Research: degree-controlled cyclic inverse bound by root reflection (no new kappa)

research/rational-circle-bound/.gitattributes

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+# Source and receipt hashes refer to exact committed bytes on every platform.
+* -text

research/rational-circle-bound/CLAIM.json

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+  "status": "prepared_for_pull_request",
+  "started_utc": "2026-10-13T05:18:28Z",
+  "claim_utc": "2026-10-13T05:27:00Z",
+  "expected_finish_utc": "2026-10-13T06:15:00Z",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "assignment": "Continuation of OS-seed-71 mathematical audit, at the user's request to try a distinct approach using the previous findings. No new random track draw.",
+  "coordination": "Only open issue #650 found; it requests Discussions and is not a coordination issue. Local claim; no reservation issue.",
+  "candidates": [
+    {
+      "idea": "Sharpen scan rounding by shared closure factors",
+      "decision": "Rejected: new PR764 already supplies a sharp inverse-square bound."
+    },
+    {
+      "idea": "Use the q-difference identity P(uz)=(1-z)P(z)+z^r/r for composite nonvanishing",
+      "decision": "Deferred: no complete quantitative consequence established in the orientation budget; PR666 already covers modular composite lifting."
+    },
+    {
+      "idea": "Reflect interior roots and use rational arc length plus Hardy coefficient control",
+      "decision": "Selected: replace a derivative-dependent inverse-gap power by a polynomial degree factor."
+    }
+  ],
+  "hypothesis": "For a degree-n polynomial P with |P|>=delta on the circle, the Laurent coefficient norm of 1/P is at most (1+4n)^2/delta; hence the same bound for P(U)^-1 when all integer powers of U have norm at most one.",
+  "bounded_experiment": "Written proof of all analytic steps; 14 exact budget comparisons; fixed rational polynomial/reflection and cyclic inverse families; finite Hilbert-matrix checks; near-circle one-pole family; explicit invalid-shortcut controls.",
+  "novelty_scope": "New application relative to the inspected repository work, not a priority claim for reflection, Blaschke products, Hardy's inequality or the Dolzhenko-Spijker inequality.",
+  "related_prs": [
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+  "verification_reserve": "Reserve the final quarter for proof audit, clean and optimized replays, source/overlap refresh, and patch packaging.",
+  "finished_utc": "2026-10-13T05:38:45.011622+00:00",
+  "final_main_unchanged": true,
+  "latest_pr_seen_final": 783,
+  "final_overlap_queries": [
+    "Blaschke",
+    "Hardy",
+    "Spijker",
+    "Fourier",
+    "cyclic inverse",
+    "circle bound"
+  ],
+  "final_related_head_unchanged": {
+    "764": "983d66f0f2e48ee500ec7eee2da9244c001c5064"
+  },
+  "result": "Degree-controlled inverse bound (1+4n)^2/delta; necessary inverse-gap exponent one; no kappa claim.",
+  "checks": {
+    "radix_budgets": 14,
+    "reflection_families": 126,
+    "boundary_identities": 756,
+    "cyclic_matrices": 546,
+    "cyclic_inverse_entries": 9516,
+    "hilbert_comparisons": 6565,
+    "near_circle_instances": 128,
+    "mathematical_controls": 21,
+    "package_tests": 8
+  },
+  "checkout": "Isolated Git worktree at local partial-snapshot commit 042d24c, with upstream identity recorded separately. Network clone unavailable; exact pinned inputs were obtained in the preceding audit.",
+  "initial_submission": "Focused patch fallback. Fresh API lookup of alephnan/integer-mult-bounds returned NOT_FOUND; no available fork-creation capability, and the session browser was unavailable. No PR created.",
+  "publication_preflight_utc": "2026-10-13T05:51:45.482843+00:00",
+  "submission": "User created the fork and authorized submission. Publication targets CrocSwap/integer-mult-bounds:main from alephnan:research/rational-circle-bound. GitHub API commit uses the complete upstream base tree and commit as parent.",
+  "latest_pr_seen_before_publication": 799,
+  "additional_overlap": {
+    "pr": 796,
+    "head": "22902155c7875139e43e4fdf15723142d6709bdd",
+    "distinction": "Nonarchimedean root valuations; explicitly not complex root moduli or a conditioning bound."
+  }
+}

research/rational-circle-bound/EXPECTED.json

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+    "epsilon": "2^-k, 1<=k<=128",
+    "one_pole_W_norm": "1",
+    "pointwise_derivative_at_1": "2^k-1",
+    "necessary_gap_power": "P=z-(1+epsilon), gap=epsilon, inverse at U=I has norm 2^k"
+  },
+  "controls": [
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+    "inadmissible inherited radix 1",
+    "inadmissible inherited radix 2",
+    "inadmissible inherited radix 6",
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+    "inadmissible inherited radix True",
+    "inadmissible inherited radix 3",
+    "unit-circle root C(re=Fraction(1, 1), im=Fraction(0, 1))",
+    "unit-circle root C(re=Fraction(0, 1), im=Fraction(1, 1))",
+    "unit-circle root C(re=Fraction(3, 5), im=Fraction(4, 5))",
+    "zero polynomial",
+    "nonunit phase",
+    "zero cycle length",
+    "reflection without conjugation",
+    "omit Blaschke factor",
+    "Taylor expansion across an interior pole",
+    "replace Hilbert constant four by one",
+    "equate scalar maximum with induced infinity norm"
+  ],
+  "new_kappa_claim": false,
+  "scope": "Exact finite checks; analytic all-length proof is PROOF.md, not formalized."
+}