Mister-Meeseeks · Nov 10, 2026 · Nov 6, 2026
diff --git a/research/p104-four-field-sharp-rank/CHECK-RESULT.json b/research/p104-four-field-sharp-rank/CHECK-RESULT.json
@@ -0,0 +1,108 @@
+{
+  "status": "PASS_SHARP_N4_FOUR_STAGE",
+  "fixtures": [
+    {
+      "label": "q1",
+      "q": 1,
+      "ranks": [
+        2,
+        4,
+        3,
+        2
+      ],
+      "total": 11,
+      "sectors": {
+        "AB": 1,
+        "BC": 1,
+        "CD": 0,
+        "ABC": 0,
+        "BCD": 1,
+        "ABCD": 1
+      },
+      "formal_coefficients": 32
+    },
+    {
+      "label": "q2",
+      "q": 2,
+      "ranks": [
+        4,
+        6,
+        6,
+        5
+      ],
+      "total": 21,
+      "sectors": {
+        "AB": 2,
+        "BC": 1,
+        "CD": 2,
+        "ABC": 0,
+        "BCD": 1,
+        "ABCD": 2
+      },
+      "formal_coefficients": 32
+    },
+    {
+      "label": "q2-reflected",
+      "q": 2,
+      "ranks": [
+        5,
+        6,
+        6,
+        4
+      ],
+      "total": 21,
+      "sectors": {
+        "AB": 2,
+        "BC": 1,
+        "CD": 2,
+        "ABC": 1,
+        "BCD": 0,
+        "ABCD": 2
+      },
+      "formal_coefficients": 32
+    },
+    {
+      "label": "q3",
+      "q": 3,
+      "ranks": [
+        6,
+        10,
+        9,
+        7
+      ],
+      "total": 32,
+      "sectors": {
+        "AB": 3,
+        "BC": 2,
+        "CD": 2,
+        "ABC": 0,
+        "BCD": 2,
+        "ABCD": 3
+      },
+      "formal_coefficients": 32
+    },
+    {
+      "label": "q4",
+      "q": 4,
+      "ranks": [
+        9,
+        12,
+        12,
+        9
+      ],
+      "total": 42,
+      "sectors": {
+        "AB": 4,
+        "BC": 2,
+        "CD": 4,
+        "ABC": 1,
+        "BCD": 1,
+        "ABCD": 4
+      },
+      "formal_coefficients": 32
+    }
+  ],
+  "hypothesis_and_omission_controls": 14,
+  "fixture_sha256": "b22d9085b8b0112b94b80ab807a11564e9a515dc8559c60b1f4a5a08ff04bb43",
+  "scope": "Finite exact fixtures and ordered reductions; universal lower bound is the written proof."
+}
diff --git a/research/p104-four-field-sharp-rank/DERIVATION-DRAFT.md b/research/p104-four-field-sharp-rank/DERIVATION-DRAFT.md
@@ -0,0 +1,13 @@
+# Independent analytic derivation draft
+
+2026-11-06. Worker B. The derivation below was completed independently before receiving worker D's rank21 result; the write tool was aborted by delivery of that result. No worker D fixture or derivation has been read at the time of this sealing.
+
+Take the selected query columns (0,1,3,4) as basis for n=4. Their Gram is K=(sign(i-j)) on four q-blocks, invertible; K^-1 has scalar rows (0,1,-1,1),(-1,0,1,-1),(1,-1,0,1),(-1,1,-1,0).
+
+A<=B and D<=C may be assumed by replacing endpoint stages A,D by AB,CD. B contains queries 0,1 and C contains queries 3,4 from actual prefix compatibility. Thus B union C=I. The joint B-only and C-only sectors are supported, in M=K P coordinates, on pairs01 and23 respectively. Any B-only sector not carrying A is supported on singleton query1, and vanishes because the corresponding K^-1 diagonal block is zero. Likewise C-only must carry D. Thus the possible nonzero sectors are AB,BC,CD,ABC,BCD,ABCD.
+
+For a two-query sector matrix M on an adjacent pair, M J M=M where J=[[0,I],[-I,0]]. Set its ordinary projection P=M J. (Order corrected during verification.) For low-cost sectors AB,BC,CD denote P_X,P_Y,P_Z. The adjacent cross maps are F=[[0,-I],[I,-I]], satisfying F^3=I. Set P=P_X, Q=F P_Y F^-1, R=F^2 P_Z F^-2. Their orthogonality constraints become PQ=QP=QR=RQ=0 and P N R=0, R(I-N)P=0, where N=[[I,0],[I,0]] is rank-q idempotent. Conversely those identities reconstruct three orthogonal sector projections in the full4q-dimensional query space.
+
+The two-sector bound: for M_X,M_Z, put v=(I,-I)^T. Their cross contractions imply rank(M_X v)+rank(v^T M_Z)<=q and rank(M_Z v)+rank(v^T M_X)<=q. Since M=M J M and J is the sum of two rank-q cross terms using v and a complement, rank(M)<=rank(Mv)+rank(v^TM). Hence rankX+rankZ<=2q, and rankB+rankC>=6q.
+
+The analogous three-sector rank bound <=2q is false as an abstract reduced projection system. At q=2, let Q=diag(0,0,0,1), P=diag(1,1,0,0), and R have top-left3x3 block [[1,1,-1],[0,1,0],[0,1,0]] and zero fourth row/column. Let N have rows [[0,0,0,0],[0,1,-1,-1],[-1,0,0,-1],[1,0,0,1]]. Then N is rank2 idempotent, P,Q,R have ranks2,1,2, and all reduced identities hold. Conjugating N to the displayed standard idempotent gives candidate low sectors of total rank5. The remaining common sector has rank3. To lift to total stage rank21, need the CD sector's last diagonal Gram block invertible, allowing a rank2 endpoint-D projector in the rank3 complement. This last condition was not checked before receipt of D's successful witness.
diff --git a/research/p104-four-field-sharp-rank/FROZEN-SHA256.txt b/research/p104-four-field-sharp-rank/FROZEN-SHA256.txt
@@ -0,0 +1,17 @@
+57e6c2e453f2586d92e19cb76be3ff6c657706a1ecfd00464893aa41463ff6c2  CHECK-RESULT.json
+ac55722bd6464d43984e1735faf438cfce297013756f0d324e63cc7397ad255d  DERIVATION-DRAFT.md
+2b307e8fce30e797ac1f3aa62c48fc55d747ca5c1293cfdcba7031ac51046e9a  ORIGINS.json
+b8cd4e7eebe7f02e94280b7d17876e0c890c16555a82e947ba4508a2431c94dd  RANK21-NOTE.md
+d845f036a97adc3a080f7e52ccda6a427d0318a2ce0a5e65fc1bbe54170ba69f  README.md
+a24e53cd62ad0d4d8ea15b929fd256063dafe0dd2a39c95ff3df09c49db590ca  REPLAY-RECEIPT.json
+2ba5d5a7aacdc45a80c1b961402ff750fa38d5dba025f69b3d67de030012b7c4  REVIEW.md
+b22d9085b8b0112b94b80ab807a11564e9a515dc8559c60b1f4a5a08ff04bb43  ROOT-RANK21.json
+41aa74357a06eab4bcd0985dbe189b43d07cfcba4f3a091a12d3a5589d15f4e3  SHARP-BOUND.md
+952c25f3f7e211887f2bfb98d426dc8f3510d291e84ea6e9668511a06a99bf3f  SOURCE6773_FOUR-STAGE-NOTE.md
+244a92888275732276f5fd9493979fea21a11e4be99fbaf792f6b80940fbc18e  SOURCE6773_ORIGINS.json
+d9162d19d3e79dc90cad3bfae9aba64095a46400bd009f8fb37dee4f11db1c6c  SOURCES.json
+03e5d6f5b062187ea6b9b78bac7222a62c95e483a7f122771037105154c8ab85  TASK.json
+ca4649e14dbd72a396e4057de5c0b5062a0932ffa2ada1d52993b1ad2b4e76bb  VERIFY.json
+dbc768c3da18cab10825a9177b5ca6b0cdd5902c3ccb6b78ff207e69b3a8f52f  check.py
+01de985a8137d4ba8a006aecc3eabc2d3cbd251a48dc7680efb1bd3c5d424133  check_rank21.py
+a5ded1111afe99ee721515b1a27660ad4285aa9520b6e18ef593add7f2c1539b  check_reduction.py
diff --git a/research/p104-four-field-sharp-rank/ORIGINS.json b/research/p104-four-field-sharp-rank/ORIGINS.json
@@ -0,0 +1,74 @@
+{
+  "schema_version": 1,
+  "canonical_base": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "predecessor_pr": 6773,
+  "files": [
+    {
+      "local_path": "SOURCE6773_FOUR-STAGE-NOTE.md",
+      "repository": "CubeJerry/integer-mult-bounds",
+      "ref": "10ecaad298e6a6af7416fbd2baca15674fcadf5b",
+      "source_path": "research/p104-four-stage-rigidity/FOUR-STAGE-NOTE.md",
+      "git_blob": "b9c8f20ef73398ef1834e237e3198ceb0173e13f",
+      "sha256": "952c25f3f7e211887f2bfb98d426dc8f3510d291e84ea6e9668511a06a99bf3f",
+      "bytes": 12562,
+      "role": "retained predecessor mathematical context and attribution"
+    },
+    {
+      "local_path": "SOURCE6773_ORIGINS.json",
+      "repository": "CubeJerry/integer-mult-bounds",
+      "ref": "10ecaad298e6a6af7416fbd2baca15674fcadf5b",
+      "source_path": "research/p104-four-stage-rigidity/ORIGINS.json",
+      "git_blob": "125ccdc83331009cdfc3418d55d6640ba5dd7a89",
+      "sha256": "244a92888275732276f5fd9493979fea21a11e4be99fbaf792f6b80940fbc18e",
+      "bytes": 3345,
+      "role": "retained predecessor mathematical context and attribution"
+    },
+    {
+      "local_path": "ROOT-RANK21.json",
+      "source": "worker D independently constructed fixture in this session",
+      "source_path": "worker-d/ROOT-RANK21.json",
+      "sha256": "b22d9085b8b0112b94b80ab807a11564e9a515dc8559c60b1f4a5a08ff04bb43",
+      "bytes": 12188,
+      "role": "sharpness construction, not predecessor code"
+    },
+    {
+      "local_path": "check_rank21.py",
+      "source": "Independently prepared in this session by workstream D",
+      "source_path": "worker-d/check_rank21.py",
+      "sha256": "01de985a8137d4ba8a006aecc3eabc2d3cbd251a48dc7680efb1bd3c5d424133",
+      "bytes": 10434,
+      "role": "independent construction and integral formal checker, workstream D"
+    },
+    {
+      "local_path": "check_reduction.py",
+      "source": "Independently prepared in this session by workstream D",
+      "source_path": "worker-d/check_reduction.py",
+      "sha256": "a5ded1111afe99ee721515b1a27660ad4285aa9520b6e18ef593add7f2c1539b",
+      "bytes": 3393,
+      "role": "independent ordered-block reduction audit, workstream D"
+    },
+    {
+      "local_path": "RANK21-NOTE.md",
+      "source": "Independently prepared in this session by workstream D",
+      "source_path": "worker-d/RANK21-NOTE.md",
+      "sha256": "b8cd4e7eebe7f02e94280b7d17876e0c890c16555a82e947ba4508a2431c94dd",
+      "bytes": 5755,
+      "role": "preserved chronological construction note, workstream D"
+    },
+    {
+      "local_path": "REVIEW.md",
+      "source": "Independently prepared in this session by workstream D",
+      "source_path": "worker-d/INDEPENDENT-REVIEW.md",
+      "sha256": "2ba5d5a7aacdc45a80c1b961402ff750fa38d5dba025f69b3d67de030012b7c4",
+      "bytes": 5928,
+      "role": "independent source-aware proof review, workstream D"
+    }
+  ],
+  "contributions": {
+    "worker_b": "analytic sharp bound and independent exact verifier",
+    "worker_d": "independent q2 rank21 construction",
+    "lead": "scalar rank11 construction and ordered-reduction review"
+  },
+  "scope": "Coefficient-space theorem and exact word; all five selected framework assumptions retained.",
+  "public_author": "Jerry Chen (@CubeJerry), with substantial OpenAI Codex assistance"
+}
diff --git a/research/p104-four-field-sharp-rank/RANK21-NOTE.md b/research/p104-four-field-sharp-rank/RANK21-NOTE.md
@@ -0,0 +1,75 @@
+# Exact eight-input rank-21 four-stage root
+
+Independent worker D construction, seed78, sealed 2026-11-06 about10:00UTC before reading worker B. Prepared with substantial OpenAI Codex assistance. Apache-2.0; human/proof-kernel review unclaimed.
+
+## Result
+
+There is a shortest five-query root on eight arbitrary independent inputs with rank-two square-zero queries and four pairwise commuting full-stage projectors at sites0,1,3,4, of ranks **(4,6,6,5), total21**. All stages support their query and commute with the actual chronological prefix. Over every field of characteristic different from two the full formal identity is
+
+    H^-1 L(S,A,B,C,D) H
+      = S diag(ABCD,ABCD,AB,AB,BCD,BC,CD,CD).
+
+S is central with S²=1; stage variables commute. Together with #6773's exclusion of total20, this establishes the exact q2,n4 minimum21. A universal11q lower bound under that coefficient-space contract is false. No input is initialized, identified or discarded.
+
+## Explicit construction
+
+Use four two-dimensional coordinate blocks W,X,Y,Z. Put I=I₂, E=diag(1,0), F=I−E. Query factors are:
+
+|i|U_i, vertical blocks|V_i, horizontal blocks|
+|---|---|---|
+|0|(I,I,0,0)^T|(-I,I,0,0)|
+|1|(I+E,E,E,-E)^T|(-E,I+E,-F,-F)|
+|3|(I,0,0,I)^T|(I,0,-2I,-I)|
+|4|(I,0,I,0)^T|(I,0,-I,0)|
+
+Set U₂=−U₀+U₁+U₃−U₄, V₂=−V₀+V₁+V₃−V₄, and R_i=U_iV_i. Exact integer multiplication gives V_iU_j=sign(i−j)I₂, R_i²=0, and (I−2R₄)…(I−2R₀)=−I₈.
+
+On the last four coordinates define
+
+    T = [[0,1,-1,1],       D0 = [[1,0,0,0],
+         [0,0, 0,0],             [0,1,0,0],
+         [0,0, 1,0],             [0,0,0,0],
+         [0,1, 0,1]],            [0,0,-1,1]].
+
+The stages are P_A=diag(I,I,0,0), P_B=diag(I,I,T), P_C=diag(I,0,I,I), P_D=diag(I,0,D0). T,D0 are commuting idempotents of ranks2,3. P_B supports both R₀,R₁; P_C supports both R₃,R₄. Thus the first two actual prefixes commute directly; the full endpoint gives B₃=−I−2R₄ and B₄=−I for the other two.
+
+Take H=diag(I,I,H0), where
+
+    H0 = [[1,0,1,-1],
+          [0,0,0, 1],
+          [0,1,0, 0],
+          [1,1,0,-1]].
+
+Its determinant is−1. The last four columns have masks BCD,BC,CD,CD; W has maskABCD, X has maskAB. The full inventory follows.
+
+## Independent derivation
+
+The #6773 endpoint-minimum branch permits ranks(A,C)=(4,6),(B,D)=(6,5). The A,C joint dimensions are AC=2,A-only=2,C-only=4. Reducing endpoints to AB,DC preserves support, so A≤B,D≤C can be imposed. On C-only normalize endpoint factors to J=(I,0)^T,K=(-I,0). The other odd site has residual U₃=(0,I)^T,V₃=(-2I,-I). The first odd residual factors can be P=(E,-E)^T,Q=(-F,-F), so QP=0.
+
+Let F0=[[1,0],[1,0]], G0=I−F0=[[0,0],[-1,1]], D0=diag(I,G0), p=(1,0,0,1)^T,q=(0,1,-1,1). Then qp=1,D0p=p,qD0=q, and T=diag(0,F0)+pq. These are orthogonal rank-one idempotents in kerD0 and imD0. Therefore T commutes with D0, while TP=P,QT=Q. The remaining query blocks are forced by the sign Gram. This is a coupled construction, not a decomposition into scalar roots.
+
+## Certificate
+
+The standalone check_rank21.py uses Fraction arithmetic without importing predecessor code. It verifies all25 sign Gram blocks, all ranks, square-zero identities, projector identities, actual prefixes, complete−I endpoint, joint masks, and all16 formal monomial coefficients at S=±1. Five query-omission and four stage-omission controls detect malformed words. Character interpolation requires char≠2; integral formal verification is being added separately.
+
+First sealed fixture SHA256: b22d9085b8b0112b94b80ab807a11564e9a515dc8559c60b1f4a5a08ff04bb43. Later verification metadata may change, not the mathematical fixture.
+
+## Pins, scope and credit
+
+Base:7c49fe32c55de02c2c873655c804b16e4c3fef0a. Predecessor @CubeJerry#6773, research/p104-four-stage-rigidity, head10ecaad298e6a6af7416fbd2baca15674fcadf5b. Its note/README were read; no executable/fixture was imported. Ali Romman and original query/controller authors retain normal-form/rank-method lineage credit; thu-nmrc#6717 and ToikY#6721/#6734 retain leakage/rank context. The explicit rank21 construction is new here.
+
+No native phase, physical ports, helper ownership, complete paid invoice, precision/compiler/analytic/all-size admission, or kappa improvement is established.
+
+## Open next questions
+
+1. Price legal unit-pair retirement opportunities with a complete native invoice; rank alone does not prove profit.
+2. Determine sharp all-q n4 bounds. Direct sums attain21q/2 for evenq; a matching lower bound is not asserted.
+3. Audit physical basis maps and all native obligations while preserving arbitrary independent inputs.
+
+## Integral formal strengthening and verifier hardening
+
+The verifier now also multiplies the literal word directly in Z[S,A,B,C,D]/(S²−1), checking all32 reduced formal coefficient matrices. This proves the formal identity over every commutative coefficient ring after specialization, including characteristic two. The characteristic-not-two lower-bound theorem and its negative-character shortest-root contract remain separately scoped. No native bit phase or physical supplier follows merely from this integral identity.
+
+H has determinant−1 and H inverse is integral. Maximum absolute entries are U,V:2; R:4; P,H,H inverse:1. These small coefficients may simplify ordinary basis operations; no physical instruction budget has been certified.
+
+Default verification is read-only and compares the unchanged committed fixture bytes with independent reconstruction. Explicit `--emit PATH` is required for generation. Five fixture-corruption controls modify U,R,P,H or an output mask and are rejected. Both normal and optimized isolated Python pass. The original fixture SHA256 remains unchanged.