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R0029-15224 · q-20261014-advance-20261030-r02-real-hilbert-space-0049
Use the same real plane with symmetric positive-definite matrix G=[[25, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).
G001 — Use the same real plane with symmetric positive-definite matrix G=[[25, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Dimension=2, norm_source=same_inner_product, metric_source=same_norm, completeness_source=finite_dimensional_theorem. Give (dimension,matrix,norm_source,metric_source,completeness_source,qualified?).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
Hilbert Gram diagonal product := mul(25, 5)
After
Hilbert Gram diagonal product = 125
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Rule: exact_rational_arithmetic · Uses: G001
Before
Hilbert Gram off-diagonal product := mul(2, 2)
After
Hilbert Gram off-diagonal product = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
Hilbert Gram determinant := sub(125, 4)
After
Hilbert Gram determinant = 121
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Hilbert completing-square slope := div(2, 25)
After
Hilbert completing-square slope = 2/25
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Hilbert completing-square positive residual := div(121, 25)
After
Hilbert completing-square positive residual = 121/25
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
complete real carrier R^2;Gram matrix=[[25, 2], [2, 5]];a=25>0;det=121>0;dimension declaration=2
After
x^T M x=25*(x0+(2/25)*x1)^2+(121/25)*x1^2;zero only at x=0
Conditions: none
Symmetry and positive principal minors give positivity for every real vector by completing the square,not just testing the displayed vectors.The triangular transformation is invertible,so this inner-product norm is equivalent to the coordinate norm on R².A Cauchy sequence has convergent coordinates and its limit remains in this same R².
Rule: hilbert_same_complete_matrix_positive_inner_product · Uses: G001, S005
Before
complete matrix=[[25, 2], [2, 5]];dimension=2;norm source=same_inner_product;metric source=same_norm;completeness source=finite_dimensional_theorem
After
same inner product norm=Yes;same norm metric=Yes;same finite-dimensional completeness source=Yes;all components qualified=Yes
Conditions: none
All three sources must refer to the same structure.A mismatch means this checklist has not supplied the full certificate;it does not prove the underlying space could never be Hilbert.
Rule: hilbert_component_binding_not_name_only · Uses: G001, S006
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 — Recompute Hilbert completing-square positive residual with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Hilbert completing-square slope with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Hilbert Gram determinant with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute Hilbert Gram off-diagonal product with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Recompute Hilbert Gram diagonal product with exact rational arithmetic, not rounded decimals.(2, [[25, 2], [2, 5]], same_inner_product, same_norm, finite_dimensional_theorem, 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: 1d3105742b5fa5f5c6d0eb004491862056c516ec24eeba37e253c41c51632065candidate record SHA-256: a0d2f38735c745fbfa76fb7d02d92c41d624eea135f12f2d36681d2355e01bc4semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117Lean: NOT_RUN · human review: NOT_PERFORMED