From 770dbd8e1b0a22437f7aa5b7500fa21997c7d504 Mon Sep 17 00:00:00 2001 From: James Reed Date: Wed, 21 Oct 2026 18:38:31 -0700 Subject: [PATCH] Record the Neumann ceiling below 10 p^2 Co-authored-by: Cursor --- research/gaussian-neumann-ten/README.md | 104 +++++++++++++++++++++ research/gaussian-neumann-ten/SOURCES.json | 24 +++++ research/gaussian-neumann-ten/TASK.json | 74 +++++++++++++++ research/gaussian-neumann-ten/VERIFY.json | 21 +++++ research/gaussian-neumann-ten/verify.py | 72 ++++++++++++++ 5 files changed, 295 insertions(+) create mode 100644 research/gaussian-neumann-ten/README.md create mode 100644 research/gaussian-neumann-ten/SOURCES.json create mode 100644 research/gaussian-neumann-ten/TASK.json create mode 100644 research/gaussian-neumann-ten/VERIFY.json create mode 100644 research/gaussian-neumann-ten/verify.py diff --git a/research/gaussian-neumann-ten/README.md b/research/gaussian-neumann-ten/README.md new file mode 100644 index 0000000000..9843ef54da --- /dev/null +++ b/research/gaussian-neumann-ten/README.md @@ -0,0 +1,104 @@ +# The Neumann ceiling U is below 10 p^2 + +**Claim type:** one rational comparison for the within-cell ceiling. +**No new κ.** The ideal-gap reading stays a hypothesis. The selected record +remains `23/10000`. + +## Question + +#4452 defines + +```text +U = 8 p^2 (D - 4) / (D - 32 p^2), D = 2^{32760 p}, +``` + +and proves `8 p^2 < U < p^6` together with `U < 16 p^2 - 1`. The tensor +chain uses the ceiling `10 p^2`. Is `U` at most `10 p^2`? + +## Lemma + +For every integer `p ≥ 2`, `D > 32 p^2` and + +```text +U < 10 p^2. +``` + +The coefficient 8 does not work: `U > 8 p^2`. This comparison is a property +of the formula. It becomes an inverse bound only when the #4452 gauge holds, +in which case the Neumann 1-norm and ∞-norm are already strictly below `U`, +hence strictly below `10 p^2`. + +## Proof + +The denominator `D - 32 p^2` is positive once `D > 32 p^2`, which follows +from the stronger estimate below. Cross-multiplying by that positive +denominator, `U < 10 p^2` is equivalent to + +```text +8(D - 4) < 10(D - 32 p^2). +``` + +The difference is + +```text +10(D - 32 p^2) - 8(D - 4) = 2D - 320 p^2 + 32 = 2(D - 160 p^2 + 16). +``` + +It is enough to show `D > 160 p^2 - 16`. Since `160 < 256 = 2^8` and +`32 < 160`, the same estimate also gives `D > 32 p^2`. + +For every integer `p ≥ 2`, `2^{2p} ≥ p^2 + 1`, so `p^2 < 2^{2p}`. The base +is `2^4 = 16 ≥ 4 + 1`. If the inequality holds at `p`, then + +```text +2^{2p+2} = 4 · 2^{2p} ≥ 4(p^2 + 1), +``` + +and `4(p^2 + 1) - ((p + 1)^2 + 1) = 3p^2 - 2p + 2`. At `p = 2` this surplus +is 10, and the step to the next integer is `6p + 1 > 0`. Thus +`160 p^2 < 2^8 p^2 < 2^{2p+8}`. + +The remaining exponent gap is `32760 p - (2p + 8) = 32758 p - 8`. At +`p = 2` it equals 65508, and it increases with `p`. Therefore +`2^{2p+8} < 2^{32760 p} = D`, and `D > 160 p^2 > 160 p^2 - 16`. + +At `p = 2` the same polynomial is `160 · 4 - 16 = 624 < 2^{10}`, while +`32760 · 2 > 10`. Replacing 10 by 8 reverses the target inequality, because +`U > 8 p^2` exactly when `32 p^2 > 4`. + +## Not covered + +- The smallest integer coefficient `c` with `U < c p^2`. +- A proof of the ideal gaps that make `U` an inverse bound. +- Editing the #4100 notes. They still say `g p^{10 g} exp(-p^2)`. +- Any new κ. + +## Reproduce + +```sh +python3.12 -B scripts/research_package.py --package research/gaussian-neumann-ten --check-only +python3.12 -B scripts/research_package.py --package research/gaussian-neumann-ten +``` + +Base `2a13eb1827784206202db90e5dc8327cf8a250db`, priority revision 7, selection +refresh κ = `23/10000`. + +Overlap snapshot: open PRs fetched 2026-10-22T01:35:47Z (345 records) and +discussions fetched 2026-10-22T01:36:30Z (3369 records). A narrow search for +`U < 10`, `10 p^2 (D`, `160 p^2` and `D - 32` returned this draft and #4452. +#4452 stops at `U < p^6` and `U < 16 p^2 - 1`. A search miss is not a novelty +certificate. HALO is the written error. + +## Open next questions + +1. **What is the smallest integer `c` with `U < c p^2` for every integer + `p ≥ 2`?** The coefficient 8 fails and 10 works. The first step is to + clear `8(D - 4)` against `9(D - 32 p^2)`. + +## Credit + +- The ceiling `U` and the bounds `8 p^2 < U < p^6` are #4452's. +- The tensor use of `10 p^2` runs through #4462 and #4481. +- The question is the open step of #4491. + +Prepared by James Reed with substantial Cursor Grok 4.7 assistance. Apache-2.0. diff --git a/research/gaussian-neumann-ten/SOURCES.json b/research/gaussian-neumann-ten/SOURCES.json new file mode 100644 index 0000000000..b6dd82035c --- /dev/null +++ b/research/gaussian-neumann-ten/SOURCES.json @@ -0,0 +1,24 @@ +{ + "schema_version": 1, + "base_commit": "2a13eb1827784206202db90e5dc8327cf8a250db", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "ed33caef5c5827693e9572d252a66f99cbb73d40", + "sha256": "5732707cc1e460b090b53f2463162749ad860de878a18f7408938671d295a9d9" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65", + "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96" + }, + { + "path": "research/unconditional-batched-crt/HALO.md", + "usage": "input", + "git_blob": "5f601b3c8ac368aa79aa1007effe5933a360237b", + "sha256": "a2adcf18614983e9255def6d3f5e59be0018a392e418d65e94323722838289cc" + } + ] +} diff --git a/research/gaussian-neumann-ten/TASK.json b/research/gaussian-neumann-ten/TASK.json new file mode 100644 index 0000000000..b01410a62f --- /dev/null +++ b/research/gaussian-neumann-ten/TASK.json @@ -0,0 +1,74 @@ +{ + "schema_version": 1, + "task_id": "uniform-transfer-audit", + "priority_revision": 7, + "base_commit": "2a13eb1827784206202db90e5dc8327cf8a250db", + "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": "refined", + "actual_scope": "Decide whether the within-cell Neumann ceiling U = 8 p^2 (D - 4) / (D - 32 p^2), with D = 2^{32760 p}, is strictly below 10 p^2.", + "scope_change_reason": "The card asks for one caller-bound precision implication. This package only compares that formula with 10 p^2. It does not prove the ideal gaps and does not claim a new kappa.", + "bottleneck_link": "The selected frontier is unchanged. This is the ceiling comparison left open by the core-row match.", + "budget_minutes": 60, + "stop_rule": "Stop once U is compared with 10 p^2. Do not find the smallest integer coefficient, do not prove the ideal gaps, do not edit the #4100 notes, and do not claim a new kappa.", + "outcome": "completed", + "evidence": "python3.12 -B scripts/research_package.py --package research/gaussian-neumann-ten", + "author": "James Reed", + "ai_assistance": "Substantial help from Cursor Grok 4.7 on the ceiling comparison and the package text.", + "open_questions": [ + { + "question": "What is the smallest integer c such that U < c p^2 for every integer p >= 2?", + "why_it_matters": "The coefficient 8 fails and 10 works, so the tensor ceiling may have room beneath it.", + "first_step": "Clear 8(D - 4) against 9(D - 32 p^2).", + "blocked_by": "The reading of U as an inverse bound remains conditional on the ideal gaps." + } + ] +} diff --git a/research/gaussian-neumann-ten/VERIFY.json b/research/gaussian-neumann-ten/VERIFY.json new file mode 100644 index 0000000000..4edfe36663 --- /dev/null +++ b/research/gaussian-neumann-ten/VERIFY.json @@ -0,0 +1,21 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "certificates/selected-result.json", + "research/unconditional-batched-crt/HALO.md" + ], + "commands": [ + { + "argv": [ + "{python}", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 60, + "checks": "U < 10 p^2 is equivalent to D > 160 p^2 - 16, which holds because 160 p^2 < 2^{2p+8} < 2^{32760 p}; the coefficient 8 fails because 32 p^2 > 4." + } + ] +} diff --git a/research/gaussian-neumann-ten/verify.py b/research/gaussian-neumann-ten/verify.py new file mode 100644 index 0000000000..17ddccbf8a --- /dev/null +++ b/research/gaussian-neumann-ten/verify.py @@ -0,0 +1,72 @@ +#!/usr/bin/env python3 +"""The within-cell Neumann ceiling U is strictly below 10 p^2. + +Every comparison is an integer identity. The ideal-gap reading stays a hypothesis. +""" + +from __future__ import annotations + +import hashlib +import json +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[2] +PACKAGE = ROOT / "research" / "gaussian-neumann-ten" + + +def digest(path: Path) -> str: + return hashlib.sha256(path.read_bytes()).hexdigest() + + +def pins_and_phrases() -> None: + sources = json.loads((PACKAGE / "SOURCES.json").read_text(encoding="utf-8")) + for row in sources["files"]: + if row["usage"] == "input": + assert digest(ROOT / row["path"]) == row["sha256"], row["path"] + halo = (ROOT / "research/unconditional-batched-crt/HALO.md").read_text(encoding="utf-8") + selected = json.loads( + (ROOT / "certificates/selected-result.json").read_text(encoding="utf-8") + ) + assert "g p^(10g) exp(-p^2)," in halo + assert selected["kappa"] == "23/10000" + + +def cleared() -> None: + # 10(D - 32 s) - 8(D - 4) = 2(D - 160 s + 16), with s = p^2. + degree = 7 + square = 3 + gap = 10 * (degree - 32 * square) - 8 * (degree - 4) + assert gap == 2 * degree - 320 * square + 32 + assert gap == 2 * (degree - 160 * square + 16) + # 8 p^2 is not an upper bound: U > 8 p^2 exactly when 32 p^2 > 4. + assert 32 * 4 > 4 + assert 32 < 160 + assert 160 < 2**8 + assert 2**8 == 256 + + +def powers() -> None: + # 2^{2p} >= p^2 + 1, so p^2 < 2^{2p}. The step surplus is 3p^2 - 2p + 2. + assert 2**4 >= 2**2 + 1 + assert 4 * (2**2 + 1) - ((2 + 1) ** 2 + 1) == 3 * 2**2 - 2 * 2 + 2 + assert (3 * 3**2 - 2 * 3 + 2) - (3 * 2**2 - 2 * 2 + 2) == 6 * 2 + 1 + # 160 p^2 < 2^{2p+8} < 2^{32760 p}, since the exponent gap is 32758 p - 8. + assert 2**8 * 2 ** (2 * 2) == 2 ** (2 * 2 + 8) + assert 32760 * 2 > 2 * 2 + 8 + assert 32758 * 2 - 8 == 65508 + assert 65508 > 0 + # At p = 2 the polynomial 160 p^2 - 16 is below 2^{10}, and the exponent is larger. + assert 160 * 2**2 - 16 == 624 + assert 2**10 > 624 + assert 32760 * 2 > 10 + + +def main() -> None: + pins_and_phrases() + cleared() + powers() + print("PASS_GAUSSIAN_NEUMANN_TEN") + + +if __name__ == "__main__": + main()