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R0028-14869 · q-20261014-advance-20261030-r05-cardinality-comparison-0094
Use the complete finite source=[180, 181, 182, 183] and target=[280, 281, 282, 283, 284] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.
G001 — Use the complete finite source=[180, 181, 182, 183] and target=[280, 281, 282, 283, 284] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
raw source list=[180, 181, 182, 183];raw target list=[280, 281, 282, 283, 284]
After
distinct finite source=[180, 181, 182, 183];target=[280, 281, 282, 283, 284];sizes=[4, 5]
Conditions: none
This family compares set cardinalities,so repetitions do not add elements.No particular mapping is supplied,and no specific collision pair is invented.
Rule: pigeonhole_set_counts_not_list_multiplicity · Uses: G001
Before
finite sizes source=4;target=5
After
source larger=No;conclusion=no_impossibility_certificate
Conditions: none
An injection has as many distinct images as source elements,all within the target,hence ns≤nt.A larger source contradicts this and excludes every injection.No means this size-based impossibility certificate does not apply,not that an unspecified map failed.
Rule: finite_pigeonhole_all_injections_not_one_map_failure · Uses: G001, S001
S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(4, 5, No, no_impossibility_certificate)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 3eca1ee7d44a5a9848b4718b9b11dbd7a6a4dd7ac15d74932e1e8aac912a286dcandidate record SHA-256: 138c9374f7e3954772168aa2adee6cdc0f19f0e1eb1692db45bbcd2ff3a882cdsemantic review receipt SHA-256: bfea39236e49aa7e44aa18bfa0d7857ff34a6acb762d0599e20b0f5af0f209aequestion-first witness SHA-256: 61c433f5c1e55ed4d84d08557fb8c117d26bbd404fdf05519ad42c09324c36e0Lean: NOT_RUN · human review: NOT_PERFORMED