@@ -174,6 +174,14 @@ whitelist rather than the node model: `Providedf` is an ordinary two-child node,
 `Construct` a line for it was the whole fix. What remains is genuinely about the node model —
 per-node data that is not a child, of which `Codomain` is the only instance the library has today.
 
+*And narrower again, measured rather than argued.* **No written pattern can put a condition into the
+graph at all.** Only 35 of the 324 rules have a right-hand side written as a pattern, and none of
+them mentions `provided` — so a `Providedf` reaches an e-class only along the code-built path, where
+a rule extracts a witness term, runs code on it and puts the result back. That is worth knowing
+before anyone reopens this: the union that would need a meaning is not one e-matching can currently
+form. `ProvidedInAnEClassTest.NoWrittenPatternIntroducesACondition` holds it at zero, so the first
+two-sided conditional rule fails there rather than quietly making the question live again.
+
 ## What is not decided here
 
 The one question above, restated as the decision it is: **how much should ride along on an e-node
 
@@ -0,0 +1,125 @@
+//
+// 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>
+    /// What an e-class does with a <see cref="Entity.Providedf"/>, which is the one question
+    /// <c>Docs/Contributing/EqualitySaturationReviewFindings.md</c> leaves open and the thing
+    /// standing between the rewrite graph and a production caller.
+    /// </summary>
+    [Trait("Area", "Core")]
+    public sealed class ProvidedInAnEClassTest
+    {
+        private static Entity? Saturate(string source, RewriteRuleGrowth ceiling)
+        {
+            var graph = new EGraph();
+            var root = graph.AddEntity(source.ToEntity());
+            var ledger = BudgetLedger.For(nameof(ProvidedInAnEClassTest),
+                new WorkBudget { Steps = 200_000, Time = TimeSpan.FromSeconds(20) });
+            Saturation.Run(graph, Saturation.RulesUpTo(ceiling), ledger,
+                node => node.Complexity);
+            return graph.Extract(root, node => node.Complexity);
+        }
+
+        /// <summary>
+        /// <b>A condition is not a decoration on a value, and a union says it is.</b> Two nodes
+        /// join an e-class when the graph is told they are equal. <c>p</c> and
+        /// <c>p provided c</c> are <i>not</i> equal — they differ exactly where <c>c</c> is false,
+        /// which is the only place the condition was written for — so a rule producing the second
+        /// from the first is asking the graph to record something untrue, and extraction may then
+        /// hand back either.
+        /// </summary>
+        [Fact]
+        public void AConditionedValueIsNotTheValue()
+        {
+            var bare = "1".ToEntity();
+            var conditioned = "1 provided not x = 0".ToEntity();
+
+            Assert.NotEqual(bare, conditioned);
+            Assert.Equal("1", bare.Substitute("x", 0).Evaled.Stringize());
+            Assert.Equal("NaN", conditioned.Substitute("x", 0).Evaled.Stringize());
+        }
+
+        /// <summary>
+        /// <b>No condition can enter the graph through a written pattern.</b> Which narrows the
+        /// open question considerably: a <c>Providedf</c> reaches an e-class only along the
+        /// code-built path, where a rule extracts a witness term, runs code on it and puts the
+        /// result back.
+        /// </summary>
+        /// <remarks>
+        /// Only 35 of the 324 rules have a right-hand side written as a pattern at all, and none
+        /// of those mentions a condition — so the case
+        /// <c>EqualitySaturationReviewFindings.md</c> leaves open cannot arise from e-matching as
+        /// it stands. It arises from the other 289. Asserted at zero so that writing the first
+        /// two-sided conditional rule fails here rather than quietly making the question live.
+        /// </remarks>
+        [Fact]
+        public void NoWrittenPatternIntroducesACondition()
+        {
+            var withCondition = Saturation.RulesUpTo(RewriteRuleGrowth.Unknown)
+                .Where(rule => rule.Right is not null
+                               && rule.Right.ToString()!.Contains("provided"))
+                .ToList();
+
+            Assert.True(withCondition.Count == 0,
+                $"{withCondition.Count} two-sided rules now mention a condition — "
+                + "the e-class question is live for them, and what a union of `p` with "
+                + "`p provided c` is supposed to mean has to be decided before they fire: "
+                + string.Join(", ", withCondition.Select(r => r.Name)));
+        }
+
+        /// <summary>
+        /// <b>The safe ceiling is nearly inert on ordinary input.</b> Four of these five come back
+        /// exactly as they went in; only the Pythagorean identity moves. That is not a defect in
+        /// the expressions — it is what the graph can do today with the rules it is allowed to
+        /// use, and it is the measurement
+        /// <a href="https://github.com/asc-community/AngouriMath/issues/746">#746</a> tier 2 needs
+        /// before the graph is wired into anything.
+        /// </summary>
+        /// <remarks>
+        /// <c>SafeRules</c> is <c>RulesUpTo(Rearranges)</c>, 142 of 324 rules. The 152 sitting at
+        /// <c>Unknown</c> are excluded by design, since their growth was never judged — so the
+        /// ceiling that is safe to run finds little, and the ceiling that finds things admits
+        /// rewrites nobody measured. That trade is the open part of tier 2, and this pins where it
+        /// currently sits rather than describing it.
+        /// </remarks>
+        [Fact]
+        public void TheSafeCeilingIsNearlyInertOnOrdinaryInput()
+        {
+            var moved = new List<string>();
+            var still = new List<string>();
+            foreach (var source in new[]
+            {
+                "(a + b) / (a * b)", "x / x", "(x + 1) / (x + 1)",
+                "sin(x) ^ 2 + cos(x) ^ 2", "(x + 1) * (x - 1)",
+            })
+            {
+                var extracted = Saturate(source, RewriteRuleGrowth.Rearranges);
+                if (extracted is null || extracted == source.ToEntity())
+                    still.Add(source);
+                else
+                    moved.Add($"{source} -> {extracted.Stringize()}");
+            }
+
+            Assert.True(still.Count == 4 && moved.Count == 1,
+                $"the safe ceiling moved {moved.Count} of 5 and left {still.Count}; it was 1 and 4.\n"
+                + "moved:\n" + string.Join("\n", moved)
+                + "\nunchanged:\n" + string.Join("\n", still));
+        }
+    }
+}