned hyperplanes modulo the verified
+   reflections. Completeness and certificate size remain open here.

research/complex-general-normal-boundary/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "ef5f39403610d231a0c27e8a6b4f08c10209bcd6",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "c05fafaa25ba95b2d5af86b1800427198e3d20eb",
+      "sha256": "89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce"
+    },
+    {
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+      "usage": "input",
+      "git_blob": "4a3063de8fe606a88bd5bdb9b7d2e5dccb70f73f",
+      "sha256": "a89530576d8a5e1ae8778d601b37140d954212cd46e06b87a3f565294ee251a2"
+    },
+    {
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+      "usage": "input",
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+      "sha256": "ba53c1d8a9bc2d935d546bde46ca7341bd65f8515013e07b073296b4db2a9a74"
+    },
+    {
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+      "usage": "input",
+      "git_blob": "57dd8993e8ac016e8a92a9a885392b58014137bb",
+      "sha256": "bdf7ee497dc04dfc634ee5b7848f6a2b39b0169668fb1a60a9715111dbd15395"
+    },
+    {
+      "path": "research/root-pair-first-drop-seed48/CERTIFICATE.json",
+      "usage": "input",
+      "git_blob": "73c0e0cac357270d7fb9510d7a0093a7d8e359d2",
+      "sha256": "a429b191d7f033864e9fcd676dc575f551a2a4093aaed0ba2a90cae1d91935ff"
+    },
+    {
+      "path": "research/root-pair-first-drop-seed48/PROOF.md",
+      "usage": "input",
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+      "sha256": "9ba97ca028405eb5e88dc9158df34de9ca1ef409de9ad72f7ae9faa34722e9c3"
+    },
+    {
+      "path": "research/root-pair-first-drop-seed48/explore.py",
+      "usage": "input",
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+    }
+  ]
+}