Scoped negative result: affine seven-cube copy floor 128 and peak tradeoff

research/seven-cube-affine-seed43/README.md

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+# Affine seven-cube decoding: copy floor 128, peak floor 30
+
+**Scoped structural negative result. No new kappa or physical supplier.**
+
+The smallest seven-port cube admits a cubic, drop-free correction geometry.
+Even after arbitrary affine recoding and any finite number of delivered
+terms, its algebraic copy-rank sum is at least **128**, equal to its port
+count. Two terms attain `64+64`; four attain `32+32+32+32`.
+
+Unlike the affine five-cube result, the sum and peak optima cannot be attained
+together. The smallest possible peak term rank is **30**, attained here by
+eight terms of total cost 240. Among total-cost-128 decoders, the smallest
+peak is **32**. We do **not** claim that 240 is the minimum total under cap 30.
+
+Under the explicitly inherited additive stage ledger, the favorable complete
+budget is `D <= (s-1)*128 - s*C <= -128` for every `s>=3`. This grants all
+gathers, source access, helpers, arithmetic, frame changes, restoration,
+recursive overhead and other nonnegative expenses zero cost. The resulting
+negative deficit already excludes positive saving in that model. No circuit
+implementation or further supplier optimization is warranted by this result.
+
+## Object and exact scope
+
+Identify the 128 ports selecting one coordinate from each of seven pairs
+with `x in F2^7`. Write `j=7-|x XOR y|` for intersection size and use
+
+```
+G[x,y] = (4/3)(j-1)(j-3)(j-5)
+       = sum_{S subset {0,...,6}, |S|<=3} (-1)^(sum_{i in S}(x_i+y_i)).
+```
+
+Thus G is an integral positive semidefinite Gram matrix of rank 64,
+realized by the 64 Walsh features of degree at most three. Distinct ports
+with odd intersection 1, 3 or 5 are orthogonal. Coordinate-star decoding
+has matrix
+
+```
+B[x,y] = j mod2 = 1+p(x)+p(y),    p(x)=x0+...+x6 in F2.
+```
+
+Adding the intersection-1/3/5 corrections gives the identity. This is
+PR231's star-family identity specialized to seven ports. The common
+coordinate-star code is exactly the eight-dimensional affine code
+`C={a dot x+c}`. The constant is available as the XOR of two complementary
+coordinate stars; no free empty-star observation is assumed.
+
+Allow every finite binary factorization `B=sum_i f_i g_i^T`, with nonzero
+`f_i,g_i in C`, including nonsymmetric and redundant factorizations.
+Equality is required on **all** 16,384 port pairs, not only on a correction
+mask. Its objective is
+
+```
+C_total = sum_i rank_Q G[support(f_i), support(g_i)].
+```
+
+This is an algebraic lower bound under the inherited per-total copy rule,
+not an invoice that realizes either witness. Changed Gram matrices,
+non-affine masks, masked identities, extra ports (`p>7`), interactive
+delivery and information retained between stages are outside the result.
+
+## Exhaustive finite evidence
+
+An affine form is encoded by an integer 1 through 255: bits 0 through 6
+are linear coefficients and bit 7 is the constant. Simultaneous cube
+translation and coordinate permutation preserve G. For linear parts a,b,
+the orbit key counts coordinates of types `(1,0),(0,1),(1,1)`; when a=b is
+nonzero it also records the XOR of the constants. Independent a,b allow
+both constants to be normalized by a translation. A zero linear part is
+necessarily the nonzero constant form. These observations cover all cases.
+
+There are **127 ordered orbits**, covering all **65,025 ordered pairs**.
+The separate checker verifies multiplicities both combinatorially and by
+enumerating the 255 forms. Both compute exact rational ranks, with no
+floating-point or finite-field acceptance decision.
+
+| Rank | Ordered pairs | Maximum diagonal coverage |
+|---:|---:|---:|
+| 20 | 56 | 0 |
+| 30 | 1,680 | 32 |
+| 32 | 644 | 32 |
+| 33 | 3,360 | 32 |
+| 36 | 4,704 | 32 |
+| 39 | 12,320 | 32 |
+| 42 | 21,000 | 64 |
+| 44 | 56 | 0 |
+| 45 | 7,840 | 32 |
+| 48 | 11,620 | 64 |
+| 54 | 1,736 | 64 |
+| 64 | 9 | 128 |
+
+The certificate also contains the full reduced factorization census:
+
+| Terms | Reader sets/multisets | Source assignments, including zeros | All factors nonzero | Minimum |
+|---:|---:|---:|---:|---:|
+| 2 | 3 | 3 | 3 | 128 |
+| 3 | 7 | 1,792 | 1,777 | 138 |
+| 4, distinct readers in a rank-three space | 2,205 | 564,480 | 494,802 | 128 |
+
+Full cost histograms and attaining words are in [CERTIFICATE.json](CERTIFICATE.json).
+The enumeration stops at the end of these exact domains; there is no
+heuristic search, numerical tolerance, timeout exclusion or novelty claim
+based on a different random seed.
+
+## Why this proves the unrestricted finite-term floor
+
+The binary rank of B is two; its row and column code is
+`U=span{1,p}`. If A and D list the n reader and source columns, respectively,
+`B=A D^T` and Sylvester's inequality gives `rank(A)+rank(D)<=n+2`.
+Each span contains U. Transposition preserves every pairing cost.
+
+For two terms, both spans equal U: the three unordered bases are exhausted.
+For three terms, one span equals U: after transposition its nonzero columns
+are a multiset from the three nonzero members of U. Exactly seven such
+multisets span U. The census includes repetitions and gives the exact
+three-term minimum 138.
+
+For four terms, one factor span has dimension at most three. Repeated
+reader columns can be merged by XORing their source columns. This cannot
+increase cost: the XOR support lies in the union of the two old supports,
+and concatenation rank is subadditive. Zero factors can then be discarded.
+The resulting two-/three-term decoder already costs at least 128. If four
+distinct reader columns remain, their span has dimension three and contains
+U. There are `(256-4)/4=63` such spaces, each with seven nonzero vectors and
+35 four-element subsets. Each subset spans its space, and each of the
+eight source coefficient columns has a one-dimensional solution freedom.
+That is exactly `63*35*256=564480` assignments. The minimum is 128.
+
+The diagonal of B is one everywhere, so the sum of the intersection sizes
+`|support(f_i) intersect support(g_i)|` must be at least 128. Using the
+table, the largest possible coverage for five terms of cost at most 127 is
+64; for six it is zero. These are deliberately optimistic type-multiset
+relaxations, checked by dynamic programming and independent enumeration.
+Seven or more terms cost at least `7*20=140`. Zero or one term cannot have binary
+rank two. All finite term counts are therefore covered.
+
+The three-/four-term enumeration is necessary here: unlike the five-cube
+argument, the seven-cube diagonal relaxation alone permits four rank-30
+terms, and the one-U-factor estimate permits three rank-42 terms. Neither
+relaxed tuple solves the full decoder identity. They are explicit controls
+against claiming the floor from counting alone.
+
+## Sharp witnesses and the peak separation
+
+The two parity totals give `B=p p^T+(1+p)(1+p)^T`, each of rank 64.
+For the four-term witness take `h=x1` (seed43 mod7) and `q=p+h`:
+
+```
+B = h q^T + q(1+h)^T + (1+h)(1+q)^T + (1+q)h^T.
+```
+
+Each term has rank 32. To attain peak 30, let
+`a=x0`, `b=x1+x2`, `c=x3+x4`, `d=x5+x6`. The eight pairs of forms are
+
+```
+(b,1+a), (1+b,a), (a,1+b), (1+a,b),
+(c,d), (d,1+c), (1+c,1+d), (1+d,c).
+```
+
+Their binary sum is B and each has rank 30. Every term of smaller rank
+has rank 20 and zero diagonal, so peak 30 is sharp.
+
+For total cost 128, the same diagonal relaxation at budget **128** excludes
+five and six terms; seven already cost 140. Two terms have peak 64, three
+terms cost at least 138, and four terms of peak at most 30 would have total
+at most 120, contradicting the proved sum floor. The rank table has no
+rank 31, and the four-term witness attains peak 32. Thus the two optima
+cannot be attained together. The exact total optimum under peak cap 30
+is not established; its diagonal/term-count relaxation gives a lower
+bound 140 and the displayed word gives an upper bound 240.
+
+## Reproduce and verification scope
+
+From the repository root, Python 3.12 and the standard library:
+
+```sh
+python3.12 -I -B scripts/research_package.py --package research/seven-cube-affine-seed43
+```
+
+This runs both implementations. Expected markers:
+
+```
+PASS_SEVEN_CUBE_AFFINE pairs=65025 orbits=127 assignments=566275 copy_floor=128 peak_floor=30 controls=10 no_new_kappa
+PASS_SEVEN_CUBE_INDEPENDENT pairs=65025 orbits=127 assignments=566275 literal_entries=49152 controls=10 no_new_kappa
+```
+
+`verify.py` uses the intersection polynomial, gcd-normalized integer
+elimination and subset-syndrome solutions. `independent.py` imports no
+generator or predecessor code: it builds Walsh features, uses Fraction
+elimination and solves coefficient equations by binary RREF. It checks
+the 64 coefficient equations for every enumerated source assignment.
+Both compare every certificate field, check the three witnesses literally
+on 49,152 port pairs, and run ten semantic or proof-boundary controls.
+Both work under `-I -O -B`. Certificate generation is explicit and refuses
+to overwrite an existing path. The optional command
+`python3.12 -I -B research/seven-cube-affine-seed43/test_package.py` checks
+both programs in a read-only relocated checkout and rejects five artifact/source
+corruptions with both implementations. [VALIDATION.json](VALIDATION.json) records
+the local environment and executed checks; hosted CI is separate.
+
+This is exact finite evidence supporting the written scoped argument,
+not proof-assistant verification, independent human review, or a proof
+of the multiplication theorem. The selected cold replay, native supplier,
+hosting, precision/tape, all-size/compiler, common-label/mutable-source,
+bit H1-H5 and analytic transfer obligations are not discharged here.
+
+## Provenance, overlap and credit
+
+Assignment seed **43**, drawn with OS randomness, selected the structural
+decoder track. Exact base: `64039f1bd3b6a523b96083fd858ad040a75e3030`.
+The selected conditional result at this base is PR1094,
+`kappa=836330098538459/500000000000000000`, with complex binding and bit
+supplier PR589. Its source manifest, actual T16 construction and maintainer
+review were followed. This package changes none of them or `upstream/`.
+
+[TASK.json](TASK.json) records the three candidate scopes and local claim;
+[SOURCES.json](SOURCES.json) pins the base, Git blobs and SHA256 digests;
+[OVERLAP.json](OVERLAP.json) records the bounded live scan and its limits.
+No pinned sprint issue existed, so the donor's local-claim fallback was used.
+The catalogue card is marked changed: this is a seven-port binary/cubic
+side result, not an E8 complex masked-decoder construction.
+
+Credit **Joel Pulikkan / GamingPuzzled (PR231)** for the star-family parity
+identity; **Ross Richardson (PR958)** for the five-port geometry and
+optimistic-budget approach; **martinxxx9 (PR1046)** for the affine five-cube
+copy floor and factor-span reduction; and **icekylinx/eumemic**, with the
+retained **DreamingOfClouds, Jacob Sussman and Zhihao Chen** copy/frame
+lineage. PR1046 is incorporated; this does not reopen its five-cube search.
+Pending PR1177, PR1222 and PR1482 treat nonlinear masks on that five-cube,
+not this seven-cube. They are overlap context, not theorem dependencies.
+
+Prepared by **Hanson Sun (@bbMisk)** with substantial **OpenAI Codex
+assistance** in mathematics, both implementations, verification and prose.
+Apache-2.0. The separate implementation is by the same AI-assisted author.
+
+## Open next questions
+
+Can a fully priced two-stage construction retain one of the four displayed
+coded totals and thereby change the additive `s*C` charge? That premise
+must change before this geometry could help the supplier. A feasible first
+step is an exact arbitrary-dirty two-stage map using the explicit four-term
+decoder, charging source access, all frame transitions and restoration.
+The blocker is the absence of such a legal physical interface; a rank-table
+optimization alone cannot remove the present negative deficit. This is an
+unclaimed next question, not evidence that retention is possible.

research/seven-cube-affine-seed43/SOURCES.json

@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "base_commit": "64039f1bd3b6a523b96083fd858ad040a75e3030",
+  "files": [
+    {
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+      "usage": "provenance",
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research/seven-cube-affine-seed43/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-masked-decoder",
+  "priority_revision": 2,
+  "base_commit": "64039f1bd3b6a523b96083fd858ad040a75e3030",
+  "original_card": {
+    "id": "complex-masked-decoder",
+    "question": "Can one specified masked or overcomplete decoder improve the complete moment budget?",
+    "starting_points": [
+      "research/complex-decoder-field-extension-seed44/README.md",
+      "research/complex-frame-pair-decoders/README.md"
+    ],
+    "methods": [
+      "Synthesize a decoder with exact coefficients and fully priced helper endpoints.",
+      "Bound the best possible budget for a fixed support alphabet before synthesis."
+    ],
+    "acceptance_gate": "Change an assumption of the eight-term all-pairs bound and count all copied information, frames and restoration.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/complex-decoder-field-extension-seed44/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_DECODER_FIELD_EXTENSION_S44",
+      "inspect": "Locate the excluded classes: masked identities, at least nine terms, and deeper punctures.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Choose one finite support alphabet and test exact decoding constraints before building a whole network.",
+    "bottleneck_link": "A cheaper fully realized decoder could increase deficit or lower recursive cost on the binding complex side.",
+    "allowed_variations": [
+      "Select a specific masked or overcomplete family with exact labels and supports."
+    ],
+    "scope_change_triggers": [
+      "Repeating the exact eight-term all-pairs search inside its proved scope."
+    ],
+    "overlap_terms": [
+      "masked decoder",
+      "overcomplete",
+      "eight-term"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Smallest seven-port cube x in F2^7; cubic Gram sum of Walsh characters of degree <=3; full parity decoder; arbitrary finite affine binary factorization",
+  "scope_change_reason": "The donor explicitly assigned structural decoder research with seed43. This is a seven-port binary/cubic geometry result, not the catalogue card's E8 complex decoder; it changes the five-cube geometry rather than reopening a closed domain.",
+  "bottleneck_link": "Scoped negative side result: affine recoding cannot make the cubic seven-cube positive under the inherited additive stage ledger. The selected complex supplier remains binding; no supplier or kappa change.",
+  "budget_minutes": 45,
+  "stop_rule": "Exhaust all 127 affine-pair orbits and the 566275 reduced source assignments; use the written rank/diagonal argument for all finite term counts. Stop construction effort if copied-rank sum is at least 128. Reserve final 12 of 45 minutes for verification and handoff.",
+  "outcome": "negative-result",
+  "evidence": "CERTIFICATE.json and README.md: sum floor 128; peak floor 30; minimum peak 32 at total 128; optimistic D <= -128.",
+  "author": "Hanson Sun (@bbMisk)",
+  "ai_assistance": "Substantial OpenAI Codex assistance in research, written argument, both separately implemented checkers, tests and prose. No independent human review.",
+  "open_questions": [
+    {
+      "question": "Can one of the four coded totals be retained legally across two consecutive stages?",
+      "why_it_matters": "The additive s*C charge must change before this geometry can help.",
+      "first_step": "Write an exact arbitrary-dirty two-stage map for the displayed four-term decoder and charge source access, every frame transition and restoration.",
+      "blocked_by": "No physical retention interface or paid implementation is supplied."
+    }
+  ],
+  "assignment": {
+    "track": "structural decoder research",
+    "assignment_seed": 43,
+    "base_commit": "64039f1bd3b6a523b96083fd858ad040a75e3030",
+    "object": "Smallest seven-port cube x in F2^7; cubic Gram sum of Walsh characters of degree <=3; full parity decoder; arbitrary finite affine binary factorization",
+    "changed_hypothesis": "Seven-port cubic geometry instead of the closed five-port quadratic case; not a larger search in the same domain",
+    "candidates": [
+      "P112 nine-term outside-code decoder (overlaps PR1224 and1320)",
+      "Seven-cube affine copied-rank floor (chosen by seed43 mod3 =1)",
+      "Dirty centre hosting (claimed in Discussion1516)"
+    ],
+    "method": "Coordinate-permutation and translation orbits of all255 nonzero affine forms; exact rational restricted-Gram ranks; rank/diagonal lower bounds and explicit attaining words",
+    "optimistic_budget": "v=128; D <= (s-1)*128 - s*C for s>=3, granting every other required cost zero; stop constructive work if C>=128",
+    "stopping_rule": "Complete finite rank orbit census, plus a written quantified factorization bound if justified; no parameter optimizer",
+    "expected_artifact": "Scoped theorem or bounded negative evidence, standalone exact replay, source pins, controls and one focused PR if warranted",
+    "expected_finish_utc": "2026-10-14T03:16:54Z",
+    "verification_start_utc": "2026-10-14T03:04:54Z",
+    "coordination": "No pinned issue; donor local-claim fallback. Live complete snapshot at02:35UTC covers763 open PRs and205 discussion records.",
+    "author": "Hanson Sun (@bbMisk)",
+    "assistance": "Substantial OpenAI Codex assistance",
+    "status": "completed locally; single PR submission pending"
+  }
+}