Exhibit complex rank-seven P104 approximations below the real 1/4 floor (no κ)

research/p104-complex-rank-seven/PROOF.md

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+# Complex rank seven beats the real quarter-error floor
+
+Prepared by apoursh with substantial OpenAI Codex assistance. Apache-2.0.
+Nick Petalas's PR6183 retains the contractive Gram question and the real
+omitted-root projector; Matarener's PR6166 retains the cited coordinate
+identification and omitted lines. The real comparison theorem is credited
+to apoursh's [PR6191](https://github.com/CrocSwap/integer-mult-bounds/blob/be3fc854aa380b3d81e134ce9a4282468e4fa999/research/p104-rank-seven-error-floor/PROOF.md).
+
+## 1. Fixed matrix and statement
+
+Use the 104 root lines retained by
+`research/p104-contractive-gram-deflation/MISSING.json`: all 56 D8 lines
+`e_i +/- e_j` and 48 odd-parity half-spin lines in eight coordinates.
+Each row of R is a root of squared norm two, with its first nonzero
+coordinate positive. Write S=2R and K=RR^T/2=SS^T/8. The sixteen omitted
+half-spin lines are part of the pinned instance, not a search parameter.
+
+There is a Hermitian orthogonal projector D over Q(i), of complex rank
+seven, such that C=RDR^T/2 has complex rank seven, satisfies
+`0 <= C <= K` in Hermitian positive-semidefinite order, and
+
+```text
+max_(j,k) |K_jk-C_jk| = 25/101 < 1/4.
+```
+
+The absolute value is the complex modulus. In contrast, every real
+104-by-104 matrix of real rank at most seven has error at least 1/4
+against this K (section 4). Thus extending the coefficient field changes
+the approximation problem even with a positive contractive carrier filter.
+The assertion `0<=D<=I` is about the eight-dimensional filter, not about
+C being a contraction on the 104-dimensional ambient space.
+
+## 2. Exact witness
+
+Take
+
+```text
+z = (5-5i, 3+9i, 5-5i, 7, 5-5i, -1-4i, -7, -7),
+N = z* z = 404,
+D = I8 - z z*/N.
+```
+
+Here * denotes conjugate transpose. Since z is nonzero, `zz*/N` is a
+rank-one Hermitian orthogonal projector. Consequently D is Hermitian,
+`D^2=D`, `0<=D<=I`, and `rank_C D=7`. R has full column rank: the
+eight retained roots `e_(2j)+e_(2j+1)` and `e_(2j)-e_(2j+1)` for
+j=0,1,2,3 form an orthogonal basis. Thus
+
+```text
+C = (RD)(RD)*/2,     rank_C C = rank_C(RD) = 7,
+K-C = (Rz)(Rz)*/(2N) = (Sz)(Sz)*/(8N).
+```
+
+Both C and K-C are positive semidefinite, proving `0<=C<=K`.
+If `m=max_j |(Sz)_j|^2`, the last rank-one formula gives
+
+```text
+max_(j,k) |K_jk-C_jk| = m/(8N).
+```
+
+Indeed each product `|(Sz)_j| |(Sz)_k|` is at most m, and a diagonal
+entry with maximal squared modulus attains m. Complete exact enumeration
+of the fixed 104 rows gives m=800, in both the D8 and retained half-spin
+parts. Therefore the error is `800/(8*404)=25/101`, whose gap below
+1/4 is exactly `1/404`.
+
+For a reproducible row ordering, enumerate pairs i<j lexicographically,
+with D8 sign -1 then +1, followed by tails in `{-1,+1}^7` lexicographically,
+first coordinate +1, odd negative parity, excluding the pinned omitted
+lines. The attaining row indices, starting at zero, are
+
+```text
+0, 3, 7, 14, 18, 29, 70, 71, 72, 77, 78, 79.
+```
+
+The primary checker also verifies every entry of the literal product
+`SDS^T/8` against the rank-one error identity. The independently written
+checker reconstructs the instance and uses separate exact arithmetic.
+Neither certificate uses floating point tolerances.
+
+## 3. Secondary witness and coefficient field
+
+A previously found second witness provides the stronger upper bound below,
+with no optimality claim:
+
+```text
+z' = (14-10i, 6+23i, 14-10i, 17+3i,
+      14-10i, -3-10i, -17-3i, -17-3i),
+N' = 2456,      max_j |(Sz')_j|^2 = 4820,
+error = 4820/(8*2456) = 1205/4912 < 1/4,
+gap = 23/4912.
+```
+
+The same proof applies, and both checkers verify its full matrix too.
+Attaining indices in the primary ordering are `17,22,24,62,69,85`.
+The small witness is the principal example because its arithmetic is
+simpler; neither witness is asserted globally optimal. No new search is
+needed to reproduce them. Discovery used bounded numerical exploration
+followed by exact integer rationalization; only the exact witnesses and
+their complete certificates supply the result.
+
+These coefficients are Gaussian-rational, not Gaussian-dyadic. For the
+first D, `D_00=177/202`; for the second, `D'_00=270/307`. The odd
+denominators cannot be called free native operations. This is an algebraic
+field separation, with no circuit or physical cost claimed.
+
+## 4. The real comparison and why real parts do not evade it
+
+The following is the real proof from PR6191, included to state the exact
+comparison without requiring its pending Git head at replay time.
+Let F be any real 104-by-104 matrix of rank at most seven. Select the
+eight columns J indexed by the orthogonal D8 basis above. There is a
+nonzero real a with `F[:,J]a=0`. Let U have those eight roots as its
+columns and set x=Ua. Then `K[:,J]a=Rx/2`.
+
+For each coordinate pair, the inverse change of basis gives
+`a_+=(x_i+x_j)/2`, `a_-=(x_i-x_j)/2`. The real identity
+
+```text
+(|s+t|+|s-t|)/2 = max(|s|,|t|)
+```
+
+implies that `||a||_1` is the sum of the larger coordinate modulus in
+each of four pairs. If t1>=...>=t8 are the ordered moduli of x, then
+`||a||_1 <= t1+t2+t3+t4 <= 2(t1+t2)`. The retained D8 rows give
+`||Rx||_infinity=t1+t2` when restricted to that subsystem. For
+`delta=max|K-F|`, therefore
+
+```text
+delta ||a||_1 >= ||(K-F)[:,J]a||_infinity
+               = ||Rx||_infinity/2 >= (t1+t2)/2,
+delta >= 1/4.
+```
+
+An omitted real half-spin root gives equality by PR6183's projection,
+so the real floor is exactly 1/4. The displayed two-coordinate identity
+is specific to real numbers: s=1,t=i already makes its left side sqrt(2)
+and right side 1. It cannot be applied to the complex kernel vector.
+
+There is also no contradiction obtained by taking real parts. Write
+z=u+iv for the small witness. Then
+
+```text
+u^T u = 232,   v^T v = 172,   u^T v = -44,
+det [[232,-44],[-44,172]] = 37968 > 0,
+Re(D) = I8 - (uu^T+vv^T)/404.
+```
+
+The real matrix `uu^T+vv^T` has two strictly positive eigenvalues whose
+sum is 404, so each is strictly less than 404. Thus Re(D) is positive
+definite and `rank_R Re(C)=8`. The complex rank-seven construction does
+not yield a seven-real-mode implementation; its usual realification has
+rank fourteen.
+
+## 5. Scope and proof obligations
+
+The mathematical object is the fixed root Gram matrix with its specified
+sixteen omitted lines. The physical identification with selected P104
+ports remains a cited premise from Matarener's PR6166; its source code,
+native operators and coordinate identification are not independently
+replayed here. Nick Petalas's PR6183 retains the Gram convention and
+contractive question, with Rohan Arun/Chafik Boukhalfa and the retained
+root/frame/native/helper/compiler/BIT lineage credited in the README.
+
+The complete finite root census proves the two stated errors for this
+instance; the projection and rank arguments are written mathematics.
+No universal complex optimum, asymptotically vanishing residual, native
+Gaussian-dyadic implementation, source/consumer installation, residual
+correction, dirty restoration, paid/target saving, new kappa or complete
+multiplication theorem follows. The selected result and its five framework
+assumptions retain their status.

research/p104-complex-rank-seven/README.md

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+# Complex rank-seven P104 approximation below 1/4
+
+**Exact finite construction and field separation; no new kappa.**
+
+Complex coefficients can beat the sharp real rank-seven error floor for the
+fixed P104 Gram matrix. This answers the released complex-field question in
+[PR6191](https://github.com/CrocSwap/integer-mult-bounds/pull/6191), at
+`be3fc854aa380b3d81e134ce9a4282468e4fa999`. It does not contradict that
+PR's theorem, whose coefficient field is real.
+
+Let R contain the 104 retained roots, each of squared norm two, and let
+`K=RR^T/2`. The explicit vector
+
+```text
+z = (5-5i, 3+9i, 5-5i, 7, 5-5i, -1-4i, -7, -7)
+```
+
+has `z* z=404`. The filter `D=I-zz*/404` is a Hermitian orthogonal
+projector of complex rank seven, hence `0<=D<=I`. Its approximation
+`C=RDR^T/2` has complex rank seven and exact maximum complex entry error
+**25/101 < 1/4**, a gap of **1/404**. A second retained witness gives the
+stronger bound **1205/4912 < 1/4**, with gap **23/4912**. Neither value is
+claimed optimal. [PROOF.md](PROOF.md) contains both vectors and the proof.
+
+The distinction matters: C is Hermitian positive semidefinite with
+`0<=C<=K`; the contraction claim applies to D, not to C relative to the
+104-dimensional identity. Both candidates have trace 91. Their real parts
+have real rank eight, so these are not seven-real-mode implementations.
+Their coefficients contain odd denominators and are not Gaussian-dyadic.
+
+## Evidence and reproduction
+
+`verify.py` authenticates the three pinned predecessor inputs, reconstructs
+all 104 roots, checks an orthogonal eight-root basis, and certifies both
+projectors and every entry of both full Gram approximations: **21,632
+entries**. It checks real-part rank eight both by exact elimination and
+by the real/imaginary Gram determinant. Nine controls cover all sixteen
+real omitted-root quarter-error witnesses, a retained-root unit error,
+dropping imaginary coordinates, missing conjugation, projector and Gram
+normalizations, wrong parity, duplicate lines and a missing line.
+
+`independent.py` was written separately, without reading or importing the
+primary checker. It reconstructs the roots in a different order, uses
+Gaussian pairs of `Fraction` values instead of a common integer denominator,
+checks every residual entry, and independently certifies complex rank seven
+and real-part rank eight through the orthogonal-root minor. Four additional
+projector controls reject wrong normalization and missing conjugation.
+
+Run from the repository root with Python 3.12 and its standard library:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-complex-rank-seven --check-only
+python3.12 -B scripts/research_package.py --package research/p104-complex-rank-seven
+```
+
+The registered commands run each checker normally and with `-O`, with
+isolated imports and warnings treated as errors. Expected markers are
+`PASS_P104_COMPLEX_RANK_SEVEN` and `PASS_INDEPENDENT_COMPLEX_RANK_SEVEN`.
+For the official Git-free source export, run these direct commands from its
+root, once normally and once with `-O` added:
+
+```sh
+python3.12 -I -B -W error research/p104-complex-rank-seven/verify.py
+python3.12 -I -B -W error research/p104-complex-rank-seven/independent.py
+```
+
+The export must include the three external inputs declared in
+[VERIFY.json](VERIFY.json). [SOURCES.json](SOURCES.json) pins their base
+commit, Git blobs and SHA256 digests. The original card and selected record
+are historical provenance; the three PR6183 files are live inputs. The
+primary verifier checks all three live hashes before arithmetic. The
+independent checker separately pins the root-data hash. Both write no files
+and preserve every mathematical check under optimized Python.
+
+## Sources, scope and coordination
+
+Canonical main was refreshed first at
+`a5bff3bf133e482541c09eba4e773884bdb2b1b7`, priority revision 8. The selected
+record identifies conditional `kappa=781451/250000000` and five remaining
+framework assumptions; neither changes here. Binding-only orientation was
+reused after confirming all 1227 bound inputs unchanged since checkpoint
+`3f0152a32eeebac0df11946dbd50cabe5a7f7839`. No selected scientific cold replay
+was run.
+
+The original `coverage-law-change` card is retained in [TASK.json](TASK.json),
+with scope explicitly changed to this algebraic field-extension question.
+This identifies a limitation of a real approximation obstruction, but does
+not meet the original completed-call and full-invoice acceptance gate.
+The [canonical claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19275237)
+records the two previously unpublished local candidates and stops after
+their exact certification. No additional numerical search was run.
+
+The local overlap helper searched 30 of 8623 notes (8593 absent in the
+sparse checkout), without a usable remote cache. Live checks covered the
+newest 20 PRs, newest 15 handoff comments, and all-state searches for
+P104/complex/rank, 6191 and Gram/contractive. Nearest results are PR6191's
+real theorem and Nick Petalas's [PR6183](https://github.com/CrocSwap/integer-mult-bounds/pull/6183)
+contractive theorem. PR6194 concerns a different paid frame-graph objective.
+No competing complex witness appeared in the checked results; this is not
+an exhaustive novelty audit.
+
+PR6183 at `eaf77790466f34b86ce72080e3ec690e26dfacab` supplies the retained
+root data, contractive question and Gram convention. Matarener's PR6166
+at `096938513138868f0ed27639061f1a5054900f96` supplies the cited P104-to-D8
+identification and omitted lines. Those physical coordinates are a cited
+premise, not independently audited here. Their archival retention does not
+constitute scientific acceptance. PR6191 remains a pending comparison
+source; its real lower-bound argument is restated with credit in PROOF.md,
+and no replay requires its pending Git objects or executable code.
+
+The proof establishes two finite counterexamples to extending the real
+quarter-error floor over Q(i). It does not give the optimum complex error,
+an asymptotically vanishing residual, native implementation, side correction,
+source/consumer installation, ownership/dirty restoration, full paid/target
+invoice or multiplication theorem. Independent human and formal proof-kernel
+review are unclaimed. No Lean, Docker or physical/native replay was run.
+
+## Open next questions
+
+- **Can a Gaussian-dyadic rank-seven filter also beat 1/4?** The odd
+  denominators leave a coefficient-field gap before native implementation.
+  A bounded first step is one exact dyadic Hermitian projector on these same
+  104 roots, checking its complete coefficient error. No such projector or
+  native circuit is supplied here; any wider search needs a justified scope.
+- **Can an actual caller use and correct this residual profitably?** A lower
+  coefficient error alone earns no independent target credit. Start with
+  one source-bound scalar word, track this exact rank-one residual through
+  every retained output, and price restoration and correction. The absent
+  caller and complete bill are blockers; no additional parameter search is
+  justified as a bound improvement before that gate.
+
+Structured questions, relevance, first steps and blockers are in TASK.json.
+
+## Credit
+
+Prepared by **apoursh**, with substantial OpenAI Codex assistance in witness
+discovery, separately written exact verification, mathematical audit, proof
+and packaging. Nick Petalas retains PR6183's contractive question and real
+attaining projection; Matarener retains PR6166's D8 identification and omitted
+lines. Rohan Arun, Chafik Boukhalfa, Ali Romman, DreamingOfClouds, Cinna Davis,
+Pierre-Loic Doulcet, icekylinx, zxv, Jacob Alan Sussman and the retained
+P104/root/frame/native/helper/compiler/BIT lineage retain their source credit.
+Hermitian projection and Gram-rank facts are classical; no general literature
+priority is claimed. New material is Apache-2.0; predecessor files are unchanged.

research/p104-complex-rank-seven/SOURCES.json

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