{"id":"streaming-fixed-point-of-solution-space","text":"Streaming simultaneously coincides with four independent characterizations of optimality — convergence attractor (lowest adoption barrier), algebraic normal form (all solutions reduce to it), universal strategy (three orthogonal extension axes), and minimal strategy (fewest prerequisites) — establishing it as the unique fixed point of the solution space under all natural orderings, contingent on streaming algorithms terminating for all valid inputs.","truth_value":"IN","source":"","source_url":"","source_hash":"","justifications":[{"type":"SL","antecedents":["convergence-attractor-coincides-with-normal-form","streaming-universality-and-minimality-coincide"],"outlist":["negative-input-causes-nontermination"],"label":"four independent optimality criteria converging on one paradigm constitutes a fixed-point claim, which requires the paradigm to actually terminate"}],"dependents":["naming-drift-structurally-inevitable-at-fixed-point"],"metadata":{"source_type":"derived","last_reviewed":"2026-06-07T22:02:22","review_result":"invalid"},"created_at":"","updated_at":"","reviewed_at":"","verified_at":"","retracted_at":"","explanation":{"steps":[{"node":"streaming-fixed-point-of-solution-space","truth_value":"IN","reason":"SL justification valid","antecedents":["convergence-attractor-coincides-with-normal-form","streaming-universality-and-minimality-coincide"],"label":"four independent optimality criteria converging on one paradigm constitutes a fixed-point claim, which requires the paradigm to actually terminate","outlist":["negative-input-causes-nontermination"]},{"node":"convergence-attractor-coincides-with-normal-form","truth_value":"IN","reason":"SL justification valid","antecedents":["abstraction-cost-predicts-convergence-strength","streaming-normal-form-is-minimal-strategy"],"label":"Connects a formal structural property (normal form minimality) with an empirical observation (convergence pattern predictable from abstraction cost) to show they identify the same strategy"},{"node":"abstraction-cost-predicts-convergence-strength","truth_value":"IN","reason":"SL justification valid","antecedents":["adoption-barrier-gradient-explains-convergence-pattern","abstraction-overhead-explains-strategy-hierarchy"],"label":"both describe the same hierarchy from different causal angles (adoption vs. infrastructure); combining reveals abstraction cost is the measurable predictor of convergence"},{"node":"adoption-barrier-gradient-explains-convergence-pattern","truth_value":"IN","reason":"SL justification valid","antecedents":["convergence-without-coordination-at-every-level","streaming-dominates-because-lowest-adoption-barrier"],"label":"The observation of multi-level convergence combined with the causal explanation for streaming dominance reveals adoption barrier as the universal gradient governing convergence strength"},{"node":"convergence-without-coordination-at-every-level","truth_value":"IN","reason":"SL justification valid","antecedents":["algorithmic-coherence-emerges-without-engineering","correctness-by-construction-not-validation"],"label":"If algorithmic paradigms and correctness strategies both converge without engineering, the convergence is a property of the problem domain, not the codebase"},{"node":"algorithmic-coherence-emerges-without-engineering","truth_value":"IN","reason":"SL justification valid","antecedents":["repo-optimized-for-submission-not-engineering","two-paradigms-cover-solution-space"],"label":"Convergence without coordination reveals domain-imposed structure rather than engineered consistency"},{"node":"repo-optimized-for-submission-not-engineering","truth_value":"IN","reason":"SL justification valid","antecedents":["leetcode-judge-optimized-not-reusable","no-consistency-enforcement-at-any-level"],"label":"Judge optimization (no validation, mutation, duplication) combines with absence of enforcement to produce a codebase where correctness is local and accidental consistency is the only kind"},{"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":"no-consistency-enforcement-at-any-level","truth_value":"IN","reason":"SL justification valid","antecedents":["isolation-enables-style-drift","test-colocation-dual-mode-inconsistent","duplication-over-shared-infrastructure"],"label":"isolation removes the forcing function, duplication removes the single point of standardization, and dual-mode testing confirms that even simple conventions (where tests live) have drifted"},{"node":"isolation-enables-style-drift","truth_value":"IN","reason":"SL justification valid","antecedents":["repo-no-cross-problem-imports","solution-class-vs-standalone-function","method-name-mismatch-pattern","leetcode-repo-mixed-function-style"],"label":"Zero coupling means zero convention enforcement; each directory evolves independently"},{"node":"solution-class-vs-standalone-function","truth_value":"IN","reason":"premise"},{"node":"method-name-mismatch-pattern","truth_value":"IN","reason":"premise"},{"node":"leetcode-repo-mixed-function-style","truth_value":"IN","reason":"premise"},{"node":"test-colocation-dual-mode-inconsistent","truth_value":"IN","reason":"SL justification valid","antecedents":["tests-colocated-in-solution-file","test-files-import-sibling-solution","some-solutions-bundle-tests-inline","repo-solution-test-convention"],"label":"Both test patterns work but the inconsistency reflects the same isolation-driven style drift as the class-vs-function split"},{"node":"tests-colocated-in-solution-file","truth_value":"IN","reason":"premise"},{"node":"test-files-import-sibling-solution","truth_value":"IN","reason":"premise"},{"node":"some-solutions-bundle-tests-inline","truth_value":"IN","reason":"premise"},{"node":"repo-solution-test-convention","truth_value":"IN","reason":"premise"},{"node":"two-paradigms-cover-solution-space","truth_value":"IN","reason":"SL justification valid","antecedents":["single-pass-streaming-dominant-shape","sort-then-two-pointer-dominant-pair-pipeline"],"label":"The two depth-2 algorithmic shapes partition the solution space nearly completely; problems not fitting either are the exceptions (binary search, closed-form, divide-and-conquer)"},{"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":"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":"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"},{"node":"sort-then-two-pointer-dominant-pair-pipeline","truth_value":"IN","reason":"SL justification valid","antecedents":["sort-preprocessing-enables-linear-scan","two-pointer-primary-linear-array-technique"],"label":"sort provides the sorted precondition that two-pointer inward sweep and sorted-pair matching require; these two depth-1 patterns co-occur in two-sum-less-than-k, array-partition, and meeting-rooms families"},{"node":"sort-preprocessing-enables-linear-scan","truth_value":"IN","reason":"SL justification valid","antecedents":["sort-then-two-pointer-pattern","meeting-rooms-sort-then-scan","array-partition-sort-greedy","subsequence-limited-sum-greedy-sort"],"label":"Sorting is the universal complexity bridge from quadratic brute-force to n-log-n solutions"},{"node":"sort-then-two-pointer-pattern","truth_value":"IN","reason":"premise"},{"node":"meeting-rooms-sort-then-scan","truth_value":"IN","reason":"premise"},{"node":"array-partition-sort-greedy","truth_value":"IN","reason":"premise"},{"node":"subsequence-limited-sum-greedy-sort","truth_value":"IN","reason":"premise"},{"node":"two-pointer-primary-linear-array-technique","truth_value":"IN","reason":"SL justification valid","antecedents":["two-pointer-convergence-linear-time","two-pointer-sorted-array-pattern","two-pointer-backward-fill-avoids-sort","sort-then-two-pointer-pattern"],"label":"Two-pointer subsumes multiple problem families into a single O(n) framework"},{"node":"two-pointer-convergence-linear-time","truth_value":"IN","reason":"premise"},{"node":"two-pointer-sorted-array-pattern","truth_value":"IN","reason":"premise"},{"node":"two-pointer-backward-fill-avoids-sort","truth_value":"IN","reason":"premise"},{"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":"streaming-dominates-because-lowest-adoption-barrier","truth_value":"IN","reason":"SL justification valid","antecedents":["streaming-is-self-sufficient-paradigm","algorithmic-coherence-emerges-without-engineering"],"label":"Causal explanation — streaming dominates because its zero-dependency property minimizes the coordination cost that the repo's isolation maximizes"},{"node":"streaming-is-self-sufficient-paradigm","truth_value":"IN","reason":"SL justification valid","antecedents":["streaming-needs-no-external-ordering","correctness-by-construction-not-validation"],"label":"Combining independence from preprocessing with independence from validation identifies streaming as uniquely self-contained"},{"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":"abstraction-overhead-explains-strategy-hierarchy","truth_value":"IN","reason":"SL justification valid","antecedents":["lookup-abstractions-instantiate-pipeline-phases","streaming-is-privileged-default-strategy"],"label":"Streaming's privilege is the inverse of alternatives' abstraction burden; combining the load-bearing role of lookup abstractions with streaming's privilege reveals the causal mechanism"},{"node":"lookup-abstractions-instantiate-pipeline-phases","truth_value":"IN","reason":"SL justification valid","antecedents":["lookup-abstractions-enable-linear-time-across-paradigms","canonical-pipeline-has-exactly-two-instantiations"],"label":"Connects the abstract pipeline decomposition (two forms) with the concrete computational mechanism (three lookup abstractions) that makes each form work"},{"node":"lookup-abstractions-enable-linear-time-across-paradigms","truth_value":"IN","reason":"SL justification valid","antecedents":["lookup-abstraction-trio-covers-all-queries","sorted-order-enables-all-efficient-search"],"label":"The trio bridges preprocessing and scanning — without these abstractions, preprocessing would not improve asymptotic complexity of the scan phase"},{"node":"lookup-abstraction-trio-covers-all-queries","truth_value":"IN","reason":"SL justification valid","antecedents":["counter-universal-frequency-primitive","set-for-o1-membership-universal","binary-search-variants-share-convergence-structure"],"label":"Three abstractions partition the query-type space with no overlap and no gaps"},{"node":"counter-universal-frequency-primitive","truth_value":"IN","reason":"SL justification valid","antecedents":["counter-dominant-frequency-tool","counter-pattern-dominates-frequency-problems","counter-missing-key-returns-zero","counter-subtraction-drops-nonpositive"],"label":"Counter's built-in semantics eliminate boilerplate that manual dicts would require"},{"node":"counter-dominant-frequency-tool","truth_value":"IN","reason":"premise"},{"node":"counter-pattern-dominates-frequency-problems","truth_value":"IN","reason":"premise"},{"node":"counter-missing-key-returns-zero","truth_value":"IN","reason":"premise"},{"node":"counter-subtraction-drops-nonpositive","truth_value":"IN","reason":"premise"},{"node":"set-for-o1-membership-universal","truth_value":"IN","reason":"SL justification valid","antecedents":["set-conversion-before-loop-for-o1-lookup","k-distant-uses-set-dedup","find-difference-set-minus-idiom","two-out-of-three-set-algebra"],"label":"Set algebra (intersection, difference, membership) replaces manual iteration wherever applicable"},{"node":"set-conversion-before-loop-for-o1-lookup","truth_value":"IN","reason":"premise"},{"node":"k-distant-uses-set-dedup","truth_value":"IN","reason":"premise"},{"node":"find-difference-set-minus-idiom","truth_value":"IN","reason":"premise"},{"node":"two-out-of-three-set-algebra","truth_value":"IN","reason":"premise"},{"node":"binary-search-variants-share-convergence-structure","truth_value":"IN","reason":"SL justification valid","antecedents":["binary-search-on-derived-quantities-pattern","binary-search-on-value-pattern","left-biased-binary-search-pattern","fixed-point-uses-leftmost-binary-search"],"label":"the four patterns are orthogonal specializations of one template — search-target and bias are independent choices, and recognizing the shared structure reveals that any new binary search problem maps to a point in this 2D design space"},{"node":"binary-search-on-derived-quantities-pattern","truth_value":"IN","reason":"premise"},{"node":"binary-search-on-value-pattern","truth_value":"IN","reason":"premise"},{"node":"left-biased-binary-search-pattern","truth_value":"IN","reason":"premise"},{"node":"fixed-point-uses-leftmost-binary-search","truth_value":"IN","reason":"premise"},{"node":"sorted-order-enables-all-efficient-search","truth_value":"IN","reason":"SL justification valid","antecedents":["sort-preprocessing-enables-linear-scan","binary-search-variants-share-convergence-structure"],"label":"Sorting is the shared upstream step; linear scan and binary search are the two downstream consumers that exploit the monotonicity it establishes, covering the full spectrum from exhaustive to targeted search"},{"node":"canonical-pipeline-has-exactly-two-instantiations","truth_value":"IN","reason":"SL justification valid","antecedents":["preprocess-then-stream-is-canonical-pipeline","two-preprocessing-paradigms-partition-problems"],"label":"The canonical pipeline and the preprocessing partition are the same dichotomy viewed from different angles — one structural, one problem-theoretic"},{"node":"preprocess-then-stream-is-canonical-pipeline","truth_value":"IN","reason":"SL justification valid","antecedents":["hash-preprocessing-universal-first-step","single-pass-streaming-dominant-shape"],"label":"Hash preprocessing (depth-2) feeds into single-pass streaming (depth-2); many solutions follow exactly this two-phase structure where the hash lookup enables the streaming pass to run in O(1) per element"},{"node":"hash-preprocessing-universal-first-step","truth_value":"IN","reason":"SL justification valid","antecedents":["counter-universal-frequency-primitive","set-for-o1-membership-universal"],"label":"Counter and set are complementary hash tools — Counter for \"how many\" and set for \"is present\" — and together cover the preprocessing needs of the majority of easy/medium problems"},{"node":"two-preprocessing-paradigms-partition-problems","truth_value":"IN","reason":"SL justification valid","antecedents":["hash-preprocessing-universal-first-step","sorted-order-enables-all-efficient-search"],"label":"Query type (lookup vs comparison) determines the preprocessing choice"},{"node":"streaming-is-privileged-default-strategy","truth_value":"IN","reason":"SL justification valid","antecedents":["streaming-is-self-sufficient-paradigm","three-strategies-cover-solution-taxonomy"],"label":"Self-sufficiency (needs nothing external) within a closed taxonomy (no alternatives are self-sufficient) establishes streaming as the default — preprocessing is the exception requiring justification"},{"node":"three-strategies-cover-solution-taxonomy","truth_value":"IN","reason":"SL justification valid","antecedents":["two-paradigms-cover-solution-space","mathematical-insight-replaces-brute-computation"],"label":"Closed-form reduction is the third paradigm that completes the two-paradigm taxonomy"},{"node":"mathematical-insight-replaces-brute-computation","truth_value":"IN","reason":"SL justification valid","antecedents":["closed-form-reduction-eliminates-iteration","greedy-algorithms-provably-optimal"],"label":"Both patterns achieve the same outcome (bypassing brute-force enumeration) via the same means (a mathematical argument that a simpler computation yields the same answer); they differ only in whether the shortcut is algebraic or algorithmic"},{"node":"closed-form-reduction-eliminates-iteration","truth_value":"IN","reason":"SL justification valid","antecedents":["max-sum-is-closed-form","leetcode-bank-closed-form","distinct-numbers-o1-mathematical-reduction","odd-subarray-count-formula"],"label":"each solution discovers that the iteration has a closed-form equivalent (Gauss sum, series formula, steady-state identity, combinatorial count), collapsing O(n) or O(k) work to O(1)"},{"node":"max-sum-is-closed-form","truth_value":"IN","reason":"premise"},{"node":"leetcode-bank-closed-form","truth_value":"IN","reason":"premise"},{"node":"distinct-numbers-o1-mathematical-reduction","truth_value":"IN","reason":"premise"},{"node":"odd-subarray-count-formula","truth_value":"IN","reason":"premise"},{"node":"greedy-algorithms-provably-optimal","truth_value":"IN","reason":"SL justification valid","antecedents":["greedy-scan-correct-for-prefix-free-codes","can-place-flowers-greedy-is-optimal","max-sum-greedy-correctness","max69-greedy-leftmost","min-time-typewriter-greedy-optimal"],"label":"Each greedy choice has a structural proof that no lookahead can improve it"},{"node":"greedy-scan-correct-for-prefix-free-codes","truth_value":"IN","reason":"premise"},{"node":"can-place-flowers-greedy-is-optimal","truth_value":"IN","reason":"premise"},{"node":"max-sum-greedy-correctness","truth_value":"IN","reason":"premise"},{"node":"max69-greedy-leftmost","truth_value":"IN","reason":"premise"},{"node":"min-time-typewriter-greedy-optimal","truth_value":"IN","reason":"premise"},{"node":"streaming-normal-form-is-minimal-strategy","truth_value":"IN","reason":"SL justification valid","antecedents":["streaming-is-solution-normal-form","streaming-is-self-sufficient-paradigm"],"label":"the universal reduction target is also the simplest strategy, forming a structural fixed point"},{"node":"streaming-is-solution-normal-form","truth_value":"IN","reason":"SL justification valid","antecedents":["all-solutions-reduce-to-adapted-streaming","taxonomy-closed-and-structurally-partitioned"],"label":"closed taxonomy + universal streaming reduction = every solution has a canonical streaming form"},{"node":"all-solutions-reduce-to-adapted-streaming","truth_value":"IN","reason":"SL justification valid","antecedents":["preprocessing-is-domain-transformation-to-streaming","streaming-is-self-sufficient-paradigm"],"label":"Preprocessing converts problems to streaming-compatible domains (depth-4) and streaming is self-sufficient (depth-4); together they reduce the entire solution architecture to streaming with optional adapters"},{"node":"preprocessing-is-domain-transformation-to-streaming","truth_value":"IN","reason":"SL justification valid","antecedents":["traversal-accumulation-universal-across-data-structures","preprocess-then-stream-is-canonical-pipeline"],"label":"Traversal-accumulation is universal across data structures including preprocessed ones; reframing preprocessing as a domain adapter into streaming's input space reveals streaming as the sole computational paradigm"},{"node":"traversal-accumulation-universal-across-data-structures","truth_value":"IN","reason":"SL justification valid","antecedents":["tree-postorder-closure-idiom","single-pass-streaming-dominant-shape"],"label":"Generalizes the dominant streaming shape beyond linear sequences to recursive structures, with closure variables as the tree analogue of scalar accumulators"},{"node":"tree-postorder-closure-idiom","truth_value":"IN","reason":"SL justification valid","antecedents":["closure-dfs-pattern","postorder-accumulator-pattern"],"label":"Closures handle cross-subtree accumulation while postorder handles intra-subtree composition — together they cover all tree computation needs"},{"node":"closure-dfs-pattern","truth_value":"IN","reason":"premise"},{"node":"postorder-accumulator-pattern","truth_value":"IN","reason":"premise"},{"node":"taxonomy-closed-and-structurally-partitioned","truth_value":"IN","reason":"SL justification valid","antecedents":["three-strategies-cover-solution-taxonomy","canonical-pipeline-has-exactly-two-instantiations"],"label":"Closure (three-strategies) plus partition (two-instantiations) jointly imply a finite, enumerable classification scheme — stronger than either alone"},{"node":"streaming-universality-and-minimality-coincide","truth_value":"IN","reason":"SL justification valid","antecedents":["streaming-extends-through-three-orthogonal-axes","streaming-normal-form-is-minimal-strategy"],"label":"One antecedent establishes universality via extensions, the other establishes minimality via normal form; their coincidence in the same paradigm is a non-obvious structural property"},{"node":"streaming-extends-through-three-orthogonal-axes","truth_value":"IN","reason":"SL justification valid","antecedents":["streaming-universal-via-specialization-and-adaptation","cursor-streaming-unifies-input-multiplicity","two-pointer-compaction-extends-streaming-to-in-place-transform"],"label":"three independent extension axes (operation, input multiplicity, output mode) exhaust the streaming variant space"},{"node":"streaming-universal-via-specialization-and-adaptation","truth_value":"IN","reason":"SL justification valid","antecedents":["streaming-universality-through-operation-specialization","all-solutions-reduce-to-adapted-streaming"],"label":"Depth-6 unification: two independently derived aspects of streaming's coverage (internal specialization vs external adaptation) are orthogonal axes that jointly explain universality"},{"node":"streaming-universality-through-operation-specialization","truth_value":"IN","reason":"SL justification valid","antecedents":["xor-instantiates-streaming-for-bit-domain","traversal-accumulation-universal-across-data-structures"],"label":"XOR streaming (depth-3) and tree traversal accumulation (depth-3) are domain-specific instantiations of the same paradigm; the streaming skeleton is data-structure-agnostic, only the accumulator operation varies"},{"node":"xor-instantiates-streaming-for-bit-domain","truth_value":"IN","reason":"SL justification valid","antecedents":["xor-universal-bit-primitive","single-pass-streaming-dominant-shape"],"label":"XOR is not merely a bit tool but the streaming paradigm specialized to the bit domain via algebraic compatibility"},{"node":"xor-universal-bit-primitive","truth_value":"IN","reason":"SL justification valid","antecedents":["xor-cancellation-finds-extra-char","xor-for-bit-diff","xor-binary-flip-pattern"],"label":"Three algebraic properties of XOR (self-inverse, positional independence, binary complement) map to three distinct algorithmic roles"},{"node":"xor-cancellation-finds-extra-char","truth_value":"IN","reason":"premise"},{"node":"xor-for-bit-diff","truth_value":"IN","reason":"premise"},{"node":"xor-binary-flip-pattern","truth_value":"IN","reason":"premise"},{"node":"cursor-streaming-unifies-input-multiplicity","truth_value":"IN","reason":"SL justification valid","antecedents":["two-pointer-is-dual-cursor-streaming","merge-scan-extends-sort-pipeline-to-dual-inputs"],"label":"Two-pointer and merge-scan are cursor-count variants of one streaming framework, not separate techniques"},{"node":"two-pointer-is-dual-cursor-streaming","truth_value":"IN","reason":"SL justification valid","antecedents":["two-pointer-primary-linear-array-technique","single-pass-streaming-dominant-shape"],"label":"both patterns share monotonic-progress + O(1)-state invariants; two-pointer adds a second cursor axis"},{"node":"merge-scan-extends-sort-pipeline-to-dual-inputs","truth_value":"IN","reason":"SL justification valid","antecedents":["merge-scan-pattern-for-sorted-pair-processing","sort-then-two-pointer-dominant-pair-pipeline"],"label":"merge-scan preserves sort-then-scan's structure while generalizing to dual sorted inputs"},{"node":"merge-scan-pattern-for-sorted-pair-processing","truth_value":"IN","reason":"SL justification valid","antecedents":["merge-alternately-linear-complexity","merge-nums-two-pointer-linear-time","two-pointer-merge-scan-for-sorted-intersection"],"label":"Depth-1 cross-problem grouping: three solutions independently instantiate the same merge-scan template with different actions at the match/advance points"},{"node":"merge-alternately-linear-complexity","truth_value":"IN","reason":"premise"},{"node":"merge-nums-two-pointer-linear-time","truth_value":"IN","reason":"premise"},{"node":"two-pointer-merge-scan-for-sorted-intersection","truth_value":"IN","reason":"premise"},{"node":"two-pointer-compaction-extends-streaming-to-in-place-transform","truth_value":"IN","reason":"SL justification valid","antecedents":["two-pointer-compaction-family","single-pass-streaming-dominant-shape"],"label":"Compaction is streaming where the accumulator is a write cursor into the input array itself"},{"node":"two-pointer-compaction-family","truth_value":"IN","reason":"premise"}]}}