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The growth ceiling is the scheduling policy, and the runaway is bounded to trig - #1194
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…ed to trig #1193 named the four rules behind sin(2x) + cos(2x) running away at the widest ceiling and left the decision open: withhold one direction of the inverse pair, or make the two Common coefficient rules confluent so the graph could absorb it. Measuring how broad the runaway is settles both, and the facts are pinned so neither branch can be re-proposed from the symptom. The coefficient rules are confluent on plain arithmetic. 2 * x * y, (1/2) * x and 2 * (x * y) / 3 saturate at the widest ceiling in a handful of nodes, with every trigonometric set present and with all five removed, and extract 2 * x * y, x / 2 and 2/3 * x * y. They were load-bearing in the runaway only because they supplied the respellings the trigonometric pair fed on; making them confluent is not a branch. The runaway is a family and it is bounded to the trigonometric sets. sin(2x), sin(x) cos(x), sin(3x) and sin(2x) cos(2x) all run away at the widest ceiling, cos(2x) does not, and with Trigonometric, ExpandMultipleAngle, ExpandTrigonometric, CollapseTrigonometricFunctions and NormalTrigonometricForm removed the exemplar saturates at seven nodes. And the safe ceiling already withholds the expanding direction. ExpandMultipleAngle's rules are declared Expands, so RulesUpTo(Rearranges) fires only the collecting one and every member of the family saturates there. That was inferred from the declarations when #1193 went in; RunawayBreadthTest asserts it now. Growth is the direction marker and the ceiling is the scheduling policy, mechanically, for every pair whose two directions are declared -- a pair whose expanding direction is unjudged is the one case it cannot protect, which is what declaring a growth for a rule is for. The RulesUpTo remark and InversePairTable.md say so. Four tests, 300 ms between them, bounded by the same 3,000-step ceiling as the ablation. Each failure message says which way the fact moved and what to do about it. Part of #746. Full suite 9578 passed, 0 failed. Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
This was referenced Sep 7, 2026
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…asured on the corpus (#1198) * The e-graph folds a rational on insertion, and the safe ceiling is measured on the corpus #746 tier 2's next item after declaring growth was to evaluate saturation on what Simplify actually sees rather than on sixteen expressions. Run over the corpus gate's forty problems and the two benchmark inputs at the safe ceiling, the graph is cheap -- every one saturates, none above 65 e-nodes, the 473-node SimplifyHard input in 12 ms -- and finds little: 6 of the 15 Simplify problems move and 3 match Simplify's answer, because what Simplify does on the rest is either fold a number, which no rule does, or fire a rule the ceiling withholds. Run over the growth corpus's 3,630 generated shapes, 54 did not saturate within 20,000 steps, and every one of the 54 was constant-only arithmetic: 2 - 0 + 0 * 2, 1/2 * 2 * sin(y), 2 + -2 + (-2) / (-1). With nothing folding a number the rearranging rules respell it for ever -- 2 * 1/2, 2 / 2 and 1/2 * 2 are three e-nodes -- so on pure constants the safe ceiling did not terminate, and two inputs overshot a two-second wall by fifteen and thirty-five times. So EGraph.Add folds an arithmetic operator over two rational leaves into that number's class, beside the neutral-element fold it already had: only where the value is itself a rational (a quotient by zero and an irrational root stay as written), only for a whole exponent of modest size (2 ^ 100000 is not computed), and never for a node carrying a codomain of its own, which is the conflation ENodeIdentityTest exists to prevent. The 54 become 7 and the run goes from more than ten minutes to twenty-four seconds; on the corpus, sqrt(2) * sqrt(3) reaches sqrt(6) and (x ^ 2) ^ 3 - x ^ 6 reaches 0, so the ceiling moves 7 of 15 and matches Simplify on 5. What is left is a stall, not a runaway, and it is pinned as one: 2 * x * 1/2, x / x * -x, x ^ 2 / x and their relatives are a rational coefficient beside a variable whose spellings -- 1/2 * x, x / 2, x * 2 ^ (-1) -- the Rearranges rules keep exchanging. Fifteen times the budget still finds no fixed point, but the graph barely grows and every one extracts the right answer. That corrects one sentence of #1194: the coefficient rules settle on the three inputs it pinned and are not confluent as a family; the breadth test, the inverse-pair table and this test now say so. ConstantFoldTest holds all of it -- what folds, what does not, that the constant-only inputs saturate, that the stall family stalls, and the corpus figures exactly, so a declaration that widens or narrows the ceiling shows up as the count it changed. Part of #746. Full suite 9601 passed, 0 failed. Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura * Record what the constant fold changes, which is almost nothing public Measured on a build of master (d9b8b37) and of this branch with one throwaway probe run on each. Transformation.CanonicalizationOverGraph runs the rule pass on either side of the graph, and that pass already folds constants, so sqrt(2) * sqrt(3), x + 3 / 3, 2 ^ 1000 and 1 / 0 answer exactly as before; the fold's effect is inside saturation. The one public change on the inputs probed is (x ^ 2) ^ 3 - x ^ 6, which was -x ^ 6 + x ^ 6 and is 0, because the graph now sees the two terms as one while saturating rather than after. Entity.Canonicalize() is unchanged. Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura --------- Co-authored-by: Claude Fable 5.1 <noreply@anthropic.com>
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…s measured (#1203) EMatching.md deferred whether Expands rules ever join SafeRules to "a separate document's question, once one exists", and Transformations.md's list of what is left said the graph had no production caller. Both were true when written and are not now: #1193 and #1194 measured that the growth ceiling is the scheduling policy and that Expands rules do not join the safe set; #1198 and #1202 found what the graph did need -- a rational folded on insertion and an extraction that is a fixed point from the leaves up -- by running the safe ceiling over the corpus; and #1201 gave the graph its first production caller. The documents point at those rather than at a question nobody is going to open. Part of #746. Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura Co-authored-by: Claude Fable 5.1 <noreply@anthropic.com>
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#1193 named the four rules behind
sin(2x) + cos(2x)running away at the widest ceiling and leftthe decision open: withhold one direction of the inverse pair, or make the two
Commoncoefficientrules confluent so the graph could absorb it. Measuring how broad the runaway is settles both, and
this pins the facts so neither branch can be re-proposed from the symptom.
2 * x * y,(1/2) * x,2 * (x * y) / 3saturate at the widest ceiling in a handful of nodes, with every trigonometric set and with all five removed, extracting2 * x * y,x / 2,2/3 * x * y. Making them confluent is not a branch.sin(2x),sin(x) cos(x),sin(3x),sin(2x) cos(2x)all run away at the widest ceiling;cos(2x)does not; with the five trigonometric sets removed the exemplar saturates at seven nodes.ExpandMultipleAngle's rules are declaredExpands, soRulesUpTo(Rearranges)fires only the collecting one, and every member of the family saturates there.The third was inferred from the declarations when #1193 went in.
RunawayBreadthTestasserts itnow: growth is the direction marker and the ceiling is the scheduling policy, mechanically, for
every pair whose two directions are declared. A pair whose expanding direction is unjudged is the
one case it cannot protect — which is what declaring a growth for a rule is for. The
RulesUpToremark and
InversePairTable.mdsay so, and point at the test.Four tests, 300 ms between them, on the same 3,000-step ceiling as the ablation. Each failure
message says which way the fact moved and what to do about it — a family member that starts
saturating is to be deleted from the list, not left.
Part of #746.
Checks
Full suite 9578 passed, 0 failed, 14 skipped.
🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura