{"id":"construction-correctness-universal-for-valid-inputs","text":"The combined construction techniques (exact arithmetic, sentinel initialization, streaming invariants, ordering independence) achieve correct output for every input within LeetCode's stated constraints.","truth_value":"IN","source":"","source_url":"","source_hash":"","justifications":[{"type":"SL","antecedents":["correctness-by-construction-not-validation","streaming-needs-no-external-ordering"],"outlist":["zero-input-returns-wrong-result","find-difference-reduce-no-initial-value"],"label":"Construction correctness plus ordering independence should imply universal correctness for constrained inputs — gated on known bugs where construction techniques fail despite valid input"}],"dependents":["within-domain-correctness-comprehensive"],"metadata":{"last_reviewed":"2026-06-07T22:02:22","review_result":"invalid"},"created_at":"","updated_at":"","reviewed_at":"","verified_at":"","retracted_at":"","explanation":{"steps":[{"node":"construction-correctness-universal-for-valid-inputs","truth_value":"IN","reason":"SL justification valid","antecedents":["correctness-by-construction-not-validation","streaming-needs-no-external-ordering"],"label":"Construction correctness plus ordering independence should imply universal correctness for constrained inputs — gated on known bugs where construction techniques fail despite valid input","outlist":["zero-input-returns-wrong-result","find-difference-reduce-no-initial-value"]},{"node":"correctness-by-construction-not-validation","truth_value":"IN","reason":"SL justification valid","antecedents":["exactness-over-performance-at-every-layer","sentinel-values-bootstrap-streaming-state","leetcode-judge-optimized-not-reusable"],"label":"Three independent construction mechanisms collectively eliminate the need for runtime validation"},{"node":"exactness-over-performance-at-every-layer","truth_value":"IN","reason":"SL justification valid","antecedents":["integer-arithmetic-avoids-float-precision","string-over-arithmetic-for-digit-ops"],"label":"Both patterns sacrifice theoretical efficiency (string ops are slower than arithmetic, isqrt has overhead) for the same reason: eliminating an entire class of precision bugs rather than reasoning about when they'd actually trigger"},{"node":"integer-arithmetic-avoids-float-precision","truth_value":"IN","reason":"SL justification valid","antecedents":["isqrt-over-sqrt-for-large-inputs","pivot-integer-isqrt-not-sqrt","sum-substitution-avoids-float-precision","percentage-floor-integer-arithmetic"],"label":"four independent solutions independently chose integer arithmetic to dodge the same class of bug (float precision near integer boundaries), indicating a deliberate defensive pattern"},{"node":"isqrt-over-sqrt-for-large-inputs","truth_value":"IN","reason":"premise"},{"node":"pivot-integer-isqrt-not-sqrt","truth_value":"IN","reason":"premise"},{"node":"sum-substitution-avoids-float-precision","truth_value":"IN","reason":"premise"},{"node":"percentage-floor-integer-arithmetic","truth_value":"IN","reason":"premise"},{"node":"string-over-arithmetic-for-digit-ops","truth_value":"IN","reason":"SL justification valid","antecedents":["str-conversion-digit-extraction-idiom","string-based-digit-check-idiom","digit-sum-via-str-conversion","bin-count-for-popcount"],"label":"String conversion is the universal digit-decomposition idiom, chosen for readability over performance"},{"node":"str-conversion-digit-extraction-idiom","truth_value":"IN","reason":"premise"},{"node":"string-based-digit-check-idiom","truth_value":"IN","reason":"premise"},{"node":"digit-sum-via-str-conversion","truth_value":"IN","reason":"premise"},{"node":"bin-count-for-popcount","truth_value":"IN","reason":"premise"},{"node":"sentinel-values-bootstrap-streaming-state","truth_value":"IN","reason":"SL justification valid","antecedents":["sentinel-initialization-encodes-boundary-conditions","o1-space-via-running-accumulators"],"label":"Sentinels and accumulators are co-dependent: accumulators need correct initial state to avoid first-iteration branching, and sentinels exist precisely to provide that state — neither pattern works well without the other"},{"node":"sentinel-initialization-encodes-boundary-conditions","truth_value":"IN","reason":"SL justification valid","antecedents":["ascending-check-uses-sentinel-minus-one","k-length-apart-sentinel-minus-one","prev-zero-sentinel","getheight-sentinel-neg1"],"label":"all four use domain-specific sentinels (-1 for \"no previous index\", 0 for \"no previous group\", -1 for \"any subtree unbalanced\") that make the first loop iteration produce the correct vacuous result without an explicit guard"},{"node":"ascending-check-uses-sentinel-minus-one","truth_value":"IN","reason":"premise"},{"node":"k-length-apart-sentinel-minus-one","truth_value":"IN","reason":"premise"},{"node":"prev-zero-sentinel","truth_value":"IN","reason":"premise"},{"node":"getheight-sentinel-neg1","truth_value":"IN","reason":"premise"},{"node":"o1-space-via-running-accumulators","truth_value":"IN","reason":"SL justification valid","antecedents":["highest-altitude-single-pass","longest-task-single-pass-o1-space","iterative-reversal-O1-space","min-tracking-pattern-shared"],"label":"Running accumulators trade re-traversal impossibility for constant memory across streaming-style problems"},{"node":"highest-altitude-single-pass","truth_value":"IN","reason":"premise"},{"node":"longest-task-single-pass-o1-space","truth_value":"IN","reason":"premise"},{"node":"iterative-reversal-O1-space","truth_value":"IN","reason":"premise"},{"node":"min-tracking-pattern-shared","truth_value":"IN","reason":"premise"},{"node":"leetcode-judge-optimized-not-reusable","truth_value":"IN","reason":"SL justification valid","antecedents":["no-validation-is-deliberate-contract","in-place-mutation-with-return-convention","duplication-over-shared-infrastructure"],"label":"each decision is rational for a judge environment (trusted input, single caller, no shared state) but would be a defect in reusable library code"},{"node":"no-validation-is-deliberate-contract","truth_value":"IN","reason":"SL justification valid","antecedents":["solutions-no-input-validation","no-input-validation-convention","solutions-trust-leetcode-preconditions","leetcode-solutions-no-validation-convention"],"label":"Validation omission is a consistent design decision, not accumulated technical debt"},{"node":"solutions-no-input-validation","truth_value":"IN","reason":"premise"},{"node":"no-input-validation-convention","truth_value":"IN","reason":"premise"},{"node":"solutions-trust-leetcode-preconditions","truth_value":"IN","reason":"premise"},{"node":"leetcode-solutions-no-validation-convention","truth_value":"IN","reason":"premise"},{"node":"in-place-mutation-with-return-convention","truth_value":"IN","reason":"SL justification valid","antecedents":["in-place-mutation-return-convention","in-place-sort-mutation-pattern","assign-cookies-mutates-inputs","fused-reverse-invert"],"label":"The mutate-and-return idiom matches LeetCode's expected interface pattern"},{"node":"in-place-mutation-return-convention","truth_value":"IN","reason":"premise"},{"node":"in-place-sort-mutation-pattern","truth_value":"IN","reason":"premise"},{"node":"assign-cookies-mutates-inputs","truth_value":"IN","reason":"premise"},{"node":"fused-reverse-invert","truth_value":"IN","reason":"premise"},{"node":"duplication-over-shared-infrastructure","truth_value":"IN","reason":"SL justification valid","antecedents":["treenode-is-de-facto-shared-via-inline-copies","tree-serialization-helpers-duplicated","per-problem-data-structure-isolation","repo-no-cross-problem-imports"],"label":"Architecture deliberately prioritizes isolation over deduplication"},{"node":"treenode-is-de-facto-shared-via-inline-copies","truth_value":"IN","reason":"premise"},{"node":"tree-serialization-helpers-duplicated","truth_value":"IN","reason":"premise"},{"node":"per-problem-data-structure-isolation","truth_value":"IN","reason":"premise"},{"node":"repo-no-cross-problem-imports","truth_value":"IN","reason":"premise"},{"node":"streaming-needs-no-external-ordering","truth_value":"IN","reason":"SL justification valid","antecedents":["single-pass-streaming-dominant-shape","sentinel-values-bootstrap-streaming-state"],"label":"Streaming is self-contained while sort-then-scan depends on external ordering"},{"node":"single-pass-streaming-dominant-shape","truth_value":"IN","reason":"SL justification valid","antecedents":["early-exit-optimizations-pervasive","o1-space-via-running-accumulators","extend-or-reset-canonical-consecutive-pattern"],"label":"these three patterns compose into a unified streaming shape — accumulators provide O(1) state, extend-or-reset handles consecutive-element logic, and early-exit bounds work to the minimum needed"},{"node":"early-exit-optimizations-pervasive","truth_value":"IN","reason":"SL justification valid","antecedents":["three-consecutive-odds-early-exit","path-crossing-early-exit","isomorphic-early-return","first-violation-sufficiency"],"label":"Early exit is the default control flow strategy, not an optimization afterthought"},{"node":"three-consecutive-odds-early-exit","truth_value":"IN","reason":"premise"},{"node":"path-crossing-early-exit","truth_value":"IN","reason":"premise"},{"node":"isomorphic-early-return","truth_value":"IN","reason":"premise"},{"node":"first-violation-sufficiency","truth_value":"IN","reason":"premise"},{"node":"extend-or-reset-canonical-consecutive-pattern","truth_value":"IN","reason":"SL justification valid","antecedents":["extend-or-reset-pattern","maxpower-eager-max-update","no-post-loop-fixup-needed"],"label":"Run-tracking with eager max avoids off-by-one errors that end-of-array special cases introduce"},{"node":"extend-or-reset-pattern","truth_value":"IN","reason":"premise"},{"node":"maxpower-eager-max-update","truth_value":"IN","reason":"premise"},{"node":"no-post-loop-fixup-needed","truth_value":"IN","reason":"premise"}]}}