diff --git a/.gitignore b/.gitignore
index 2325f1ab..7c38440c 100644
--- a/.gitignore
+++ b/.gitignore
@@ -14,5 +14,12 @@ __pycache__/
 # Virtual environments
 .venv/
 
-# R cache lock
+# R cache lock and fresh-regeneration backup
 tests/_r_cache.lock
+tests/_r_cache.json.bak
+
+# Vendored R NNS local-install build artifacts
+tools/NNS/src/*.o
+
+# Stray R plotting artifact from headless regeneration
+Rplots.pdf
diff --git a/docs/r13_cache_regeneration.md b/docs/r13_cache_regeneration.md
new file mode 100644
index 00000000..229f2b64
--- /dev/null
+++ b/docs/r13_cache_regeneration.md
@@ -0,0 +1,170 @@
+# R NNS 13.0 parity cache regeneration provenance
+
+This document records the fresh, from-empty regeneration of
+`tests/_r_cache.json` against live vendored R NNS 13.0.
+
+## Environment
+
+| Item | Value |
+| --- | --- |
+| Date of fresh regeneration | 2026-06-13 (UTC) |
+| OS | Ubuntu 24.04.4 LTS (Linux 6.18.5 x86_64) |
+| Python | 3.11.15 |
+| R | R version 4.3.3 (2024-02-29) "Angel Food Cake" |
+| Rscript | Rscript (R) version 4.3.3 (2024-02-29) |
+| Vendored NNS source | `tools/NNS` (extracted package directory, preferred over `tools/NNS_13.0.tar.gz`) |
+| `packageVersion("NNS")` | `13.0` |
+
+R NNS itself was installed exclusively from the vendored repository source via
+`scripts/install_local_r_nns.py` (`R CMD INSTALL tools/NNS`), never from CRAN.
+R package dependencies required to load vendored NNS (`data.table`,
+`doParallel`, `foreach`, `Rcpp`, `RcppParallel`, `rgl`, `xts`, `zoo`,
+`jsonlite`) were installed as Ubuntu binary packages; `Rfast` (plus its `zigg`
+dependency) was built from the upstream GitHub release source
+`RfastOfficial/Rfast` tag `v2.1.5.1-apollo` because CRAN was unreachable in
+the regeneration environment. Only dependencies came from external archives;
+NNS came from `tools/NNS`.
+
+## Commands run
+
+```bash
+python scripts/install_local_r_nns.py
+Rscript -e "suppressPackageStartupMessages(library(NNS)); cat(as.character(packageVersion('NNS')))"
+# printed: 13.0
+
+# Fresh regeneration: moves tests/_r_cache.json to tests/_r_cache.json.bak,
+# starts from an EMPTY cache, and repopulates every entry with a live R call.
+python scripts/regenerate_r_cache.py --fresh -- -n 0 tests/parity
+
+# Replay checks after regeneration
+PYNNS_R_CACHE_ONLY=1 python -m pytest -q -n 0 tests/parity
+python -m pytest -q tests/parity/test_r13_smoke.py
+python -m pytest -q tests/invariants
+ruff check .
+mypy
+python -m build
+```
+
+All cache-only/offline toggles (`PYNNS_R_CACHE_ONLY`, `NNS_R_CACHE_ONLY`,
+`PYNNS_OFFLINE`, `NNS_OFFLINE`, `CI`) were unset for the regeneration run.
+The `--fresh` mode added to `scripts/regenerate_r_cache.py` in this change
+refuses to run in CI, verifies a live local R NNS 13.0 install before touching
+anything, then moves the existing cache aside so no existing entry can be
+reused during regeneration.
+
+The fresh regeneration was run twice: once to surface the true Python parity
+gaps against live R 13.0 (216 test failures), and a second time from an empty
+cache after the Python fixes below, confirming every parity test passes
+against entries produced exclusively by live R calls.
+
+## Results
+
+| Item | Value |
+| --- | --- |
+| `nns_version` | `13.0` |
+| `schema_version` | `1` |
+| Final entry count | 2385 |
+| Prior entry count | 2406 |
+
+The entry count dropped from 2406 to 2385 because 21 entries in the old
+committed cache are stale: an instrumented cache-only replay confirmed that no
+current parity test computes those 21 keys, so a from-empty regeneration never
+recreates them. All 2385 keys requested by the current test suite were
+regenerated from live R.
+
+### Deterministic cache entries that changed
+
+456 of the 2384 shared keys changed value relative to the previously committed
+cache, summarized by NNS function (mapped by instrumenting the cache-key
+computation during a full replay):
+
+| Function | Changed entries |
+| --- | --- |
+| `PM.matrix` | 216 |
+| `NNS.reg` | 147 |
+| `NNS.boost` (numeric harness) | 25 |
+| `LPM.VaR` | 12 |
+| `UPM.VaR` | 12 |
+| `NNS.ANOVA` (custom harness) | 12 |
+| `NNS.M.reg` | 10 |
+| `NNS.stack` (numeric harness) | 6 |
+| `NNS.dep` | 3 |
+| `NNS.caus` | 3 |
+| `NNS.moments` | 2 |
+| `dy.d` (scalar harnesses) | 4 |
+| factor-predictor harnesses (boost/stack/reg) | 4 |
+
+The fresh live R 13.0 values were treated as authoritative in every case.
+
+### Python parity fixes required
+
+Fresh regeneration surfaced 216 failing parity tests. All of them traced to
+four deterministic divergences between NNS Python and vendored R NNS 13.0, and
+Python was fixed to match R in each case:
+
+1. **`LPM.VaR` / `UPM.VaR` integer degrees 1–4** (`src/nns/var.py`):
+   R 13.0 replaced the `optimize()` search with an exact polynomial
+   root-finding inversion on the located order-statistic interval
+   (`.NNS_LPM_VaR_integer`). Python now ports that algorithm (prefix power
+   sums, break-ratio interval location, Brent root with
+   `tol = .Machine$double.eps^0.5`, and the same boundary/fallback handling).
+   This also fixed the `NNS.reg`/`NNS.M.reg` confidence intervals and
+   stack/boost prediction intervals built on these helpers.
+2. **Distance-kernel lognormal rank weight** (`src/nns/distance.py`):
+   R 13.0's bulk path kernels (`NNS_distance_path_single_parallel_cpp`, used
+   for fitted values when `n.best > 1` and for multi-point estimates) use the
+   population sd of ranks `sqrt((k^2 - 1) / 12)` in the lognormal weight,
+   while the single-point `NNS_distance_cpp` kernel keeps the sample sd.
+   Python now provides `nns_distance_path_single_bulk` mirroring the bulk
+   kernel and keeps `nns_distance` on the single-point formulation.
+3. **`NNS.M.reg` out-of-hull multi-point extrapolation**
+   (`src/nns/multivariate_regression.py`): R 13.0 vectorized the multi-point
+   outsider path (bulk kernel estimates, `pmax(distance, 1e-10)` gradient
+   guards) and removed the old dims-dropping behavior for a single outsider
+   row that Python previously emulated. Python now mirrors the new path.
+4. **`NNS.ARMA` numeric multi-lag seasonal weighting** (`src/nns/arma.py`):
+   R weights each numeric seasonal factor by reversing the series with the
+   factor's *position* in the `seasonal.factor` vector
+   (`variable[seq(length(variable), 1, -i)]`), not its lag value. Python now
+   matches, which resolved the two previously `xfail`-ed Sunspots ARMA and
+   macro-like VAR practical examples; both now pass against live R and the
+   `xfail` markers were removed.
+
+Structural test updates justified by fresh R output: R 13.0's `NNS.boost`
+returns only `results`, `pred.int`, `feature.weights`, and
+`feature.frequency` (no `n.best`), so the Python return dictionary and the
+test assertions that expected `n.best` were updated, and the NaN handling of
+final boost estimates now matches R (`stack` falls back to `reg` only when
+absent; NaNs are filled with the gravity of the remaining estimates).
+
+The only remaining `xfail` is the balanced Iris `NNS.boost` diagnostic, which
+is documented as a stochastic sampling gap (R RNG-driven CV-index, feature
+subset, and up/down-sampling draws cannot be reproduced bit-for-bit with
+NumPy's RNG). No deterministic parity gap is excluded from the suite.
+
+### Manual comparison harness
+
+No `scripts/compare_nns.py` / `scripts/compare_nns_r13.py` diagnostic scripts
+exist in this repository, so there was no `$RPM`-based univariate `NNS.reg`
+extraction to fix. The univariate regression-point diagnostic lives in
+`tests/parity/test_r13_smoke.py` and already extracts regression points via
+`NNS.reg(..., multivariate.call = TRUE)$y`, matching the supported R-side
+extraction.
+
+### ARMA nonseasonal nonlinear reconciliation
+
+The suspected live mismatch (Python `[125.25, 107.75, 158.75, 213.66]` vs R
+`[128.50, 113.50, 155.50, 213.66]`) was re-run against live vendored R NNS
+13.0 before regeneration:
+
+- Live R `NNS.ARMA(series, h = 4, seasonal.factor = FALSE, method = "nonlin")`
+  on the 24-point AirPassengers-style series returned
+  `128.5, 113.5, 155.5, 213.6666666667`.
+- Python `nns_arma(series, h=4, seasonal_factor=False, method="nonlin")`
+  returned `128.5, 113.5, 155.5, 213.66666667`.
+
+Python matches live R 13.0 exactly; the previously reported divergent Python
+values could not be reproduced with the current implementation. The hardcoded
+smoke expectations in `tests/parity/test_r13_smoke.py` (LPM/UPM, copulas,
+regression points, seasonal ARMA, nonseasonal nonlinear ARMA, seeded stack)
+were each verified against this fresh live R 13.0 run and required no changes.
diff --git a/scripts/regenerate_r_cache.py b/scripts/regenerate_r_cache.py
index 6277da52..67235180 100644
--- a/scripts/regenerate_r_cache.py
+++ b/scripts/regenerate_r_cache.py
@@ -3,21 +3,43 @@
 
 CI should not run this script. It intentionally clears cache-only/offline toggles
 and invokes pytest so tests/_r.py can refresh tests/_r_cache.json as needed.
+
+Usage::
+
+    python scripts/regenerate_r_cache.py [--fresh] [-- PYTEST_ARGS...]
+
+By default, existing cache entries are reused and only cache misses call R.
+With ``--fresh``, the existing ``tests/_r_cache.json`` is moved aside to
+``tests/_r_cache.json.bak`` and regeneration starts from an empty cache, so
+every entry is produced by a live R call. ``--fresh`` refuses to run in CI and
+requires a working local R NNS install reporting ``packageVersion("NNS")`` of
+13.0 (see ``scripts/install_local_r_nns.py``).
 """
 
 from __future__ import annotations
 
+import argparse
 import json
 import os
+import shutil
 import subprocess
 import sys
 from pathlib import Path
 from typing import Any
 
 _CACHE_PATH = Path(__file__).resolve().parents[1] / "tests" / "_r_cache.json"
+_CACHE_BACKUP_PATH = _CACHE_PATH.with_suffix(".json.bak")
 _NNS_VERSION = "13.0"
 _SCHEMA_VERSION = 1
 
+_OFFLINE_TOGGLES = (
+    "PYNNS_R_CACHE_ONLY",
+    "NNS_R_CACHE_ONLY",
+    "PYNNS_OFFLINE",
+    "NNS_OFFLINE",
+    "CI",
+)
+
 
 def _validate_cache() -> int:
     if not _CACHE_PATH.exists():
@@ -71,15 +93,95 @@ def _validate_cache() -> int:
         )
         return 1
 
+    print(f"OK: {_CACHE_PATH} contains {len(entries)} entries for NNS {_NNS_VERSION}.")
+    return 0
+
+
+def _running_in_ci() -> bool:
+    return os.environ.get("CI", "").lower() in {"1", "true", "yes"} or bool(
+        os.environ.get("GITHUB_ACTIONS")
+    )
+
+
+def _verify_live_r_nns() -> int:
+    """Confirm a local R NNS install reports the expected version before a fresh run."""
+
+    rscript = shutil.which("Rscript")
+    if rscript is None:
+        print(
+            "ERROR: --fresh requires Rscript on PATH; "
+            "run scripts/install_local_r_nns.py first.",
+            file=sys.stderr,
+        )
+        return 1
+    probe = subprocess.run(
+        [
+            rscript,
+            "-e",
+            "suppressPackageStartupMessages(library(NNS)); "
+            "cat(as.character(packageVersion('NNS')))",
+        ],
+        capture_output=True,
+        text=True,
+        check=False,
+    )
+    if probe.returncode != 0:
+        print(
+            "ERROR: --fresh could not load R NNS:\n" + probe.stderr,
+            file=sys.stderr,
+        )
+        return 1
+    version = probe.stdout.strip()
+    if version != _NNS_VERSION:
+        print(
+            f"ERROR: --fresh requires R NNS {_NNS_VERSION!r}; "
+            f"installed version is {version!r}.",
+            file=sys.stderr,
+        )
+        return 1
+    print(f"OK: live R NNS {version} detected for fresh regeneration.")
     return 0
 
 
 def main() -> int:
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument(
+        "--fresh",
+        action="store_true",
+        help=(
+            "Move the existing cache to tests/_r_cache.json.bak and regenerate every "
+            "entry from a live R call. Refuses to run in CI."
+        ),
+    )
+    parser.add_argument(
+        "pytest_args",
+        nargs="*",
+        help="Arguments passed to pytest after an optional '--' separator.",
+    )
+    parsed = parser.parse_args()
+
+    if parsed.fresh:
+        if _running_in_ci():
+            print(
+                "ERROR: --fresh must not run in CI; it deletes the committed cache "
+                "and requires a local R NNS install.",
+                file=sys.stderr,
+            )
+            return 1
+        verify_status = _verify_live_r_nns()
+        if verify_status:
+            return verify_status
+        if _CACHE_PATH.exists():
+            _CACHE_PATH.replace(_CACHE_BACKUP_PATH)
+            print(f"Moved existing cache to {_CACHE_BACKUP_PATH}; starting from empty cache.")
+        else:
+            print("No existing cache found; starting from empty cache.")
+
     env = os.environ.copy()
-    for name in ("PYNNS_R_CACHE_ONLY", "PYNNS_OFFLINE", "CI"):
+    for name in _OFFLINE_TOGGLES:
         env.pop(name, None)
 
-    args = sys.argv[1:]
+    args = parsed.pytest_args
     if args[:1] == ["--"]:
         args = args[1:]
     if not args:
diff --git a/src/nns/arma.py b/src/nns/arma.py
index bc6a0dd4..e3fb50f6 100644
--- a/src/nns/arma.py
+++ b/src/nns/arma.py
@@ -753,8 +753,10 @@ def _numeric_seasonal_weights(
     lags: NDArray[np.int64],
 ) -> NDArray[np.float64]:
     output = np.empty(lags.size, dtype=np.float64)
-    for index, lag in enumerate(lags):
-        rev_var = variable[:: -int(lag)]
+    for index in range(lags.size):
+        # R NNS 13.0 reverses by the candidate's POSITION in seasonal.factor
+        # (variable[seq(length(variable), 1, -i)]), not by its lag value.
+        rev_var = variable[:: -(index + 1)]
         with np.errstate(invalid="ignore", divide="ignore"):
             output[index] = abs(
                 np.float64(np.std(rev_var, ddof=1)) / np.float64(np.mean(rev_var))
diff --git a/src/nns/boost.py b/src/nns/boost.py
index 133fafcc..da4c9019 100644
--- a/src/nns/boost.py
+++ b/src/nns/boost.py
@@ -282,9 +282,11 @@ def _nns_boost_core(
         type=type_value,
         pred_int=pred_int,
     )
-    estimates = np.asarray(final_fit["stack"], dtype=np.float64)
-    if estimates.size == 0 or np.any(np.isnan(estimates)):
+    stack_estimates = final_fit["stack"]
+    if stack_estimates is None or np.asarray(stack_estimates, dtype=np.float64).size == 0:
         estimates = np.asarray(final_fit["reg"], dtype=np.float64)
+    else:
+        estimates = np.asarray(stack_estimates, dtype=np.float64)
     estimates = _fill_nan_with_gravity(estimates)
     if type_value == "class":
         estimates = _round_clamp_classes(estimates, y_train)
@@ -294,7 +296,6 @@ def _nns_boost_core(
         "pred.int": final_fit["pred.int"],
         "feature.weights": weights[order_idx],
         "feature.frequency": counts[order_idx],
-        "n.best": final_fit["NNS.reg.n.best"],
     }
 
 
diff --git a/src/nns/distance.py b/src/nns/distance.py
index e3450cf7..a96f6cda 100644
--- a/src/nns/distance.py
+++ b/src/nns/distance.py
@@ -41,6 +41,45 @@ def nns_distance(
     return float(np.dot(selected_y, weights))
 
 
+def nns_distance_path_single_bulk(
+    rpm: NDArray[np.float64],
+    x_test: NDArray[np.float64],
+    k: KValue,
+    class_: object | None = None,
+) -> NDArray[np.float64]:
+    """Return R's NNS.distance.path.single.bulk predictions for many target rows.
+
+    This mirrors the ``NNS_distance_path_single_parallel_cpp`` kernel, which
+    differs from the single-point ``NNS.distance`` kernel in its lognormal rank
+    weight (population sd of ranks instead of sample sd).
+    """
+    features, y_hat = _split_rpm(rpm)
+    tests = _as_matrix(x_test, "x_test")
+    if tests.shape[1] != features.shape[1]:
+        raise ValueError("x_test column count must match rpm feature column count.")
+
+    k_value = _resolve_k(k, features.shape[0])
+    out = np.empty(tests.shape[0], dtype=np.float64)
+    for row_index, dest in enumerate(tests):
+        scaled_features, scaled_dest = _rescale_joint(features, dest)
+        distances = _distance_sum(scaled_features, scaled_dest, zero_eps=1e-10)
+        indices = np.argsort(distances, kind="mergesort")
+        selected = indices[:k_value]
+        selected_distances = distances[selected]
+        selected_y = y_hat[selected]
+
+        if k_value == 1:
+            out[row_index] = float(selected_y[0])
+            continue
+
+        weights = _combined_weights(selected_distances, rank_sd_population=True)
+        if class_ is not None:
+            out[row_index] = _weighted_mode(selected_y, weights)
+        else:
+            out[row_index] = float(np.dot(selected_y, weights))
+    return out
+
+
 def nns_distance_bulk(
     rpm: NDArray[np.float64],
     x_test: NDArray[np.float64],
@@ -100,7 +139,11 @@ def _distance_sum(
     return distances
 
 
-def _combined_weights(distances: NDArray[np.float64]) -> NDArray[np.float64]:
+def _combined_weights(
+    distances: NDArray[np.float64],
+    *,
+    rank_sd_population: bool = False,
+) -> NDArray[np.float64]:
     from scipy import stats  # type: ignore[import-untyped]
 
     count = distances.size
@@ -115,7 +158,12 @@ def _combined_weights(distances: NDArray[np.float64]) -> NDArray[np.float64]:
 
     lognormal = np.zeros(count, dtype=np.float64)
     if count >= 2:
-        sd_ranks = float(np.std(ranks, ddof=1))
+        # R's single-point NNS.distance kernel uses the sample sd of ranks; the
+        # bulk path kernels use the population sd sqrt((k^2 - 1) / 12).
+        if rank_sd_population:
+            sd_ranks = float(np.sqrt((count * count - 1.0) / 12.0))
+        else:
+            sd_ranks = float(np.std(ranks, ddof=1))
         lognormal = np.abs(stats.lognorm.logpdf(ranks, s=sd_ranks, scale=1.0))[::-1]
         lognormal = _normalized(lognormal)
 
diff --git a/src/nns/multivariate_regression.py b/src/nns/multivariate_regression.py
index 6a185f7d..7918b096 100644
--- a/src/nns/multivariate_regression.py
+++ b/src/nns/multivariate_regression.py
@@ -8,7 +8,7 @@
 
 from nns.central_tendencies import nns_mode
 from nns.dependence import _gravity
-from nns.distance import KValue, nns_distance
+from nns.distance import KValue, nns_distance, nns_distance_path_single_bulk
 from nns.part import NoiseReduction
 from nns.regression import Order, _normalize_type, _round_clamp_classes, nns_reg
 from nns.regression import _nns_copula_matrix as _copula_matrix
@@ -78,10 +78,7 @@ def nns_m_reg(
 
     k = _resolve_n_best(n_best, x_values, y_values, rpm)
     if _k_as_count(k, rpm.shape[0]) > 1 and not point_only:
-        fitted_y = np.array(
-            [nns_distance(rpm, row, k, type_value) for row in x_values],
-            dtype=np.float64,
-        )
+        fitted_y = nns_distance_path_single_bulk(rpm, x_values, k, type_value)
         if type_value == "class":
             fitted_y = _round_clamp_classes(fitted_y, y_values)
         residuals = fitted_y - y_values
@@ -336,30 +333,46 @@ def _predict_points(
     minimums = np.min(x, axis=0)
     maximums = np.max(x, axis=0)
     central = np.array([_gravity(rpm[:, col]) for col in range(x.shape[1])], dtype=np.float64)
-    out = np.empty(point_est.shape[0], dtype=np.float64)
-    outsider_rows = np.flatnonzero(np.any((point_est < minimums) | (point_est > maximums), axis=1))
-    for row_index, point in enumerate(point_est):
-        outsiders = (point < minimums) | (point > maximums)
-        if not np.any(outsiders):
-            out[row_index] = nns_distance(rpm, point, k, class_)
-            continue
-        if point_is_matrix and outsider_rows.size == 1:
-            # Installed R drops dimensions for one outsider row in the multi-point path:
-            # apply(as.matrix(point.est[i, ]), 1, f) passes scalar elements to f and
-            # vector assignment keeps the first result. Match that behavior.
-            scalar_point = np.full(point.shape, point[0], dtype=np.float64)
-            out[row_index] = _outside_prediction(
-                scalar_point,
-                minimums,
-                maximums,
-                central,
-                rpm,
-                k,
-                class_,
+    # NaN comparisons are already False, matching R's outsiders[is.na(outsiders)] <- 0.
+    outsider_mask = (point_est < minimums) | (point_est > maximums)
+
+    if not point_is_matrix:
+        point = point_est[0]
+        if not np.any(outsider_mask[0]):
+            return np.array(
+                [nns_distance(rpm, point, k, class_)],
+                dtype=np.float64,
             )
-            continue
-        out[row_index] = _outside_prediction(point, minimums, maximums, central, rpm, k, class_)
-    return out if point_is_matrix else out[:1]
+        return np.array(
+            [_outside_prediction(point, minimums, maximums, central, rpm, k, class_)],
+            dtype=np.float64,
+        )
+
+    out = nns_distance_path_single_bulk(rpm, point_est, k, class_)
+    outsider_rows = np.flatnonzero(np.any(outsider_mask, axis=1))
+    if outsider_rows.size:
+        outside_points = point_est[outsider_rows, :]
+        boundary = np.minimum(np.maximum(outside_points, minimums), maximums)
+        mid = (boundary + central) / 2.0
+        mid_2 = (boundary + mid) / 2.0
+
+        d1 = np.sqrt(np.sum((boundary - central) ** 2, axis=1))
+        d2 = np.sqrt(np.sum((boundary - mid) ** 2, axis=1))
+        d3 = np.sqrt(np.sum((boundary - mid_2) ** 2, axis=1))
+
+        boundary_est = nns_distance_path_single_bulk(rpm, boundary, k, class_)
+        mid_est = nns_distance_path_single_bulk(rpm, mid, k, class_)
+        mid_2_est = nns_distance_path_single_bulk(rpm, mid_2, k, class_)
+        central_est = nns_distance(rpm, central, k, class_)
+
+        g1 = (boundary_est - central_est) / np.maximum(d1, 1e-10)
+        g2 = (boundary_est - mid_est) / np.maximum(d2, 1e-10)
+        g3 = (boundary_est - mid_2_est) / np.maximum(d3, 1e-10)
+
+        last_gradient = (g1 * 3.0 + g2 * 2.0 + g3 * 1.0) / 6.0
+        last_distance = np.sqrt(np.sum((outside_points - boundary) ** 2, axis=1))
+        out[outsider_rows] = last_distance * last_gradient + boundary_est
+    return out
 
 
 def _outside_prediction(
@@ -378,10 +391,13 @@ def _outside_prediction(
     gradients = []
     for compare in (central, mid, mid_2):
         distance = float(np.sqrt(np.sum((boundary - compare) ** 2)))
-        if distance == 0.0:
-            gradients.append(0.0)
-        else:
-            gradients.append((boundary_est - nns_distance(rpm, compare, k, class_)) / distance)
+        with np.errstate(invalid="ignore", divide="ignore"):
+            gradients.append(
+                float(
+                    np.float64(boundary_est - nns_distance(rpm, compare, k, class_))
+                    / np.float64(distance)
+                )
+            )
     last_gradient = float(np.dot(np.asarray(gradients), np.array([3.0, 2.0, 1.0])) / 6.0)
     last_distance = float(np.sqrt(np.sum((point - boundary) ** 2)))
     return last_distance * last_gradient + boundary_est
diff --git a/src/nns/var.py b/src/nns/var.py
index 65badf59..1f05fc2e 100644
--- a/src/nns/var.py
+++ b/src/nns/var.py
@@ -567,6 +567,8 @@ def lpm_var(percentile: float, degree: float, x: NDArray[np.float64]) -> float:
     pct = min(max(float(percentile), 0.0), 1.0)
     if degree == 0:
         return float(np.quantile(values, pct, method="linear"))
+    if _is_supported_integer_degree(degree):
+        return _lpm_var_integer(pct, int(degree), values)
     xmin = float(np.min(values))
     xmax = float(np.max(values))
     if xmin == xmax:
@@ -589,6 +591,9 @@ def upm_var(percentile: float, degree: float, x: NDArray[np.float64]) -> float:
     pct = min(max(float(percentile), 0.0), 1.0)
     if degree == 0:
         return float(np.quantile(values, 1.0 - pct, method="linear"))
+    if _is_supported_integer_degree(degree):
+        # UPM.ratio(t) = p is equivalent to LPM.ratio(t) = 1 - p.
+        return _lpm_var_integer(1.0 - pct, int(degree), values)
     xmin = float(np.min(values))
     xmax = float(np.max(values))
     if xmin == xmax:
@@ -605,6 +610,119 @@ def upm_var(percentile: float, degree: float, x: NDArray[np.float64]) -> float:
     return float(result.x)
 
 
+_UNIROOT_TOL = float(np.finfo(float).eps ** 0.5)
+
+
+def _is_supported_integer_degree(degree: float) -> bool:
+    return (
+        math.isfinite(float(degree))
+        and float(degree) == int(degree)
+        and 1 <= int(degree) <= 4
+    )
+
+
+def _lpm_var_integer(pct: float, degree: int, values: NDArray[np.float64]) -> float:
+    """Exact integer-degree LPM VaR inversion matching R NNS 13.0.
+
+    Mirrors R's ``.NNS_LPM_VaR_integer``: within an interval between sorted
+    unique observations, LPM_d(t) and UPM_d(t) are degree-d polynomials in t,
+    so VaR solves ``(1 - p) * LPM_d(t) - p * UPM_d(t) = 0`` exactly on the
+    located order-statistic interval.
+    """
+    n = values.size
+    x_sorted = np.sort(values)
+    # Center at the sorted median: (t - x) is translation-invariant, so results
+    # are mathematically identical, but prefix powers are built at deviation
+    # scale, eliminating catastrophic cancellation for level-shifted data.
+    shift = float(x_sorted[(n + 1) // 2 - 1])
+    x_sorted = x_sorted - shift
+    x_min = float(x_sorted[0])
+    x_max = float(x_sorted[-1])
+    if n == 1 or x_min == x_max:
+        return x_min + shift
+
+    unique_x, counts = np.unique(x_sorted, return_counts=True)
+    k_break = np.cumsum(counts)
+
+    prefix_power = np.zeros((degree + 1, unique_x.size), dtype=np.float64)
+    total_power = np.zeros(degree + 1, dtype=np.float64)
+    prefix_power[0] = k_break
+    total_power[0] = float(n)
+    for j in range(1, degree + 1):
+        x_power = x_sorted**j
+        prefix_power[j] = np.cumsum(x_power)[k_break - 1]
+        total_power[j] = float(np.sum(x_power))
+
+    lpm = np.zeros(unique_x.size, dtype=np.float64)
+    upm = np.zeros(unique_x.size, dtype=np.float64)
+    for j in range(degree + 1):
+        coef = float(math.comb(degree, j))
+        lpm += coef * (-1.0) ** j * unique_x ** (degree - j) * prefix_power[j]
+        suffix_power = total_power[j] - prefix_power[j]
+        upm += coef * (-1.0) ** (degree - j) * unique_x ** (degree - j) * suffix_power
+
+    with np.errstate(invalid="ignore", divide="ignore"):
+        ratio_break = lpm / (lpm + upm)
+    ratio_break[np.isnan(ratio_break)] = 0.0
+    ratio_break = np.clip(ratio_break, 0.0, 1.0)
+    # Protect interval location from tiny floating-point non-monotonicity.
+    ratio_break = np.maximum.accumulate(ratio_break)
+    ratio_break[0] = 0.0
+    ratio_break[-1] = 1.0
+
+    if pct <= 0.0:
+        return x_min + shift
+    if pct >= 1.0:
+        return x_max + shift
+
+    interval = int(np.searchsorted(ratio_break, pct, side="right"))
+    interval = max(interval, 1)
+    interval = min(interval, unique_x.size - 1)
+
+    lower = float(unique_x[interval - 1])
+    upper = float(unique_x[interval])
+    if lower == upper:
+        return lower + shift
+
+    interval_prefix = prefix_power[:, interval - 1]
+
+    def root_value(t: float) -> float:
+        lpm_raw = 0.0
+        upm_raw = 0.0
+        for j in range(degree + 1):
+            coef = float(math.comb(degree, j))
+            lpm_raw += coef * (-1.0) ** j * t ** (degree - j) * float(interval_prefix[j])
+            suffix = float(total_power[j]) - float(interval_prefix[j])
+            upm_raw += coef * (-1.0) ** (degree - j) * t ** (degree - j) * suffix
+        return (1.0 - pct) * lpm_raw - pct * upm_raw
+
+    f_lower = root_value(lower)
+    f_upper = root_value(upper)
+    tol = _UNIROOT_TOL
+
+    if math.isfinite(f_lower) and abs(f_lower) <= tol:
+        return lower + shift
+    if math.isfinite(f_upper) and abs(f_upper) <= tol:
+        return upper + shift
+
+    if math.isfinite(f_lower) and math.isfinite(f_upper) and f_lower * f_upper <= 0.0:
+        from scipy.optimize import brentq
+
+        root = float(brentq(root_value, lower, upper, xtol=tol))
+        return min(max(root, lower), upper) + shift
+
+    # Numerical safety fallback, mirroring R's optimize() on |f|.
+    from scipy.optimize import minimize_scalar
+
+    result = minimize_scalar(
+        lambda t: abs(root_value(t)),
+        bounds=(lower, upper),
+        method="bounded",
+        options={"xatol": _R_OPTIMIZE_TOL},
+    )
+    return min(max(float(result.x), lower), upper) + shift
+
+
 def _finite_values(x: NDArray[np.float64]) -> NDArray[np.float64]:
     values = np.asarray(x, dtype=np.float64)
     finite = values[np.isfinite(values)]
diff --git a/tests/_r_cache.json b/tests/_r_cache.json
index 085a0f23..487fca90 100644
--- a/tests/_r_cache.json
+++ b/tests/_r_cache.json
@@ -540,7 +540,7 @@
         ]
       ]
     },
-    "0093e53ac14848de438403f6eadb762e58f1b57b26d486abf3573478c7949559": 9.9999296475848,
+    "0093e53ac14848de438403f6eadb762e58f1b57b26d486abf3573478c7949559": 10.0,
     "009f5e861e4854ceae1ebad7c90aaf52bc4e84fc5c4de56f386fb67f1148abc0": {
       "clpm": [
         [
@@ -692,10 +692,10 @@
       "cov.matrix": [
         [
           3.57045497669587,
-          0.0543276341920025
+          0.0543276341920024
         ],
         [
-          0.0543276341920025,
+          0.0543276341920024,
           3.67897177996954
         ]
       ],
@@ -730,7 +730,7 @@
         ]
       ]
     },
-    "010675734c0ed3fe523f650e10f70fdbf1dd59eeebf8341227519ad2f2247137": -2.44230156439077,
+    "010675734c0ed3fe523f650e10f70fdbf1dd59eeebf8341227519ad2f2247137": -2.44230769230769,
     "0106f6fc0fcd996cb1939d138136607e31d8540f14b898f2d386b0bdf355104b": {
       "Fitted.xy": {
         "NNS.ID": [
@@ -778,26 +778,26 @@
           0.976853636186311
         ],
         "conf.int.neg": [
-          1.21276405323794,
-          0.212764053237943,
-          0.212764053237943,
-          0.212764053237943,
-          2.21276405323794,
-          1.21276405323794,
-          1.21276405323794,
-          1.21276405323794,
-          2.21276405323794
+          1.21276595744681,
+          0.212765957446809,
+          0.212765957446809,
+          0.212765957446809,
+          2.21276595744681,
+          1.21276595744681,
+          1.21276595744681,
+          1.21276595744681,
+          2.21276595744681
         ],
         "conf.int.pos": [
-          2.78723594676206,
-          1.78723594676206,
-          1.78723594676206,
-          1.78723594676206,
-          3.78723594676206,
-          2.78723594676206,
-          2.78723594676206,
-          2.78723594676206,
-          3.78723594676206
+          2.78723404255319,
+          1.78723404255319,
+          1.78723404255319,
+          1.78723404255319,
+          3.78723404255319,
+          2.78723404255319,
+          2.78723404255319,
+          2.78723404255319,
+          3.78723404255319
         ],
         "residuals": [
           0.0,
@@ -874,12 +874,12 @@
       },
       "pred.int": {
         "lower.pred.int": [
-          1.21276405323794,
-          0.212764053237943
+          1.21276595744681,
+          0.212765957446809
         ],
         "upper.pred.int": [
-          2.78723594676206,
-          1.78723594676206
+          2.78723404255319,
+          1.78723404255319
         ]
       },
       "rhs.partitions": {
@@ -992,44 +992,44 @@
     },
     "027c8ffa64dd540ae2976abeb04238218d41b26acbd208a3ca9981831a5e1335": {
       "First": [
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-        -7.05945180730579
+        -5.9405219051417,
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+        -7.2204624069754,
+        -7.17442567828375,
+        -6.96194703651745,
+        -7.05945524110244
       ],
       "Second": [
-        1.49515465074063,
-        1.5890007254594,
-        -0.0451257927865479,
-        0.935907242564785,
-        1.14715623346015,
-        3.70841183469597,
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-        -1.42806012902205,
-        -3.37613110980975,
-        -3.77025313745222,
-        -3.76316354971138
+        1.49521615376457,
+        1.58905935625217,
+        -0.0450947097154234,
+        0.935912824541486,
+        1.1471465703165,
+        3.70838392082871,
+        2.03288555173985,
+        5.34516794708414,
+        6.32067738125457,
+        7.492255166365,
+        3.62122176664185,
+        1.87891917806957,
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+        -3.25866060306883,
+        -1.42799816385594,
+        -3.37609635188798,
+        -3.77025170125934,
+        -3.76318565895692
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@@ -1194,7 +1194,7 @@
           0.412513583734238,
           0.0156029191021663,
           0.0295873518570758,
-          0.0784536248958598
+          0.0784536248958597
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           0.270902015957033,
@@ -1209,15 +1209,15 @@
           -0.0120396965766335,
           0.202529455848334,
           3.75456730255261,
-          -0.132506332345094,
+          -0.132506332345095,
           0.0822587304106916,
           -0.0986307274544219,
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+          0.0137527640753658
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           0.412513583734238,
           0.317493272791804,
-          -0.132506332345094,
+          -0.132506332345095,
           3.73594759063517,
           0.24103119725226,
           0.213200469385454,
@@ -1242,9 +1242,9 @@
           0.250422148239398
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-          0.0784536248958598,
+          0.0784536248958597,
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+          0.0137527640753658,
           0.102027229039864,
           0.179314631705325,
           0.250422148239398,
@@ -1502,108 +1502,108 @@
           "q2222"
         ],
         "conf.int.neg": [
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@@ -6214,7 +6214,6 @@
         0.5,
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       ],
-      "n.best": 1.0,
       "pred.int": null,
       "results": [
         -2.3323575907919,
@@ -6285,108 +6284,108 @@
           "q2"
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           0.185664659750584,
           2.66378426155088
@@ -885461,7 +845616,7 @@
           0.418188964444404,
           0.420720240271034,
           1.43383495483722,
-          0.192933662263965,
+          0.192933662263966,
           0.268722258687502,
           0.201601093538659,
           0.175688346828612
@@ -885469,7 +845624,7 @@
         [
           0.484372187623575,
           0.381053059673361,
-          0.192933662263965,
+          0.192933662263966,
           1.15691333593557,
           0.249192429949431,
           0.211199120256049,
@@ -892723,7 +852878,7 @@
       "cov.matrix": [
         [
           3.76609867652436,
-          0.834075668107259,
+          0.83407566810726,
           -0.249380202692651
         ],
         [
diff --git a/tests/invariants/test_boost.py b/tests/invariants/test_boost.py
index 038559ae..6b0c5d59 100644
--- a/tests/invariants/test_boost.py
+++ b/tests/invariants/test_boost.py
@@ -88,7 +88,7 @@ def test_nns_boost_multiple_factor_predictors_are_positional(features_only: bool
     assert set(result) == (
         {"feature.weights", "feature.frequency"}
         if features_only
-        else {"results", "pred.int", "feature.weights", "feature.frequency", "n.best"}
+        else {"results", "pred.int", "feature.weights", "feature.frequency"}
     )
     assert np.sum(result["feature.weights"]) == pytest.approx(1.0)
     if not features_only:
diff --git a/tests/parity/test_boost.py b/tests/parity/test_boost.py
index 2dc855ce..06921d69 100644
--- a/tests/parity/test_boost.py
+++ b/tests/parity/test_boost.py
@@ -294,7 +294,6 @@ def test_nns_boost_stochastic_epoch_path_matches_r_structure() -> None:
     )
     assert np.asarray(actual["feature.weights"], dtype=np.float64).ndim == 1
     assert np.asarray(actual["feature.frequency"], dtype=np.float64).ndim == 1
-    assert np.asarray(actual["n.best"], dtype=np.float64).size > 0
     assert actual["pred.int"] is None
 
 
@@ -337,7 +336,6 @@ def test_nns_boost_stochastic_epoch_ts_test_matches_r_structure() -> None:
     )
     assert np.asarray(actual["feature.weights"], dtype=np.float64).ndim == 1
     assert np.asarray(actual["feature.frequency"], dtype=np.float64).ndim == 1
-    assert np.asarray(actual["n.best"], dtype=np.float64).size > 0
     assert actual["pred.int"] is None
 
 
@@ -547,7 +545,7 @@ def test_nns_boost_binary_class_matches_r(depth: int | None) -> None:
         feature_importance=False,
     )
 
-    _assert_boost_matches(actual, expected, exact_n_best=False)
+    _assert_boost_matches(actual, expected)
 
 
 @pytest.mark.parity
@@ -581,7 +579,7 @@ def test_nns_boost_binary_class_pred_int_matches_r(depth: int) -> None:
         feature_importance=False,
     )
 
-    _assert_boost_matches(actual, expected, exact_n_best=False)
+    _assert_boost_matches(actual, expected)
     assert isinstance(actual["pred.int"], dict)
     assert set(actual["pred.int"]) == {"lower.pred.int", "upper.pred.int"}
     assert all(values.shape == actual["results"].shape for values in actual["pred.int"].values())
@@ -615,7 +613,7 @@ def test_nns_boost_multiclass_matches_r(depth: int) -> None:
         feature_importance=False,
     )
 
-    _assert_boost_matches(actual, expected, exact_n_best=False)
+    _assert_boost_matches(actual, expected)
 
 
 @pytest.mark.parity
@@ -681,7 +679,7 @@ def test_nns_boost_factor_like_class_matches_r() -> None:
         feature_importance=False,
     )
 
-    _assert_boost_matches(actual, expected, exact_n_best=False)
+    _assert_boost_matches(actual, expected)
 
 
 @pytest.mark.parity
@@ -724,8 +722,6 @@ def test_nns_boost_class_stable_metadata_matches_r_when_n_best_is_structural() -
         expected_dict["feature.frequency"],
         atol=COMPOUND,
     )
-    assert np.asarray(actual["n.best"], dtype=np.float64).size > 0
-    assert np.asarray(expected_dict["n.best"], dtype=np.float64).size > 0
 
 
 @pytest.mark.parity
@@ -981,19 +977,10 @@ def test_nns_boost_raw_character_class_raises() -> None:
         nns_boost(variable, labels, variable[:3], type="class", cv_size=0.25)
 
 
-def _assert_boost_matches(
-    actual: dict[str, Any],
-    expected: Any,
-    *,
-    exact_n_best: bool = True,
-) -> None:
+def _assert_boost_matches(actual: dict[str, Any], expected: Any) -> None:
     assert isinstance(expected, dict)
     assert set(actual) == set(expected)
     for key in actual:
-        if key == "n.best" and not exact_n_best:
-            assert np.asarray(actual[key], dtype=np.float64).size > 0
-            assert np.asarray(expected[key], dtype=np.float64).size > 0
-            continue
         _assert_nested_numeric_close(actual[key], expected[key])
 
 
@@ -1043,5 +1030,3 @@ def _assert_boost_class_structure(
     assert np.asarray(expected["feature.weights"], dtype=np.float64).ndim == 1
     assert np.asarray(actual["feature.frequency"], dtype=np.float64).ndim == 1
     assert np.asarray(expected["feature.frequency"], dtype=np.float64).ndim == 1
-    assert np.asarray(actual["n.best"], dtype=np.float64).size > 0
-    assert np.asarray(expected["n.best"], dtype=np.float64).size > 0
diff --git a/tests/parity/test_practical_examples.py b/tests/parity/test_practical_examples.py
index 55ec5804..25ba9df4 100644
--- a/tests/parity/test_practical_examples.py
+++ b/tests/parity/test_practical_examples.py
@@ -215,8 +215,11 @@ def test_iris_stack_classification_vignette_predicts_holdout_class() -> None:
 @pytest.mark.practical
 @pytest.mark.xfail(
     reason=(
-        "Installed R NNS 13.0 and NNS Python balanced Iris boost remain a true "
-        "diagnostic parity gap; both miss the all-class-3 holdout."
+        "Stochastic sampling gap: balanced NNS.boost draws CV indices, feature "
+        "subsets, and up/down-sampled class rows from R's RNG, which NNS Python "
+        "cannot reproduce bit-for-bit with NumPy's RNG. Both implementations "
+        "miss the all-class-3 holdout; deterministic boost paths are covered "
+        "by the strict parity tests in test_boost.py."
     ),
     strict=True,
 )
@@ -238,19 +241,11 @@ def test_iris_boost_classification_vignette_gap_is_explicit() -> None:
     assert not np.array_equal(_array(actual["results"]), _array(expected_boost["results"]))
     assert not np.array_equal(_array(actual["results"]), y_test)
     assert not np.array_equal(_array(expected_boost["results"]), y_test)
-    assert set(actual) == {"results", "feature_weights", "feature_frequency", "n_best"}
+    assert set(actual) == {"results", "feature_weights", "feature_frequency"}
 
 
 @pytest.mark.parity
 @pytest.mark.practical
-@pytest.mark.xfail(
-    reason=(
-        "Intentional ARMA weighting divergence: installed R weights numeric "
-        "multi-lag seasonal factors using reverse steps 1:length(lags), while "
-        "NNS Python weights each candidate using its actual lag."
-    ),
-    strict=True,
-)
 def test_sunspots_arma_example_matches_installed_r() -> None:
     expected = _r_sunspots_arma_example()
     actual = nns_arma(
@@ -260,21 +255,11 @@ def test_sunspots_arma_example_matches_installed_r() -> None:
         method="lin",
     )
 
-    # This documents the installed-R compatibility delta, not a target fix.
-    # NNS Python uses the actual seasonal factors when estimating lag strength;
-    # installed R uses the seasonal factor's position in the input vector.
     np.testing.assert_allclose(actual, _array(expected["estimates"]), atol=COMPOUND)
 
 
 @pytest.mark.parity
 @pytest.mark.practical
-@pytest.mark.xfail(
-    reason=(
-        "Remaining macro-like NNS.VAR difference is inherited from the "
-        "documented ARMA numeric multi-lag weighting divergence."
-    ),
-    strict=True,
-)
 def test_var_macro_like_example_matches_installed_r() -> None:
     expected = _r_var_macro_like_example()
     actual = nns_var(_matrix(expected["variables"]), h=4, tau=3, ncores=1, status=False)
@@ -317,7 +302,6 @@ def _iris_boost_diagnostics(expected: dict[str, Any]) -> dict[str, object]:
         "results": np.asarray(boost["results"], dtype=np.float64),
         "feature_weights": np.asarray(boost["feature.weights"], dtype=np.float64),
         "feature_frequency": np.asarray(boost["feature.frequency"], dtype=np.float64),
-        "n_best": float(boost["n.best"]),
     }
 
 
@@ -562,8 +546,7 @@ def _r_iris_classification_vignette() -> dict[str, Any]:
           boost = list(
             results = as.numeric(boost$results),
             feature_weights = as.numeric(boost$feature.weights),
-            feature_frequency = as.numeric(boost$feature.frequency),
-            n_best = as.numeric(boost$n.best)
+            feature_frequency = as.numeric(boost$feature.frequency)
           ),
           stack = list(
             results = as.numeric(stacked$stack),