Classify polynomial two-gap insertions with exact dimension 64d+50 (no κ)

research/polynomial-two-gap-insertions/README.md

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+# Exact polynomial pairs across two stage-0 gaps
+
+**Claim type: scoped written proof with exact finite corroboration.**
+For the two literal gaps in unpaidmoon's [PR #7103][7103], replacing constant
+insertion matrices by matrices over `K[x]` permits previously excluded
+off-diagonal couplings through coordinate 1. Over every field `K`, all pairs
+of entry degree at most `d >= 0` form an affine space of dimension
+**`64d + 50`**. In particular, the degree-one dimension is **114**. The proof
+gives an explicit bijection, hence exactly `q^(64d+50)` pairs over `F_q`.
+The checker enumerates small entrywise factors of that count; it does not
+enumerate the entire degree-one space.
+
+This answers Open next question 2 of #7103. Its constant-entry classification
+is a cited predecessor, not the deliverable. No new kappa, paid-rank gain,
+native caller, supplier admission or complete multiplication theorem is claimed.
+
+## Literal word and target
+
+Use zero-based matrix indices `0,...,7` and ordinary written matrix-product
+order, exactly as in #7103. Work in `K[x,y,z][e]/(e^2)`, with `e` central.
+For stage `s=0,1,2`, write `x_s=x,y,z`, respectively. Its three diagonal gates,
+in order, are
+
+```text
+I + x_s E00,  I + x_s E11,
+I + x_s (E_(2+2s,2+2s) + E_(3+2s,3+2s)).
+```
+
+Concatenate the stages to obtain `G0,...,G8`, and let `P=G0...G8`.
+Insert `I+eR` just after `G0` and `I+eS` just after `G1`. The question is
+
+```text
+G0 (I+eR) G1 (I+eS) G2 ... G8 = (1+e) P.             (1)
+```
+
+Here `R,S` have entries in `K[x]`. Their pure term always remains `P`.
+Equation (1) requires the coefficient of `e` to equal **`P`**, rather than
+the identity. Matrix equality means the endpoint holds on every initial
+input vector; it does not establish a physical workspace contract.
+
+## Classification and proof
+
+Put `t=1+x`, `B=G1=diag(1,t,1,1,1,1,1,1)` and `H=G2...G8`.
+Expanding the left side of (1), using centrality and `e^2=0`, gives
+
+```text
+P + e G0 (R B + B S) H.
+```
+
+Every diagonal entry of `G0` and `H` is a nonzero polynomial. Since
+`K[x,y,z]` is an integral domain, entrywise cancellation is valid even
+though these matrices are not invertible over the polynomial ring.
+Thus (1) is equivalent to
+
+```text
+R B + B S = B.                                      (2)
+```
+
+Write `b_1=t` and `b_i=1` for `i != 1`. Equation (2) at `(i,j)` is
+`b_j R_ij + b_i S_ij = b_i delta_ij`. Solving each independent entry gives
+the complete list:
+
+| Position | All solutions |
+|---|---|
+| `i=j` | `R_ii=a_i`, `S_ii=1-a_i` |
+| `i!=j`, both different from 1 | `R_ij=a_ij`, `S_ij=-a_ij` |
+| `i=1`, `j!=1` | `R_1j=-t p_j`, `S_1j=p_j` |
+| `j=1`, `i!=1` | `R_i1=p_i`, `S_i1=-t p_i` |
+
+All parameters are arbitrary polynomials. Necessity follows directly from
+the entry equation; substitution proves sufficiency. Each parameter can be
+recovered from the displayed pair, so the parametrization is injective as
+well as surjective. In characteristic two the displayed minus signs become
+plus signs.
+
+Now impose entry degree at most `d`. Each of the eight diagonal parameters
+and the `7*6=42` off-diagonal parameters missing coordinate 1 has degree at
+most `d`. Each of the fourteen crossing parameters has degree at most `d-1`:
+for nonzero `p`, the monic polynomial `t` satisfies `deg(tp)=deg(p)+1`.
+When `d=0`, those fourteen parameters must be zero. The solution space is
+therefore explicitly affine-isomorphic to
+
+```text
+(K[x]_(<=d))^50  x  (K[x]_(<=d-1))^14,
+```
+
+where `K[x]_(<=-1)={0}`. Its dimension is
+`50(d+1)+14d=64d+50`. Over a field with `q` elements the bijection gives
+the stated exact count. This is a parameter count, not a claim that the
+checker visited each full pair.
+
+The simplest new crossing solution, valid even over the integers, is
+
+```text
+R = I - (1+x) E10,       S = E10.
+```
+
+Indeed, `E10 B=E10` and `B E10=t E10`, so `RB+BS=B`.
+It has degree one. Degree zero cannot realize any nonzero crossing entry,
+by the degree restriction just proved. This settles the minimum degree for
+opening such a coupling. Reversing the two occurrences of `t` does not work;
+the written product order matters.
+
+## Evidence and reproduction
+
+The proof above supplies the quantifiers over every field and every `d`.
+The standalone [verifier](verify.py) uses only the Python 3.12 standard library.
+It derives both entry scales from the literal nine gates, then:
+
+- computes coefficient-system ranks for `d=1,2,3,4` over `F2,F3,F5,F7`;
+  the resulting affine dimensions are `114,178,242,306`, respectively;
+- exhausts **56,960 entry pairs** in the four domains `(q,d)=(2,1),(2,2),
+  (3,1),(3,2)`, compares the solved criterion with the literal scales, and
+  multiplies the entrywise counts using their Cartesian independence;
+- multiplies the full eleven-factor operator for **56 integer crossing
+  witnesses**, covering both crossing directions, all seven other coordinates
+  and parameters `1,x,x^2,x^3`, plus the affine origin `(I,0)`;
+- rejects four false witnesses: reversed crossing orientation, a missing
+  `x` term, an omitted second insertion, and `(I,I)`.
+
+It prints `PASS_POLYNOMIAL_TWO_GAP_INSERTIONS` and a generated JSON receipt
+binding its results to SHA256 hashes of all five package files. The receipt
+is emitted to stdout, never maintained by hand.
+
+From the repository root, the exact local interpreter is
+`~/git/integer-mult-bounds/.venv/bin/python3.12` (CPython 3.12.15):
+
+```sh
+~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/polynomial-two-gap-insertions/verify.py
+~/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/polynomial-two-gap-insertions/verify.py
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/polynomial-two-gap-insertions
+~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/polynomial-two-gap-insertions --check-source-history
+```
+
+Both registered checks have 30-second limits. Explicit exceptions keep the
+checks active under `-O`. [TASK.json](TASK.json) records the changed card
+scope and open question. [SOURCES.json](SOURCES.json) pins the historical
+catalogue and selected record, and separately lists the immutable external
+citation. No external file or predecessor verifier is loaded at runtime;
+the word and convention used here are fully specified above.
+
+The verifier has no recursive calls or unbounded searches. Width is fixed
+at eight, degree at four, field samples at seven, the largest enumeration
+at 729 polynomial pairs per entry, and the longest word at eleven factors.
+Python requires dynamic allocation; bounded finite collections replace the
+fixed-allocation rule. Warning-strict normal and optimized checks provide
+runtime diagnostics. No additional static type checker is required by this
+package. These checks corroborate a written proof, not a formal-kernel proof.
+
+## Limitations
+
+The allowed insertion entries change from constants to `K[x]`. Their
+implementation, interaction with the native stage operator, physical
+restoration, precision, hosting and complete signed invoice are unproved.
+The displayed witness has no established cost advantage over `(I,0)`.
+Other gap pairs, non-idempotent stage factors and the larger paired-star
+callers are outside scope. No predecessor scientific replay, Lean, Docker,
+selected-frontier replay or complete multiplication implementation was run.
+The selected result is unchanged.
+
+## Open next questions
+
+Can a polynomial crossing pair reduce a complete legal invoice after every
+`x`-dependent entry is implemented and restoration is charged? Start with
+`R=I-(1+x)E10, S=E10` and compare its physical realization with `(I,0)`.
+The blocker is a legal, fully priced implementation of those entries. This
+is necessary before the new algebraic freedom can advance a supplier.
+The structured question is also in [TASK.json](TASK.json).
+
+## Credit
+
+This result was produced by Nick Petalas. New work is Apache-2.0.
+
+Credit **unpaidmoon** for #7103, exact head
+`f3af1384aeda8c75be58f09cb07da3941ba7aba3`, which poses the polynomial-entry
+question and fixes the gap convention; #7027, head
+`1ba2ac5adfe58fbe9dba34d50d4e5d30633ce3d0`, supplies the diagonal word;
+#7039, head `e4ddcd2fde8c46b7e528e8806ab72d9bf7690978`, distinguishes the
+one-gap coefficient targets. Credit **Ali Romman** for the paired-star
+contract in #5747 and its retained #5673, #5631 and #5617 lineage. Their
+source attributions remain in their original packages. The new step here
+is the polynomial classification, its exact degree filtration and its
+explicit crossing witnesses.
+
+[7103]: https://github.com/CrocSwap/integer-mult-bounds/pull/7103

research/polynomial-two-gap-insertions/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "75d77b328ea02ded535e7f23aee5b1e55d0bb1a8",
+      "sha256": "154c3d30455d778fd3343b6b4851f0fa4665f0f7625b0adb3675e2bafbbf6bf1"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    }
+  ],
+  "external_citations": [
+    {
+      "pr": 7103,
+      "commit": "f3af1384aeda8c75be58f09cb07da3941ba7aba3",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/blob/f3af1384aeda8c75be58f09cb07da3941ba7aba3/research/two-gap-constant-insertion/PROOF.md",
+      "role": "Cited mathematical predecessor, not loaded by the verifier. The literal word and gap order are stated completely in README.md."
+    }
+  ]
+}

research/polynomial-two-gap-insertions/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "joint-query-controller",
+  "priority_revision": 8,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "original_card": {
+    "id": "joint-query-controller",
+    "question": "Can a changed query demand or caller remove real work while satisfying native chronology and restoration?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "A source-bound arbitrary-input identity and legal producer/consumer boundary with the complete paid and productive target invoice, including restoration, ownership, native alignment and extra charges. Price under the actual current model; historical ceiling.py assumptions must be established before use. A favorable optimistic relaxation keeps a route open but is not a completed supplier. A scoped local result remains useful; expansion requires the family-level budget gate.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Choose an exact caller and a new demand/interface hypothesis. Read the scoped #5617/#5631/#5649/#5747 and #5720/#5745 objections before rebuilding an excluded schedule. Compare the complete matched source/consumer protocols including all retained helper outputs. For an active-mode injection, first check #5745: its canonical delivery claim repays the eight removed centre jobs. Audit that exact claim or change its channel/feedback hypothesis; do not repeat the same free-injection search. Stop if the full favorable budget fails or the required restored output is unobservable.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "query demand",
+      "active mode",
+      "5720",
+      "5745",
+      "5747",
+      "caller"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4166: author-checked installed producer/service/return/bridge connection; native-component comparison loses.",
+        "evidence": "research/qq16-caller-install/README.md",
+        "remaining": "Complete matched physical protocol bill, or changed source-compatible demand/circuit; no affordable supplier yet.",
+        "status": "author-checked",
+        "source_commit": "0b082d22a0e9a442d264ab0633e35ef26c790630",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Classify every R,S in M8(K[x]) of entry degree at most d in the two adjacent stage-0 gaps of PR7103, over every field K, with central e^2=0 and coefficient of e equal to the literal pure product.",
+  "scope_change_reason": "The card asks for a changed query-demand construction. This package takes the explicit polynomial-entry question in PR7103 and proves a local algebraic interface classification; it supplies no complete caller or invoice gain.",
+  "bottleneck_link": "Polynomial insertion freedom changes which off-diagonal couplings are algebraically possible across a native-call boundary. Physical realization, native legality and a favorable complete signed bill remain open; this is a scoped interface result.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop at the first original proved classification and its exact finite corroboration; do not extend to new gaps, words or paid-rank optimization.",
+  "outcome": "completed",
+  "evidence": "Scoped written proof in README.md: explicit bijection with K[x] degree-bounded parameter spaces. Exact independent coefficient elimination, exhaustive entry-pair checks and full integer polynomial operator witnesses in verify.py corroborate the proof. Not a formal-kernel proof or a complete multiplication theorem.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Can a polynomial crossing pair reduce a complete legal paid invoice after its x-dependent entries are implemented and all restoration costs are charged?",
+      "why_it_matters": "The new coefficient freedom alone does not save recursive work or improve kappa.",
+      "first_step": "Implement the explicit R=I-(1+x)E10, S=E10 pair in the actual native interface and compare its full bill with R=I, S=0.",
+      "blocked_by": "No paid implementation or native legality for the polynomial insertions is provided."
+    }
+  ],
+  "predecessors": [
+    {
+      "pr": 7103,
+      "author": "unpaidmoon",
+      "head": "f3af1384aeda8c75be58f09cb07da3941ba7aba3",
+      "role": "Question source, literal two-gap convention and constant-entry predecessor; cited, not audited."
+    },
+    {
+      "pr": 7027,
+      "author": "unpaidmoon",
+      "head": "1ba2ac5adfe58fbe9dba34d50d4e5d30633ce3d0",
+      "role": "Literal diagonal nine-gate word."
+    },
+    {
+      "pr": 7039,
+      "author": "unpaidmoon",
+      "head": "e4ddcd2fde8c46b7e528e8806ab72d9bf7690978",
+      "role": "One-gap predecessor and distinction between coefficient targets I and P."
+    },
+    {
+      "pr": 5747,
+      "author": "aliromman",
+      "role": "Paired-star contract and retained PR5673, PR5631 and PR5617 lineage."
+    }
+  ]
+}

research/polynomial-two-gap-insertions/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 30,
+      "checks": "Exact finite polynomial coefficient systems, exhaustive entry-pair checks, complete integer operator witnesses and four false witnesses."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 30,
+      "checks": "Repeat the same real verifier with isolation and assertions disabled; proof-relevant checks use explicit exceptions."
+    }
+  ]
+}