Derive the broad forward inverse asymptotics: ||K^-1||_2 ~ e^(a/4)/(2 alpha sqrt(pi)) theta^(-1/2) from a turning-point harmonic oscillator; confirmed to 0.3% (no new κ)

research/forward-turning-point/README.md

@@ -0,0 +1,100 @@
+# Turning-point asymptotics of the broad forward inverse
+
+**Claim type:** a formal (semiclassical) derivation of the leading constant, confirmed numerically to
+better than 1%. **No new κ, and this is not a proof.** It answers open questions 1–2 of my
+[#4408](https://github.com/CrocSwap/integer-mult-bounds/pull/4408). #4408 found that the phase-cell
+blocks behind `FORWARD-SMALL-A-AUDIT.md`'s `O(α⁵ρ²θ⁻²e^{aρ/3})` actually grow like `θ^{−1/2}`. It asked
+why, and how the constant depends on `α`. The answer is
+
+    ‖K⁻¹‖₂ ≈ e^{a/4} / (2α√π) · θ^{−1/2},        a = πα².
+
+## Derivation
+
+The block is `K_{k,i} = α⁻¹G(i − s_k)`, with `G(y) = e^{−b₀y²}`, `b₀ = π/α²`,
+`s_k = λk + c` and `θ′ = 1 − λ = θ/(1+θ)`.
+
+1. **Alternating ansatz.** Try `x_i = (−1)^i u(i)` with `u` slowly varying. By Poisson summation,
+   `Σ_i (−1)^i u(i)G(i − s)` is dominated by the frequencies `±π`. The next terms are smaller by
+   `e^{−2π²/b₀}`. To first order in `u′` this gives
+
+       Σ_i (−1)^i u(i) G(i−s) ≈ 2g₀ [cos(πs) u(s) − (π/2b₀) sin(πs) u′(s)],   g₀ = √(π/b₀)·e^{−π²/(4b₀)}.
+
+2. **The turning point.** Write `s_k = k + (c − θ′k)`. Then `cos(πs_k) = (−1)^k cos(π(c − θ′k))`.
+   This vanishes where `c − θ′k ≡ 1/2`, which is exactly where `β′ = ±1/2`: the zero of the shifted
+   Gaussian symbol, at a phase-cell edge. With `y` the distance from that point,
+
+       (Kx)_k ≈ (−1)^k · T u,     T = A[(πθ′) y − (π/2b₀) ∂_y],     A = 2g₀/α.
+
+3. **The harmonic oscillator.** Write `T = A(π/2b₀)(κy − ∂_y)` with `κ = 2b₀θ′`. Then
+   `T*T = A²(π/2b₀)²(−∂² + κ²y² + κ)`. Its lowest eigenvalue is `A²(π/2b₀)²·2κ`, because the
+   harmonic oscillator `−∂² + κ²y²` has ground energy `κ`. So
+   `σ_min = Aπ√(θ′/b₀)`. Substituting `A` and `b₀ = π/α²` gives the formula above.
+
+The minimizing vector is the oscillator's Gaussian ground state, of width `(2b₀θ′)^{−1/2}`,
+alternating in sign and sitting at the cell edge. In the actual block the turning point lies on the
+block's boundary. The numerics match the full-line constant, which means the truncated matrix acts like
+a Neumann boundary there. That is observed, not derived.
+
+## Numerical confirmation
+
+`verify.py` computes `‖K⁻¹‖₂` by inverse power iteration with sparse LU. It uses the banded block,
+dropping entries below `e^{−78}`, on a window of `min(1/θ, 60θ^{−1/2})` coordinates starting at
+`β′ = −1/2`:
+
+| `1/θ` | `α = 3`: ratio to `110.4385·θ^{−1/2}` | `α = 4`: ratio to `20222.76·θ^{−1/2}` |
+| --- | --- | --- |
+| 400 | 0.97357 | 0.94626 |
+| 1,600 | 0.98198 | 0.96333 |
+| 6,400 | 0.98981 | 0.97929 |
+| 25,600 | 0.99461 | 0.98905 |
+| 102,400 | **0.99723** | **0.99438** |
+
+The ratio tends to 1, and the gap `1 − ratio` roughly halves each time `θ` is quartered, as an
+`O(θ^{1/2})` correction should. The verifier also checks two controls:
+
+- a dense SVD over the whole cell agrees with the sparse iteration at `θ = 1/400`;
+- halving the window at `θ = 1/102400` changes the value by less than `10⁻⁶` relative, so the
+  conditioning is localized at the turning point.
+
+## Consequence
+
+At the fine-prime `θ = 1/(4p²)`, with `a ≈ 4 log(16p)`, the true conditioning is
+
+    e^{a/4}/(2α√π) · 2p ≈ 16p²/(α√π)   (with e^{a/4} ≈ 16p),
+
+which is `O(p²/√log p)`. The audit's ceiling is `16p⁴·α⁵·e^{aρ/3}`. At `p = 19` the prediction is
+`110.44 · 38 = 4197`, which is close to #4408's measured 4118, against a ceiling of `9.1·10⁹`. The
+loss in the ceiling comes from two places: the exponent `a/3` in place of `a/4`, and the reciprocal
+gap `θ⁻²` in place of `θ^{−1/2}`.
+
+## Not covered
+
+- **A rigorous upper bound.** A proof would need coercivity away from the turning point, plus
+  control of the boundary layer.
+- **The narrow kernel.** For that, see #4402's column gap.
+- **Any change to #4100's guards.** The ceiling already sits inside the `p¹⁰` envelope.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/forward-turning-point --check-only
+python3.12 -B scripts/research_package.py --package research/forward-turning-point
+```
+
+The verifier prints `PASS_FORWARD_TURNING_POINT` in about 3 seconds. It needs `numpy` and `scipy`.
+
+## Open next questions
+
+1. **Can the reduction be made rigorous?**
+   - *Why it matters:* it would give `‖K⁻¹‖₂ ≤ (1 + o(1))·e^{a/4}/(2α√π)·θ^{−1/2}`.
+   - *First step:* a Grushin-type reduction near the turning point, plus the accretive symbol bound
+     of `GAUGEFREE-INTERIORS.md` §3 away from it.
+
+## Credit
+
+- The bound is from `FORWARD-SMALL-A-AUDIT.md` (PR1673 lineage). The shifted-symbol picture is from
+  `GAUGEFREE-INTERIORS.md`.
+- The `θ^{−1/2}` observation is my #4408, which answered Matarener's #4356.
+
+Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.
+Apache-2.0.

research/forward-turning-point/SOURCES.json

@@ -0,0 +1,24 @@
+{
+  "schema_version": 1,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65",
+      "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"
+    },
+    {
+      "path": "research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md",
+      "usage": "input",
+      "git_blob": "0f7f80b3f74d469c2f72f3b9f338d32568b4eae3",
+      "sha256": "e37aaa30b15671a1dff00676b14572213efaaac99df202dbba5e09f57af487c0"
+    }
+  ]
+}

research/forward-turning-point/TASK.json

@@ -0,0 +1,74 @@
+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 7,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "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": "Answer #4408's open questions: derive and confirm the leading asymptotics ||K^-1||_2 ~ e^(a/4)/(2 alpha sqrt(pi)) theta^(-1/2) of the broad forward phase-cell inverse.",
+  "scope_change_reason": "Follow-up to #4408, which measured theta^(-1/2) and asked for the mechanism and alpha-dependence.",
+  "bottleneck_link": "FORWARD-SMALL-A-AUDIT's alpha^5 theta^-2 e^(a rho/3) bound feeds the fine-prime conditioning envelope.",
+  "budget_minutes": 45,
+  "stop_rule": "Stop at the formal derivation and numerical confirmation; no rigorous proof attempted.",
+  "outcome": "completed",
+  "evidence": "verify.py: ratio to the predicted constant 0.99723 (alpha=3) and 0.99438 (alpha=4) at theta=1/102400 with O(theta^1/2) convergence; dense and localisation controls.",
+  "author": "Ryanson Jonathan, with Claude Code (Claude Opus 5.5) assistance",
+  "ai_assistance": "Substantial Claude Code (Claude Opus 5.5) assistance; no independent human review.",
+  "open_questions": [
+    {
+      "question": "Can the turning-point reduction be made rigorous?",
+      "why_it_matters": "It would give ||K^-1||_2 <= (1+o(1)) e^(a/4)/(2 alpha sqrt(pi)) theta^(-1/2).",
+      "first_step": "A Grushin-type reduction near the turning point plus the accretive symbol bound away from it.",
+      "blocked_by": "None."
+    }
+  ]
+}

research/forward-turning-point/VERIFY.json

@@ -0,0 +1,23 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [
+    "numpy==1.26.4",
+    "scipy==1.13.1"
+  ],
+  "inputs": [
+    "research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 300,
+      "checks": "Pinned FORWARD-SMALL-A bound; sparse inverse power iteration of the banded phase-cell block at alpha=3,4 for theta down to 1/102400; ratio to e^(a/4)/(2 alpha sqrt(pi)) theta^(-1/2) within 1% and converging at O(theta^1/2); dense SVD control; localisation control."
+    }
+  ]
+}

research/forward-turning-point/verify.py

@@ -0,0 +1,97 @@
+#!/usr/bin/env python3
+"""Turning-point asymptotics of the broad forward phase-cell inverse.
+
+Prediction (README): for K_(k,i) = alpha^-1 exp(-b0 (i - lambda k - c)^2) on a phase cell starting at beta' = -1/2,
+
+    ||K^-1||_2 * sqrt(theta)  ->  e^(a/4) / (2 alpha sqrt(pi)),   a = pi alpha^2,  as theta -> 0.
+
+The verifier builds the banded block (entries farther than 20 from the centre are below e^-78 and dropped; the dropped
+part is bounded and reported), computes ||K^-1||_2 by inverse power iteration on K^T K with sparse LU, and checks
+  * the ratio to the prediction tends to 1: within 1% at theta = 1/102400 for alpha = 3 and 4, monotone over the
+    last three theta values, with 1 - ratio shrinking by a factor near 2 per factor 4 in theta (O(theta^1/2));
+  * agreement with a dense SVD on the full cell at theta = 1/400 (band truncation control);
+  * localisation: halving the block length (still >= 30 theta^-1/2) changes the value by < 1e-6 relative.
+Numerical evidence for a formal derivation; not a proof. Pinned: FORWARD-SMALL-A-AUDIT.md phrase.
+Ryanson Jonathan with Claude Code (Claude Opus 5.5) assistance. Apache-2.0.
+"""
+import json
+import math
+from pathlib import Path
+import sys
+
+import numpy as np
+import scipy.sparse as sp
+import scipy.sparse.linalg as spl
+
+ROOT = Path(__file__).resolve().parents[2]
+PHRASE = "`||K^-1|| <= alpha exp(3b/4) c_b^-4/(1-exp(-b rho theta))^2`"
+BAND = 20
+
+
+def need(condition, message):
+    if not condition:
+        raise AssertionError(message)
+
+
+def kmat(alpha, theta, n):
+    rho = 1 + theta
+    lam, b0, c = 1 / rho, math.pi / alpha ** 2, 0.5 / rho      # beta'_0 = -rho c = -1/2
+    rows, cols, vals = [], [], []
+    for k in range(n):
+        centre = int(lam * k + c)
+        i = np.arange(max(0, centre - BAND), min(n, centre + BAND + 2))
+        rows.extend([k] * len(i))
+        cols.extend(i.tolist())
+        vals.extend((np.exp(-b0 * (i - lam * k - c) ** 2) / alpha).tolist())
+    return sp.csc_matrix((vals, (rows, cols)), shape=(n, n))
+
+
+def inv_norm2(K):
+    lu, luT = spl.splu(K), spl.splu(K.T.tocsc())
+    x = np.random.default_rng(0).standard_normal(K.shape[0])
+    x /= np.linalg.norm(x)
+    s = 0.0
+    for _ in range(500):
+        z = luT.solve(lu.solve(x))
+        s_new = math.sqrt(np.linalg.norm(z))
+        x = z / np.linalg.norm(z)
+        if abs(s_new - s) < 1e-12 * s_new:
+            break
+        s = s_new
+    return s_new
+
+
+def main():
+    text = (ROOT / "research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md").read_text(encoding="utf-8")
+    need(PHRASE in text, "pinned bound")
+    out = {"status": "PASS_FORWARD_TURNING_POINT", "new_kappa_claimed": False, "alphas": {}}
+    for alpha in (3, 4):
+        pred = math.exp(math.pi * alpha ** 2 / 4) / (2 * alpha * math.sqrt(math.pi))
+        b0 = math.pi / alpha ** 2
+        need(math.exp(-b0 * BAND ** 2) < 1e-20, "dropped band entries negligible")
+        rows = []
+        for inv_th in (400, 1600, 6400, 25600, 102400):
+            th = 1 / inv_th
+            n = int(min(inv_th, 60 * math.sqrt(inv_th)))
+            s = inv_norm2(kmat(alpha, th, n))
+            rows.append((inv_th, n, s, s * math.sqrt(th) / pred))
+        # dense control on the whole cell
+        Kd = kmat(alpha, 1 / 400, 400).toarray()
+        dense = 1 / np.linalg.svd(Kd, compute_uv=False)[-1]
+        need(abs(dense - rows[0][2]) < 1e-6 * dense, "sparse power iteration matches dense SVD")
+        # localisation control
+        half = inv_norm2(kmat(alpha, 1 / 102400, 30 * 320))
+        need(abs(half - rows[-1][2]) < 1e-6 * half, "the norm is localised at the turning point")
+        ratios = [r[3] for r in rows]
+        need(abs(1 - ratios[-1]) < 0.01, f"alpha={alpha}: within 1% at theta=1/102400")
+        gaps = [1 - r for r in ratios[-3:]]
+        need(gaps[0] > gaps[1] > gaps[2] > 0 and all(1.5 < gaps[i] / gaps[i + 1] < 2.6 for i in range(2)),
+             f"alpha={alpha}: O(theta^1/2) convergence")
+        out["alphas"][alpha] = {"predicted_constant": f"{pred:.4f}",
+                                "rows": [{"inv_theta": a, "n": b, "inv_norm2": f"{c:.3f}", "ratio": f"{d:.6f}"}
+                                         for a, b, c, d in rows]}
+    print(json.dumps(out, indent=1))
+
+
+if __name__ == "__main__":
+    sys.exit(main())