The growth ceiling is the scheduling policy, and the runaway is bounded to trig

Sources/AngouriMath/Core/Transformations/Saturation.cs

@@ -65,6 +65,16 @@ internal static class Saturation
         /// classified may fire at all. Naming that as the widest setting makes it a request a
         /// caller has to make on purpose, which is the most this can honestly do about it.
         /// </para>
+        /// <para>
+        /// <b>And the ceiling is the scheduling policy, not a stand-in for one.</b> When both
+        /// directions of an inverse pair are declared, <c>RulesUpTo(Rearranges)</c> keeps the
+        /// collecting one and withholds the expanding one, mechanically, for every such pair.
+        /// <c>sin(2x)</c> and <c>2 sin x cos x</c> run away at the widest ceiling and saturate at
+        /// the safe one because <c>ExpandMultipleAngle</c> declares
+        /// <see cref="RewriteRuleGrowth.Expands"/>. Growth is the direction marker. A pair whose
+        /// expanding direction is unjudged is the one case this cannot protect, and that is what
+        /// declaring a growth for a rule is for. <c>RunawayBreadthTest</c> holds this.
+        /// </para>
         /// </remarks>
         internal static IReadOnlyList<MatchedRule> RulesUpTo(RewriteRuleGrowth widest)
             => MatchedRules.All

Sources/AngouriMath/Docs/Contributing/InversePairTable.md

@@ -141,6 +141,18 @@ What this is and is not:
   held by `SaturationAblationTest` in both directions. It is also the measured answer to what
   #746 tier 2 called "a scheduling policy for `Expands`/`Unknown` rules": on this input the
   runaway is not a tier, it is one pair.
+
+  And the pair is bounded. The two `Common` coefficient rules are confluent on plain arithmetic —
+  `2 * x * y`, `(1/2) * x` and `2 * (x * y) / 3` saturate in a handful of nodes at the widest
+  ceiling, with the trigonometric sets present and with all of them removed — so making them
+  confluent is not a branch. The runaway is the sine family (`sin(2x)`, `sin(x) cos(x)`, `sin(3x)`,
+  `sin(2x) cos(2x)`; not `cos(2x)`), and it does not exist without the trigonometric sets.
+  **The growth ceiling already withholds one direction of it.** `ExpandMultipleAngle`'s rules are
+  declared `Expands`, so `Saturation.SafeRules` — `RulesUpTo(Rearranges)` — fires only the
+  collecting direction, and every member of the family saturates there. Growth is the direction
+  marker and the ceiling is the scheduling policy, for every pair whose two directions are
+  declared; a pair whose expanding direction is unjudged is the one case it cannot protect. Held by
+  `RunawayBreadthTest`.
 - **It needs no reversibility mechanism at all**, which is the whole of what makes it available
   today: it is behaviour observed at the top of the pipeline, not a property computed about a rule.
 - **It answers the question equality saturation actually asks.** The consumer in the opening quote

Sources/Tests/UnitTests/Core/Transformations/RunawayBreadthTest.cs

@@ -0,0 +1,161 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using System;
+using System.Collections.Generic;
+using System.Linq;
+using AngouriMath.Core;
+using AngouriMath.Core.Budgets;
+using AngouriMath.Core.Transformations;
+using AngouriMath.Core.Transformations.Matching;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Core.Transformations
+{
+    /// <summary>
+    /// How broad the saturation runaway is, and where it is not. <see cref="SaturationAblationTest"/>
+    /// names the four rules behind one input; this holds the facts that decide what to do about
+    /// them, so that a branch already measured dead cannot be re-proposed.
+    /// </summary>
+    /// <remarks>
+    /// <para>
+    /// Two branches were on the table for the inverse pair <c>sin(2x)</c> / <c>2 sin x cos x</c>:
+    /// withhold one direction from saturation, or make the two <c>Common</c> coefficient rules
+    /// confluent so the graph could absorb the pair. The second assumed those rules were the
+    /// problem. <b>They are not.</b> On plain arithmetic — <c>2 * x * y</c>, <c>(1/2) * x</c>,
+    /// <c>2 * (x * y) / 3</c> — the widest ceiling saturates in a handful of nodes, with every
+    /// trigonometric set present and with all of them removed. The coefficient rules were
+    /// load-bearing in the runaway only because they supplied the respellings the trigonometric
+    /// pair fed on.
+    /// </para>
+    /// <para>
+    /// <b>The runaway is a family, and it is bounded to the trigonometric sets.</b> Every sine
+    /// input reachable from a product loops at the widest ceiling — <c>sin(2x)</c>,
+    /// <c>sin(x) cos(x)</c>, <c>sin(3x)</c>, <c>sin(2x) cos(2x)</c> — and <c>cos(2x)</c> does not.
+    /// With the five trigonometric sets removed, the exemplar <c>sin(2x) + cos(2x)</c> saturates
+    /// at seven nodes.
+    /// </para>
+    /// <para>
+    /// <b>And the safe ceiling already withholds the expanding direction.</b>
+    /// <c>ExpandMultipleAngle</c>'s rules are declared <see cref="RewriteRuleGrowth.Expands"/>,
+    /// and <see cref="Saturation.SafeRules"/> is <c>RulesUpTo(Rearranges)</c>, so at that ceiling
+    /// only the collecting direction fires and every member of the family saturates. That is the
+    /// measured answer to <a href="https://github.com/asc-community/AngouriMath/issues/746">#746</a>
+    /// tier 2's "scheduling policy" item: the growth ceiling <i>is</i> the policy. When both
+    /// directions of an inverse pair are declared, it keeps the collecting one and withholds the
+    /// expanding one, mechanically, for every such pair — and it can only protect pairs whose
+    /// directions are declared, which is what declaring growth for a code-built rule is for.
+    /// </para>
+    /// </remarks>
+    [Trait("Area", "Core")]
+    public sealed class RunawayBreadthTest
+    {
+        private static readonly WorkBudget Quick = new() { Steps = 3_000, Time = TimeSpan.FromSeconds(1) };
+
+        private static readonly string[] TrigonometricSets =
+        {
+            "Trigonometric", "ExpandMultipleAngle", "ExpandTrigonometric",
+            "CollapseTrigonometricFunctions", "NormalTrigonometricForm",
+        };
+
+        private static IReadOnlyList<MatchedRule> Rules(RewriteRuleGrowth ceiling, Func<MatchedRuleSet, bool> keep) =>
+            MatchedRules.All
+                .Where(keep)
+                .SelectMany(set => set.Rules)
+                .Where(rule => rule.Soundness is Soundness.Sound or Soundness.SoundUnderAssumptions)
+                .Where(rule => rule.Growth <= ceiling)
+                .ToList();
+
+        private static (bool Saturated, int Nodes, string Extracted) Saturate(string source, IReadOnlyList<MatchedRule> rules)
+        {
+            var graph = new EGraph();
+            var root = graph.AddEntity(source.ToEntity());
+            var ledger = BudgetLedger.For(nameof(RunawayBreadthTest), Quick);
+            var saturated = Saturation.Run(graph, rules, ledger, node => node.Complexity);
+            return (saturated, graph.NodeCount, graph.Extract(root, node => node.Complexity)?.Stringize() ?? "(nothing)");
+        }
+
+        private static readonly (string Input, string Extracts)[] PlainArithmetic =
+        {
+            ("2 * x * y", "2 * x * y"),
+            ("(1/2) * x", "x / 2"),
+            ("2 * (x * y) / 3", "2/3 * x * y"),
+        };
+
+        [Fact]
+        public void TheCoefficientRulesAreConfluentOnPlainArithmetic()
+        {
+            var failures = new List<string>();
+            foreach (var (input, extracts) in PlainArithmetic)
+                foreach (var (label, rules) in new[]
+                {
+                    ("all sets", Rules(RewriteRuleGrowth.Unknown, _ => true)),
+                    ("no trigonometric sets", Rules(RewriteRuleGrowth.Unknown, set => !TrigonometricSets.Contains(set.Name))),
+                })
+                {
+                    var (saturated, nodes, extracted) = Saturate(input, rules);
+                    if (!saturated || extracted != extracts)
+                        failures.Add($"{input} with {label}: saturated={saturated} nodes={nodes} => {extracted}");
+                }
+
+            Assert.True(failures.Count == 0,
+                "the coefficient rules no longer settle on plain arithmetic — making them confluent "
+                + "stopped being a phantom branch:\n" + string.Join("\n", failures));
+        }
+
+        private static readonly string[] Family =
+            { "sin(2 * x)", "sin(x) * cos(x)", "sin(3 * x)", "sin(2 * x) * cos(2 * x)" };
+
+        [Fact]
+        public void TheWidestCeilingRunsAwayOnTheWholeFamilyAndNotOnCosine()
+        {
+            var all = Rules(RewriteRuleGrowth.Unknown, _ => true);
+            var settled = Family.Where(input => Saturate(input, all).Saturated).ToList();
+            Assert.True(settled.Count == 0,
+                "these members of the family saturate at the widest ceiling now; delete them from "
+                + "the list rather than leave them: " + string.Join(", ", settled));
+
+            var cosine = Saturate("cos(2 * x)", all);
+            Assert.True(cosine.Saturated,
+                $"cos(2 * x) runs away at the widest ceiling now ({cosine.Nodes} nodes); the family "
+                + "has grown a cosine side and the remark above is wrong about the asymmetry");
+        }
+
+        [Fact]
+        public void TheFamilyIsBoundedToTheTrigonometricSets()
+        {
+            var noTrig = Rules(RewriteRuleGrowth.Unknown, set => !TrigonometricSets.Contains(set.Name));
+            var (saturated, nodes, _) = Saturate("sin(2 * x) + cos(2 * x)", noTrig);
+            Assert.True(saturated,
+                $"the exemplar runs away ({nodes} nodes) with every trigonometric set removed, so the "
+                + "runaway is no longer bounded to them");
+        }
+
+        /// <summary>
+        /// The claim that makes the ceiling the policy. Asserted, not inferred: at
+        /// <see cref="RewriteRuleGrowth.Rearranges"/> the expanding direction is withheld and every
+        /// member of the family must settle.
+        /// </summary>
+        [Fact]
+        public void TheSafeCeilingSaturatesTheWholeFamily()
+        {
+            var safe = Rules(RewriteRuleGrowth.Rearranges, _ => true);
+            var runsAway = new List<string>();
+            foreach (var input in Family.Append("sin(2 * x) + cos(2 * x)"))
+            {
+                var (saturated, nodes, extracted) = Saturate(input, safe);
+                if (!saturated) runsAway.Add($"{input}: {nodes} nodes => {extracted}");
+            }
+
+            Assert.True(runsAway.Count == 0,
+                "these run away at the safe ceiling, so the growth ceiling is not withholding the "
+                + "expanding direction and is not the policy the remark says it is:\n"
+                + string.Join("\n", runsAway));
+        }
+    }
+}