xiangfu0 commented on code in PR #19795:
URL: https://github.com/apache/pinot/pull/19795#discussion_r4242261489


##########
pinot-segment-local/src/main/java/org/apache/pinot/segment/local/customobject/tdigest/TDigestCodec.java:
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@@ -0,0 +1,849 @@
+/**
+ * Licensed to the Apache Software Foundation (ASF) under one
+ * or more contributor license agreements.  See the NOTICE file
+ * distributed with this work for additional information
+ * regarding copyright ownership.  The ASF licenses this file
+ * to you under the Apache License, Version 2.0 (the
+ * "License"); you may not use this file except in compliance
+ * with the License.  You may obtain a copy of the License at
+ *
+ *   http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing,
+ * software distributed under the License is distributed on an
+ * "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+ * KIND, either express or implied.  See the License for the
+ * specific language governing permissions and limitations
+ * under the License.
+ */
+package org.apache.pinot.segment.local.customobject.tdigest;
+
+import java.lang.invoke.MethodHandles;
+import java.lang.invoke.VarHandle;
+import java.nio.BufferUnderflowException;
+import java.nio.ByteBuffer;
+import java.nio.ByteOrder;
+import java.util.Arrays;
+
+/// Compatibility helpers for the serialized t-digest format shared by 
t-digest 3.2 and 3.3.
+///
+/// Version 3.3 requires the first and last centroid to have unit weight 
whenever a digest is recompressed. Version
+/// 3.2 did not maintain that invariant, so an otherwise valid 3.2 digest can 
fail an assertion after being read by
+/// 3.3. Pure reads retain the legacy centroids and interpolation. Mutation 
and serialization repair their boundaries
+/// only when unit endpoints can preserve their total weight and first moment; 
otherwise unchanged historical bytes
+/// remain readable and mutation fails explicitly.
+/// Serialization uses the compact form when 3.3's larger low-compression 
centroid buffer would not fit in a 3.2
+/// default buffer and every centroid is exactly representable in that format. 
Otherwise, it merges the nearest
+/// interior centroids until the verbose representation fits the 3.2 capacity. 
This keeps intermediate values readable
+/// in both directions during a rolling upgrade without silently narrowing 
double-precision values to floats.
+///
+/// Pinot owns the in-memory K1 implementation and retains these legacy bytes 
for stored columns and mixed-version
+/// intermediate results. Decoding checks lengths and numerical state before 
allocating centroid arrays.
+/// NaN means at an identifiable infinite tail are restored to that infinite 
bound without changing their weights.
+/// Other poisoned states remain byte-readable with NaN statistics because 
their distribution cannot be recovered.
+/// They retain their original bytes; attempting to mutate them or mix them 
with another distribution fails explicitly.
+/// Opaque poisoned payloads retain their original centroids even if an old 
reader's default array cannot hold them;
+/// these historical exceptions remain readable by Pinot, but are not 
guaranteed to survive a reverse upgrade.
+/// Historical fractional boundary masses below one, and singleton masses 
between one and two, retain their original
+/// bytes. Fresh serialization rejects these boundaries: t-digest 3.3's 
unit-endpoint assertion cannot be satisfied
+/// without changing their mass. Fractional interior weights remain supported.
+public final class TDigestCodec {
+  public static final int VERBOSE_ENCODING = 1;
+  public static final int SMALL_ENCODING = 2;
+  private static final int LEGACY_CAPACITY_PADDING = 10;
+  public static final int VERBOSE_HEADER_SIZE = 32;
+  public static final int VERBOSE_CENTROID_SIZE = 16;
+  public static final int SMALL_HEADER_SIZE = 30;
+  public static final int SMALL_CENTROID_SIZE = 8;
+  public static final int DEFAULT_MERGE_BUFFER_MULTIPLIER = 5;
+
+  private TDigestCodec() {
+  }
+
+  /// Reads the compression from a validated header without decoding centroid 
arrays.
+  public static double readCompression(byte[] bytes) {
+    return readSerializedHeader(ByteBuffer.wrap(bytes)).compression();
+  }
+
+  /// Rejects non-finite compression while retaining the legacy clamping of 
finite settings below ten.
+  public static void validateCompression(double compression) {
+    if (!Double.isFinite(compression)) {
+      throw new IllegalArgumentException("Invalid TDigest compression: " + 
compression);
+    }
+  }
+
+  /// Applies the legacy minimum compression to query, ingestion and 
serialized-header settings.
+  public static double normalizeCompression(double compression) {
+    validateCompression(compression);
+    return Math.max(compression, 10.0);
+  }
+
+  /// Writes validated centroid arrays in the standard verbose encoding 
without sorting or recompression.
+  /// The caller retains the arrays; only the first `count` entries are 
written, with normalized compression.
+  public static byte[] serializeCentroids(double compression, double min, 
double max, double[] means,
+      double[] weights, int count) {
+    if (count < 0 || count > means.length || count > weights.length) {
+      throw new IllegalArgumentException("Invalid TDigest centroid count: " + 
count);
+    }
+    ByteBuffer encoded = ByteBuffer.allocate(Math.addExact(VERBOSE_HEADER_SIZE,
+        Math.multiplyExact(VERBOSE_CENTROID_SIZE, count)));
+    
encoded.putInt(VERBOSE_ENCODING).putDouble(min).putDouble(max).putDouble(normalizeCompression(compression));
+    encoded.putInt(count);
+    for (int i = 0; i < count; i++) {
+      encoded.putDouble(weights[i]).putDouble(means[i]);
+    }
+    return encoded.array();
+  }
+
+  /// Encodes validated, sorted native centroids and selects a 
legacy-compatible representation.
+  static byte[] serializeCompatibleCentroids(double compression, double min, 
double max, double[] means,
+      double[] weights, int count, boolean inheritedFractionalBoundaries) {
+    if (count < 0 || count > 0 && (count > means.length || count > 
weights.length || !(min <= max))) {
+      throw new IllegalArgumentException("Invalid TDigest centroid count or 
extrema");
+    }
+    ByteBuffer buffer = ByteBuffer.allocate(
+        Math.addExact(VERBOSE_HEADER_SIZE,
+            Math.multiplyExact(VERBOSE_CENTROID_SIZE, count)));
+    buffer.putInt(VERBOSE_ENCODING);
+    buffer.putDouble(min);
+    buffer.putDouble(max);
+    buffer.putDouble(compression);
+    buffer.putInt(count);
+    double encodedWeight = 0.0;
+    boolean hasNonFiniteMeans = false;
+    boolean fractionalWeights = false;
+    for (int i = 0; i < count; i++) {
+      if (!(weights[i] > 0.0) || !Double.isFinite(weights[i]) || !(means[i] >= 
min && means[i] <= max)) {
+        throw new IllegalArgumentException("Invalid TDigest centroid");
+      }
+      buffer.putDouble(weights[i]);
+      buffer.putDouble(means[i]);
+      encodedWeight += weights[i];
+      hasNonFiniteMeans |= !Double.isFinite(means[i]);
+      fractionalWeights |= weights[i] != Math.rint(weights[i]);
+    }
+    if (!Double.isFinite(encodedWeight)) {
+      throw new IllegalArgumentException("TDigest total weight exceeds the 
supported range");
+    }
+    boolean weightedBoundaries = count == 1 ? weights[0] >= 2.0
+        : count > 1 && (weights[0] > 1.0 || weights[count - 1] > 1.0);
+    // Collect exact wire metadata during the write rather than decoding and 
validating our fresh bytes again.
+    SerializedTDigestMetadata metadata = new 
SerializedTDigestMetadata(VERBOSE_ENCODING, min, max,
+        compression, count,
+        VERBOSE_HEADER_SIZE, VERBOSE_CENTROID_SIZE, buffer.capacity(), 
encodedWeight,
+        hasNonFiniteMeans, false, false, fractionalWeights, 
weightedBoundaries, false, compression, Double.NaN,
+        Double.NaN, Double.NaN);
+    return makeLegacyCompatible(buffer.array(), metadata, 
inheritedFractionalBoundaries);
+  }
+
+  /// Rejects fresh boundary masses that legacy readers cannot recompress 
without inventing mass.
+  /// Call after boundary normalization and removing zero-weight centroids; 
fractional interior mass is supported.
+  static void checkLegacyBoundaryWeights(int count, double firstWeight, double 
lastWeight) {
+    if (count > 0 && (firstWeight < 1.0 || lastWeight < 1.0
+        || count == 1 && firstWeight != 1.0 && firstWeight < 2.0)) {
+      throw new IllegalArgumentException("Cannot serialize fractional TDigest 
boundary mass for legacy readers");
+    }
+  }
+
+  /// Allows only unsupported endpoint means inherited from a validated legacy 
source. This is shared by native,
+  /// enclosing and generic writers; new fractional global extrema do not 
receive the historical exception.
+  /// Exact equality is intentional: endpoint repair preserves these means, 
and admitting nearby means would grant
+  /// the exception to fresh extrema. Fresh fractional input at or beyond a 
recorded endpoint permanently excludes
+  /// that endpoint, even if its historical mass was merged away or the new 
weight equals the historical weight.
+  static boolean hasInheritedFractionalBoundaryEncoding(TDigest source, int 
count, double firstMean,
+      double firstWeight, double lastMean, double lastWeight) {
+    if (count <= 0 || !source.hasValidStatistics() || !(firstWeight > 0.0) || 
!(lastWeight > 0.0)
+        || !Double.isFinite(firstWeight) || !Double.isFinite(lastWeight)) {
+      return false;
+    }
+    boolean singleton = count == 1 && firstWeight != 1.0 && firstWeight < 2.0;
+    boolean firstUnsupported = firstWeight < 1.0 || singleton;
+    boolean lastUnsupported = lastWeight < 1.0 || singleton;
+    return (firstUnsupported || lastUnsupported)
+        && (!firstUnsupported || firstMean == 
source.getHistoricalFractionalBoundaryMean(true)
+            && !(source.getFreshFractionalBoundaryMean(true) <= firstMean))
+        && (!lastUnsupported || lastMean == 
source.getHistoricalFractionalBoundaryMean(false)
+            && !(source.getFreshFractionalBoundaryMean(false) >= lastMean));
+  }
+
+  private static void checkSerializedBoundaryWeights(byte[] bytes, 
SerializedTDigestMetadata metadata) {
+    int count = metadata.centroidCount();
+    if (count > 0) {
+      ByteBuffer encoded = ByteBuffer.wrap(bytes);
+      int firstOffset = metadata.centroidOffset();
+      int lastOffset = firstOffset + (count - 1) * metadata.centroidSize();
+      double firstWeight = metadata.encoding() == VERBOSE_ENCODING
+          ? encoded.getDouble(firstOffset) : encoded.getFloat(firstOffset);
+      double lastWeight = metadata.encoding() == VERBOSE_ENCODING
+          ? encoded.getDouble(lastOffset) : encoded.getFloat(lastOffset);
+      checkLegacyBoundaryWeights(count, firstWeight, lastWeight);
+    }
+  }
+
+  /// Converts a verbose `MergingDigest` payload to a representation readable 
by t-digest 3.2 and 3.3.
+  ///
+  /// The input must use the standard verbose `MergingDigest` encoding. The 
returned payload stays verbose when it
+  /// already fits both legacy default capacities, uses compact encoding only 
when every narrowed field is exact, and
+  /// otherwise reduces interior centroids without narrowing doubles. 
Compression below ten is normalized in the
+  /// output header because the 3.2 verbose reader sizes its arrays from the 
raw header before applying that minimum.
+  /// Opaque degraded state is retained without reducing its unknown 
distribution, including over-capacity payloads.
+  /// Fresh fractional endpoints that legacy unit-boundary repair cannot 
represent are rejected without reweighting.
+  public static byte[] makeLegacyCompatible(byte[] verboseBytes) {
+    return makeLegacyCompatible(verboseBytes, 
inspectSerialized(ByteBuffer.wrap(verboseBytes), false));
+  }
+
+  /// Serializes an unchanged, independently retained payload using metadata 
validated at ingestion.
+  ///
+  /// The metadata must describe these exact bytes. This avoids another 
centroid validation walk for lazy digests.
+  public static byte[] makeLegacyCompatible(byte[] verboseBytes, 
SerializedTDigestMetadata metadata) {
+    return makeLegacyCompatible(verboseBytes, metadata, false);
+  }
+
+  /// Retains exact verbose weights for inherited historical fractional 
endpoints that cannot become unit endpoints.
+  /// All other compatibility checks apply. These payloads keep their 
historical assertion-enabled 3.3 read/merge
+  /// limitation; callers must establish inherited endpoint provenance before 
enabling this exception.
+  static byte[] makeLegacyCompatible(byte[] verboseBytes, 
SerializedTDigestMetadata metadata,
+      boolean inheritedFractionalBoundaries) {
+    if (metadata.encoding() != VERBOSE_ENCODING) {
+      throw new IllegalArgumentException("Expected verbose TDigest encoding");
+    }
+    if (metadata.needsLegacyFallback()) {
+      return verboseBytes;
+    }
+    if (!Double.isNaN(metadata.recoveredInfinityMean())) {
+      double[] means = new double[metadata.centroidCount()];
+      double[] weights = new double[means.length];
+      decodeSerializedCentroids(ByteBuffer.wrap(verboseBytes), metadata, 
means, weights);
+      if (!inheritedFractionalBoundaries) {
+        return serializeRecoveredCentroids(metadata, means, weights);
+      }
+      verboseBytes = serializeCentroids(metadata.compression(), 
metadata.min(), metadata.max(), means, weights,
+          means.length);
+      metadata = inspectSerialized(ByteBuffer.wrap(verboseBytes), false);
+    }
+    if (metadata.hasZeroWeightCentroids()) {
+      verboseBytes = removeZeroWeightCentroids(verboseBytes, metadata);
+      metadata = inspectSerialized(ByteBuffer.wrap(verboseBytes), false);
+    }
+    if (!inheritedFractionalBoundaries) {
+      checkSerializedBoundaryWeights(verboseBytes, metadata);
+    }
+    double compression = metadata.compression();
+    if (metadata.encodedCompression() != compression) {
+      verboseBytes = verboseBytes.clone();
+      ByteBuffer.wrap(verboseBytes).putDouble(20, compression);
+    }
+    int centroidCount = metadata.centroidCount();
+    int legacyCapacity =
+        Math.min(getLegacyDefaultCentroidCapacity(compression), 
getDefaultCentroidCapacity(compression));
+    if (centroidCount <= legacyCapacity) {
+      return verboseBytes;
+    }
+    int mainCapacity = getDefaultCentroidCapacity(compression);
+    long bufferCapacity = DEFAULT_MERGE_BUFFER_MULTIPLIER * (long) 
mainCapacity;
+    if (centroidCount > mainCapacity || inheritedFractionalBoundaries || 
(double) (float) compression != compression
+        || centroidCount > Short.MAX_VALUE
+        || mainCapacity > Short.MAX_VALUE || bufferCapacity > Short.MAX_VALUE) 
{
+      return reduceVerboseCentroids(verboseBytes, legacyCapacity);
+    }
+    ByteBuffer verbose = ByteBuffer.wrap(verboseBytes);
+    ByteBuffer centroids = verbose.duplicate();
+    centroids.position(metadata.centroidOffset());
+    for (int i = 0; i < centroidCount; i++) {
+      double weight = centroids.getDouble();
+      double mean = centroids.getDouble();
+      if ((double) (float) weight != weight || (double) (float) mean != mean) {
+        return reduceVerboseCentroids(verboseBytes, legacyCapacity);
+      }
+    }
+
+    ByteBuffer small = ByteBuffer.allocate(SMALL_HEADER_SIZE + 
SMALL_CENTROID_SIZE * centroidCount);
+    verbose.rewind();
+    verbose.getInt();
+    small.putInt(SMALL_ENCODING);
+    small.putDouble(verbose.getDouble());
+    small.putDouble(verbose.getDouble());
+    verbose.getDouble();
+    verbose.getInt();
+    small.putFloat((float) compression);
+    small.putShort((short) mainCapacity);
+    small.putShort((short) bufferCapacity);
+    small.putShort((short) centroidCount);
+    for (int i = 0; i < centroidCount; i++) {
+      small.putFloat((float) verbose.getDouble());
+      small.putFloat((float) verbose.getDouble());
+    }
+    return small.array();
+  }
+
+  private static byte[] serializeRecoveredCentroids(SerializedTDigestMetadata 
metadata, double[] means,
+      double[] weights) {
+    PercentileTDigestAccumulator digest = 
PercentileTDigestAccumulator.forLegacyAggregation(metadata.compression());
+    digest.addHistoricalCentroids(means, weights, means.length, 
metadata.min(), metadata.max(), true);
+    
digest.inheritHistoricalFractionalBoundaries(metadata.historicalFractionalMinMean(),
+        metadata.historicalFractionalMaxMean());
+    return digest.serialize();
+  }
+
+  private static byte[] removeZeroWeightCentroids(byte[] verboseBytes, 
SerializedTDigestMetadata metadata) {
+    ByteBuffer source = ByteBuffer.wrap(verboseBytes);
+    source.position(metadata.centroidOffset());
+    ByteBuffer filtered = ByteBuffer.allocate(metadata.encodedLength());
+    filtered.put(verboseBytes, 0, VERBOSE_HEADER_SIZE);
+    int count = 0;
+    for (int i = 0; i < metadata.centroidCount(); i++) {
+      double weight = source.getDouble();
+      double mean = source.getDouble();
+      if (weight != 0.0) {
+        filtered.putDouble(weight).putDouble(mean);
+        count++;
+      }
+    }
+    filtered.putInt(28, count);
+    return Arrays.copyOf(filtered.array(), filtered.position());
+  }
+
+  /// Validates serialized lengths and centroid values without allocating 
centroid arrays.
+  public static double validateSerialized(byte[] bytes) {
+    return validateSerialized(ByteBuffer.wrap(bytes));
+  }
+
+  /// Validates one legacy digest without changing `input` and returns its 
total centroid weight.
+  ///
+  /// Trailing bytes remain available to the caller for intermediate values 
containing several objects.
+  public static double validateSerialized(ByteBuffer input) {
+    return validateSerialized(input, true);
+  }
+
+  private static double validateSerialized(ByteBuffer input, boolean 
checkCapacity) {
+    return inspectSerialized(input, checkCapacity).totalWeight();
+  }
+
+  /// Header and numerical metadata for one legacy byte payload. The centroid 
bytes remain owned by the caller.
+  /// Header-only inspection leaves total weight as NaN until centroid 
inspection validates the numerical fields.
+  ///
+  /// Legacy writers can emit poisoned numerical state after overflow or 
broken fractional boundary repair.
+  /// Such state is retained with degraded quantiles, rather than being passed 
to the sorted K1 kernel.
+  public record SerializedTDigestMetadata(int encoding, double min, double 
max, double compression,
+      int centroidCount, int centroidOffset, int centroidSize,
+      int encodedLength, double totalWeight, boolean hasNonFiniteMeans, 
boolean needsLegacyFallback,
+      boolean unorderedMeans, boolean fractionalWeights, boolean 
weightedBoundaries, boolean hasZeroWeightCentroids,
+      double encodedCompression, double recoveredInfinityMean, double 
historicalFractionalMinMean,
+      double historicalFractionalMaxMean) {
+    /// Whether the encoded centroid view can be used unchanged by healthy 
pending-state readers and writers.
+    public boolean isCanonicalPayload() {
+      return !needsLegacyFallback && !weightedBoundaries && !unorderedMeans && 
!hasZeroWeightCentroids
+          && Double.isNaN(recoveredInfinityMean);
+    }
+  }
+
+  /// Invocation-local parser state. Primitive byte-array reads avoid per-row 
buffer and metadata allocations.
+  /// Public metadata and retained pending payloads receive immutable 
snapshots instead of sharing this state.
+  static final class SerializedTDigestState {
+    private static final VarHandle SHORT_VIEW =
+        MethodHandles.byteArrayViewVarHandle(short[].class, 
ByteOrder.BIG_ENDIAN);
+    private static final VarHandle INT_VIEW = 
MethodHandles.byteArrayViewVarHandle(int[].class, ByteOrder.BIG_ENDIAN);
+    private static final VarHandle LONG_VIEW = 
MethodHandles.byteArrayViewVarHandle(long[].class, ByteOrder.BIG_ENDIAN);
+
+    int _encoding;
+    double _min;
+    double _max;
+    double _compression;
+    int _centroidCount;
+    int _centroidOffset;
+    int _centroidSize;
+    int _encodedLength;
+    double _totalWeight;
+    boolean _hasNonFiniteMeans;
+    boolean _needsLegacyFallback;
+    boolean _unorderedMeans;
+    boolean _fractionalWeights;
+    boolean _weightedBoundaries;
+    boolean _hasZeroWeightCentroids;
+    double _encodedCompression;
+    double _recoveredInfinityMean;
+    double _historicalFractionalMinMean;
+    double _historicalFractionalMaxMean;
+    private byte[] _bytes;
+    private ByteBuffer _buffer;
+    private int _position;
+    private int _limit;
+
+    void reset(byte[] bytes) {
+      _bytes = bytes;
+      _buffer = null;
+      _position = 0;
+      _limit = bytes.length;
+    }
+
+    void reset(ByteBuffer input) {
+      _bytes = null;
+      _buffer = input.slice().order(ByteOrder.BIG_ENDIAN);
+      _position = 0;
+      _limit = _buffer.limit();
+    }
+
+    void copyFrom(SerializedTDigestMetadata metadata) {
+      _encoding = metadata.encoding();
+      _min = metadata.min();
+      _max = metadata.max();
+      _compression = metadata.compression();
+      _centroidCount = metadata.centroidCount();
+      _centroidOffset = metadata.centroidOffset();
+      _centroidSize = metadata.centroidSize();
+      _encodedLength = metadata.encodedLength();
+      _totalWeight = metadata.totalWeight();
+      _hasNonFiniteMeans = metadata.hasNonFiniteMeans();
+      _needsLegacyFallback = metadata.needsLegacyFallback();
+      _unorderedMeans = metadata.unorderedMeans();
+      _fractionalWeights = metadata.fractionalWeights();
+      _weightedBoundaries = metadata.weightedBoundaries();
+      _hasZeroWeightCentroids = metadata.hasZeroWeightCentroids();
+      _encodedCompression = metadata.encodedCompression();
+      _recoveredInfinityMean = metadata.recoveredInfinityMean();
+      _historicalFractionalMinMean = metadata.historicalFractionalMinMean();
+      _historicalFractionalMaxMean = metadata.historicalFractionalMaxMean();
+    }
+
+    SerializedTDigestMetadata snapshot() {
+      return new SerializedTDigestMetadata(_encoding, _min, _max, 
_compression, _centroidCount, _centroidOffset,
+          _centroidSize, _encodedLength, _totalWeight, _hasNonFiniteMeans, 
_needsLegacyFallback, _unorderedMeans,
+          _fractionalWeights, _weightedBoundaries, _hasZeroWeightCentroids, 
_encodedCompression, _recoveredInfinityMean,
+          _historicalFractionalMinMean, _historicalFractionalMaxMean);
+    }
+
+    private int readPosition(int length) {
+      if (length > _limit - _position) {
+        throw new BufferUnderflowException();
+      }
+      int position = _position;
+      _position += length;
+      return position;
+    }
+
+    private int readInt() {
+      int position = readPosition(Integer.BYTES);
+      return _bytes != null ? (int) INT_VIEW.get(_bytes, position) : 
_buffer.getInt(position);
+    }
+
+    private short readShort() {
+      int position = readPosition(Short.BYTES);
+      return _bytes != null ? (short) SHORT_VIEW.get(_bytes, position) : 
_buffer.getShort(position);
+    }
+
+    private double readDouble() {
+      int position = readPosition(Double.BYTES);
+      return _bytes != null
+          ? Double.longBitsToDouble((long) LONG_VIEW.get(_bytes, position))
+          : _buffer.getDouble(position);
+    }
+
+    private float readFloat() {
+      int position = readPosition(Float.BYTES);
+      return _bytes != null ? Float.intBitsToFloat((int) INT_VIEW.get(_bytes, 
position)) : _buffer.getFloat(position);
+    }
+  }
+
+  /// Inspects one payload without mutating `input`, scanning the centroid 
values once.
+  public static SerializedTDigestMetadata inspectSerialized(ByteBuffer input) {
+    return inspectSerialized(input, true);
+  }
+
+  /// Inspects a payload, optionally enforcing the legacy decoder's declared 
centroid capacity.
+  public static SerializedTDigestMetadata inspectSerialized(ByteBuffer input, 
boolean checkCapacity) {
+    SerializedTDigestState state = new SerializedTDigestState();
+    state.reset(input);
+    readSerializedHeader(state, checkCapacity);
+    if (Double.isNaN(state._totalWeight)) {
+      inspectSerialized(state, null, null);
+    }
+    return state.snapshot();
+  }
+
+  /// Reads the fixed-size header without allocating centroid arrays; 
oversized historical state is also inspected.
+  public static SerializedTDigestMetadata readSerializedHeader(ByteBuffer 
input) {
+    return readSerializedHeader(input, true);
+  }
+
+  /// Reads a fixed-size header, optionally enforcing the declared centroid 
capacity.
+  ///
+  /// NaN or inverted nonempty extrema mark legacy degraded state. Oversized 
payloads require numerical inspection:
+  /// historical degraded state remains readable even if an old decoder's 
arrays could not hold its centroids.
+  public static SerializedTDigestMetadata readSerializedHeader(ByteBuffer 
input, boolean checkCapacity) {
+    SerializedTDigestState state = new SerializedTDigestState();
+    state.reset(input);
+    readSerializedHeader(state, checkCapacity);
+    return state.snapshot();
+  }
+
+  static void readSerializedHeader(SerializedTDigestState state, boolean 
checkCapacity) {
+    int encoding = state.readInt();
+    if (encoding != VERBOSE_ENCODING && encoding != SMALL_ENCODING) {
+      throw new IllegalStateException("Invalid format for serialized 
histogram");
+    }
+    double min = state.readDouble();
+    double max = state.readDouble();
+    double compression;
+    double encodedCompression;
+    int mainCapacity;
+    int bufferCapacity = 0;
+    int centroidCount;
+    int centroidSize;
+    if (encoding == VERBOSE_ENCODING) {
+      encodedCompression = state.readDouble();
+      compression = normalizeCompression(encodedCompression);
+      mainCapacity = Math.max(getDefaultCentroidCapacity(compression), 
getLegacyDefaultCentroidCapacity(compression));
+      centroidCount = state.readInt();
+      centroidSize = VERBOSE_CENTROID_SIZE;
+    } else {
+      encodedCompression = state.readFloat();
+      compression = normalizeCompression(encodedCompression);
+      mainCapacity = state.readShort();
+      bufferCapacity = state.readShort();
+      checkSmallArrayCapacity(mainCapacity);
+      checkSmallArrayCapacity(bufferCapacity);
+      if (mainCapacity == -1) {
+        mainCapacity = getDefaultCentroidCapacity(compression);
+      }
+      centroidCount = state.readShort();
+      centroidSize = SMALL_CENTROID_SIZE;
+    }
+    if (centroidCount < 0) {
+      throw new IllegalArgumentException("Invalid negative TDigest centroid 
count: " + centroidCount);
+    }
+    if (centroidCount > (state._limit - state._position) / centroidSize) {
+      throw new BufferUnderflowException();
+    }
+    int centroidOffset = state._position;
+    int encodedLength = centroidOffset + centroidCount * centroidSize;
+    state._encoding = encoding;
+    state._min = min;
+    state._max = max;
+    state._compression = compression;
+    state._centroidCount = centroidCount;
+    state._centroidOffset = centroidOffset;
+    state._centroidSize = centroidSize;
+    state._encodedLength = encodedLength;
+    state._totalWeight = Double.NaN;
+    state._hasNonFiniteMeans = false;
+    state._needsLegacyFallback = Double.isNaN(min) || Double.isNaN(max) || 
centroidCount > 0 && min > max;
+    state._unorderedMeans = false;
+    state._fractionalWeights = false;
+    state._weightedBoundaries = false;
+    state._hasZeroWeightCentroids = false;
+    state._encodedCompression = encodedCompression;
+    state._recoveredInfinityMean = Double.NaN;
+    state._historicalFractionalMinMean = Double.NaN;
+    state._historicalFractionalMaxMean = Double.NaN;
+    if (checkCapacity && centroidCount > mainCapacity) {
+      inspectSerialized(state, null, null);
+      if (!state._needsLegacyFallback) {
+        throw new IllegalArgumentException("TDigest centroid count exceeds 
capacity: " + centroidCount);
+      }
+    }
+  }
+
+  /// Validates centroids in one pass and optionally decodes their means and 
weights into reusable arrays.
+  ///
+  /// The header must describe this exact input, and each supplied array must 
hold all encoded centroids. Endpoint
+  /// rounding inherited from compact re-encoding or old merge arithmetic is 
restored to the double header bounds.
+  public static SerializedTDigestMetadata inspectSerialized(ByteBuffer input, 
SerializedTDigestMetadata header,
+      double[] means, double[] weights) {
+    SerializedTDigestState state = new SerializedTDigestState();
+    state.reset(input);
+    state.copyFrom(header);
+    inspectSerialized(state, means, weights);
+    return state.snapshot();
+  }
+
+  static void inspectSerialized(SerializedTDigestState state, double[] means, 
double[] weights) {
+    state._position = state._centroidOffset;
+    int centroidCount = state._centroidCount;
+    int centroidSize = state._centroidSize;
+    double min = state._min;
+    double max = state._max;
+    double totalWeight = 0.0;
+    double previousMean = Double.NEGATIVE_INFINITY;
+    boolean hasNonFiniteMeans = false;
+    boolean needsLegacyFallback = state._needsLegacyFallback;
+    boolean unorderedMeans = false;
+    boolean fractionalWeights = false;
+    boolean hasZeroWeightCentroids = false;
+    int positiveCentroidCount = 0;
+    double firstPositiveWeight = 0.0;
+    double lastPositiveWeight = 0.0;
+    double lowerPositiveMean = Double.POSITIVE_INFINITY;
+    double upperPositiveMean = Double.NEGATIVE_INFINITY;
+    double lowerPositiveWeight = 0.0;
+    double upperPositiveWeight = 0.0;
+    boolean withinBounds = true;
+    // Infer only a matching one-sided tail, or a distribution consisting 
entirely of one infinity. A NaN
+    // between finite centroids or in a +/-Infinity mixture has lost 
information and remains opaque.
+    double infinityMean = min == max && Double.isInfinite(min) ? min
+        : Double.isFinite(min) && max == Double.POSITIVE_INFINITY ? max
+        : min == Double.NEGATIVE_INFINITY && Double.isFinite(max) ? min : 
Double.NaN;
+    boolean hasNaNMeans = false;
+    boolean hasFiniteMeans = false;
+    boolean positiveInfinityTailStarted = false;
+    boolean negativeInfinityTailEnded = false;
+    boolean identifiableInfinityTail = true;
+    double roundedMin = (double) (float) min;
+    double roundedMax = (double) (float) max;
+    double lowerAdjacent = Math.nextDown(min);
+    double upperAdjacent = Math.nextUp(max);
+    for (int i = 0; i < centroidCount; i++) {
+      double weight = centroidSize == VERBOSE_CENTROID_SIZE ? 
state.readDouble() : state.readFloat();
+      double mean = centroidSize == VERBOSE_CENTROID_SIZE ? state.readDouble() 
: state.readFloat();
+      // Legacy readers admitted raw non-finite weights. Their mass cannot be 
recovered, so preserve the bytes
+      // as opaque state while keeping the same finite-weight validation for 
newly created distributions.
+      needsLegacyFallback |= !Double.isFinite(weight);
+      // Legacy compact writers can narrow a finite double endpoint outside 
the float range to infinity.
+      // Recover that identifiable rounding before classifying non-finite 
state; verbose infinity remains opaque.
+      if (state._encoding == SMALL_ENCODING) {
+        if (mean == Double.NEGATIVE_INFINITY && Double.isFinite(min) && 
roundedMin == mean) {
+          mean = min;
+        } else if (mean == Double.POSITIVE_INFINITY && Double.isFinite(max) && 
roundedMax == mean) {
+          mean = max;
+        }
+      }
+      if (means != null) {
+        means[i] = Math.max(min, Math.min(mean, max));
+      }
+      if (weights != null) {
+        weights[i] = weight;
+      }
+      // A zero-mass centroid has no distribution to merge, even when its 
unused mean is NaN or stale.
+      if (weight == 0.0) {
+        hasZeroWeightCentroids = true;
+        continue;
+      }
+      if (weight > 0.0) {
+        if (positiveCentroidCount++ == 0) {
+          firstPositiveWeight = weight;
+        }
+        lastPositiveWeight = weight;
+      }
+      fractionalWeights |= weight != Math.rint(weight);
+      boolean nanMean = Double.isNaN(mean);
+      hasNaNMeans |= nanMean;
+      hasFiniteMeans |= Double.isFinite(mean);
+      if (nanMean && !Double.isNaN(infinityMean)) {
+        mean = infinityMean;
+      }
+      if (weight > 0.0) {
+        double boundaryMean = Math.max(min, Math.min(mean, max));
+        // Stable sorting keeps the first lower endpoint and the last upper 
endpoint among equal means.
+        if (lowerPositiveWeight == 0.0 || boundaryMean < lowerPositiveMean) {
+          lowerPositiveMean = boundaryMean;
+          lowerPositiveWeight = weight;
+        }
+        if (upperPositiveWeight == 0.0 || boundaryMean >= upperPositiveMean) {
+          upperPositiveMean = boundaryMean;
+          upperPositiveWeight = weight;
+        }
+      }
+      if (infinityMean == Double.POSITIVE_INFINITY) {
+        if (mean == infinityMean) {
+          positiveInfinityTailStarted = true;
+        } else if (positiveInfinityTailStarted) {
+          identifiableInfinityTail = false;
+        }
+      } else if (infinityMean == Double.NEGATIVE_INFINITY) {
+        if (mean == infinityMean && negativeInfinityTailEnded) {
+          identifiableInfinityTail = false;
+        } else if (mean != infinityMean) {
+          negativeInfinityTailEnded = true;
+        }
+      }
+      if (min == max && Double.isInfinite(min) && mean != infinityMean) {
+        identifiableInfinityTail = false;
+      }
+      // Negative residuals and unidentifiable NaN means still have an unknown 
distribution. Never normalize
+      // or sort them into plausible statistics, even if another centroid 
could be repaired as an infinite tail.
+      needsLegacyFallback |= weight < 0.0 || Double.isNaN(mean)
+          || !Double.isFinite(mean) && (mean < min || mean > max);
+      hasNonFiniteMeans |= !Double.isFinite(mean);
+      unorderedMeans |= mean < previousMean;
+      if (!Double.isNaN(mean)) {
+        // A compact digest can be re-emitted as verbose without recovering 
its double endpoint means. Old
+        // merge arithmetic can also drift by one ULP. Permit only these 
identifiable endpoint roundings.
+        boolean lowerRounded = mean == roundedMin || mean == lowerAdjacent;
+        boolean upperRounded = mean == roundedMax || mean == upperAdjacent;
+        withinBounds &= (mean >= min || lowerRounded) && (mean <= max || 
upperRounded);
+        previousMean = mean;
+      }
+      totalWeight += weight;
+      needsLegacyFallback |= !Double.isFinite(totalWeight);
+    }
+    // Identifiable endpoint rounding can be restored safely. Other historical 
out-of-extrema values have an
+    // unknown distribution: retain their original bytes and NaN statistics 
instead of inventing clamped quantiles.
+    needsLegacyFallback |= !withinBounds

Review Comment:
   Confirmed with the actual pre-PR Pinot accumulator at `dd2c3cdf`, which 
differs from the tdunning merge path examined earlier. Valid finite, 
positive-weight input can produce means 6, 26 and 410 endpoint ULPs below the 
saved minimum while the old reader returns numeric percentiles. Compact input 
can also leave a mean between the float-rounded endpoint and the double header.
   
   Fixed in `3d319ff14d`: the decoder clamps finite displacement within four 
ULPs of the finite extrema scale, plus the finite compact-rounding interval. 
This uses the old merge arithmetic's operand scale; 410 ULPs at 0.1 is only 
about 0.05 ULP at that case's maximum of 1000. Negative/non-finite mass, 
ambiguous NaNs and excursions outside the allowance retain their existing 
rejection/opaque behavior. The bound concerns value displacement, not a 
universal rank-error guarantee.
   
   Regressions reconstruct the actual old output, reflect it for upper bounds, 
and check the acceptance/rejection boundary, wide ranges and overflow edges. 
All 180 old-accumulator probe cases now stay readable (76 were opaque before); 
545 affected tests and the real 3.2/3.3 compatibility gate pass.



##########
compatibility-verifier/tdigest-compatibility/GenerateRankErrors.java:
##########
@@ -0,0 +1,162 @@
+/**
+ * Licensed to the Apache Software Foundation (ASF) under one
+ * or more contributor license agreements.  See the NOTICE file
+ * distributed with this work for additional information
+ * regarding copyright ownership.  The ASF licenses this file
+ * to you under the Apache License, Version 2.0 (the
+ * "License"); you may not use this file except in compliance
+ * with the License.  You may obtain a copy of the License at
+ *
+ *   http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing,
+ * software distributed under the License is distributed on an
+ * "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+ * KIND, either express or implied.  See the License for the
+ * specific language governing permissions and limitations
+ * under the License.
+ */
+
+import com.tdunning.math.stats.Centroid;
+import com.tdunning.math.stats.TDigest;
+import java.util.Arrays;
+import java.nio.ByteBuffer;
+import java.security.MessageDigest;
+import java.util.HexFormat;
+import java.util.SplittableRandom;
+import org.apache.pinot.segment.local.utils.TDigestUtils;
+
+
+/// Independent legacy rank oracle, run sequentially with the pinned baseline 
utility and t-digest 3.3 only.
+public final class GenerateRankErrors {
+  private static final double[] QUANTILES = {0, 0.5, 0.75, 0.95, 0.99, 1};
+  private static final String[] DISTRIBUTIONS = {"UNIFORM", "SKEWED", 
"HEAVY_TAIL", "BIMODAL", "DUPLICATE_HEAVY"};
+  private static final String[] ORDERS = {"ORIGINAL", "REVERSED", 
"RANDOMIZED"};
+  private static final String[] INPUTS = {"SERVER_LOCAL", "DISTRIBUTED"};
+
+  private GenerateRankErrors() {
+  }
+
+  public static void main(String[] args)
+      throws Exception {
+    System.out.println("# mergeOrders=" + String.join(",", ORDERS));
+    System.out.println("# reducerInputs=" + String.join(",", INPUTS));
+    for (int fanIn : new int[]{8, 32, 128}) {
+      for (int compression : new int[]{20, 100, 1000}) {
+        for (String distribution : DISTRIBUTIONS) {
+          for (String order : ORDERS) {
+            for (String input : INPUTS) {
+              Reference reference = referenceCase(fanIn, compression, 
distribution, order, input);
+              System.out.print(fanIn + "/" + compression + "/" + distribution 
+ "/" + order + "/" + input
+                  + "," + reference.inputSha256());
+              for (double error : reference.rankErrors()) {
+                System.out.print("," + error);
+              }
+              System.out.println();
+            }
+          }
+        }
+      }
+    }
+  }
+
+  private record Reference(String inputSha256, double[] rankErrors) {
+  }
+
+  private static Reference referenceCase(int fanIn, int compression, String 
distribution, String order, String input)
+      throws Exception {
+    TDigest[] sources = new TDigest[fanIn];
+    double[] raw = new double[fanIn * 64];
+    for (int i = 0; i < fanIn; i++) {
+      TDigest digest = TDigestUtils.createMergingDigest(compression);
+      SplittableRandom random = new SplittableRandom(0x5EEDL + i);
+      for (int j = 0; j < 64; j++) {
+        double value = value(distribution, random.nextDouble());
+        raw[64 * i + j] = value;
+        digest.add(value);
+      }
+      sources[i] = input.equals("DISTRIBUTED") ? 
TDigestUtils.deserialize(TDigestUtils.serialize(digest)) : digest;
+    }
+    TDigest result = TDigestUtils.createMergingDigest(compression);
+    int[] sourceIndexes = indexes(fanIn, order);
+    for (int index : sourceIndexes) {
+      if (result.size() == 0) {
+        result = sources[index];
+      } else {
+        result.add(sources[index]);
+      }
+    }
+    // Match the final public-compression phase in the historical test before 
measuring quantiles.
+    for (Centroid centroid : result.centroids()) {

Review Comment:
   Checked against the released 3.3 sources and binary: 
[`MergingDigest.centroids()` calls 
`compress()`](https://github.com/tdunning/t-digest/blob/8d5c1523c3d46925e9b3979a8d63c0b9d004ed1c/core/src/main/java/com/tdunning/math/stats/MergingDigest.java#L791-L795),
 which [forces public 
compression](https://github.com/tdunning/t-digest/blob/8d5c1523c3d46925e9b3979a8d63c0b9d004ed1c/core/src/main/java/com/tdunning/math/stats/MergingDigest.java#L549-L552).
 `centroidCount()` is the internal-compression accessor. The validation loop 
therefore already performs the final public pass before measuring quantiles.
   
   Clarified that side effect in `3d319ff14d` without adding a second forced 
pass. Regeneration using the actual 3.3 JAR reproduces all 270 frozen rows 
exactly in normal and interpreted mode. Removing the centroid-validation loop 
as a negative control changes 194 rows, confirming the corpus detects a 
genuinely missing public pass. The CSV remains unchanged; the full oracle 
verification also passes.



##########
pinot-segment-local/src/main/java/org/apache/pinot/segment/local/customobject/tdigest/NonFiniteAwareTDigest.java:
##########
@@ -0,0 +1,665 @@
+/**
+ * Licensed to the Apache Software Foundation (ASF) under one
+ * or more contributor license agreements.  See the NOTICE file
+ * distributed with this work for additional information
+ * regarding copyright ownership.  The ASF licenses this file
+ * to you under the Apache License, Version 2.0 (the
+ * "License"); you may not use this file except in compliance
+ * with the License.  You may obtain a copy of the License at
+ *
+ *   http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing,
+ * software distributed under the License is distributed on an
+ * "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+ * KIND, either express or implied.  See the License for the
+ * specific language governing permissions and limitations
+ * under the License.
+ */
+package org.apache.pinot.segment.local.customobject.tdigest;
+
+import java.util.ArrayList;
+import java.util.Collection;
+import java.util.List;
+import javax.annotation.Nullable;
+import 
org.apache.pinot.segment.local.customobject.tdigest.TDigestCodec.SerializedTDigestMetadata;
+
+
+/// Stores non-finite values as tail masses while delegating finite values to 
an owned native digest.
+/// Emits legacy t-digest bytes and preserves historical fractional endpoint 
provenance across merges and copies.
+/// Instances are mutable and require external synchronization.
+public final class NonFiniteAwareTDigest extends TDigest {
+  private PercentileTDigestAccumulator _finiteDigest;
+  private double _negativeInfinityWeight;
+  private double _positiveInfinityWeight;
+  private List<Centroid> _finiteCentroids;
+  private byte[] _originalFractionalBytes;
+  private boolean _pendingHistoricalBoundaryRepair;
+  private double _historicalMin;
+  private double _historicalMax;
+
+  public static NonFiniteAwareTDigest forLegacyAggregation(double compression) 
{
+    return new 
NonFiniteAwareTDigest(PercentileTDigestAccumulator.forLegacyAggregation(compression),
 0.0, 0.0);
+  }
+
+  private NonFiniteAwareTDigest(PercentileTDigestAccumulator finiteDigest, 
double negativeInfinityWeight,
+      double positiveInfinityWeight) {
+    _finiteDigest = finiteDigest;
+    _negativeInfinityWeight = negativeInfinityWeight;
+    _positiveInfinityWeight = positiveInfinityWeight;
+  }
+
+  /// Snapshots the aggregation without flushing or compressing the source's 
buffered inputs.
+  public NonFiniteAwareTDigest copy() {
+    NonFiniteAwareTDigest copy =
+        new NonFiniteAwareTDigest(_finiteDigest.copy(), 
_negativeInfinityWeight, _positiveInfinityWeight);
+    copy._originalFractionalBytes = _originalFractionalBytes == null ? null : 
_originalFractionalBytes.clone();
+    copy._pendingHistoricalBoundaryRepair = _pendingHistoricalBoundaryRepair;
+    copy._historicalMin = _historicalMin;
+    copy._historicalMax = _historicalMax;
+    return copy;
+  }
+
+  public static NonFiniteAwareTDigest fromBytes(byte[] bytes) {
+    SerializedTDigestInput validated = new SerializedTDigestInput();
+    validated.reset(bytes);
+    SerializedTDigestMetadata metadata = validated.getMetadata();
+    if (!metadata.hasNonFiniteMeans() || metadata.needsLegacyFallback()) {
+      return new 
NonFiniteAwareTDigest(PercentileTDigestAccumulator.fromBytes(validated), 0.0, 
0.0);
+    }
+    validated.decode();
+    if (!PercentileTDigestAccumulator.canNormalizeBoundaries(validated._means, 
validated._weights,
+        validated._numCentroids, metadata.min(), metadata.max())) {
+      return new 
NonFiniteAwareTDigest(PercentileTDigestAccumulator.fromBytes(validated), 0.0, 
0.0);
+    }
+    double encodedMin = metadata.min();
+    double encodedMax = metadata.max();
+    double compression = metadata.compression();
+    int centroidCount = validated._numCentroids;
+    double negativeInfinityWeight = 0.0;
+    double positiveInfinityWeight = 0.0;
+    double[] finiteMeans = validated._means;
+    double[] finiteWeights = validated._weights;
+    int finiteCount = 0;
+    double finiteMin = Double.POSITIVE_INFINITY;
+    double finiteMax = Double.NEGATIVE_INFINITY;
+    for (int i = 0; i < centroidCount; i++) {
+      double weight = finiteWeights[i];
+      double mean = finiteMeans[i];
+      if (weight == 0.0) {
+        continue;
+      }
+      if (mean == Double.NEGATIVE_INFINITY) {
+        negativeInfinityWeight += weight;
+      } else if (mean == Double.POSITIVE_INFINITY) {
+        positiveInfinityWeight += weight;
+      } else {
+        finiteWeights[finiteCount] = weight;
+        finiteMeans[finiteCount] = mean;
+        finiteMin = Math.min(finiteMin, mean);
+        finiteMax = Math.max(finiteMax, mean);
+        finiteCount++;
+      }
+    }
+
+    PercentileTDigestAccumulator finiteDigest =
+        PercentileTDigestAccumulator.forLegacyAggregation(compression);
+    if (negativeInfinityWeight > 0.0 || positiveInfinityWeight > 0.0) {
+      finiteDigest.deferBoundaryNormalization();
+    }
+    if (finiteCount > 0) {
+      finiteMin = Double.isFinite(encodedMin) ? encodedMin : finiteMin;
+      finiteMax = Double.isFinite(encodedMax) ? encodedMax : finiteMax;
+      finiteDigest.addHistoricalCentroids(finiteMeans, finiteWeights, 
finiteCount, finiteMin, finiteMax, true);
+    }
+    validated.recordHistoricalFractionalBoundaries(finiteDigest);
+    NonFiniteAwareTDigest wrapped =
+        new NonFiniteAwareTDigest(finiteDigest, negativeInfinityWeight, 
positiveInfinityWeight);
+    wrapped._pendingHistoricalBoundaryRepair = metadata.weightedBoundaries();
+    wrapped._historicalMin = encodedMin;
+    wrapped._historicalMax = encodedMax;
+    if (metadata.fractionalWeights()) {
+      wrapped._originalFractionalBytes = bytes.clone();
+    }
+    return wrapped;
+  }
+
+  @Override
+  public void add(double value) {
+    add(value, 1);
+  }
+
+  @Override
+  public void add(double value, double weight) {
+    if (Double.isNaN(value)) {
+      throw new IllegalArgumentException("Cannot add NaN to t-digest");
+    }
+    if (!(weight > 0.0) || !Double.isFinite(weight)) {
+      throw new IllegalArgumentException("TDigest weight must be positive: " + 
weight);
+    }
+    PercentileTDigestAccumulator.checkTotalWeight(getTotalWeight() + weight);
+    double infinityWeight = 0.0;
+    if (value == Double.NEGATIVE_INFINITY) {
+      addInfinityWeight(_negativeInfinityWeight, weight);
+      infinityWeight = 
addInfinityWeight(boundaryPreparedView()._negativeInfinityWeight, weight);
+    } else if (value == Double.POSITIVE_INFINITY) {
+      addInfinityWeight(_positiveInfinityWeight, weight);
+      infinityWeight = 
addInfinityWeight(boundaryPreparedView()._positiveInfinityWeight, weight);
+    }
+    requireMutable();
+    if (value == Double.NEGATIVE_INFINITY) {
+      _finiteDigest.deferBoundaryNormalization();
+      _finiteDigest.recordFreshFractionalMean(value, weight);
+      _negativeInfinityWeight = infinityWeight;
+    } else if (value == Double.POSITIVE_INFINITY) {
+      _finiteDigest.deferBoundaryNormalization();
+      _finiteDigest.recordFreshFractionalMean(value, weight);
+      _positiveInfinityWeight = infinityWeight;
+    } else {
+      _finiteDigest.add(value, weight);
+    }
+    mutated();
+  }
+
+  @Override
+  public void add(TDigest other) {
+    if (other.isEmpty()) {
+      return;
+    }
+    if (!hasValidStatistics()) {
+      throw PercentileTDigestAccumulator.corruptedMutation();
+    }
+    if (!other.hasValidStatistics() && !isEmpty()) {
+      throw PercentileTDigestAccumulator.corruptedMutation();
+    }
+    if (other.hasValidStatistics()) {
+      PercentileTDigestAccumulator.checkTotalWeight(getTotalWeight() + 
other.getTotalWeight());
+    }
+    if (other instanceof NonFiniteAwareTDigest) {
+      NonFiniteAwareTDigest wrapped = (NonFiniteAwareTDigest) other;
+      addInfinityWeight(_negativeInfinityWeight, 
wrapped._negativeInfinityWeight);
+      addInfinityWeight(_positiveInfinityWeight, 
wrapped._positiveInfinityWeight);
+      NonFiniteAwareTDigest preparedReceiver = boundaryPreparedView();
+      NonFiniteAwareTDigest preparedSource = wrapped.boundaryPreparedView();
+      double negativeInfinityWeight =
+          addInfinityWeight(preparedReceiver._negativeInfinityWeight, 
preparedSource._negativeInfinityWeight);
+      double positiveInfinityWeight =
+          addInfinityWeight(preparedReceiver._positiveInfinityWeight, 
preparedSource._positiveInfinityWeight);
+      requireMutable();
+      wrapped.prepareHistoricalBoundaries();
+      if (negativeInfinityWeight > 0.0 || positiveInfinityWeight > 0.0) {
+        _finiteDigest.deferBoundaryNormalization();
+      }
+      try {
+        if (!wrapped._finiteDigest.isEmpty()) {
+          _finiteDigest.add(wrapped._finiteDigest);
+          wrapped.invalidateDerivedCaches();
+        }
+      } catch (RuntimeException | Error e) {
+        restoreBoundaryNormalizationAfterFailedMerge();
+        throw e;
+      }
+      if (wrapped.hasValidStatistics()) {
+        _finiteDigest.inheritHistoricalFractionalBoundaries(wrapped);
+      }
+      _negativeInfinityWeight = negativeInfinityWeight;
+      _positiveInfinityWeight = positiveInfinityWeight;
+      mutated();
+      return;
+    }
+    if (!other.hasValidStatistics() || Double.isFinite(other.getMin()) && 
Double.isFinite(other.getMax())) {
+      requireMutable();
+      _finiteDigest.add(other);
+      mutated();
+      return;
+    }
+    // Read precise centroid weights directly; generic sources need no full 
digest serialization for their tails.
+    if (other instanceof PercentileTDigestAccumulator) {
+      PercentileTDigestAccumulator nativeDigest = 
(PercentileTDigestAccumulator) other;
+      if (isEmpty() && nativeDigest.hasUnrepresentableBoundaryPayload()) {
+        _finiteDigest = nativeDigest.copy();
+        mutated();
+        return;
+      }
+    }
+    Collection<Centroid> centroids = other.centroids();
+    double[] finiteMeans = new double[Math.addExact(centroids.size(), 2)];
+    double[] finiteWeights = new double[finiteMeans.length];
+    int count = 0;
+    double sourceMin = other.getMin();
+    double sourceMax = other.getMax();
+    double negativeInfinityWeight = 0.0;
+    double positiveInfinityWeight = 0.0;
+    for (Centroid centroid : centroids) {
+      double mean = centroid.mean();
+      double weight = centroid.weight();
+      if (weight == 0.0) {
+        continue;
+      }
+      if (!(weight > 0.0) || !Double.isFinite(weight) || !(mean >= sourceMin 
&& mean <= sourceMax)) {
+        throw new IllegalArgumentException("Invalid TDigest centroid");
+      }
+      if (mean == Double.NEGATIVE_INFINITY) {
+        negativeInfinityWeight = other instanceof PercentileTDigestAccumulator
+            ? negativeInfinityWeight + weight
+            : addInfinityWeight(negativeInfinityWeight, weight);
+      } else if (mean == Double.POSITIVE_INFINITY) {
+        positiveInfinityWeight = other instanceof PercentileTDigestAccumulator
+            ? positiveInfinityWeight + weight
+            : addInfinityWeight(positiveInfinityWeight, weight);
+      }
+      finiteMeans[count] = mean;
+      finiteWeights[count++] = weight;
+    }
+    // Original contributions exclude unit endpoints introduced by boundary 
repair.
+    addInfinityWeight(_negativeInfinityWeight, negativeInfinityWeight);
+    addInfinityWeight(_positiveInfinityWeight, positiveInfinityWeight);
+    PercentileTDigestAccumulator.sortCentroids(finiteMeans, finiteWeights, 
count);
+    count = PercentileTDigestAccumulator.normalizeSerializedBoundaries(
+        finiteMeans, finiteWeights, count, sourceMin, sourceMax);
+    negativeInfinityWeight = 0.0;
+    positiveInfinityWeight = 0.0;
+    int finiteCount = 0;
+    double finiteMin = Double.POSITIVE_INFINITY;
+    double finiteMax = Double.NEGATIVE_INFINITY;
+    double totalWeight = getTotalWeight();
+    for (int i = 0; i < count; i++) {
+      double mean = finiteMeans[i];
+      double weight = finiteWeights[i];
+      totalWeight += weight;
+      if (mean == Double.NEGATIVE_INFINITY) {
+        negativeInfinityWeight += weight;
+      } else if (mean == Double.POSITIVE_INFINITY) {
+        positiveInfinityWeight += weight;
+      } else {
+        finiteMeans[finiteCount] = mean;
+        finiteWeights[finiteCount++] = weight;
+        finiteMin = Math.min(finiteMin, mean);
+        finiteMax = Math.max(finiteMax, mean);
+      }
+    }
+    PercentileTDigestAccumulator.checkTotalWeight(totalWeight);
+    // Repair may relocate tail mass into a finite header endpoint; merge only 
the remaining tail mass.
+    NonFiniteAwareTDigest preparedReceiver = boundaryPreparedView();
+    negativeInfinityWeight = 
addInfinityWeight(preparedReceiver._negativeInfinityWeight, 
negativeInfinityWeight);
+    positiveInfinityWeight = 
addInfinityWeight(preparedReceiver._positiveInfinityWeight, 
positiveInfinityWeight);
+    requireMutable();
+    if (negativeInfinityWeight > 0.0 || positiveInfinityWeight > 0.0) {
+      _finiteDigest.deferBoundaryNormalization();
+    }
+    try {
+      if (finiteCount > 0) {
+        finiteMin = Double.isFinite(other.getMin()) ? other.getMin() : 
finiteMin;
+        finiteMax = Double.isFinite(other.getMax()) ? other.getMax() : 
finiteMax;
+        if (other instanceof PercentileTDigestAccumulator) {
+          _finiteDigest.addHistoricalCentroids(finiteMeans, finiteWeights, 
finiteCount, finiteMin, finiteMax, false);
+        } else {
+          _finiteDigest.addCentroids(finiteMeans, finiteWeights, finiteCount, 
finiteMin, finiteMax, false);
+        }
+      }
+    } catch (RuntimeException | Error e) {
+      restoreBoundaryNormalizationAfterFailedMerge();
+      throw e;
+    }
+    if (!(other instanceof PercentileTDigestAccumulator)) {
+      for (Centroid centroid : centroids) {
+        if (Double.isInfinite(centroid.mean())) {
+          _finiteDigest.recordFreshFractionalMean(centroid.mean(), 
centroid.weight());
+        }
+      }
+    }
+    _finiteDigest.inheritHistoricalFractionalBoundaries(other);
+    _negativeInfinityWeight = negativeInfinityWeight;
+    _positiveInfinityWeight = positiveInfinityWeight;
+    mutated();
+  }
+
+  private static double addInfinityWeight(double currentWeight, double 
addedWeight) {
+    double totalWeight = currentWeight + addedWeight;
+    if (currentWeight > 0.0 && addedWeight > 0.0
+        && (totalWeight == currentWeight || totalWeight == addedWeight)) {
+      throw new IllegalArgumentException("TDigest infinity-tail weight 
addition would absorb positive mass");
+    }
+    return totalWeight;
+  }
+
+  private void requireMutable() {
+    if (!hasValidStatistics()) {
+      throw PercentileTDigestAccumulator.corruptedMutation();
+    }
+    if (_negativeInfinityWeight == 0.0 && _positiveInfinityWeight == 0.0) {
+      _finiteDigest.prepareForMutation();
+    } else {
+      prepareHistoricalBoundaries();
+    }
+  }
+
+  private void prepareHistoricalBoundaries() {
+    if (!_pendingHistoricalBoundaryRepair) {
+      return;
+    }
+    double encodedMin = getMin();
+    double encodedMax = getMax();
+    List<Centroid> centroids = getFiniteCentroids();
+    double[] means = new double[Math.addExact(centroids.size(), 4)];
+    double[] weights = new double[means.length];
+    int count = 0;
+    if (_negativeInfinityWeight > 0.0) {
+      means[count] = Double.NEGATIVE_INFINITY;
+      weights[count++] = _negativeInfinityWeight;
+    }
+    for (Centroid centroid : centroids) {
+      means[count] = centroid.mean();
+      weights[count++] = centroid.weight();
+    }
+    if (_positiveInfinityWeight > 0.0) {
+      means[count] = Double.POSITIVE_INFINITY;
+      weights[count++] = _positiveInfinityWeight;
+    }
+    count = PercentileTDigestAccumulator.normalizeSerializedBoundaries(means, 
weights, count, encodedMin, encodedMax);
+    int finiteCount = 0;
+    double min = Double.POSITIVE_INFINITY;
+    double max = Double.NEGATIVE_INFINITY;
+    double negativeInfinityWeight = 0.0;
+    double positiveInfinityWeight = 0.0;
+    for (int i = 0; i < count; i++) {
+      if (means[i] == Double.NEGATIVE_INFINITY) {
+        negativeInfinityWeight += weights[i];
+      } else if (means[i] == Double.POSITIVE_INFINITY) {
+        positiveInfinityWeight += weights[i];
+      } else {
+        means[finiteCount] = means[i];
+        weights[finiteCount++] = weights[i];
+        min = Math.min(min, means[i]);
+        max = Math.max(max, means[i]);
+      }
+    }
+    PercentileTDigestAccumulator finiteDigest = 
PercentileTDigestAccumulator.forLegacyAggregation(compression());
+    finiteDigest.deferBoundaryNormalization();
+    min = Double.isFinite(encodedMin) ? encodedMin : min;
+    max = Double.isFinite(encodedMax) ? encodedMax : max;
+    finiteDigest.addHistoricalCentroids(means, weights, finiteCount, min, max, 
true);
+    finiteDigest.inheritHistoricalFractionalBoundaries(_finiteDigest);
+    _finiteDigest = finiteDigest;
+    _negativeInfinityWeight = negativeInfinityWeight;
+    _positiveInfinityWeight = positiveInfinityWeight;
+    _pendingHistoricalBoundaryRepair = false;
+    invalidateDerivedCaches();
+  }
+
+  private NonFiniteAwareTDigest boundaryPreparedView() {
+    if (!_pendingHistoricalBoundaryRepair) {
+      return this;
+    }
+    NonFiniteAwareTDigest prepared = copy();
+    prepared.prepareHistoricalBoundaries();
+    return prepared;
+  }
+
+  private void restoreBoundaryNormalizationAfterFailedMerge() {
+    if (_negativeInfinityWeight == 0.0 && _positiveInfinityWeight == 0.0) {
+      _finiteDigest.resetBoundaryNormalization();
+    }
+  }
+
+  @Nullable
+  PercentileTDigestAccumulator finiteDigestIfNoTails() {
+    if (_negativeInfinityWeight == 0.0 && _positiveInfinityWeight == 0.0) {
+      invalidateDerivedCaches();
+      return _finiteDigest;
+    }
+    return null;
+  }
+
+  private void mutated() {
+    _originalFractionalBytes = null;
+    invalidateDerivedCaches();
+  }
+
+  @Override
+  public void compress() {
+    _finiteDigest.compress();
+    invalidateDerivedCaches();
+  }
+
+  @Override
+  public double getTotalWeight() {
+    return _finiteDigest.getTotalWeight() + _negativeInfinityWeight + 
_positiveInfinityWeight;
+  }
+
+  @Override
+  double getHistoricalFractionalBoundaryMean(boolean lowerBoundary) {
+    return _finiteDigest.getHistoricalFractionalBoundaryMean(lowerBoundary);
+  }
+
+  @Override
+  double getFreshFractionalBoundaryMean(boolean lowerBoundary) {
+    return _finiteDigest.getFreshFractionalBoundaryMean(lowerBoundary);
+  }
+
+  @Override
+  public boolean hasValidStatistics() {
+    return _finiteDigest.hasValidStatistics();
+  }
+
+  @Override
+  public double cdf(double value) {
+    if (Double.isNaN(value) || Double.isInfinite(value)) {
+      throw new IllegalArgumentException(String.format("Invalid value: %f", 
value));
+    }
+    double size = getTotalWeight();
+    if (size == 0.0 || !hasValidStatistics()) {
+      return Double.NaN;
+    }
+    double finiteSize = _finiteDigest.getTotalWeight();
+    double finiteWeight = finiteSize == 0L ? 0.0 : _finiteDigest.cdf(value) * 
finiteSize;
+    return (_negativeInfinityWeight + finiteWeight) / size;
+  }
+
+  @Override
+  public double quantile(double quantile) {
+    if (Double.isNaN(quantile) || quantile < 0.0 || quantile > 1.0) {
+      throw new IllegalArgumentException("q should be in [0,1], got " + 
quantile);
+    }
+    double size = getTotalWeight();
+    if (size == 0.0 || !hasValidStatistics()) {
+      return Double.NaN;
+    }
+    double index = quantile * size;
+    if (_negativeInfinityWeight > 0L && index < _negativeInfinityWeight) {
+      return Double.NEGATIVE_INFINITY;
+    }
+    if (_positiveInfinityWeight > 0L && index >= size - 
_positiveInfinityWeight) {
+      return Double.POSITIVE_INFINITY;
+    }
+    double finiteSize = _finiteDigest.getTotalWeight();
+    if (finiteSize == 0L) {
+      return _negativeInfinityWeight > 0L ? Double.NEGATIVE_INFINITY : 
Double.POSITIVE_INFINITY;
+    }
+    double finiteQuantile = (index - _negativeInfinityWeight) / finiteSize;
+    return _finiteDigest.quantile(Math.max(0.0, Math.min(finiteQuantile, 
1.0)));
+  }
+
+  @Override
+  public Collection<Centroid> centroids() {
+    return getAllCentroids();
+  }
+
+  @Override
+  public double compression() {
+    return _finiteDigest.compression();
+  }
+
+  @Override
+  public int maxSerializedByteSize() {
+    if (_originalFractionalBytes != null) {
+      return _originalFractionalBytes.length;
+    }
+    if (_negativeInfinityWeight == 0.0 && _positiveInfinityWeight == 0.0) {
+      return _finiteDigest.maxSerializedByteSize();
+    }
+    int finiteByteSize = _finiteDigest.maxSerializedByteSize();
+    PercentileTDigestAccumulator finiteDigest = _finiteDigest;
+    if (finiteDigest.hasOriginalFractionalPayload()) {
+      // Infinite tails can make retained fractional boundaries interior; 
their finite view may split endpoints.
+      finiteByteSize = Math.max(finiteByteSize,
+          TDigestCodec.getMaxLegacyCompatibleByteSize(compression(), 
finiteDigest.getCentroidCountUpperBound()));
+    }
+    boolean hasFiniteValues = _finiteDigest.getTotalWeight() > 0.0;
+    int negativeInfinityCount = 
getInfinityCentroidCount(_negativeInfinityWeight, true,
+        !hasFiniteValues && _positiveInfinityWeight == 0.0);
+    int positiveInfinityCount = 
getInfinityCentroidCount(_positiveInfinityWeight,
+        !hasFiniteValues && _negativeInfinityWeight == 0.0, true);
+    int tailCount = Math.addExact(negativeInfinityCount, 
positiveInfinityCount);
+    int tailByteSize = Math.multiplyExact(TDigestCodec.VERBOSE_CENTROID_SIZE, 
tailCount);
+    long finiteCount = hasFiniteValues ? 
finiteDigest.getCentroidCountUpperBound() : 0L;
+    int compatibleBound =
+        TDigestCodec.getMaxLegacyCompatibleByteSize(compression(), 
Math.addExact(finiteCount, tailCount));
+    return Math.min(Math.addExact(finiteByteSize, tailByteSize), 
compatibleBound);
+  }
+
+  @Override
+  public int centroidCount() {
+    return getAllCentroids().size();
+  }
+
+  @Override
+  public double getMin() {
+    if (_pendingHistoricalBoundaryRepair) {
+      return _historicalMin;
+    }
+    if (_negativeInfinityWeight > 0L) {
+      return Double.NEGATIVE_INFINITY;
+    }
+    if (!_finiteDigest.isEmpty()) {
+      return _finiteDigest.getMin();
+    }
+    return Double.POSITIVE_INFINITY;
+  }
+
+  @Override
+  public double getMax() {
+    if (_pendingHistoricalBoundaryRepair) {
+      return _historicalMax;
+    }
+    if (_positiveInfinityWeight > 0L) {
+      return Double.POSITIVE_INFINITY;
+    }
+    if (!_finiteDigest.isEmpty()) {
+      return _finiteDigest.getMax();
+    }
+    return Double.NEGATIVE_INFINITY;
+  }
+
+  private List<Centroid> getFiniteCentroids() {
+    if (_finiteCentroids == null) {
+      // Centroid reads use the native working-compression view; only 
serialization applies public compression.
+      _finiteCentroids = new ArrayList<>(_finiteDigest.centroids());
+    }
+    return _finiteCentroids;
+  }
+
+  private List<Centroid> getAllCentroids() {
+    List<Centroid> finiteCentroids = getFiniteCentroids();
+    boolean hasFiniteValues = _finiteDigest.getTotalWeight() > 0.0;
+    List<Centroid> centroids = new ArrayList<>(finiteCentroids.size() + 6);
+    appendInfinityCentroids(centroids, Double.NEGATIVE_INFINITY, 
_negativeInfinityWeight, true,
+        !hasFiniteValues && _positiveInfinityWeight == 0L);
+    centroids.addAll(finiteCentroids);
+    appendInfinityCentroids(centroids, Double.POSITIVE_INFINITY, 
_positiveInfinityWeight,
+        !hasFiniteValues && _negativeInfinityWeight == 0L, true);
+    return centroids;
+  }
+
+  @Override
+  public byte[] serialize() {
+    if (_originalFractionalBytes != null) {
+      return _originalFractionalBytes.clone();
+    }
+    if (_pendingHistoricalBoundaryRepair) {
+      return boundaryPreparedView().serialize();
+    }
+    if (_negativeInfinityWeight == 0.0 && _positiveInfinityWeight == 0.0) {
+      byte[] bytes = _finiteDigest.serialize();
+      invalidateDerivedCaches();
+      return bytes;
+    }
+    // Repair boundaries only after enclosing infinity tails establish the 
complete distribution's extrema.
+    _finiteDigest.compress();
+    invalidateDerivedCaches();
+    int finiteCentroidCount = _finiteDigest.centroidCount();
+    boolean hasFiniteValues = _finiteDigest.getTotalWeight() > 0.0;
+    int negativeInfinityCentroidCount = 
getInfinityCentroidCount(_negativeInfinityWeight, true,
+        !hasFiniteValues && _positiveInfinityWeight == 0.0);
+    int positiveInfinityCentroidCount = 
getInfinityCentroidCount(_positiveInfinityWeight,
+        !hasFiniteValues && _negativeInfinityWeight == 0.0, true);
+    int centroidCount = Math.addExact(finiteCentroidCount,
+        Math.addExact(negativeInfinityCentroidCount, 
positiveInfinityCentroidCount));
+    double[] means = new double[Math.addExact(centroidCount, 2)];
+    double[] weights = new double[means.length];
+    _finiteDigest.copyCentroids(means, weights, negativeInfinityCentroidCount);
+    appendInfinityCentroids(means, weights, 0, Double.NEGATIVE_INFINITY, 
_negativeInfinityWeight, true,
+        !hasFiniteValues && _positiveInfinityWeight == 0.0);
+    appendInfinityCentroids(means, weights, negativeInfinityCentroidCount + 
finiteCentroidCount,
+        Double.POSITIVE_INFINITY, _positiveInfinityWeight,
+        !hasFiniteValues && _negativeInfinityWeight == 0.0, true);
+    centroidCount = PercentileTDigestAccumulator.normalizeSerializedBoundaries(
+        means, weights, centroidCount, getMin(), getMax());
+    return TDigestCodec.serializeCompatibleCentroids(compression(), getMin(), 
getMax(), means, weights, centroidCount,
+        _finiteDigest.hasInheritedFractionalBoundaryEncoding(means, weights, 
centroidCount));
+  }
+
+  private void invalidateDerivedCaches() {
+    _finiteCentroids = null;
+  }
+
+  private static void appendInfinityCentroids(List<Centroid> centroids, double 
value, double weight,
+      boolean unitWeightAtStart, boolean unitWeightAtEnd) {
+    appendInfinityCentroids(centroids, null, null, 0, value, weight, 
unitWeightAtStart, unitWeightAtEnd);
+  }
+
+  // A null destination counts the same split without allocating centroids or 
temporary arrays.
+  private static int appendInfinityCentroids(@Nullable List<Centroid> 
centroids, @Nullable double[] means,
+      @Nullable double[] weights, int offset, double value, double weight, 
boolean unitWeightAtStart,
+      boolean unitWeightAtEnd) {
+    if (!(weight > 0.0)) {
+      return 0;
+    }
+    int count = 0;
+    if (unitWeightAtStart && weight >= (unitWeightAtEnd ? 2.0 : 1.0)) {
+      appendInfinityCentroid(centroids, means, weights, offset + count++, 
value, 1.0);
+      weight -= 1.0;

Review Comment:
   Fixed in `3d319ff14d` with a shared exact residual check: `weight - residual 
== removedWeight`. Checking the rounded sum in the opposite direction can hide 
the error. Subtracting a unit from 2^53 itself is exact; 2^53 + 2 is the first 
counterexample, and at 2^54 the subtraction can become a no-op.
   
   Fresh unsupported tail additions/merges reject before changing the receiver. 
Native boundary normalization uses the same check so importing a source cannot 
lose mass before reaching the tail writer. Untouched historical payloads remain 
readable/copyable and retain their original bytes, including 2^63 tails; 
mutation or nonempty merging fails when exact normalization is impossible. The 
migration note documents this limit.
   
   The regression checks exact centroid and wire mass with BigDecimal for both 
infinity signs and finite mixtures, supported 2^53, rejected 2^53 + 2 / 2^63 
updates, and unchanged historical bytes/state after rejection. 545 affected 
tests, legacy compatibility and independent review pass.



##########
pinot-segment-local/src/main/java/org/apache/pinot/segment/local/customobject/tdigest/SerializedTDigest.java:
##########
@@ -42,6 +41,6 @@ public int compareTo(SerializedTDigest other) {
 
   @Override
   public String toString() {
-    return 
BytesUtils.toHexString(CustomSerDeUtils.TDIGEST_SER_DE.serialize(_tDigest));
+    return BytesUtils.toHexString(_tDigest.serialize());

Review Comment:
   Fixed in `3d319ff14d`: `SerializedTDigest` now owns the finalized bytes and 
a cached comparison key computed from those bytes. Comparisons and rendering 
therefore use the same distribution, and later changes to the source digest 
cannot affect this final result. This also handles compatibility serialization 
that reduces centroids without mutating the source.
   
   Regressions cover the original comparison sign flip after rendering, a 
51-centroid/low-compression compatibility rewrite, repeated 
rendering/comparison, source mutation, and retained historical 
fractional/boundary states under different requested compression. The existing 
compact-capacity byte-retention test is unchanged and passes. This adds one 
decode per final RAW result; no throughput improvement is claimed. 545 affected 
tests, legacy compatibility and independent review pass.



##########
pinot-segment-local/src/main/java/org/apache/pinot/segment/local/customobject/tdigest/PercentileTDigestAccumulator.java:
##########
@@ -0,0 +1,1703 @@
+/**
+ * Licensed to the Apache Software Foundation (ASF) under one
+ * or more contributor license agreements.  See the NOTICE file
+ * distributed with this work for additional information
+ * regarding copyright ownership.  The ASF licenses this file
+ * to you under the Apache License, Version 2.0 (the
+ * "License"); you may not use this file except in compliance
+ * with the License.  You may obtain a copy of the License at
+ *
+ *   http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing,
+ * software distributed under the License is distributed on an
+ * "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+ * KIND, either express or implied.  See the License for the
+ * specific language governing permissions and limitations
+ * under the License.
+ */
+/*
+ * Portions of the CDF, quantile, weighted-average and K1 weight-limit 
implementation are adapted from
+ * MergingDigest, AbstractTDigest and ScaleFunction in t-digest 3.3 
(https://github.com/tdunning/t-digest).
+ * Modified for Pinot's primitive centroid storage, lazy serialized inputs and 
legacy boundary handling.
+ * Original license notice for these portions:
+ *
+ * Licensed to Ted Dunning under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *     http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+package org.apache.pinot.segment.local.customobject.tdigest;
+
+import java.nio.ByteBuffer;
+import java.util.ArrayList;
+import java.util.Arrays;
+import java.util.Collection;
+import java.util.List;
+import 
org.apache.pinot.segment.local.customobject.tdigest.TDigestCodec.SerializedTDigestMetadata;
+
+
+
+/// Accumulates raw values and serialized TDigests into primitive centroids 
using Pinot's K1 weight-limit rule.
+///
+/// Raw values are sorted in small batches; serialized centroids retain their 
existing sorted order. Both are
+/// linearly merged with accumulated centroids. K1 preserves Pinot's existing 
middle-quantile behavior, and the
+/// legacy verbose and small encodings preserve stored data and mixed-version 
wire compatibility.
+///
+/// Serialized state keeps its first digest pending and allocates centroid 
buffers only when another input must be
+/// merged or the digest is queried. Raw-value buffers remain unallocated 
until a raw value is added.
+///
+/// Historical corrupted payloads remain opaque and byte-exact, with NaN 
statistics. Their reported historical mass
+/// may be invalid; they cannot be mutated or mixed with another distribution. 
Such an operation fails before
+/// changing aggregate mass, because neither invented statistics nor synthetic 
NaN bytes are safe for old readers.
+/// Unchanged historical fractional boundaries retain their original bytes, 
including when configured compression
+/// differs. After a merge, inherited fractional endpoints retain precise 
verbose weights and finite statistics when
+/// unit endpoints cannot preserve their mass. These keep the historical 
t-digest 3.3 assertion-enabled reader and
+/// recompression limitation. Newly introduced unsupported fractional 
endpoints are rejected.
+///
+/// Instances are externally serialized per aggregation or group key and are 
not safe for concurrent mutation.
+public final class PercentileTDigestAccumulator extends TDigest {
+  private static final int MIN_RAW_BUFFER_SIZE = 256;
+  private static final int MAX_RAW_BUFFER_SIZE = 10_000;
+  private static final int MIN_COMPRESSION = 10;
+  private static final int TWO_LEVEL_COMPRESSION_MULTIPLIER = 2;
+  private static final int MIN_INCREMENTAL_COMPRESSION = 50;
+
+  private final double _compression;
+  private final boolean _useTwoLevelRawCompression;
+  private final double _workingCompression;
+  private final double _publicMaxWeightScale;
+  private final double _workingMaxWeightScale;
+  private final int _centroidCapacity;
+  private final int _pendingCentroidCapacity;
+  private double[] _rawValues;
+  private double[] _centroidMeans;
+  private double[] _centroidWeights;
+  private double[] _outputMeans;
+  private double[] _outputWeights;
+  private double[] _incomingMeans;
+  private double[] _incomingWeights;
+  private double[] _sortedIncomingMeans;
+  private double[] _sortedIncomingWeights;
+  private SerializedTDigestInput _serializedTDigestInput;
+  private byte[] _pendingSerializedTDigest;
+  private byte[] _originalFractionalBytes;
+  private byte[] _originalBoundaryBytes;
+  private double _historicalFractionalMinMean = Double.NaN;
+  private double _historicalFractionalMaxMean = Double.NaN;
+  private double _freshFractionalMinMean = Double.NaN;
+  private double _freshFractionalMaxMean = Double.NaN;
+  private SerializedTDigestMetadata _pendingSerializedMetadata;
+  private boolean _legacyDegraded;
+  private boolean _hasFractionalWeights;
+  private boolean _deferBoundaryNormalization;
+
+  private int _numRawValues;
+  private int _numCentroids;
+  private int _numIncomingCentroids;
+  private int _mergeCount;
+  private boolean _publiclyCompressed;
+  private boolean _incomingCentroidsSorted = true;
+  private double _totalWeight;
+  private double _incomingWeight;
+  private double _min = Double.POSITIVE_INFINITY;
+  private double _max = Double.NEGATIVE_INFINITY;
+
+  public PercentileTDigestAccumulator(int compression) {
+    this((double) compression, true, true);
+  }
+
+  public PercentileTDigestAccumulator(double compression) {
+    this(compression, true, true);
+  }
+
+  private PercentileTDigestAccumulator(double compression, boolean 
allocateRawBuffer, boolean allocateMergeBuffers) {
+    this(compression, allocateRawBuffer, allocateMergeBuffers, false);
+  }
+
+  private PercentileTDigestAccumulator(double compression, boolean 
allocateRawBuffer, boolean allocateMergeBuffers,
+      boolean useTwoLevelRawCompression) {
+    TDigestCodec.validateCompression(compression);
+    _useTwoLevelRawCompression = useTwoLevelRawCompression;
+    _compression = Math.max(MIN_COMPRESSION, compression);
+    _workingCompression = TWO_LEVEL_COMPRESSION_MULTIPLIER * _compression;
+    _publicMaxWeightScale = calculateK1MaxWeightScale(_compression);
+    _workingMaxWeightScale = calculateK1MaxWeightScale(_workingCompression);
+    long roundedCompression = (long) Math.ceil(_compression);
+    _centroidCapacity = TDigestCodec.getDefaultCentroidCapacity(_compression);
+    _pendingCentroidCapacity = getPendingCentroidCapacity(_centroidCapacity, 
useTwoLevelRawCompression);
+    if (allocateRawBuffer) {
+      _rawValues = new double[getInitialRawBufferSize(roundedCompression)];
+    }
+    if (allocateMergeBuffers) {
+      int initialCapacity = Math.min(_centroidCapacity, MIN_RAW_BUFFER_SIZE);
+      _centroidMeans = new double[initialCapacity];
+      _centroidWeights = new double[initialCapacity];
+      _outputMeans = new double[initialCapacity];
+      _outputWeights = new double[initialCapacity];
+    }
+  }
+
+  /// Creates a lazy digest with the legacy two-level compression used by 
Star-tree and general digest aggregation.
+  public static PercentileTDigestAccumulator forLegacyAggregation(double 
compression) {
+    return new PercentileTDigestAccumulator(compression, false, false, true);
+  }
+
+  /// Reads and retains one legacy digest, preserving the general aggregation 
compression policy.
+  public static PercentileTDigestAccumulator fromBytes(byte[] bytes) {
+    return fromBytes(ByteBuffer.wrap(bytes));
+  }
+
+  /// Reads one payload and advances the buffer past it.
+  public static PercentileTDigestAccumulator fromBytes(ByteBuffer bytes) {
+    SerializedTDigestInput input = new SerializedTDigestInput();
+    input.reset(bytes);
+    return fromBytes(input);
+  }
+
+  /// Retains an already validated input without rescanning its centroid bytes.
+  public static PercentileTDigestAccumulator fromBytes(SerializedTDigestInput 
input) {
+    PercentileTDigestAccumulator digest = 
forLegacyAggregation(input.getCompression());
+    digest.addSerializedTDigest(input);
+    return digest;
+  }
+
+  public static PercentileTDigestAccumulator forReduction(double compression) {
+    return new PercentileTDigestAccumulator(compression, false, false);
+  }
+
+  /// Snapshots buffered inputs and centroid state without materializing or 
compressing the source. Mutable buffers
+  /// are copied; immutable retained bytes and metadata are shared. Merge and 
sort scratch buffers remain lazy.
+  PercentileTDigestAccumulator copy() {
+    PercentileTDigestAccumulator copy =
+        new PercentileTDigestAccumulator(_compression, false, false, 
_useTwoLevelRawCompression);
+    copy._rawValues = _rawValues != null ? _rawValues.clone() : null;
+    copy._centroidMeans = _numCentroids > 0 ? Arrays.copyOf(_centroidMeans, 
_numCentroids) : null;
+    copy._centroidWeights = _numCentroids > 0 ? 
Arrays.copyOf(_centroidWeights, _numCentroids) : null;
+    copy._incomingMeans = _numIncomingCentroids > 0 ? 
Arrays.copyOf(_incomingMeans, _numIncomingCentroids) : null;
+    copy._incomingWeights = _numIncomingCentroids > 0 ? 
Arrays.copyOf(_incomingWeights, _numIncomingCentroids) : null;
+    copy._pendingSerializedTDigest = _pendingSerializedTDigest;
+    copy._pendingSerializedMetadata = _pendingSerializedMetadata;
+    copy._originalFractionalBytes = _originalFractionalBytes;
+    copy._originalBoundaryBytes = _originalBoundaryBytes;
+    copy._historicalFractionalMinMean = _historicalFractionalMinMean;
+    copy._historicalFractionalMaxMean = _historicalFractionalMaxMean;
+    copy._freshFractionalMinMean = _freshFractionalMinMean;
+    copy._freshFractionalMaxMean = _freshFractionalMaxMean;
+    copy._legacyDegraded = _legacyDegraded;
+    copy._hasFractionalWeights = _hasFractionalWeights;
+    copy._deferBoundaryNormalization = _deferBoundaryNormalization;
+    copy._numRawValues = _numRawValues;
+    copy._numCentroids = _numCentroids;
+    copy._numIncomingCentroids = _numIncomingCentroids;
+    copy._mergeCount = _mergeCount;
+    copy._publiclyCompressed = _publiclyCompressed;
+    copy._incomingCentroidsSorted = _incomingCentroidsSorted;
+    copy._totalWeight = _totalWeight;
+    copy._incomingWeight = _incomingWeight;
+    copy._min = _min;
+    copy._max = _max;
+    return copy;
+  }
+
+  /// Copies primitive centroids without encoding and decoding them. Zero 
weights are ignored, and unordered
+  /// means are sorted when initializing a stored, already compressed 
distribution; caller arrays are not mutated.
+  /// Subsequent inputs use the normal buffered merge. Positive weights and 
consistent extrema are required.
+  public void addCentroids(double[] means, double[] weights, int count, double 
min, double max,
+      boolean alreadyCompressed) {
+    addCentroids(means, weights, count, min, max, alreadyCompressed, false);
+  }
+
+  /// Adds validated legacy centroids without marking their existing 
fractional mass as fresh input.
+  void addHistoricalCentroids(double[] means, double[] weights, int count, 
double min, double max,
+      boolean alreadyCompressed) {
+    addCentroids(means, weights, count, min, max, alreadyCompressed, true);
+  }
+
+  private void addCentroids(double[] means, double[] weights, int count, 
double min, double max,
+      boolean alreadyCompressed, boolean historical) {
+    requireMutable();
+    if (count < 0 || count > means.length || count > weights.length) {
+      throw new IllegalArgumentException("Invalid TDigest centroid count: " + 
count);
+    }
+    double totalWeight = 0.0;
+    int positiveCount = 0;
+    boolean sorted = true;
+    boolean fractionalWeights = false;
+    double previousMean = Double.NEGATIVE_INFINITY;
+    for (int i = 0; i < count; i++) {
+      double weight = weights[i];
+      if (weight == 0.0) {
+        continue;
+      }
+      double mean = means[i];
+      if (!(weight > 0.0) || !Double.isFinite(weight) || !(mean >= min && mean 
<= max)) {
+        throw new IllegalArgumentException("Invalid TDigest centroid");
+      }
+      totalWeight += weight;
+      sorted &= mean >= previousMean;
+      previousMean = mean;
+      fractionalWeights |= weight != Math.rint(weight);
+      positiveCount++;
+    }
+    checkTotalWeight(getTotalWeight() + totalWeight);
+    if (positiveCount == 0) {
+      return;
+    }
+    prepareForMutation();
+    _originalFractionalBytes = null;
+    _hasFractionalWeights |= fractionalWeights;
+    if (!historical && fractionalWeights) {
+      for (int i = 0; i < count; i++) {
+        recordFreshFractionalMean(means[i], weights[i]);
+      }
+    }
+    if (hasNoInputs() && alreadyCompressed) {
+      ensureCentroidCapacity(Math.addExact(positiveCount, 2));
+      for (int i = 0; i < count; i++) {
+        if (weights[i] > 0.0) {
+          _centroidMeans[_numCentroids] = means[i];
+          _centroidWeights[_numCentroids++] = weights[i];
+        }
+      }
+      if (!sorted) {
+        sortCentroids(_centroidMeans, _centroidWeights, _numCentroids);
+      }
+      _totalWeight = totalWeight;
+      _min = min;
+      _max = max;
+      if (!historical) {
+        normalizeBoundaryCentroids();
+      }
+      _publiclyCompressed = true;
+    } else {
+      if (sorted && positiveCount == count) {
+        bufferIncomingCentroids(means, weights, count, totalWeight, min, max);
+      } else {
+        // Caller-owned arrays are copied only when filtering or sorting is 
necessary.
+        double[] sortedMeans = new double[positiveCount];
+        double[] sortedWeights = new double[positiveCount];
+        int index = 0;
+        for (int i = 0; i < count; i++) {
+          if (weights[i] > 0.0) {
+            sortedMeans[index] = means[i];
+            sortedWeights[index++] = weights[i];
+          }
+        }
+        if (!sorted) {
+          sortCentroids(sortedMeans, sortedWeights, positiveCount);
+        }
+        bufferIncomingCentroids(sortedMeans, sortedWeights, positiveCount, 
totalWeight, min, max);
+      }
+      _min = Math.min(_min, min);
+      _max = Math.max(_max, max);
+    }
+  }
+
+  public void add(double[] values, int from, int toExclusive) {
+    if (from >= toExclusive) {
+      return;
+    }
+    requireMutable();
+    prepareForMutation();
+    while (from < toExclusive) {
+      ensureRawBuffer();
+      if (_numRawValues == _rawValues.length
+          || _numRawValues + _numIncomingCentroids == getPendingInputLimit()) {
+        flush();
+      }
+      int numValues = Math.min(toExclusive - from, Math.min(_rawValues.length 
- _numRawValues,
+          getPendingInputLimit() - _numRawValues - _numIncomingCentroids));
+      int rawOffset = _numRawValues;
+      for (int i = 0; i < numValues; i++) {
+        double value = values[from + i];
+        if (Double.isNaN(value)) {
+          throw new IllegalArgumentException("Cannot add NaN to t-digest");
+        }
+        _rawValues[rawOffset + i] = value;
+      }
+      _originalFractionalBytes = null;
+      from += numValues;
+      _numRawValues += numValues;
+      if (_numRawValues == _rawValues.length
+          && (!_useTwoLevelRawCompression || _rawValues.length == 
_pendingCentroidCapacity)
+          || _numRawValues + _numIncomingCentroids == getPendingInputLimit()) {
+        flush();
+      }
+    }
+  }
+
+  @Override
+  public void add(double value) {
+    if (Double.isNaN(value)) {
+      throw new IllegalArgumentException("Cannot add NaN to t-digest");
+    }
+    requireMutable();
+    prepareForMutation();
+    ensureRawBuffer();
+    if (_numRawValues == _rawValues.length
+        || _numRawValues + _numIncomingCentroids == getPendingInputLimit()) {
+      flush();
+    }
+    _originalFractionalBytes = null;
+    _rawValues[_numRawValues++] = value;
+  }
+
+  @Override
+  public void add(double value, double weight) {
+    if (weight == 1) {
+      add(value);
+      return;
+    }
+    if (Double.isNaN(value)) {
+      throw new IllegalArgumentException("Cannot add NaN to t-digest");
+    }
+    if (!(weight > 0.0) || !Double.isFinite(weight)) {
+      throw new IllegalArgumentException("TDigest weight must be positive: " + 
weight);
+    }
+    requireMutable();
+    prepareForMutation();
+    bufferIncomingCentroid(value, weight);
+    _originalFractionalBytes = null;
+    _hasFractionalWeights |= weight != Math.rint(weight);
+    recordFreshFractionalMean(value, weight);
+    _min = Math.min(_min, value);
+    _max = Math.max(_max, value);
+  }
+
+  @Override
+  public void add(TDigest other) {
+    if (other.hasValidStatistics() && other.getTotalWeight() == 0.0) {
+      return;
+    }
+    requireMutable();
+    if (other instanceof PercentileTDigestAccumulator) {
+      addAccumulator((PercentileTDigestAccumulator) other);
+      return;
+    }
+    if (other instanceof NonFiniteAwareTDigest) {
+      PercentileTDigestAccumulator finiteDigest = ((NonFiniteAwareTDigest) 
other).finiteDigestIfNoTails();
+      if (finiteDigest != null) {
+        addAccumulator(finiteDigest);
+        return;
+      }
+    }
+    if (!other.hasValidStatistics()) {
+      materializePendingSerializedTDigest();
+      SerializedTDigestInput input = getSerializedTDigestInput();
+      input.reset(ByteBuffer.wrap(other.serialize()), false);
+      addSerializedTDigest(input);
+      return;
+    }
+    double otherWeight = other.getTotalWeight();
+    checkTotalWeight(getTotalWeight() + otherWeight);
+    Collection<Centroid> centroids = other.centroids();
+    boolean wrapped = other instanceof NonFiniteAwareTDigest;
+    int capacity = wrapped ? Math.addExact(centroids.size(), 2) : 
centroids.size();
+    double[] means = new double[capacity];
+    double[] weights = new double[capacity];
+    int count = 0;
+    for (Centroid centroid : centroids) {
+      double weight = centroid.weight();
+      if (weight == 0.0) {
+        continue;
+      }
+      if (!(weight > 0.0) || !Double.isFinite(weight) || 
Double.isNaN(centroid.mean())) {
+        throw new IllegalArgumentException("Invalid TDigest centroid");
+      }
+      means[count] = centroid.mean();
+      weights[count++] = weight;
+    }
+    if (wrapped) {
+      count = normalizeSerializedBoundaries(means, weights, count, 
other.getMin(), other.getMax());
+    }
+    // Validate all generic centroids before changing the target, then use the 
same whole-source primitive path.
+    addCentroids(means, weights, count, other.getMin(), other.getMax(), false, 
wrapped);
+    inheritHistoricalFractionalBoundaries(other);
+  }
+
+  private void addAccumulator(PercentileTDigestAccumulator other) {
+    if (other == this) {
+      compress();
+      prepareForMutation();
+      if (_numCentroids == 0) {
+        return;
+      }
+      // Snapshot only self-merges; fresh fractional mass need not be 
representable by a legacy wire payload.
+      addHistoricalCentroids(Arrays.copyOf(_centroidMeans, _numCentroids),
+          Arrays.copyOf(_centroidWeights, _numCentroids), _numCentroids, _min, 
_max, false);
+      return;
+    }
+    if (_deferBoundaryNormalization && !other._deferBoundaryNormalization) {
+      other.prepareForMutation();
+    }
+    if (other._originalBoundaryBytes != null) {
+      if (getTotalWeight() != 0.0) {
+        throw unrepresentableBoundaryMutation();
+      }
+      addSerializedTDigest(other._originalBoundaryBytes);
+      return;
+    }
+    if (other._originalFractionalBytes != null && getTotalWeight() == 0.0) {
+      materializePendingSerializedTDigest();
+      SerializedTDigestInput input = getSerializedTDigestInput();
+      input.reset(other._originalFractionalBytes);
+      input._retainedBytes = other._originalFractionalBytes;
+      addSerializedTDigest(input);
+      return;
+    }
+    if (other._pendingSerializedTDigest != null) {
+      // Materialize before resetting the reusable reader: materialization 
uses that same reader.
+      materializePendingSerializedTDigest();
+      SerializedTDigestInput input = getSerializedTDigestInput();
+      input.reset(other._pendingSerializedTDigest, 
other._pendingSerializedMetadata);
+      input._retainedBytes = other._pendingSerializedTDigest;
+      addSerializedTDigest(input);
+      return;
+    }
+    if (other._legacyDegraded) {
+      addSerializedTDigest(other.serialize());
+      return;
+    }
+
+    // MergingDigest compressed merge sources in place. Aggregation owns these 
mutable inputs exclusively.
+    other.compress();
+    other.prepareForMutation();
+    prepareForMutation();
+    checkTotalWeight(_totalWeight + _incomingWeight + _numRawValues + 
other._totalWeight);
+    _hasFractionalWeights |= other._hasFractionalWeights;
+    inheritHistoricalFractionalBoundaries(other);
+    bufferIncomingCentroids(other._centroidMeans, other._centroidWeights, 
other._numCentroids, other._totalWeight,
+        other._min, other._max);
+    if (other._totalWeight > 0.0) {
+      _originalFractionalBytes = null;
+    }
+    _min = Math.min(_min, other._min);
+    _max = Math.max(_max, other._max);
+  }
+
+  public void addSerializedTDigest(byte[] bytes) {
+    if (_legacyDegraded || _originalBoundaryBytes != null) {
+      SerializedTDigestInput input = getSerializedTDigestInput();
+      input.reset(bytes);
+      addSerializedTDigest(input);
+      return;
+    }
+    requireMutable();
+    materializePendingSerializedTDigest();
+    SerializedTDigestInput input = getSerializedTDigestInput();
+    input.reset(bytes);
+    addSerializedTDigest(input);
+  }
+
+  private SerializedTDigestInput getSerializedTDigestInput() {
+    if (_serializedTDigestInput == null) {
+      _serializedTDigestInput = new SerializedTDigestInput();
+    }
+    return _serializedTDigestInput;
+  }
+
+  public void addSerializedTDigest(SerializedTDigestInput input) {
+    if (ignoreEmptyInputForDegradedState(input)) {
+      return;
+    }
+    requireMutable();
+    if (prepareSerializedTDigest(input)) {
+      mergeSerializedTDigest(input, false);
+    }
+  }
+
+  public void addSerializedTDigestDirect(SerializedTDigestInput input) {
+    if (ignoreEmptyInputForDegradedState(input)) {
+      return;
+    }
+    requireMutable();
+    if (prepareSerializedTDigest(input)) {
+      mergeSerializedTDigest(input, true);
+    }
+  }
+
+  private boolean ignoreEmptyInputForDegradedState(SerializedTDigestInput 
input) {
+    if (!_legacyDegraded && _originalBoundaryBytes == null
+        && (_pendingSerializedMetadata == null || 
!_pendingSerializedMetadata.weightedBoundaries())) {
+      return false;
+    }
+    input.inspectMetadata();
+    return !input._metadata._needsLegacyFallback && 
input._metadata._totalWeight == 0.0;
+  }
+
+  private boolean prepareSerializedTDigest(SerializedTDigestInput input) {
+    if (_pendingSerializedTDigest == null && hasNoInputs()) {
+      input.inspectMetadata();
+      _hasFractionalWeights |= input._metadata._fractionalWeights;
+      _pendingSerializedTDigest = input.retainBytes();
+      _pendingSerializedMetadata = input.getMetadata();
+      if (input.recordHistoricalFractionalBoundaries(this)) {
+        // Alias the retained snapshot; historical fractional bytes remain 
byte-exact after read-only materialization.
+        _originalFractionalBytes = _pendingSerializedTDigest;
+      }
+      if (input._metadata._needsLegacyFallback) {
+        _legacyDegraded = true;
+      }
+      return false;
+    }
+    materializePendingSerializedTDigest();
+    input.decode();
+    if (input._metadata._needsLegacyFallback) {
+      if (hasNoInputs()) {
+        _pendingSerializedTDigest = input.retainBytes();
+        _pendingSerializedMetadata = input.getMetadata();
+        _legacyDegraded = true;
+        return false;
+      }
+      throw corruptedMutation();
+    }
+    if (!_deferBoundaryNormalization) {
+      input.normalizeBoundaries();
+    }
+    prepareForMutation();
+    checkTotalWeight(_totalWeight + _incomingWeight + _numRawValues + 
input._metadata._totalWeight);
+    input.recordHistoricalFractionalBoundaries(this);
+    if (input._metadata._totalWeight > 0.0) {
+      _originalFractionalBytes = null;
+    }
+    _hasFractionalWeights |= input._metadata._fractionalWeights;
+    return true;
+  }
+
+  private boolean hasNoInputs() {
+    return _numCentroids == 0 && _totalWeight == 0.0 && _numRawValues == 0 && 
_numIncomingCentroids == 0;
+  }
+
+  private void mergeSerializedTDigest(SerializedTDigestInput input, boolean 
direct) {
+    if (input._metadata._totalWeight == 0.0) {
+      return;
+    }
+    if (hasNoInputs()) {
+      if (input._metadata._totalWeight != 0.0) {
+        ensureCentroidCapacity(input._numCentroids);
+        System.arraycopy(input._means, 0, _centroidMeans, 0, 
input._numCentroids);
+        System.arraycopy(input._weights, 0, _centroidWeights, 0, 
input._numCentroids);
+        _numCentroids = input._numCentroids;
+        _totalWeight = input._metadata._totalWeight;
+        _min = input._metadata._min;
+        _max = input._metadata._max;
+        // Stored centroids already went through the writer's public 
compression. Reading or repairing their
+        // endpoints needs another K1 pass only when the output compression 
differs from the stored compression.
+        _publiclyCompressed = _compression == input._metadata._compression;
+      }
+      return;
+    }
+    if (direct) {
+      flush();
+      boolean runBackwards = (_mergeCount++ & 1) != 0;
+      if (!mergeSorted(input._means, input._weights, input._numCentroids, 
input._metadata._totalWeight,
+          getIncrementalCompression(), runBackwards)) {
+        return;
+      }
+    } else {
+      bufferIncomingCentroids(input._means, input._weights, 
input._numCentroids, input._metadata._totalWeight,
+          input._metadata._min, input._metadata._max);
+    }
+    _min = Math.min(_min, input._metadata._min);
+    _max = Math.max(_max, input._metadata._max);
+  }
+
+  @Override
+  public void compress() {
+    if (_legacyDegraded) {
+      return;
+    }
+    materializePendingSerializedTDigest();
+    if (_useTwoLevelRawCompression) {
+      // Public compression and serialization both retain the legacy working 
pass before final compression.
+      flush();
+    }
+    if (_numRawValues > 0 || _numIncomingCentroids > 0) {
+      // SQL reduction combines pending values and existing centroids directly 
at public compression, matching
+      // MergingDigest 3.3. Its writer avoids an extra working-compression 
pass that can increase reducer rank error.
+      flush(_compression);
+      _publiclyCompressed = true;
+    } else if (!_publiclyCompressed && _numCentroids > 0) {
+      recompressCentroids(_compression);
+      _publiclyCompressed = true;
+    }
+  }
+
+  @Override
+  public boolean hasValidStatistics() {
+    return !_legacyDegraded;
+  }
+
+  @Override
+  public double getTotalWeight() {
+    return _pendingSerializedTDigest != null ? 
_pendingSerializedMetadata.totalWeight()
+        : _totalWeight + _incomingWeight + _numRawValues;
+  }
+
+  /// Bounds serialized output centroids from buffered inputs, including up to 
two boundary splits before a merge
+  /// and two more before writing, without flushing or using the potentially 
huge input mass.
+  public long getCentroidCountUpperBound() {
+    return _pendingSerializedTDigest != null ? (long) 
_pendingSerializedMetadata.centroidCount() + 4L
+        : (long) _numCentroids + _numRawValues + _numIncomingCentroids + 4L;
+  }
+
+  /// Whether unchanged historical fractional boundaries still require their 
original byte representation.
+  public boolean hasOriginalFractionalPayload() {
+    return _originalFractionalBytes != null;
+  }
+
+  @Override
+  public double cdf(double value) {
+    if (Double.isNaN(value) || Double.isInfinite(value)) {
+      throw new IllegalArgumentException(String.format("Invalid value: %f", 
value));
+    }
+    if (_legacyDegraded) {
+      return Double.NaN;
+    }
+    materializePendingSerializedTDigest();
+    flush();
+    if (_numCentroids == 0) {
+      return Double.NaN;
+    }
+    if (_numCentroids == 1) {
+      double width = _max - _min;
+      if (value < _min) {
+        return 0.0;
+      } else if (value > _max) {
+        return 1.0;
+      } else if (value - _min <= width) {
+        return 0.5;
+      } else {
+        return (value - _min) / width;
+      }
+    }
+    if (value < _min) {
+      return 0.0;
+    }
+    if (value > _max) {
+      return 1.0;
+    }
+    if (value < _centroidMeans[0]) {
+      double width = _centroidMeans[0] - _min;
+      if (width > 0.0) {
+        if (_centroidWeights[0] < 1.0) {
+          return (_centroidWeights[0] / 2.0 * (value - _min) / width) / 
_totalWeight;
+        }
+        return value == _min ? 0.5 / _totalWeight
+            : (1.0 + (value - _min) / width * (_centroidWeights[0] / 2.0 - 
1.0)) / _totalWeight;
+      }
+      return 0.0;
+    }
+    int lastIndex = _numCentroids - 1;
+    if (value > _centroidMeans[lastIndex]) {
+      double width = _max - _centroidMeans[lastIndex];
+      if (width > 0.0) {
+        if (_centroidWeights[lastIndex] < 1.0) {
+          return 1.0 - (_centroidWeights[lastIndex] / 2.0 * (_max - value) / 
width) / _totalWeight;
+        }
+        return value == _max ? 1.0 - 0.5 / _totalWeight
+            : 1.0 - (1.0 + (_max - value) / width * 
(_centroidWeights[lastIndex] / 2.0 - 1.0))
+                / _totalWeight;
+      }
+      return 1.0;
+    }
+
+    double weightSoFar = 0.0;
+    for (int i = 0; i < lastIndex; i++) {
+      if (_centroidMeans[i] == value) {
+        double equalWeight = 0.0;
+        int equalIndex = i;
+        while (equalIndex < _numCentroids && _centroidMeans[equalIndex] == 
value) {
+          equalWeight += _centroidWeights[equalIndex];
+          equalIndex++;
+        }
+        return (weightSoFar + equalWeight / 2.0) / _totalWeight;
+      }
+      if (_centroidMeans[i] <= value && _centroidMeans[i + 1] > value) {
+        if (!Double.isFinite(_centroidMeans[i]) || 
!Double.isFinite(_centroidMeans[i + 1])) {
+          if (_centroidMeans[i] == Double.NEGATIVE_INFINITY) {
+            return (weightSoFar + _centroidWeights[i]) / _totalWeight;
+          }
+          if (_centroidWeights[i] == 1.0 && _centroidWeights[i + 1] == 1.0) {
+            return (weightSoFar + 1.0) / _totalWeight;
+          }
+          double leftExcludedWeight = _centroidWeights[i] == 1.0 ? 0.5 : 0.0;
+          double rightExcludedWeight = _centroidWeights[i + 1] == 1.0 ? 0.5 : 
0.0;
+          double interpolatedWeight = (_centroidWeights[i] + 
_centroidWeights[i + 1]) / 2.0
+              - leftExcludedWeight - rightExcludedWeight;
+          double base = weightSoFar + _centroidWeights[i] / 2.0 + 
leftExcludedWeight;
+          return (base + interpolatedWeight * (value - _centroidMeans[i])
+              / (_centroidMeans[i + 1] - _centroidMeans[i])) / _totalWeight;
+        }
+        double width = _centroidMeans[i + 1] - _centroidMeans[i];
+        if (width > 0.0) {
+          double leftExcludedWeight = getPointMassHalfWeight(i);
+          double rightExcludedWeight = getPointMassHalfWeight(i + 1);
+          double halfWeight = (_centroidWeights[i] + _centroidWeights[i + 1]) 
/ 2.0;
+          double interpolatedWeight = halfWeight - leftExcludedWeight - 
rightExcludedWeight;
+          double base = weightSoFar + _centroidWeights[i] / 2.0 + 
leftExcludedWeight;
+          return (base + interpolatedWeight * (value - _centroidMeans[i]) / 
width) / _totalWeight;
+        }
+        return (weightSoFar + (_centroidWeights[i] + _centroidWeights[i + 1]) 
/ 2.0) / _totalWeight;
+      }
+      weightSoFar += _centroidWeights[i];
+    }
+    if (value == _centroidMeans[lastIndex]) {
+      return 1.0 - Math.min(1.0, _centroidWeights[lastIndex]) / 2.0 / 
_totalWeight;
+    }
+    throw new IllegalStateException("Unable to compute TDigest CDF");
+  }
+
+  @Override
+  public double quantile(double quantile) {
+    if (Double.isNaN(quantile) || quantile < 0.0 || quantile > 1.0) {
+      throw new IllegalArgumentException("q should be in [0,1], got " + 
quantile);
+    }
+    if (_legacyDegraded) {
+      return Double.NaN;
+    }
+    materializePendingSerializedTDigest();
+    flush();
+    if (_numCentroids == 0) {
+      return Double.NaN;
+    }
+    if (_numCentroids == 1) {
+      return _centroidMeans[0];
+    }
+    if (quantile == 0.0) {
+      return _min;
+    }
+    if (quantile == 1.0) {
+      return _max;
+    }
+
+    double index = quantile * _totalWeight;
+    double firstWeight = _centroidWeights[0];
+    if (firstWeight < 1.0 && index < firstWeight / 2.0) {
+      return weightedAverage(_min, firstWeight / 2.0 - index, 
_centroidMeans[0], index);
+    }
+    if (firstWeight >= 1.0 && index < 1.0) {
+      return _min;
+    }
+    if (_centroidWeights[0] > 1.0 && index < _centroidWeights[0] / 2.0) {
+      return _min + (index - 1.0) / (_centroidWeights[0] / 2.0 - 1.0) * 
(_centroidMeans[0] - _min);
+    }
+    int lastIndex = _numCentroids - 1;
+    double lastWeight = _centroidWeights[lastIndex];
+    if (lastWeight < 1.0 && index >= _totalWeight - lastWeight / 2.0) {
+      return weightedAverage(_centroidMeans[lastIndex], _totalWeight - index, 
_max,
+          index - (_totalWeight - lastWeight / 2.0));
+    }
+    if (lastWeight >= 1.0 && index > _totalWeight - 1.0) {
+      return _max;
+    }
+    if (_centroidWeights[lastIndex] > 1.0
+        && _totalWeight - index <= _centroidWeights[lastIndex] / 2.0) {
+      return _max - (_totalWeight - index - 1.0) / 
(_centroidWeights[lastIndex] / 2.0 - 1.0)
+          * (_max - _centroidMeans[lastIndex]);
+    }
+
+    double weightSoFar = _centroidWeights[0] / 2.0;
+    for (int i = 0; i < lastIndex; i++) {
+      double halfWeight = (_centroidWeights[i] + _centroidWeights[i + 1]) / 
2.0;
+      if (weightSoFar + halfWeight > index) {
+        double leftExcludedWeight = getPointMassHalfWeight(i);
+        if (index - weightSoFar < leftExcludedWeight) {
+          return _centroidMeans[i];
+        }
+        double rightExcludedWeight = getPointMassHalfWeight(i + 1);
+        if (weightSoFar + halfWeight - index <= rightExcludedWeight) {
+          return _centroidMeans[i + 1];
+        }
+        double leftWeight = index - weightSoFar - leftExcludedWeight;
+        double rightWeight = weightSoFar + halfWeight - index - 
rightExcludedWeight;
+        return weightedAverage(_centroidMeans[i], rightWeight, 
_centroidMeans[i + 1], leftWeight);
+      }
+      weightSoFar += halfWeight;
+    }
+    double weightToMax = index - _totalWeight - _centroidWeights[lastIndex] / 
2.0;
+    double weightFromLastCentroid = _centroidWeights[lastIndex] / 2.0 - 
weightToMax;
+    return weightedAverage(_centroidMeans[lastIndex], weightToMax, _max, 
weightFromLastCentroid);
+  }
+
+  private double getPointMassHalfWeight(int index) {
+    // Adjacent equal means establish a repeated-value plateau. Keep its 
complete mass out of interpolation at
+    // either edge; an isolated weighted centroid still represents an unknown 
distribution around its mean.
+    double mean = _centroidMeans[index];
+    boolean pointMass = _centroidWeights[index] == 1.0 || index > 0 && 
_centroidMeans[index - 1] == mean
+        || index + 1 < _numCentroids && _centroidMeans[index + 1] == mean;
+    return pointMass ? _centroidWeights[index] / 2.0 : 0.0;
+  }
+
+  private static double weightedAverage(double firstValue, double firstWeight, 
double secondValue,
+      double secondWeight) {
+    if (firstValue > secondValue) {
+      return weightedAverage(secondValue, secondWeight, firstValue, 
firstWeight);
+    }
+    if (firstWeight == 0.0) {
+      return secondValue;
+    }
+    if (secondWeight == 0.0 || firstValue == secondValue) {
+      return firstValue;
+    }
+    if (!Double.isFinite(firstValue)) {
+      return firstValue;
+    }
+    if (!Double.isFinite(secondValue)) {
+      return secondValue;
+    }
+    double average = (firstValue * firstWeight + secondValue * secondWeight) / 
(firstWeight + secondWeight);
+    return Math.max(firstValue, Math.min(average, secondValue));
+  }
+
+  @Override
+  public Collection<Centroid> centroids() {
+    if (_legacyDegraded) {
+      byte[] bytes = serialize();
+      ByteBuffer encoded = ByteBuffer.wrap(bytes);
+      SerializedTDigestMetadata metadata = 
TDigestCodec.readSerializedHeader(encoded);
+      encoded.position(metadata.centroidOffset());
+      List<Centroid> centroids = new ArrayList<>(metadata.centroidCount());
+      for (int i = 0; i < metadata.centroidCount(); i++) {
+        double weight = metadata.centroidSize() == 
TDigestCodec.VERBOSE_CENTROID_SIZE
+            ? encoded.getDouble() : encoded.getFloat();
+        double mean = metadata.centroidSize() == 
TDigestCodec.VERBOSE_CENTROID_SIZE
+            ? encoded.getDouble() : encoded.getFloat();
+        centroids.add(new Centroid(mean, weight));
+      }
+      return centroids;
+    }
+    materializePendingSerializedTDigest();
+    flush();
+    List<Centroid> centroids = new ArrayList<>(_numCentroids);
+    for (int i = 0; i < _numCentroids; i++) {
+      centroids.add(new Centroid(_centroidMeans[i], _centroidWeights[i]));
+    }
+    return centroids;
+  }
+
+  /// Copies a valid distribution's primitive centroid arrays after flushing 
buffered inputs. This does not apply
+  /// public compression or serialize a finite sub-distribution; callers can 
include surrounding infinity mass.
+  public void copyCentroids(double[] means, double[] weights, int offset) {
+    if (_legacyDegraded) {
+      throw corruptedMutation();
+    }
+    materializePendingSerializedTDigest();
+    flush();
+    if (means == weights || offset < 0 || offset > means.length - _numCentroids
+        || offset > weights.length - _numCentroids) {
+      throw new IllegalArgumentException("Insufficient distinct TDigest 
centroid arrays");
+    }
+    if (_numCentroids == 0) {
+      return;
+    }
+    System.arraycopy(_centroidMeans, 0, means, offset, _numCentroids);
+    System.arraycopy(_centroidWeights, 0, weights, offset, _numCentroids);
+  }
+
+  @Override
+  public double compression() {
+    return _compression;
+  }
+
+  /// Bounds compatible output without flushing. Unchanged historical bytes 
and retained pending payloads can
+  /// exceed the newly encoded legacy-capacity bound and retain their actual 
length.
+  @Override
+  public int maxSerializedByteSize() {
+    if (_legacyDegraded) {
+      return _pendingSerializedTDigest.length;
+    }
+    if (_originalFractionalBytes != null) {
+      return _originalFractionalBytes.length;
+    }
+    if (_originalBoundaryBytes != null) {
+      return _originalBoundaryBytes.length;
+    }
+    long centroidCount = getCentroidCountUpperBound();
+    if (!_hasFractionalWeights) {
+      centroidCount = (long) Math.min(centroidCount, 
Math.ceil(getTotalWeight()));
+    }
+    int maximumBytes = 
TDigestCodec.getMaxLegacyCompatibleByteSize(_compression, centroidCount);
+    return _pendingSerializedTDigest != null ? Math.max(maximumBytes, 
_pendingSerializedTDigest.length) : maximumBytes;
+  }
+
+  /// Returns the centroid view's count, including repaired endpoints and 
excluding zero-mass serialized entries.
+  @Override
+  public int centroidCount() {
+    if (_pendingSerializedTDigest != null) {
+      if (_legacyDegraded) {
+        return _pendingSerializedMetadata.centroidCount();
+      }
+      if (_pendingSerializedMetadata.isCanonicalPayload()) {
+        return _pendingSerializedMetadata.centroidCount();
+      }
+      materializePendingSerializedTDigest();
+    }
+    flush();
+    return _numCentroids;
+  }
+
+  @Override
+  public double getMin() {
+    if (_pendingSerializedTDigest != null) {
+      return !_legacyDegraded && _pendingSerializedMetadata.totalWeight() == 
0.0
+          ? Double.POSITIVE_INFINITY : _pendingSerializedMetadata.min();
+    }
+    flush();
+    return _min;
+  }
+
+  @Override
+  public double getMax() {
+    if (_pendingSerializedTDigest != null) {
+      return !_legacyDegraded && _pendingSerializedMetadata.totalWeight() == 
0.0
+          ? Double.NEGATIVE_INFINITY : _pendingSerializedMetadata.max();
+    }
+    flush();
+    return _max;
+  }
+
+  @Override
+  public byte[] serialize() {
+    if (_legacyDegraded) {
+      return _pendingSerializedTDigest.clone();
+    }
+    if (_originalFractionalBytes != null) {
+      return _originalFractionalBytes.clone();
+    }
+    if (_originalBoundaryBytes != null) {
+      return _originalBoundaryBytes.clone();
+    }
+    if (_pendingSerializedTDigest != null) {
+      if (!_pendingSerializedMetadata.isCanonicalPayload()) {
+        materializePendingSerializedTDigest();
+      } else {
+        if (ByteBuffer.wrap(_pendingSerializedTDigest).getInt() == 
TDigestCodec.VERBOSE_ENCODING) {
+          byte[] serialized = 
TDigestCodec.makeLegacyCompatible(_pendingSerializedTDigest, 
_pendingSerializedMetadata);
+          return serialized == _pendingSerializedTDigest ? serialized.clone() 
: serialized;
+        }
+        return _pendingSerializedTDigest.clone();
+      }
+    }
+    if (_originalBoundaryBytes != null) {
+      return _originalBoundaryBytes.clone();
+    }
+    compress();
+    normalizeBoundaryCentroids();
+    checkFractionalBoundaryEncoding();
+    return TDigestCodec.serializeCompatibleCentroids(_compression, _min, _max, 
_centroidMeans, _centroidWeights,
+        _numCentroids, hasInheritedFractionalBoundaryEncoding());
+  }
+
+  private void recompressCentroids(double compression) {
+    normalizeBoundaryCentroids();
+    double[] means = _centroidMeans;
+    double[] weights = _centroidWeights;
+    int numCentroids = _numCentroids;
+    double totalWeight = _totalWeight;
+    _numCentroids = 0;
+    _totalWeight = 0.0;
+    boolean runBackwards = (_mergeCount++ & 1) != 0;
+    if (!mergeSorted(means, weights, numCentroids, totalWeight, compression, 
runBackwards)) {
+      throw new IllegalStateException("Cannot recompress an empty TDigest");
+    }
+  }
+
+  private void checkFractionalBoundaryEncoding() {
+    if (_numCentroids > 0 && !hasInheritedFractionalBoundaryEncoding()) {
+      TDigestCodec.checkLegacyBoundaryWeights(_numCentroids, 
_centroidWeights[0], _centroidWeights[_numCentroids - 1]);
+    }
+  }
+
+  private boolean hasInheritedFractionalBoundaryEncoding() {
+    return hasInheritedFractionalBoundaryEncoding(_centroidMeans, 
_centroidWeights, _numCentroids);
+  }
+
+  /// Applies the canonical inherited-endpoint check to a complete digest's 
sorted centroid view. Enclosing writers
+  /// use this after adding their infinity tails; unsupported fresh global 
endpoints never inherit permission.
+  boolean hasInheritedFractionalBoundaryEncoding(double[] means, double[] 
weights, int count) {
+    if (count < 0 || count > 0 && (count > means.length || count > 
weights.length)) {
+      throw new IllegalArgumentException("Invalid TDigest centroid count: " + 
count);
+    }
+    if (count == 0) {
+      return false;
+    }
+    return TDigestCodec.hasInheritedFractionalBoundaryEncoding(this, count, 
means[0], weights[0],
+        means[count - 1], weights[count - 1]);
+  }
+
+  @Override
+  double getHistoricalFractionalBoundaryMean(boolean lowerBoundary) {
+    return lowerBoundary ? _historicalFractionalMinMean : 
_historicalFractionalMaxMean;
+  }
+
+  @Override
+  double getFreshFractionalBoundaryMean(boolean lowerBoundary) {
+    return lowerBoundary ? _freshFractionalMinMean : _freshFractionalMaxMean;
+  }
+
+  void recordFreshFractionalMean(double mean, double weight) {
+    if (weight != Math.rint(weight)) {
+      inheritFreshFractionalMeans(mean, mean);
+    }
+  }
+
+  private void inheritFreshFractionalMeans(double minMean, double maxMean) {
+    if (!Double.isNaN(minMean)) {
+      _freshFractionalMinMean = Double.isNaN(_freshFractionalMinMean) ? minMean
+          : Math.min(_freshFractionalMinMean, minMean);
+    }
+    if (!Double.isNaN(maxMean)) {
+      _freshFractionalMaxMean = Double.isNaN(_freshFractionalMaxMean) ? maxMean
+          : Math.max(_freshFractionalMaxMean, maxMean);
+    }
+  }
+
+  /// Propagates provenance from a validated source without exposing arbitrary 
endpoint setters.
+  void inheritHistoricalFractionalBoundaries(TDigest source) {
+    requireMutable();
+    inheritFreshFractionalMeans(source.getFreshFractionalBoundaryMean(true),
+        source.getFreshFractionalBoundaryMean(false));
+    
inheritHistoricalFractionalBoundaries(source.getHistoricalFractionalBoundaryMean(true),
+        source.getHistoricalFractionalBoundaryMean(false));
+  }
+
+  void inheritHistoricalFractionalBoundaries(double minMean, double maxMean) {
+    if (!Double.isNaN(minMean)) {
+      _historicalFractionalMinMean = 
Double.isNaN(_historicalFractionalMinMean) ? minMean
+          : Math.min(_historicalFractionalMinMean, minMean);
+    }
+    if (!Double.isNaN(maxMean)) {
+      _historicalFractionalMaxMean = 
Double.isNaN(_historicalFractionalMaxMean) ? maxMean
+          : Math.max(_historicalFractionalMaxMean, maxMean);
+    }
+  }
+
+  private void flush() {
+    // SQL accumulators merge raw inputs incrementally at configured 
compression. General digest aggregation keeps
+    // the legacy two-level raw merge, preserving sparse Star-tree query 
accuracy. Incremental buffer exhaustion or
+    // centroid/statistics reads use working compression for reducer inputs; 
final serialization bypasses that pass.
+    flush(_numIncomingCentroids == 0 && !_useTwoLevelRawCompression
+        ? getIncrementalCompression()
+        : _workingCompression);
+  }
+
+  private double getIncrementalCompression() {
+    // Very low compression leaves too little centroid headroom for 
incremental public-compression merges to retain
+    // t-digest 3.3's accuracy. Keep the two-level strategy for those uncommon 
configurations.
+    return _compression >= MIN_INCREMENTAL_COMPRESSION ? _compression : 
_workingCompression;
+  }
+
+  private void flush(double compression) {
+    if (_numIncomingCentroids > 0) {
+      flushIncoming(compression);
+      return;
+    }
+    if (_numRawValues == 0) {
+      return;
+    }
+
+    Arrays.sort(_rawValues, 0, _numRawValues);
+    double rawMin = _rawValues[0];
+    double rawMax = _rawValues[_numRawValues - 1];
+    boolean runBackwards = (_mergeCount++ & 1) != 0;
+    mergeSorted(_rawValues, null, _numRawValues, _numRawValues, compression, 
runBackwards);
+    _numRawValues = 0;
+    _min = Math.min(_min, rawMin);
+    _max = Math.max(_max, rawMax);
+  }
+
+  private void flushIncoming(double compression) {
+    // Match mergeSorted's association before rearranging raw or paired 
centroid buffers.
+    checkTotalWeight(_totalWeight + (_incomingWeight + _numRawValues));
+    double rawMin = Double.POSITIVE_INFINITY;
+    double rawMax = Double.NEGATIVE_INFINITY;
+    if (_numRawValues > 0) {
+      Arrays.sort(_rawValues, 0, _numRawValues);
+      rawMin = _rawValues[0];
+      rawMax = _rawValues[_numRawValues - 1];
+      ensureIncomingCapacity(Math.addExact(_numIncomingCentroids, 
_numRawValues));
+      System.arraycopy(_incomingMeans, 0, _incomingMeans, _numRawValues, 
_numIncomingCentroids);
+      System.arraycopy(_incomingWeights, 0, _incomingWeights, _numRawValues, 
_numIncomingCentroids);
+      for (int i = 0; i < _numRawValues; i++) {
+        _incomingMeans[i] = _rawValues[i];
+        _incomingWeights[i] = 1.0;
+      }
+      _numIncomingCentroids += _numRawValues;
+      _incomingWeight += _numRawValues;
+      _numRawValues = 0;
+      _incomingCentroidsSorted = false;
+    }
+
+    int incomingCount = _numIncomingCentroids;
+    double incomingWeight = _incomingWeight;
+    double[] incomingMeans = _incomingMeans;
+    double[] incomingWeights = _incomingWeights;
+    if (!_incomingCentroidsSorted) {
+      ensureSortedIncomingCapacity(incomingCount);
+      stableSortIncoming(incomingCount);
+      incomingMeans = _incomingMeans;
+      incomingWeights = _incomingWeights;
+      _incomingCentroidsSorted = true;
+    }
+    boolean runBackwards = (_mergeCount & 1) != 0;
+    if (!mergeSorted(incomingMeans, incomingWeights, incomingCount, 
incomingWeight, compression, runBackwards)) {
+      throw new IllegalStateException("Cannot flush an empty TDigest input 
buffer");
+    }
+    _mergeCount++;
+    _numIncomingCentroids = 0;
+    _incomingWeight = 0.0;
+    _incomingCentroidsSorted = true;
+    _min = Math.min(_min, rawMin);
+    _max = Math.max(_max, rawMax);
+  }
+
+  private boolean mergeSorted(double[] incomingMeans, double[] 
incomingWeights, int incomingCount,
+      double incomingWeight, double compression, boolean runBackwards) {
+    if (incomingWeight == 0.0) {
+      return false;
+    }
+
+    double newTotalWeight = _totalWeight + incomingWeight;
+    checkTotalWeight(newTotalWeight);
+    normalizeBoundaryCentroids();
+    int preferredOutputCapacity = getPreferredOutputCapacity(incomingCount);
+    ensureOutputCapacity(1);
+    double totalWeightNormalizer = 1.0 / newTotalWeight;
+    double weightNormalizer = totalWeightNormalizer / 
getK1MaxWeightScale(compression);
+    int incomingIndex = runBackwards ? incomingCount - 1 : 0;
+    int centroidIndex = runBackwards ? _numCentroids - 1 : 0;
+    int incomingLimit = runBackwards ? -1 : incomingCount;
+    int centroidLimit = runBackwards ? -1 : _numCentroids;
+    int direction = runBackwards ? -1 : 1;
+    int numInputs = incomingCount + _numCentroids;
+    int inputIndex = 0;
+    int numOutputCentroids = 0;
+    double weightSoFar = 0.0;
+    double firstInputMean = _numCentroids == 0 ? incomingMeans[0]
+        : Math.min(incomingMeans[0], _centroidMeans[0]);
+    double lastInputMean = _numCentroids == 0 ? incomingMeans[incomingCount - 
1]
+        : Math.max(incomingMeans[incomingCount - 1], 
_centroidMeans[_numCentroids - 1]);
+    boolean allFiniteMeans = Double.isFinite(firstInputMean) && 
Double.isFinite(lastInputMean);
+    boolean sameSignMeans = firstInputMean > 0.0 || lastInputMean < 0.0;
+    while (incomingIndex != incomingLimit || centroidIndex != centroidLimit) {
+      double mean;
+      double weight;
+      boolean takeIncoming = centroidIndex == centroidLimit || (incomingIndex 
!= incomingLimit
+          && (runBackwards ? incomingMeans[incomingIndex] > 
_centroidMeans[centroidIndex]
+              : incomingMeans[incomingIndex] <= 
_centroidMeans[centroidIndex]));
+      if (takeIncoming) {
+        mean = incomingMeans[incomingIndex];
+        weight = incomingWeights == null ? 1.0 : 
incomingWeights[incomingIndex];
+        incomingIndex += direction;
+      } else {
+        mean = _centroidMeans[centroidIndex];
+        weight = _centroidWeights[centroidIndex];
+        centroidIndex += direction;
+      }
+      if (numOutputCentroids == 0) {
+        _outputMeans[0] = mean;
+        _outputWeights[0] = weight;
+        numOutputCentroids = 1;
+      } else {
+        int currentIndex = numOutputCentroids - 1;
+        double proposedWeight = _outputWeights[currentIndex] + weight;
+        double q0 = weightSoFar * totalWeightNormalizer;
+        double q2 = (weightSoFar + proposedWeight) * totalWeightNormalizer;
+        double normalizedWeight = proposedWeight * weightNormalizer;
+        double normalizedWeightSquared = normalizedWeight * normalizedWeight;
+        boolean canMerge = (allFiniteMeans || 
canMergeMeans(_outputMeans[currentIndex], mean))
+            && normalizedWeightSquared <= q0 * (1.0 - q0)
+            && normalizedWeightSquared <= q2 * (1.0 - q2);
+        if (inputIndex == 1 || inputIndex == numInputs - 1) {
+          canMerge = false;
+        }
+        if (canMerge) {
+          _outputWeights[currentIndex] = proposedWeight;
+          _outputMeans[currentIndex] =
+              TDigestCodec.weightedMean(_outputMeans[currentIndex], 
proposedWeight - weight, mean, weight,
+                  proposedWeight, sameSignMeans);
+        } else {
+          weightSoFar += _outputWeights[currentIndex];
+          ensureOutputCapacity(Math.max(preferredOutputCapacity, 
numOutputCentroids + 1));
+          _outputMeans[numOutputCentroids] = mean;
+          _outputWeights[numOutputCentroids] = weight;
+          numOutputCentroids++;
+        }
+      }
+      inputIndex++;
+    }
+    if (runBackwards) {
+      reverse(_outputMeans, numOutputCentroids);
+      reverse(_outputWeights, numOutputCentroids);
+    }
+    finishMerge(numOutputCentroids, newTotalWeight, compression);
+    return true;
+  }
+
+  private void finishMerge(int numOutputCentroids, double totalWeight, double 
compression) {
+    double[] temporary = _centroidMeans;
+    _centroidMeans = _outputMeans;
+    _outputMeans = temporary;
+    temporary = _centroidWeights;
+    _centroidWeights = _outputWeights;
+    _outputWeights = temporary;
+    _numCentroids = numOutputCentroids;
+    _totalWeight = totalWeight;
+    _publiclyCompressed = compression == _compression;
+  }
+
+  private void ensureIncomingCapacity(int capacity) {
+    if (_incomingMeans == null || capacity > _incomingMeans.length) {
+      int newCapacity = _incomingMeans == null
+          ? Math.max(capacity, Math.min(_centroidCapacity, 
_pendingCentroidCapacity))
+          : Math.max(capacity, (int) Math.min(_pendingCentroidCapacity,
+              (long) _incomingMeans.length * 
TDigestCodec.DEFAULT_MERGE_BUFFER_MULTIPLIER));
+      _incomingMeans = _incomingMeans == null ? new double[newCapacity] : 
Arrays.copyOf(_incomingMeans, newCapacity);
+      _incomingWeights =
+          _incomingWeights == null ? new double[newCapacity] : 
Arrays.copyOf(_incomingWeights, newCapacity);
+    }
+  }
+
+  private void stableSortIncoming(int count) {

Review Comment:
   Consolidated in `3d319ff14d`: incoming flushes and stored-centroid repair 
now use one paired stable-merge implementation. Flushes retain reusable scratch 
arrays and avoid copy-back by swapping buffers. The two required ordering 
policies remain explicit: numeric-equal stability (including signed zero) for 
the hot merge path, and total ordering for stored repair, including NaNs.
   
   The regression checks stable weight/mean pairing, signed zeros, NaNs and 
untouched suffixes; existing repeated-flush tests remain. Independent 
validation covered 6,156 sort cases across both policies. 545 affected tests, 
precommit checks and independent Pinot review pass.



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