Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 4, 2, 1], positive source_scale=18 and target_scale=19. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).
G001 — Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 3, 4, 2, 1], positive source_scale=18 and target_scale=19. In the scaled-discrete metrics each distance is0 on the diagonal and its supplied scale off the diagonal. Give (vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary).
Definition and scope
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.
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field names are not mathematical evidence,and No retains the operation's specific claim scope.
original complete carrier Fin(5);original TOTAL source-indexed candidate map=[0, 3, 4, 2, 1]
After
all target values distinct and complete;explicit inverse=[0, 4, 3, 1, 2];inverse(map(x))=x and map(inverse(y))=y for EVERY carrier point
Conditions: none
Accept a bijection only when every target appears exactly once in the complete source list.Build the inverse by original indices and retain both inverse identities rather than trusting a family name.
positive radii=[9, 19/2] give singleton balls,so BOTH metric topologies discrete and map homeomorphic;equal SCALE parameters=No
Conditions: none
Half each positive scale gives singleton balls,hence every subset is open and both metric topologies are discrete.This supplied bijection is a homeomorphism.The third field compares scale parameters only;No is not an unqualified disproof of isometry.
actual distinct source pair[0,1];distinct images=[0, 3];source distance=18;target distance=19
After
in THIS at-least-two-point scaled-discrete model,isometry iff scales equal=No;no differentiability or congruence structure supplied
Conditions: none
With at least two points this pair witnesses a nonzero distance.Unequal scales refute isometry in this model;no differentiability or geometric-congruence structure has been supplied.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S004 · premise_and_result_audit — Recompute homeo target singleton-ball radius with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute homeo source singleton-ball radius with exact rational arithmetic, not rounded decimals.
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.
Result
(5, Yes, No, 18, 19, homeomorphism_does_not_imply_isometry_diffeomorphism_or_congruence)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
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.