Count optimal second callees for a two-dimensional Jordan first map

research/jordan-second-callee-count/PROOF.md

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+# Exact optimal-callee counts for a two-dimensional Jordan block
+
+Prepared for NicholasStone with OpenAI Codex assistance. Apache-2.0.
+
+## Result
+
+Let F = F_q be any finite field, let lambda be nonzero, and put
+
+    N = [[0,1],[0,0]],   P = lambda I + N.
+
+Use the unchanged ordinary wrapper with chronological operations
+
+    D -= H; F_P; D += H; F_Q; D = -D; D += H,
+    F_T(H,D) = (T D, T^-1 H).
+
+The second map Q must be invertible. Call it optimal when the rank of the
+wrapper error relative to interchange J(H,D)=(D,H) is minimal over all such Q.
+
+**Theorem.** The exact number of optimal second maps is
+
+    1          if lambda^2 = 1;
+    q^2        if lambda^2 != 1 and lambda^2 = -1;
+    q(q-1)     if lambda^2 != 1 and lambda^2 != -1.
+
+The order of these cases matters in characteristic 2, where 1=-1. In that
+characteristic every lambda != 0,1 belongs to the third case. In the regular
+case lambda^2 != 1, write A=P^2-I. All optimal second maps are uniquely
+
+    Q = P + A S,
+    S = [[a,b],[c,-a]],   a^2+bc = 0,
+    lambda^2 + (lambda^2+1)c != 0.                    (1)
+
+If lambda^2+1 is nonzero, the excluded parameters are exactly the q matrices
+
+    c0 = -lambda^2/(lambda^2+1),
+    S_a = [[a, -a^2/c0], [c0, -a]],   a in F.        (2)
+
+Each regular-case optimal Q agrees with interchange on q^2 of the q^4
+input pairs. This is optimal agreement, not agreement on all inputs.
+
+## Attribution and the inherited reduction
+
+Nick Petalas's [PR5228](https://github.com/CrocSwap/integer-mult-bounds/pull/5228),
+`research/optimal-second-callees/PROOF.md`, classifies all optimal second
+callees using a square-zero endomorphism and an invertibility filter.
+Its README explicitly leaves nonscalar filtered counting open, beginning
+with one nonsemisimple block. Jerry Chen / CubeJerry's
+[PR2950](https://github.com/CrocSwap/integer-mult-bounds/pull/2950) supplies
+the already-known optimum and the attainer Q=P. These are predecessor
+results, not new claims here.
+
+For clarity, the small amount of inherited algebra needed here is recorded.
+Put R=Q-P, x=D-H and y=P^-1 H+P x. Direct execution gives
+
+    H' = Q P^-1 H + Q P x,
+    D' = H' - Q^-1 P x.
+
+For e1=H'-D and e2=D'-H this yields
+
+    e1 = A x + R y,      Q(e2-e1)=R x.
+
+Both the input change to (x,y) and the output change to (e1,Q(e2-e1)) are
+invertible. Therefore the error rank equals the rank of
+
+    M = [[A,R],[R,0]].                                (3)
+
+The minimum is rank A, since A is a submatrix and Q=P attains that lower
+bound. If A is invertible, block elimination gives
+
+    rank M = 2 + rank(R A^-1 R).
+
+Writing R=A S gives R A^-1 R=A S^2. Thus optimality is equivalent to
+S^2=0, subject to Q being invertible. Since A is invertible, different S
+give different Q. This is the invertible-A specialization of PR5228.
+
+## Square-zero matrices over every characteristic
+
+For a two-by-two matrix S, S^2=0 is equivalent to
+
+    S = [[a,b],[c,-a]],    a^2+bc=0.                  (4)
+
+Indeed, im S is contained in ker S, so det S=0. Cayley-Hamilton gives
+S^2-(tr S)S=0; for nonzero S with S^2=0 this forces tr S=0, and S=0 has
+zero trace as well. Conversely the square of the displayed matrix is
+(a^2+bc)I. No division by 2 is used.
+
+For each fixed c there are exactly q solutions (a,b):
+
+* If c=0, a^2=0 forces a=0, while b is arbitrary.
+* If c is nonzero, a is arbitrary and b=-a^2/c is uniquely determined.
+
+Consequently there are q^2 square-zero matrices, and the lower-left entry c
+has q preimages for every field element. This uniform fiber count is the
+counting step used below.
+
+## The determinant filter
+
+Assume lambda^2 != 1, so A is invertible. Since N^2=0,
+
+    P^-1 = lambda^-1 I - lambda^-2 N,
+    B := P^-1 A = P-P^-1
+       = (lambda-lambda^-1)I + (1+lambda^-2)N.
+
+For square-zero S, both tr S and det S vanish, and tr(NS)=c. The
+two-dimensional identity det(I+T)=1+tr T+det T therefore gives
+
+    det Q = det P det(I+B S)
+          = lambda^2 [1 + (1+lambda^-2)c]
+          = lambda^2 + (lambda^2+1)c.                (5)
+
+If lambda^2+1=0, this determinant is the fixed nonzero value lambda^2, so
+all q^2 parameters survive. Otherwise (5) vanishes at precisely the nonzero
+value c0 in (2). The fiber over c0 has exactly q elements, yielding q^2-q
+survivors and the complete rejection family (2).
+
+In fact, when lambda^2+1 != 0 the determinant map from the entire square-zero
+parameter set to F is uniform: every determinant value has exactly q
+preimages. Thus each prescribed nonzero determinant occurs for q optimal Q.
+In the exceptional case every optimal Q has determinant lambda^2.
+
+For an alternative direct calculation, with d=lambda^2-1 one has
+
+    Q = [[lambda+d a+2 lambda c, 1+d b-2 lambda a],
+         [d c,                   lambda-d a]].
+
+Expanding this determinant and substituting a^2+bc=0 also gives (5).
+
+## The boundary lambda^2=1
+
+Here A=2 lambda N. In characteristic 2, A=0 and the minimum error rank is
+zero; (3) has rank zero only when R=0. Thus Q=P is the unique optimum.
+
+In odd characteristic, A has rank one. Equality rank M=rank A requires both
+im R contained in im A and ker A contained in ker R: otherwise adjoining R
+to A horizontally or vertically would already raise the rank. These two
+containments force R=beta N. Deleting the zero rows and columns of (3)
+leaves
+
+    [[2 lambda,beta],[beta,0]].
+
+If beta is nonzero its determinant is -beta^2 and its rank is two, exceeding
+the minimum one. Hence beta=0 and again Q=P is the unique optimum.
+This completes the theorem for every nonzero lambda over every finite field.
+
+## Scope
+
+The contribution is the explicit filtered count, determinant distribution
+and rejection family for one two-dimensional nonscalar Jordan block. The
+result is also invariant under simultaneous change of basis: conjugating P
+and Q preserves wrapper error rank and gives a bijection of optimal maps.
+The generic rational-canonical-data counting question and Jordan blocks of
+size at least three remain open here. The finite verifier corroborates the
+written proof; it is not a proof-kernel formalization. No implementation cost,
+supplier, improved kappa or complete multiplication theorem is claimed.
+
+## Open next questions
+
+* For P=J_3(lambda), count the filtered square-zero parameters, first with
+  lambda^2 != 1. Every nonzero square-zero 3x3 matrix has rank one, so writing
+  S=u v^T with v^T u=0 may make the determinant filter tractable.
+* Determine how this filtered count combines across several Jordan blocks;
+  off-diagonal parts of S couple the blocks and prevent a naive product rule.

research/jordan-second-callee-count/README.md

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+# Two-dimensional Jordan first map: solved filtered callee count
+
+This package resolves the two-dimensional nonsemisimple starting case of
+the nonscalar counting question in Nick Petalas's PR5228. It does not close
+that question for arbitrary rational canonical data.
+
+For P=J_2(lambda) over F_q with lambda nonzero, the number of optimal
+invertible second maps in the fixed ordinary wrapper is:
+
+| Condition | Count |
+| --- | ---: |
+| lambda^2=1 | 1 |
+| lambda^2!=1 and lambda^2=-1 | q^2 |
+| lambda^2!=1 and lambda^2!=-1 | q(q-1) |
+
+For lambda^2!=1 the parameters are Q=P+(P^2-I)S with S^2=0. Write
+S=[[a,b],[c,-a]], a^2+bc=0. The determinant is exactly
+lambda^2+(lambda^2+1)c. Each c has q square-zero preimages, giving both the
+count and a complete description of singular rejections.
+
+Examples: over F4 both eligible lambda give 12 optimal maps; over F5,
+lambda=2,3 both give 25; over F7, lambda=2,3,4,5 each give 42. These examples
+also show why the exceptional branch must be retained.
+
+See [PROOF.md](PROOF.md) for the all-field proof and predecessor credit.
+
+## Reproduce
+
+Standard library only; Python 3.12 or newer:
+
+```sh
+python3 -I -W error -B research/jordan-second-callee-count/verify.py
+python3 -I -O -W error -B research/jordan-second-callee-count/verify.py
+```
+
+The script prints a generated JSON receipt and writes no files. All checks
+use explicit exceptions and remain active under Python -O.
+
+The fields are q=2,3,4,5,7,8,9,11,13,16. Polynomial-basis extension fields
+are used for nonprime q; these are not integers modulo q. The script verifies
+the finite field tables, enumerates every 2x2 matrix to find S^2=0 without
+using the claimed parameterization, then compares both sets and all
+determinant fibers. For each regular lambda it checks the rejection family
+and the accepted count. For every q<=9 and every nonzero lambda, it separately
+enumerates all invertible Q and runs the six literal wrapper operations on
+all four basis inputs. The resulting error-rank-optimal set must equal the
+predicted set, including the lambda^2=1 boundary. This second path does not
+use the block-matrix reduction to calculate the wrapper error rank.
+
+`VALIDATION.json` records the actual interpreter and completed runs.
+`SOURCES.json` records the pinned predecessor source. The predecessor documents are pinned at their canonical repository paths;
+all predecessor attribution is retained in PROOF.md.
+
+## Open next questions
+
+The next narrow target is J_3(lambda); arbitrary block structures remain open.
+See PROOF.md for the rank-one starting representation and the block-coupling
+issue. This algebraic result does not change the selected multiplication bound.

research/jordan-second-callee-count/SOURCES.json

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research/jordan-second-callee-count/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 7,
+  "base_commit": "13be6ac685845c1ff671f5d26e7166165abd998e",
+  "original_card": {
+    "id": "uniform-transfer-audit",
+    "question": "Does one quantified step of the proposed #4100 all-size construction hold at its actual caller?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/unconditional-batched-crt/README.md"
+    ],
+    "methods": [
+      "Prove one missing implication with explicit quantified hypotheses.",
+      "Find an exact counterexample at the claimed interface boundary."
+    ],
+    "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read PROOF.md and AUDIT.md in #4100. Select one distinct obligation: CRT total repair, native/free-algebra lifting, precision/conditioning, or complete fixed-tape cost. This is an author-submitted proof, not an accepted unconditional theorem.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Check live handoffs; name one proposition, its caller, quantified domain and dependency closure. Independently prove the implication or exhibit a precise gap/counterexample. Prioritize CRT.md total repair or COMPLEX.md free-algebra lift only if unclaimed. Stop at the chosen obligation; do not substitute finite samples or root arithmetic for an all-size proof, or launch the complete package by default.",
+    "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.",
+    "allowed_variations": [
+      "One CRT repair, native lift, precision/conditioning or tape-cost proposition, with its exact caller."
+    ],
+    "scope_change_triggers": [
+      "Replacing a quantified obligation with another finite smoke test.",
+      "Treating archive intake, an author PASS receipt or a finite composed fixture as discharge of the universal hypotheses."
+    ],
+    "overlap_terms": [
+      "uniform transfer",
+      "all-size",
+      "precision",
+      "4100",
+      "unconditional-batched-crt"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.",
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.",
+        "status": "author-checked",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "scope_change_reason": "Explicit user-selected follow-up to PR5228: solve the nonscalar filtered-count question for one two-dimensional Jordan block. No task picker reroll.",
+  "actual_scope": "Count every optimal invertible second callee for P=J_2(lambda) over every finite field, including characteristic two and lambda squared equal to one.",
+  "bottleneck_link": "Resolves the first nonsemisimple case of the optimal-callee counting handoff; no paid supplier or bound improvement follows.",
+  "stop_rule": "Stop after proof, finite corroboration and a focused PR. Higher Jordan blocks remain next questions.",
+  "outcome": "completed",
+  "evidence": "PROOF.md, verify.py and VALIDATION.json",
+  "author": "Nicholas Stone / NicholasStone",
+  "ai_assistance": "Substantial OpenAI Codex assistance. No independent human or proof-kernel review.",
+  "open_questions": [
+    {
+      "question": "Count the optimal second callees for J_3(lambda).",
+      "why_it_matters": "Extends the first nonscalar block count.",
+      "first_step": "Represent nonzero square-zero parameters as rank-one u v^T with v^T u=0.",
+      "blocked_by": "The determinant filter is not derived here for dimension three."
+    }
+  ]
+}