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R0058-29776 · q-20261006-advance-20261007-r07-geometric-transformations-0076
Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(0)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.
G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. For the origin-centered map H(P)=(0)P on all of R², classify the whole-plane map as a degenerate constant map, whole-plane isometry, or non-isometric similarity, retaining the positive length scale in the last case.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
source roles = [qf-transform-uniform-scale-class, uniform_scale_class]
After
derived = [[WHOLEplane_map_class_label_positive_length_scale]]
Conditions: none
Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention.
Rule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001
Before
source roles = [0, euclidean_orthonormal_equal_units]
After
derived = [0, ||cP-cQ||=|c|*||P-Q||, degenerate constant map]
Conditions: none
For ANY P,Q,the difference scales by c,its squared norm by c²,and the distance by|c|.At0 the whole plane collapses to the origin;|c|=1 preserves ALL distances;other nonzero c gives positive scale|c| but not isometry for distinct points.One coincident pair is not a global certificate;c=-1 is an origin180-degree rotation,not a line reflection.
Rule: Transform_WHOLE_plane_distance_scaling_identity_ALL_P_Q · Uses: G001, S001
S002 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.degenerate constant map
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: e93cb743d630690d58128d1aefb6888cbfb810637549202dba157f733347e2fdcandidate record SHA-256: a127390fbf84aa0ed9a5c094ad88c3aa5765888b0c01d28cbec5f3a79eaeb992semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05dfquestion-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7Lean: NOT_RUN · human review: NOT_PERFORMED