Construct an exact dyadic rank-seven P104 projector (no new kappa)

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

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+# An exact Gaussian-dyadic Hermitian projection below the real P104 floor
+
+Jerry Chen (@CubeJerry), substantial OpenAI Codex assistance. Apache-2.0.
+
+## Statement and scope
+
+Let R contain one squared-norm-two representative from each of PR6211's
+104 root lines: all 56 D8 lines and the 48 odd half-spin lines remaining
+after the sixteen omissions in the pinned MISSING.json. Set S=2R and
+K=RR^T/2=SS^T/8. There is a Hermitian orthogonal projector D over
+Z[i,1/2] with complex rank seven such that
+
+    C = R D R^T / 2,
+    rank_C C = 7,       0 <= C <= K,
+    max_(j,k) |K_jk-C_jk| = 16193/65536 < 1/4.
+
+The modulus is the complex modulus. The contraction 0<=D<=I concerns the
+eight-dimensional carrier, not C relative to the 104-dimensional identity.
+This answers PR6211's first open question, the exact Gaussian-dyadic
+Hermitian projector. It is not an optimality or physical-supplier theorem.
+
+## Preserve the projection identity before approximating its coefficients
+
+Take the Gaussian-integer vector
+
+    z = (73-53i, 31+118i, 73-51i, 88+16i,
+         73-51i, -15-51i, -87-15i, -88-17i).
+
+Its squared norm is N=z* z=65536=2^16. Put P=zz*/N and D=I8-P.
+Since P*=P and P^2=z(z*z)z*/N^2=P, P is a rank-one Hermitian orthogonal
+projector. Thus D*=D, D^2=D, rank_C D=7 and 0<=D<=I. All its entries
+are Gaussian-dyadic with common denominator 2^16. This exact identity
+would not follow from entrywise dyadic rounding of an old matrix.
+
+The eight retained roots e_0 +/- e_1, e_2 +/- e_3, e_4 +/- e_5 and
+e_6 +/- e_7 form an orthogonal basis, so R has column rank eight.
+Multiplication on the left by R and on the right by R^T preserves the
+rank of D: these maps have left and right inverses respectively. Also
+C=(RD)(RD)*/2 and K-C=(Rz)(Rz)*/(2N). Hence C has rank seven and
+0<=C<=K. Since R is dyadic, every C coefficient is Gaussian-dyadic.
+The sufficient common denominator is 8N; the largest reduced component
+denominator is actually 2^17, as the complete certificate checks.
+
+## Complete entry-error certificate
+
+Write w=Sz. Then K-C=ww*/(8N). For any complex vector w,
+max_(j,k)|w_j conjugate(w_k)|=max_j|w_j|^2: a diagonal entry attains
+the product bound. Exact enumeration of every retained row gives:
+
+| Root family | Rows | Maximum squared response |
+| --- | ---: | ---: |
+| D8 | 56 | 129544 |
+| Odd half-spin | 48 | 128596 |
+| Complete instance | 104 | 129544 |
+
+Use PR6211's order: D8 pairs lexicographic, signs minus then plus,
+followed by lexicographic half-spin tails. The unique maximum is row 24,
+S_24=2(e_1-e_7), where (Sz)_24=238+270i. Therefore
+
+    max |K-C| = (238^2+270^2)/(8*65536) = 16193/65536,
+    1/4 - max |K-C| = 191/65536 > 0.
+
+The primary verifier reconstructs the root set and checks all 10,816
+entries of the literal product SDS^T/8 against the residual formula.
+It does not trust the search maximum. Rational Gaussian elimination
+separately checks the rank of D and an eight-root principal minor of C.
+
+The second arithmetic path imports no primary or predecessor executable.
+It uses separately ordered roots and Fraction arithmetic on the 16-by-16
+realification, checking symmetry, idempotence and rank fourteen. It
+independently evaluates every residual entry using real and imaginary
+outer products. Both paths are by the same contributor with AI assistance;
+this is not independent human or formal-kernel review.
+
+## Real and operator-norm boundaries
+
+Writing z=u+iv gives
+
+    u^T u=40230, u^T v=-2683, v^T v=25306,
+    det Gram(u,v)=1010861891 > 0.
+
+The two nonzero eigenvalues of uu^T+vv^T are positive and sum to N,
+so each is less than N. Thus Re(D)=I-(uu^T+vv^T)/N is positive
+definite and Re(C) has real rank eight. This does not yield a
+seven-real-mode approximation or contradict PR6191's real theorem.
+
+For this root set S^T S=104 I, checked exactly. Thus ||w||_2^2=104N
+and ||K-C||_2=104N/(8N)=13. The largest eigenvalue of K is also 13.
+Smaller maximum coefficient error therefore does not supply vanishing
+operator error: worst-case relative spectral error is one. This is a
+scope calculation, not a new optimality result.
+
+Exact reconstruction of K through a linear seven-dimensional intermediate
+is impossible, since rank K=8. Retaining the lost scalar response gives
+the elementary seven-plus-one reconstruction, with eight independent
+pieces of information before acquisition, transport or restoration costs.
+No free residual correction or completed recursive invoice is assumed.
+
+## Bounded discovery
+
+The guide is PR6211's stronger Gaussian-rational witness, written as
+sixteen interleaved real coordinates:
+
+    a=(14,-10,6,23,14,-10,17,3,14,-10,-3,-10,-17,-3,-17,-3),
+    sum a_i^2=2456.
+
+For k=4,...,9 in increasing order, let
+q_i=floor(2^k a_i/sqrt(2456)), restrict y_i to {q_i-1,q_i,q_i+1},
+and require sum y_i^2=4^k. Floors are calculated with integer square
+roots and sign-aware exact comparisons. On that sphere, minimizing
+squared distance from 2^k a/sqrt(2456) is equivalent to maximizing a dot y.
+
+The DP keeps the largest dot product at each partial squared norm.
+The remaining coordinate choices depend only on the norm and position,
+so a lower-scoring prefix cannot produce a better completion. Remaining
+minimum/maximum squared norms discard only infeasible completions.
+The nonce 0b7fc9054fc70396e2d971f95342f0a4 rotates the coordinate order
+by four; tied scores take the lexicographically largest tuple in that
+order. One selected nearest vector at each scale is tested for its error.
+
+| k | Error of the selected nearest vector | Strictly below 1/4? |
+| ---: | --- | --- |
+| 4 | 37/128 | no |
+| 5 | 17/64 | no |
+| 6 | 4121/16384 | no |
+| 7 | 257/1024 | no |
+| 8 | 16193/65536 | yes; stop |
+
+The search stops at k=8, leaving k=9 unsearched. Each completed box has
+3^16 possible tuples before the norm constraint; the DP compresses them
+by exact sufficient state. WITNESS.json retains all five selected vectors,
+scores, state counts and transitions. search.py reproduces this record.
+The certificate's validity does not depend on DP completeness.
+
+These are not error optima or smaller-denominator exclusions. Another
+vector in the same box might have lower error. No larger search was run.
+The initial exploratory root parser failed to canonicalize seven omitted
+representatives and produced 111 rows; those outputs are invalid, not
+search evidence. The corrected 104-row loader also reproduces PR6211's
+exact guide maximum 4820 before any accepted search or verification.
+
+## New result and inherited obligations
+
+Armin Pourshafeie / apoursh's PR6211 already supplies complex Hermitian
+rank-seven examples below 1/4. The present error lies between its
+25/101 and 1205/4912 witnesses; the contribution is Gaussian-dyadic
+coefficient feasibility, not a new best complex error. Nick Petalas's
+PR6183 supplies the contractive question, real attaining projector and
+omission data; apoursh's PR6191 supplies the sharp real comparison.
+Matarener's PR6166 supplies the cited physical P104/root identification,
+which is not independently audited here. Exact input pins and full
+predecessor heads are recorded in SOURCES.json and README.md.
+
+Dyadic entries are not a native implementation. Gate synthesis, a caller,
+source access, correction, dirty restoration and the complete paid/target
+bill remain open. No inverse for this singular projection is claimed.
+No selected assumption is discharged, no full multiplication theorem is
+proved, and no new kappa or literature priority is claimed.

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

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+# A dyadic rank-seven P104 projector below the real quarter-error floor
+
+**An exact Gaussian-dyadic Hermitian projector answers PR6211's first
+open next question. No new κ.**
+
+For the same fixed 104-root Gram matrix K=RR^T/2, construct
+D=I-zz*/65536 with complex rank seven and coefficients in Z[i,1/2].
+Its approximation C=RDR^T/2 has exact maximum complex entry error
+**16193/65536**, below **1/4** by **191/65536**. Both C and K-C are
+positive semidefinite. The complete witness is
+
+```text
+z = (73-53i, 31+118i, 73-51i, 88+16i,
+     73-51i, -15-51i, -87-15i, -88-17i),   z* z = 2^16.
+```
+
+PR6211 already shows complex coefficients can beat the sharp real floor.
+Its examples contain odd denominators; this result closes that specific
+coefficient-ring gap. It does not improve on PR6211's stronger error
+witness or claim optimal error/minimal denominator. Rounding the old
+matrix is insufficient: the retained control has rank eight and fails
+exact projector idempotence.
+
+## Reproduce
+
+From the repository root, with CPython 3.12 and the standard library:
+
+```sh
+python3 -B scripts/research_package.py --package research/p104-dyadic-rank-seven
+```
+
+Expected: `PASS_REGISTERED_RESEARCH_PACKAGE`, with normal and optimized
+copies of `PASS_P104_DYADIC_RANK_SEVEN`, `PASS_FRACTION_REALIFICATION`
+and `PASS_BOUNDED_DYADIC_SEARCH`. Six bounded commands are registered.
+All four external inputs are pinned in SOURCES.json. Frozen inputs,
+shared code, selected records and upstream/ remain unchanged.
+
+The principal check also runs directly and reads files only:
+
+```sh
+python3 -I -B -W error research/p104-dyadic-rank-seven/verify.py
+```
+
+[PROOF.md](PROOF.md) gives the exact argument and scope; WITNESS.json
+records the vector, expected receipt and completed discovery coverage.
+The same checks remain active under optimized Python (`-O`).
+
+## Evidence
+
+- All 10,816 entries of the literal Gram product are checked with exact
+  integer Gaussian arithmetic. The largest squared root response is
+  129544, uniquely at zero-based row 24. Gaussian-rational elimination
+  checks the projector and an eight-root minor independently of the
+  projection rank formula.
+- A second same-author checker imports no primary or predecessor code.
+  Fraction realification checks rank fourteen, real-part rank eight,
+  and all 10,816 residual entries with a different root order.
+- Nine semantic rejecting controls cover wrong normalization, missing
+  conjugation, old odd denominators, a changed vector, false error,
+  naive rounding and malformed root omissions. Four source-byte controls
+  reject changed inputs. Sixteen real attaining projectors, global
+  conjugation and sign changes of omitted representatives are positive
+  scope controls.
+- The bounded DP solves five closest-vector boxes on dyadic spheres,
+  guided by PR6211's stronger witness, and stops at k=8. It does not
+  minimize coefficient error. Failures at k=4..7 concern only the
+  selected vectors, not all vectors or denominators. Domain, nonce,
+  stopping rule and completed coverage are retained in PROOF/WITNESS.
+
+Actual execution and the initial root-parser correction are recorded in
+TASK.json and VALIDATION.json. This is exact finite arithmetic plus a
+written proof, not independent human or formal-kernel review. No selected
+scientific cold replay or physical supplier replay is claimed.
+
+## Sources, overlap and scope
+
+Base: `2485805a6d47f36ea83ddde6327d453ac82d44dc`, priority revision 8.
+The selected record identifies PR6030, κ=781451/250000000. Its binding-only
+orientation passed. The older number in the priority narrative was not
+used as the frontier.
+
+| Predecessor | Exact source head | Retained contribution |
+| --- | --- | --- |
+| [apoursh #6211](https://github.com/CrocSwap/integer-mult-bounds/pull/6211) | `369ad15647ad0df0b0ef6f4fcb060509b67f1823` | Complex witnesses, guide and explicit dyadic-projector handoff |
+| [apoursh #6191](https://github.com/CrocSwap/integer-mult-bounds/pull/6191) | `be3fc854aa380b3d81e134ce9a4282468e4fa999` | Sharp real rank-seven comparison |
+| [Nick Petalas #6183](https://github.com/CrocSwap/integer-mult-bounds/pull/6183) | `eaf77790466f34b86ce72080e3ec690e26dfacab` | Contractive problem, real projector and omission data |
+| [Matarener #6166](https://github.com/CrocSwap/integer-mult-bounds/pull/6166) | `096938513138868f0ed27639061f1a5054900f96` | Cited P104-to-D8 physical identification |
+
+All four are archived as partially reviewed in the base ledger; archival
+inclusion is not scientific acceptance. Physical coordinate identification
+remains a cited premise. Replay inputs are pinned to their exact copies at
+the base commit; no pending branch or predecessor executable is required.
+
+The repository handoff-first process led to an explicitly changed
+coverage-law card for this local algebraic question. Bounded all-state
+PR searches for 6191/6211, complex rank approximations and dyadic
+projectors, the ledger and newest public handoffs found no competing
+dyadic witness. Local overlap was sparse (79/8722 notes), and Discussion
+coverage was incomplete. These checks do not certify novelty.
+No pinned sprint issue was found, and connected tools cannot post to
+Discussions; the advance claim and completion are recorded in TASK.json.
+
+The real part of C has rank eight, and the residual has spectral norm 13.
+Smaller individual coefficient error is not vanishing operator error.
+An exact linear seven-channel reconstruction would lose one required
+direction. Source access, native realization, correction, dirty
+restoration, caller installation and the complete paid/target invoice
+remain open. No completed supplier or paid-work saving is claimed.
+All five framework assumptions and the selected κ are unchanged.
+
+## Open next questions
+
+Can an actual caller use this projection and handle its nonzero residual
+at a favorable complete cost? Start with one source/consumer, trace the
+exact rank-one residual through every required output, and charge
+acquisition, delivery, native realization and restoration. The missing
+caller and complete invoice are the blockers. Another approximation or
+denominator search alone would not resolve them. TASK.json records this
+question structurally; no broader search is reserved.
+
+## Credit and license
+
+New witness, bounded construction, proof and verification by **Jerry Chen
+(@CubeJerry)** with substantial OpenAI Codex assistance. Armin Pourshafeie /
+apoursh, Nick Petalas and Matarener retain the specific predecessor credit
+above. 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
+contributions. Hermitian projection, Gram-rank and dynamic programming
+are classical; no literature-priority claim. Apache-2.0.

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

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+{
+  "schema_version": 1,
+  "base_commit": "2485805a6d47f36ea83ddde6327d453ac82d44dc",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/p104-complex-rank-seven/PROOF.md",
+      "usage": "input",
+      "git_blob": "78c8c0019b8d3d3f8435685102bbf9d541fb611f",
+      "sha256": "2e1d28966cc50faa857afe425c27cd0bd4142cb77e54d1b289200ac3c6dcfbee"
+    },
+    {
+      "path": "research/p104-complex-rank-seven/README.md",
+      "usage": "input",
+      "git_blob": "fa6ac74b970edfdc1916c5844cc0079758f6c599",
+      "sha256": "a7298296c2c1b74a6e70dfb8cbbf3328719977bed53889b1f6676dcf4bca28e1"
+    },
+    {
+      "path": "research/p104-contractive-gram-deflation/MISSING.json",
+      "usage": "input",
+      "git_blob": "d5f76681db5edb99899ce6509913b0db60f7280c",
+      "sha256": "9ddfcf087912e02c46722d5a82eea5f1221a4a33d5037a0d90d392dfcd3c06d4"
+    },
+    {
+      "path": "research/p104-rank-seven-error-floor/PROOF.md",
+      "usage": "input",
+      "git_blob": "08778cc91fac7f668ac9c2f6074e79e9ba1ed0db",
+      "sha256": "2681e0879e0a0395aa9107f1db24a47a711573ce595ce04c706723bd4ac25305"
+    }
+  ]
+}