Question 15287: Check Orthogonality before Using the Inner-Product Pythagorean Identity

Original question and givens

Use the same real plane with symmetric positive-definite matrix G=[[13, 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. Vectors x=[1, -1], y=[2, -13]. 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=[[13, 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. Vectors x=[1, -1], y=[2, -13]. 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

Step-by-step derivation

Hilbert Gram diagonal product := mul(13, 5)

Hilbert Gram diagonal product = 65

Hilbert Gram off-diagonal product := mul(2, 2)

Hilbert Gram off-diagonal product = 4

Hilbert Gram determinant := sub(65, 4)

Hilbert Gram determinant = 61

Hilbert completing-square slope := div(2, 13)

Hilbert completing-square slope = 2/13

Hilbert completing-square positive residual := div(61, 13)

Hilbert completing-square positive residual = 61/13

complete real carrier R^2;Gram matrix=[[13, 2], [2, 5]];a=13>0;det=61>0;dimension declaration=2

x^T M x=13*(x0+(2/13)*x1)^2+(61/13)*x1^2;zero only at x=0

Hilbert cross inner product coefficient(0,0) times left coordinate := mul(13, 1)

Hilbert cross inner product coefficient(0,0) times left coordinate = 13

Hilbert cross inner product term(0,0) := mul(13, 2)

Hilbert cross inner product term(0,0) = 26

Hilbert cross inner product coefficient(0,1) times left coordinate := mul(2, 1)

Hilbert cross inner product coefficient(0,1) times left coordinate = 2

Hilbert cross inner product term(0,1) := mul(2, -13)

Hilbert cross inner product term(0,1) = -26

Hilbert cross inner product coefficient(1,0) times left coordinate := mul(2, -1)

Hilbert cross inner product coefficient(1,0) times left coordinate = -2

Hilbert cross inner product term(1,0) := mul(-2, 2)

Hilbert cross inner product term(1,0) = -4

Hilbert cross inner product coefficient(1,1) times left coordinate := mul(5, -1)

Hilbert cross inner product coefficient(1,1) times left coordinate = -5

Hilbert cross inner product term(1,1) := mul(-5, -13)

Hilbert cross inner product term(1,1) = 65

Hilbert cross inner product complete bilinear sum := sum(26, -26, -4, 65)

Hilbert cross inner product complete bilinear sum = 61

Hilbert vector sum coordinate0 := sum(1, 2)

Hilbert vector sum coordinate0 = 3

Hilbert vector sum coordinate1 := sum(-1, -13)

Hilbert vector sum coordinate1 = -14

Hilbert norm square of sum coefficient(0,0) times left coordinate := mul(13, 3)

Hilbert norm square of sum coefficient(0,0) times left coordinate = 39

Hilbert norm square of sum term(0,0) := mul(39, 3)

Hilbert norm square of sum term(0,0) = 117

Hilbert norm square of sum coefficient(0,1) times left coordinate := mul(2, 3)

Hilbert norm square of sum coefficient(0,1) times left coordinate = 6

Hilbert norm square of sum term(0,1) := mul(6, -14)

Hilbert norm square of sum term(0,1) = -84

Hilbert norm square of sum coefficient(1,0) times left coordinate := mul(2, -14)

Hilbert norm square of sum coefficient(1,0) times left coordinate = -28

Hilbert norm square of sum term(1,0) := mul(-28, 3)

Hilbert norm square of sum term(1,0) = -84

Hilbert norm square of sum coefficient(1,1) times left coordinate := mul(5, -14)

Hilbert norm square of sum coefficient(1,1) times left coordinate = -70

Hilbert norm square of sum term(1,1) := mul(-70, -14)

Hilbert norm square of sum term(1,1) = 980

Hilbert norm square of sum complete bilinear sum := sum(117, -84, -84, 980)

Hilbert norm square of sum complete bilinear sum = 929

Hilbert first norm square coefficient(0,0) times left coordinate := mul(13, 1)

Hilbert first norm square coefficient(0,0) times left coordinate = 13

Hilbert first norm square term(0,0) := mul(13, 1)

Hilbert first norm square term(0,0) = 13

Hilbert first norm square coefficient(0,1) times left coordinate := mul(2, 1)

Hilbert first norm square coefficient(0,1) times left coordinate = 2

Hilbert first norm square term(0,1) := mul(2, -1)

Hilbert first norm square term(0,1) = -2

Hilbert first norm square coefficient(1,0) times left coordinate := mul(2, -1)

Hilbert first norm square coefficient(1,0) times left coordinate = -2

Hilbert first norm square term(1,0) := mul(-2, 1)

Hilbert first norm square term(1,0) = -2

Hilbert first norm square coefficient(1,1) times left coordinate := mul(5, -1)

Hilbert first norm square coefficient(1,1) times left coordinate = -5

Hilbert first norm square term(1,1) := mul(-5, -1)

Hilbert first norm square term(1,1) = 5

Hilbert first norm square complete bilinear sum := sum(13, -2, -2, 5)

Hilbert first norm square complete bilinear sum = 14

Hilbert second norm square coefficient(0,0) times left coordinate := mul(13, 2)

Hilbert second norm square coefficient(0,0) times left coordinate = 26

Hilbert second norm square term(0,0) := mul(26, 2)

Hilbert second norm square term(0,0) = 52

Hilbert second norm square coefficient(0,1) times left coordinate := mul(2, 2)

Hilbert second norm square coefficient(0,1) times left coordinate = 4

Hilbert second norm square term(0,1) := mul(4, -13)

Hilbert second norm square term(0,1) = -52

Hilbert second norm square coefficient(1,0) times left coordinate := mul(2, -13)

Hilbert second norm square coefficient(1,0) times left coordinate = -26

Hilbert second norm square term(1,0) := mul(-26, 2)

Hilbert second norm square term(1,0) = -52

Hilbert second norm square coefficient(1,1) times left coordinate := mul(5, -13)

Hilbert second norm square coefficient(1,1) times left coordinate = -65

Hilbert second norm square term(1,1) := mul(-65, -13)

Hilbert second norm square term(1,1) = 845

Hilbert second norm square complete bilinear sum := sum(52, -52, -52, 845)

Hilbert second norm square complete bilinear sum = 793

Hilbert sum of separate norm squares := sum(14, 793)

Hilbert sum of separate norm squares = 807

Hilbert cross term twice inner product := mul(2, 61)

Hilbert cross term twice inner product = 122

Hilbert expanded square including cross term := sum(807, 122)

Hilbert expanded square including cross term = 929

same inner product=61;norm square of sum=929;separate squares sum=807;twice cross term=122

bilinear expansion=929;orthogonality premise=No;Pythagorean conclusion status=orthogonality_premise_false

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, -1], [2, -13], 61, 929, 807, 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.