# Question 19083: Equal Scale Parameters Are Not an Unqualified Isometry Test [Return to catalogue page 96](https://ocme.evemisslab.com/en/questions/page/96/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261105-r05-homeomorphism-foundations/#q-20261014-advance-20261105-r05-homeomorphism-0083) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0037-19083 · q-20261014-advance-20261105-r05-homeomorphism-0083 ## Original question and givens Use discrete spaces on Fin(5)={0,…,4}, bijection map=[0, 2, 1, 4, 3], positive source_scale=8 and target_scale=9. 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, 2, 1, 4, 3], positive source_scale=8 and target_scale=9. 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. - [mko-homeomorphism](https://ocme.evemisslab.com/en/mko/mko-homeomorphism/) ## Step-by-step derivation S001 · impliesBefore source operation=homeo-metric-nonimplication;original snapshot and mathematics retained After explicit ordered source output fields=[vertex_count, homeomorphic, equal_scale_parameters, source_scale, target_scale, boundary] Conditions: none 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. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore original complete carrier Fin(5);original TOTAL source-indexed candidate map=[0, 2, 1, 4, 3] After all target values distinct and complete;explicit inverse=[0, 2, 1, 4, 3];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. Step identifiers and dependenciesRule: homeo_complete_bijection_not_name_assumption · Uses: G001, S001 S003 · calculationBefore homeo source singleton-ball radius := div(8, 2) After homeo source singleton-ball radius = 4 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore homeo target singleton-ball radius := div(9, 2) After homeo target singleton-ball radius = 9/2 Conditions: none Substitute original roles and established intermediates exactly,retaining parameter,outcome,target and denominator roles without replacing the claim. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore actual n=5;positive source off-diagonal distance=8;target off-diagonal distance=9;complete bijection=[0, 2, 1, 4, 3] After positive radii=[4, 9/2] 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. Step identifiers and dependenciesRule: homeo_scaled_discrete_topology_not_unqualified_isometry_flag · Uses: G001, S004 S006 · impliesBefore actual distinct source pair[0,1];distinct images=[0, 2];source distance=8;target distance=9 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. Step identifiers and dependenciesRule: homeo_actual_nonzero_distance_witness_not_extra_geometry · Uses: G001, S005 ## Reverse checks - 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, 8, 9, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: ef644968dc9d44c7e0c622aaa5fe449cc94870a68560804ddeca3079238e3e4c - candidate record SHA-256: 76b6e91f400e696e31f6b9e9bafa4e9df020bdc32155fb316f667610164b06d7 - semantic review receipt SHA-256: 065507b602ceee60cf47a700d387a51940e0c6724c35a3336a7a7948a199ed80 - question-first witness SHA-256: 7203f33b030792559673b0fee520a8c0e70903c4dd970b35293aa40852f333d6 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r05-homeomorphism-0083.json)