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R0025-13438 · q-20261014-advance-20261028-r03-real-banach-space-0063
Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[0, 2, -2], right=[0, 2, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).
G001 — Use the explicit real coordinate space R^n of dimension=3, with the named norm=l1, limited to l1 or linfinity. A Banach certificate must bind the carrier, vector operations, norm, the metric induced by that same norm, and completeness of that same space. The named finite-dimensional real normed-space theorem is a universal mathematical premise, not a new Lean receipt or sampled-sequence proof. No Hilbert, compactness or arbitrary infinite-dimensional conclusion is inferred. left=[0, 2, -2], right=[0, 2, -2]. Give (norm_name,left,right,left−right,distance,distance_evaluation_not_completeness_proof).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
Banach coordinate distance difference coordinate0 := sub(0, 0)
After
Banach coordinate distance difference coordinate0 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001
Before
Banach coordinate distance difference coordinate1 := sub(2, 2)
After
Banach coordinate distance difference coordinate1 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Banach coordinate distance difference coordinate2 := sub(-2, -2)
After
Banach coordinate distance difference coordinate2 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Banach coordinate distance:norm=l1;left=[0, 2, -2];right=[0, 2, -2]
After
signed left-right=[0, 0, 0];use same named norm
Conditions: none
Use the same stated norm on the same real coordinate space.The derived difference can exceed original input coordinate bounds and must not be clipped or rejected as a new raw input.
Rule: norm_distance_same_space_and_direction · Uses: G001, S003
Before
Banach coordinate distance:coordinate=0;signed value=0
After
absolute value=0
Conditions: none
Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.
Rule: named_norm_coordinate_absolute_value · Uses: G001, S004
Before
Banach coordinate distance:coordinate=1;signed value=0
After
absolute value=0
Conditions: none
Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.
Rule: named_norm_coordinate_absolute_value · Uses: G001, S005
Before
Banach coordinate distance:coordinate=2;signed value=0
After
absolute value=0
Conditions: none
Take each whole coordinate's absolute value while preserving the original signed vector.Do not use a signed maximum.
Rule: named_norm_coordinate_absolute_value · Uses: G001, S006
Before
Banach coordinate distance l1 sum := sum(0, 0, 0)
After
Banach coordinate distance l1 sum = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining coefficients,coordinates,order and signs.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
fixed carrier R^3;named norm=l1;distance=0
After
distance_evaluation_not_completeness_proof
Conditions: none
One distance evaluation quantifies over no Cauchy sequences and is not itself a completeness proof.
Rule: Banach_distance_evaluation_not_all_Cauchy_claim · Uses: G001, S008
S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Recompute Banach coordinate distance l1 sum with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute Banach coordinate distance difference coordinate2 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Banach coordinate distance difference coordinate1 with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Recompute Banach coordinate distance difference coordinate0 with exact rational arithmetic, not rounded decimals.(l1, [0, 2, -2], [0, 2, -2], [0, 0, 0], 0, distance_evaluation_not_completeness_proof)
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: 4ba8d9f42ca47168cbe115fccb80ef3494ac8406b888671e9e8798467ed59afacandidate record SHA-256: c49143bf41859d5d9dc043e0f52c913afcfc86c853c94f9b2a1b2e116a905c75semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8aLean: NOT_RUN · human review: NOT_PERFORMED