# Question 15286: Check Orthogonality before Using the Inner-Product Pythagorean Identity [Return to catalogue page 77](https://ocme.evemisslab.com/en/questions/page/77/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261030-r02-real-hilbert-space-foundations/#q-20261014-advance-20261030-r02-real-hilbert-space-0086) 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. R0029-15286 · q-20261014-advance-20261030-r02-real-hilbert-space-0086 ## Original question and givens Use the same real plane with symmetric positive-definite matrix G=[[12, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -2], y=[1, -12]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). - G001 — Use the same real plane with symmetric positive-definite matrix G=[[12, 1], [1, 2]] and inner product =x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Vectors x=[1, -2], y=[1, -12]. Give (x,y,inner_product,squared_norm_of_sum,sum_of_squared_norms,premise_checked_identity). ## 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-real-hilbert-space](https://ocme.evemisslab.com/en/mko/mko-real-hilbert-space/) ## Step-by-step derivation S001 · calculationBefore Hilbert Gram diagonal product := mul(12, 2) After Hilbert Gram diagonal product = 24 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore Hilbert Gram off-diagonal product := mul(1, 1) After Hilbert Gram off-diagonal product = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Hilbert Gram determinant := sub(24, 1) After Hilbert Gram determinant = 23 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Hilbert completing-square slope := div(1, 12) After Hilbert completing-square slope = 1/12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Hilbert completing-square positive residual := div(23, 12) After Hilbert completing-square positive residual = 23/12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore complete real carrier R^2;Gram matrix=[[12, 1], [1, 2]];a=12>0;det=23>0;dimension declaration=2 After x^T M x=12*(x0+(1/12)*x1)^2+(23/12)*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. Step identifiers and dependenciesRule: hilbert_same_complete_matrix_positive_inner_product · Uses: G001, S005 S007 · calculationBefore Hilbert cross inner product coefficient(0,0) times left coordinate := mul(12, 1) After Hilbert cross inner product coefficient(0,0) times left coordinate = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Hilbert cross inner product term(0,0) := mul(12, 1) After Hilbert cross inner product term(0,0) = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Hilbert cross inner product coefficient(0,1) times left coordinate := mul(1, 1) After Hilbert cross inner product coefficient(0,1) times left coordinate = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Hilbert cross inner product term(0,1) := mul(1, -12) After Hilbert cross inner product term(0,1) = -12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Hilbert cross inner product coefficient(1,0) times left coordinate := mul(1, -2) After Hilbert cross inner product coefficient(1,0) times left coordinate = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Hilbert cross inner product term(1,0) := mul(-2, 1) After Hilbert cross inner product term(1,0) = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Hilbert cross inner product coefficient(1,1) times left coordinate := mul(2, -2) After Hilbert cross inner product coefficient(1,1) times left coordinate = -4 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Hilbert cross inner product term(1,1) := mul(-4, -12) After Hilbert cross inner product term(1,1) = 48 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Hilbert cross inner product complete bilinear sum := sum(12, -12, -2, 48) After Hilbert cross inner product complete bilinear sum = 46 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Hilbert vector sum coordinate0 := sum(1, 1) After Hilbert vector sum coordinate0 = 2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Hilbert vector sum coordinate1 := sum(-2, -12) After Hilbert vector sum coordinate1 = -14 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Hilbert norm square of sum coefficient(0,0) times left coordinate := mul(12, 2) After Hilbert norm square of sum coefficient(0,0) times left coordinate = 24 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Hilbert norm square of sum term(0,0) := mul(24, 2) After Hilbert norm square of sum term(0,0) = 48 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Hilbert norm square of sum coefficient(0,1) times left coordinate := mul(1, 2) After Hilbert norm square of sum coefficient(0,1) times left coordinate = 2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Hilbert norm square of sum term(0,1) := mul(2, -14) After Hilbert norm square of sum term(0,1) = -28 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Hilbert norm square of sum coefficient(1,0) times left coordinate := mul(1, -14) After Hilbert norm square of sum coefficient(1,0) times left coordinate = -14 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Hilbert norm square of sum term(1,0) := mul(-14, 2) After Hilbert norm square of sum term(1,0) = -28 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Hilbert norm square of sum coefficient(1,1) times left coordinate := mul(2, -14) After Hilbert norm square of sum coefficient(1,1) times left coordinate = -28 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Hilbert norm square of sum term(1,1) := mul(-28, -14) After Hilbert norm square of sum term(1,1) = 392 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Hilbert norm square of sum complete bilinear sum := sum(48, -28, -28, 392) After Hilbert norm square of sum complete bilinear sum = 384 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Hilbert first norm square coefficient(0,0) times left coordinate := mul(12, 1) After Hilbert first norm square coefficient(0,0) times left coordinate = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Hilbert first norm square term(0,0) := mul(12, 1) After Hilbert first norm square term(0,0) = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Hilbert first norm square coefficient(0,1) times left coordinate := mul(1, 1) After Hilbert first norm square coefficient(0,1) times left coordinate = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore Hilbert first norm square term(0,1) := mul(1, -2) After Hilbert first norm square term(0,1) = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore Hilbert first norm square coefficient(1,0) times left coordinate := mul(1, -2) After Hilbert first norm square coefficient(1,0) times left coordinate = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Hilbert first norm square term(1,0) := mul(-2, 1) After Hilbert first norm square term(1,0) = -2 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Hilbert first norm square coefficient(1,1) times left coordinate := mul(2, -2) After Hilbert first norm square coefficient(1,1) times left coordinate = -4 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Hilbert first norm square term(1,1) := mul(-4, -2) After Hilbert first norm square term(1,1) = 8 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore Hilbert first norm square complete bilinear sum := sum(12, -2, -2, 8) After Hilbert first norm square complete bilinear sum = 16 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Hilbert second norm square coefficient(0,0) times left coordinate := mul(12, 1) After Hilbert second norm square coefficient(0,0) times left coordinate = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Hilbert second norm square term(0,0) := mul(12, 1) After Hilbert second norm square term(0,0) = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore Hilbert second norm square coefficient(0,1) times left coordinate := mul(1, 1) After Hilbert second norm square coefficient(0,1) times left coordinate = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore Hilbert second norm square term(0,1) := mul(1, -12) After Hilbert second norm square term(0,1) = -12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore Hilbert second norm square coefficient(1,0) times left coordinate := mul(1, -12) After Hilbert second norm square coefficient(1,0) times left coordinate = -12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Hilbert second norm square term(1,0) := mul(-12, 1) After Hilbert second norm square term(1,0) = -12 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore Hilbert second norm square coefficient(1,1) times left coordinate := mul(2, -12) After Hilbert second norm square coefficient(1,1) times left coordinate = -24 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore Hilbert second norm square term(1,1) := mul(-24, -12) After Hilbert second norm square term(1,1) = 288 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore Hilbert second norm square complete bilinear sum := sum(12, -12, -12, 288) After Hilbert second norm square complete bilinear sum = 276 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore Hilbert sum of separate norm squares := sum(16, 276) After Hilbert sum of separate norm squares = 292 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore Hilbert cross term twice inner product := mul(2, 46) After Hilbert cross term twice inner product = 92 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Hilbert expanded square including cross term := sum(292, 92) After Hilbert expanded square including cross term = 384 Conditions: none Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · impliesBefore same inner product=46;norm square of sum=384;separate squares sum=292;twice cross term=92 After bilinear expansion=384;orthogonality premise=No;Pythagorean conclusion status=orthogonality_premise_false Conditions: none The universal identity includes2〈x,y〉.Remove the cross term only when the inner product is zero.A nonzero result reports a failed premise,not a counterexample to the Pythagorean theorem. Step identifiers and dependenciesRule: hilbert_pythagorean_requires_orthogonality_not_counterexample · Uses: G001, S047 ## Reverse checks - S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S047 · premise_and_result_audit — Recompute Hilbert expanded square including cross term with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Hilbert cross term twice inner product with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Hilbert sum of separate norm squares with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Hilbert second norm square complete bilinear sum with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Hilbert second norm square term(1,1) with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Hilbert second norm square coefficient(1,1) times left coordinate with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute Hilbert second norm square term(1,0) with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Hilbert second norm square coefficient(1,0) times left coordinate with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Hilbert second norm square term(0,1) with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Hilbert second norm square coefficient(0,1) times left coordinate with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Hilbert second norm square term(0,0) with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Hilbert second norm square coefficient(0,0) times left coordinate with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Hilbert first norm square complete bilinear sum with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Hilbert first norm square term(1,1) with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Hilbert first norm square coefficient(1,1) times left coordinate with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Hilbert first norm square term(1,0) with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Hilbert first norm square coefficient(1,0) times left coordinate with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Hilbert first norm square term(0,1) with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Hilbert first norm square coefficient(0,1) times left coordinate with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Hilbert first norm square term(0,0) with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Hilbert first norm square coefficient(0,0) times left coordinate with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Hilbert norm square of sum complete bilinear sum with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Hilbert norm square of sum term(1,1) with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Hilbert norm square of sum coefficient(1,1) times left coordinate with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Hilbert norm square of sum term(1,0) with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Hilbert norm square of sum coefficient(1,0) times left coordinate with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Hilbert norm square of sum term(0,1) with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Hilbert norm square of sum coefficient(0,1) times left coordinate with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Hilbert norm square of sum term(0,0) with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Hilbert norm square of sum coefficient(0,0) times left coordinate with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Hilbert vector sum coordinate1 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Hilbert vector sum coordinate0 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Hilbert cross inner product complete bilinear sum with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Hilbert cross inner product term(1,1) with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Hilbert cross inner product coefficient(1,1) times left coordinate with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Hilbert cross inner product term(1,0) with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Hilbert cross inner product coefficient(1,0) times left coordinate with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Hilbert cross inner product term(0,1) with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Hilbert cross inner product coefficient(0,1) times left coordinate with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Hilbert cross inner product term(0,0) with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Hilbert cross inner product coefficient(0,0) times left coordinate with exact rational arithmetic, not rounded decimals. - 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. ## Result ([1, -2], [1, -12], 46, 384, 292, orthogonality_premise_false) 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: a1d0eae231c8e71a142a586e1ad9a2aaec7882f62105297620c555dbe9e051ff - candidate record SHA-256: 9cca4b198a2c73384bba16b38c7002e47d6799a3d8d65b615d7a2c58f0c0dc58 - semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725 - question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261030-r02-real-hilbert-space-0086.json)