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R0015-08151 · q-20261012-advance-20261017-r02-inverse-function-0026
Use the complete total map Fin(0)=∅→Fin(0) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(0)→Fin(0) has g=[]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.
G001 — Use the complete total map Fin(0)=∅→Fin(0) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Candidate g:Fin(0)→Fin(0) has g=[]. Give (images, per-cell g type mask, whole-g eligibility, g∘f values, f∘g values, first identity?, second identity?). If any g cell is untyped, use not-evaluated in the last four fields.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
n=0;m=0;images=[]
After
complete typed total map;inverse is not reciprocal
Conditions: none
Validate the whole original total table first.Equal finite cardinalities do not ensure this particular table is bijective,and a zero function value does not prohibit an inverse.
Rule: inverse_typed_map_scope · Uses: G001
Before
n=0;m=0;fibers=[]
After
bijective=Yes
Conditions: none
For the already total map,exactly one preimage per codomain target is equivalent to bijectivity.The empty-to-empty map qualifies;checking n=m alone is insufficient.
Rule: inverse_bijection_gate · Uses: G001, S001
Before
candidate g=[];target Fin(0)
After
mask=[];whole table eligible=Yes
Conditions: none
Every candidate cell must be typed,including cells not reached by f.If any cell is invalid,neither composition is evaluated;do not skip that cell.
Rule: whole_inverse_candidate_type_guard · Uses: G001, S002
Before
gof=[];fog=[]
After
left identity=Yes;right identity=Yes
Conditions: none
Check each identity over its own complete index set.A one-sided identity does not imply the other;conditions over an empty index set are true.
Rule: inverse_two_identities_separate · Uses: G001, S003
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([], [], Yes, [], [], Yes, Yes)
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: cb04835c25e7c41729eb3f8fa49f00201d39525ce34c1f2f5bb3ead71a2e6ac0candidate record SHA-256: ad2c1e7314e6c5735048ad687a6da1524e484a2a7abb66e1b8fdbfb88ba155eesemantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1bLean: NOT_RUN · human review: NOT_PERFORMED