Require repeated axes in seven-source four-root connectors (no κ)

research/seven-source-distinct-axis-obstruction/PROOF.md

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+# Seven distinct source axes cannot supply the seven-source connector
+
+This scoped obstruction was produced by Nick Petalas. Apache-2.0.
+The finite exclusion below is new. The normalized-axis and mandatory-mask
+methods are credited to conorholds, Mason Hale and Arham Tahir.
+
+## 1. Contract and conclusion
+
+There are seven immutable source rows x_j, four arbitrary dirty root rows z_p,
+and separate output rows y_m for every nonzero seven-bit mask m. Source calls
+add B_j x_j to one physical root. Output calls add A_m z_p to their own target.
+Every B_j and A_m occurs exactly twice. The compatible binary linear maps are
+arbitrary, including singular and noncommuting maps. Outputs are not observed
+or used as scratch. All roots and sources restore. Child scratch is private
+and restores under its inherited contract.
+
+Allow any finite number of invertible scalar binary root-frame changes, any
+physical port on each call, and no bound on the cost of those changes. Granting
+these operations only enlarges a paid-XOR interface.
+
+**Theorem.** A universal connector under this contract cannot have seven
+pairwise distinct normalized source axes. If it supplies the all-ones output
+and its normalized axes are distinct, at least 22 other requested masks fail
+the mandatory-mask necessary condition. The number 22 is the exact minimum
+of that deficit in the complete reduced domain. It is not an exact maximum
+for the total set of outputs a legal circuit can supply.
+
+Consequently a universal seven-source connector with four dirty roots must
+repeat an axis. This is a restriction within the known optimal root count,
+not a new root-count theorem or a transfer-count minimum.
+
+## 2. Necessary mask equation
+
+Let R_t be the invertible matrix carrying the four independent initial dirt
+rows into the current roots. In normalized coordinates a source call at port p
+injects the nonzero column R_t^{-1} e_p. Its two calls must have the same
+normalized column v_j, since no other call can remove that source coefficient
+from the restored roots. Final dirt restoration also forces R_final=I.
+
+Let q_t be the seven-bit parity of source calls before a cut. A physical root
+whose dirty covector is r then contains
+
+    r z_initial + sum_j (r v_j) q_t[j] B_j x_j.
+
+Two calls of an arbitrary A_m cancel initial dirt only when their dirty
+covectors agree. Their source mask is therefore
+
+    m = h AND d,    h = r V,    d = q_s XOR q_t,    V=[v_0 ... v_6].
+
+Write H=rowspace(V) and D(w)={q_s XOR q_t} for the paired source word w.
+Thus every supplied mask belongs to {h AND d: h in H, d in D(w)}.
+This implication is the normalized-axis method of PR2182 and the code
+formulation of PR2795. The calculation above states the exact premise used
+here; neither predecessor's census or attaining construction is reverified.
+
+The implication is necessary already for identity source/output maps on
+independent scalar inputs. Hence it applies at every positive row length,
+without inferring arbitrary-size behavior from a finite dense fixture.
+
+## 3. Complete domain for distinct axes
+
+The all-ones mask requires both h=127 and d=127. Some nonzero covector r_*
+therefore has r_* v_j=1 for every j. The axes lie in an eight-point affine
+hyperplane of F2^4. Under the distinctness hypothesis they occupy seven of
+its points and span F2^4.
+
+A change of root basis takes this hyperplane to the points (1,u), u in F2^3.
+The affine transformations u -> Au+b act transitively on its omitted point
+and are induced by invertible linear maps on (1,u). We may choose the omitted
+point to be u=7. Coordinate assignments to the seven sources remain arbitrary.
+Therefore every possible H is a coordinate permutation of
+
+    H_0 = { ((r dot (1,j)))_(j=0,...,6) : r in F2^4 }.
+
+Here (1,j) is encoded by 1|(j<<1). The primary generator visits all 7!
+coordinate permutations, takes their exact row spaces, and deduplicates them.
+There are exactly 30 distinct codes, each with sixteen words. No source-order
+symmetry is assumed after this code inventory is formed.
+
+For every proper codeword h != 127 define its supported subsets. A nonzero
+mask m is mandatory when no such h contains its support. If m=h AND d is
+mandatory, h must be 127, so d=m. Thus
+
+    M(H) subset D(w)
+
+is necessary for universality. Each of the thirty codes has exactly 57
+mandatory masks, as directly enumerated in both checkers.
+
+## 4. Complete source-order reduction
+
+The equality d=127 means an interval of the paired source word contains each
+source an odd number of times. Since there are only two calls per source,
+that interval contains each source exactly once and has length seven.
+Rotate the closed source word to start at that interval and rename sources
+in its order. The resulting word is
+
+    0,1,2,3,4,5,6, permutation(0,1,2,3,4,5,6).
+
+Rotation translates all prefix parities by the same vector and preserves D.
+Renaming coordinates permutes H as well as D. Enumerating all 5,040 second
+orders and all thirty codes is therefore complete for this necessary test.
+This is the all-ones interval reduction of PR2795. It does not assert that
+physical frame changes can be rotated without paying for them.
+
+## 5. New finite certificate
+
+For every reduced pair the checker computes the full bitset M(H) minus D(w).
+The two engines agree on every one of the 151,200 bitsets. Their minimum
+cardinality is 22, attained by four reduced pairs. There are no zero-deficit
+pairs. The generated report contains the complete deficit histogram and a
+SHA256 digest of all bitsets in lexicographic (source order, code) order.
+
+One attaining deficit witness has second order
+
+    1,4,0,3,6,2,5
+
+and code
+
+    0,11,22,29,37,46,51,56,71,76,81,90,98,105,116,127.
+
+Its missing mandatory masks are
+
+    23,45,53,54,55,57,58,61,75,78,79,85,86,87,92,94,102,
+    113,117,118,119,121.
+
+The primary engine uses coordinate permutations of the affine-cube code,
+pairwise prefix XORs and support containment. The second enumerates all
+11,811 three-dimensional binary RREF row spaces, retains the thirty whose
+nonzero vectors all have weight four, and takes orthogonal complements.
+It forms D by contiguous interval parities and M by enumerating proper
+codeword subsets. The full code inventories and every deficit agree.
+No predecessor executable, stored numerical receipt or search solver is an
+acceptance input.
+
+## 6. Consequence and limits
+
+The 22 absent mandatory masks cannot be recovered by more scalar frame
+changes, different physical read ports or a more expensive frame itinerary.
+Thus the complete 127-mask endpoint is impossible with distinct axes. If the
+all-ones mask is supplied, at most 105 nonzero masks can pass this necessary
+test. Attainment of that total-output bound is unclaimed.
+
+Repeated axes escape the premise. As a positive control, the credited star's
+axes (1,1,1,1,2,4,8) and source word
+
+    0,1,0,1,4,5,6,2,3,2,3,4,5,6
+
+pass the full mask-intersection criterion. This is a tiny control on the
+scope, not a new verification claim for its physical construction or invoice.
+The predecessor's four-root, 56-X star and root-minimum proof remain credited
+premises. No smaller paid word was established here.
+
+Extra source/output calls, source mutation, output feedback, clean roots,
+matrix-valued root transformations and nonlinear updates change the contract.
+Physical hosting, native realization, tape/precision/stock costs and supplier
+admission are not supplied. All five selected framework assumptions remain.
+The finite certificate supports this scoped written proof; independent human
+and formal proof-kernel review are unclaimed.

research/seven-source-distinct-axis-obstruction/README.md

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+# Seven-source connectors must repeat a normalized source axis
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+A universal seven-source connector with four arbitrary dirty roots cannot
+assign seven distinct normalized axes to its twice-called source maps. This
+is a scoped obstruction with an exhaustive finite certificate. It excludes
+an axis-diversity route toward Mason Hale's open transfer-cost question,
+without changing the selected frontier or kappa.
+
+The complete reduced domain contains thirty affine-cube codes and 5,040
+source orders. Every one of the 151,200 pairs misses at least 22 of its 57
+mandatory masks. The minimum deficit 22 is attained by four pairs. It is
+sharp for this necessary-condition census, not an exact total-output capacity.
+If the all-ones output is supplied, at most 105 nonzero masks pass the test.
+
+[PROOF.md](PROOF.md) gives the arbitrary-input reduction, complete domain and
+an attaining deficit witness. The program grants arbitrarily many scalar
+binary root frames and every physical read port, so unpaid frame restrictions
+do not cause the exclusion. Every source and output map must occur exactly
+twice; sources and roots restore and outputs remain unobserved.
+
+## Evidence and reproduction
+
+From the repository root, using CPython 3.12 and its standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/seven-source-distinct-axis-obstruction/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/seven-source-distinct-axis-obstruction
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/seven-source-distinct-axis-obstruction --check-source-history
+```
+
+Another CPython 3.12 path may replace the local interpreter. Direct replay
+prints a generated JSON receipt with status
+`PASS_SEVEN_SOURCE_DISTINCT_AXIS_OBSTRUCTION`, the full deficit histogram,
+minimum witness, census digest and hashes of every package file. It writes
+nothing. Normal and optimized modes are registered.
+
+Two separately implemented code generators, mandatory-set constructions and
+word-difference engines agree on every missing-mask bitset. The secondary
+code generator examines all 11,811 three-dimensional binary RREF spaces.
+Controls cover fourteen cyclic rotations, all 21 coordinate transpositions,
+the repeated-axis star positive case, two invalid-word rejections and four
+false-variant boundaries.
+The two cited main-tree documents are hash authenticated before enumeration.
+These are two implementations by one contributor, not independent review.
+
+The current selected binding control and the card's 945-pairing positive
+control pass. No predecessor scientific census is replayed as this result.
+
+## Source, scope and credit
+
+Base `7c49fe32c55de02c2c873655c804b16e4c3fef0a`, priority revision 8;
+selected conditional kappa remains 0.003125804. Exact document pins are in
+[SOURCES.json](SOURCES.json). The original card and changed scope are in
+[TASK.json](TASK.json).
+
+- Mason Hale / [PR2282](https://github.com/CrocSwap/integer-mult-bounds/pull/2282)
+  poses the four-root transfer question and proves the seven-source root minimum.
+- Arham Tahir / [PR2795](https://github.com/CrocSwap/integer-mult-bounds/pull/2795)
+  supplies the code formulation, mandatory masks and all-ones interval reduction.
+  Its eight-source exclusion and seven-source three-root calibration are different
+  finite domains from the four-root distinct-axis domain here.
+- conorholds / [PR2182](https://github.com/CrocSwap/integer-mult-bounds/pull/2182)
+  supplies normalized axes, the dirty-root contract and the 56-X attaining star.
+  Ross Richardson / PR934/943, Douglas Colkitt / PR1185 and their retained
+  connector, common-factor and compiler lineage retain credit.
+
+The [advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19372499)
+aimed for fewer transfers. The [finding](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19372766)
+records the delivered narrower obstruction. No shorter word or global transfer
+minimum was established. Construction search stopped at this first proved step.
+
+The main/merged/open searches used the source object, distinct normalized axes,
+affine-cube codes and mandatory-mask deficits. The overlap cache covered 550
+open PRs and 6,574 Discussion records; sparse local notes and unpublished work
+remain limits. No global literature-priority claim is made for classical
+binary code geometry or the credited reductions.
+
+## Open next questions
+
+- Can a repeated-axis four-root program use fewer than 56 paid XORs while
+  retaining all minimum map incidences? First choose one repeated-axis code
+  and emit its complete frame/source/readout itinerary. A cheaper legal word
+  and its complete physical port bill remain missing.
+- Can a structural argument replace this 151,200-case distinct-axis census?
+  Start with the 57 mandatory supports and a double-permutation source word.
+  A uniform interval argument, rather than sampled orders, is needed.
+
+No new supplier, physical caller, complete multiplication theorem, independent
+human review or formal-kernel proof is claimed. Native, compiler, ownership,
+stock, precision/tape and analytic transfer obligations remain open.
+
+## Implementation bounds
+
+Every loop is bounded by a fixed finite domain: seven coordinates, sixteen
+codewords, 128 masks, 5,040 orders, thirty codes or the 11,811 RREF spaces.
+Functions are under sixty lines; checks use exceptions and survive `-O`.
+There is no recursion, solver, runtime network call or generated executable.
+The two census lists have exactly 151,200 integers of at most 128 bits each.
+
+Power-of-Ten rule 3 has a managed-runtime deviation for Python allocation in
+these bounded lists, sets and exact integers. The interpreter allocator is not
+inspected. Fixed domain sizes bound live mathematical storage; both lists are
+released on process exit. A fixed-capacity implementation would be needed
+before reuse in a long-lived service. Warning-strict normal and optimized
+commands provide the applicable interpreted-code diagnostics.

research/seven-source-distinct-axis-obstruction/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
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+    },
+    {
+      "path": "research/eight-source-root-code/README.md",
+      "usage": "input",
+      "git_blob": "ce5d5add72a7b57bb17cc339d9fda4bb0592160c",
+      "sha256": "8a9100409033acfc0ede9fa260c0172c958e5103a244b1de97b7e132659038d0"
+    },
+    {
+      "path": "research/seven-source-root-minimum/README.md",
+      "usage": "input",
+      "git_blob": "2fac5ab937d94856e55fd753d067f2f6f8ada111",
+      "sha256": "17554797cf3a8752ef3f8b65b0d7fc9e98f7f99f3afb5dbb1b0952fbf9f1fd58"
+    }
+  ],
+  "predecessor_context": [
+    {
+      "pr": 2282,
+      "author": "mason-hale",
+      "original_head": "74e57135608dade29029a1107fdfceec8419c932",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2282",
+      "usage": "Bibliography only. The replay inputs above are retained main-tree documents, not these separate Git objects."
+    },
+    {
+      "pr": 2795,
+      "author": "Arham-Tahir64",
+      "original_head": "fad52f9528cfea1c3bcfc0094ff56abf945d31a7",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2795",
+      "usage": "Bibliography only. The replay inputs above are retained main-tree documents, not these separate Git objects."
+    },
+    {
+      "pr": 2182,
+      "author": "conorholds",
+      "original_head": "dc252de1cc594829c3031abdb4de104713bc474a",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2182",
+      "usage": "Bibliography only. The replay inputs above are retained main-tree documents, not these separate Git objects."
+    }
+  ]
+}