Question 15295: Check Orthogonality before Using the Inner-Product Pythagorean Identity

Original question and givens

Use the same real plane with symmetric positive-definite matrix G=[[21, 1], [1, 2]] 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, 2], y=[1, -21]. 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=[[21, 1], [1, 2]] 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, 2], y=[1, -21]. 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(21, 2)

Hilbert Gram diagonal product = 42

Hilbert Gram off-diagonal product := mul(1, 1)

Hilbert Gram off-diagonal product = 1

Hilbert Gram determinant := sub(42, 1)

Hilbert Gram determinant = 41

Hilbert completing-square slope := div(1, 21)

Hilbert completing-square slope = 1/21

Hilbert completing-square positive residual := div(41, 21)

Hilbert completing-square positive residual = 41/21

complete real carrier R^2;Gram matrix=[[21, 1], [1, 2]];a=21>0;det=41>0;dimension declaration=2

x^T M x=21*(x0+(1/21)*x1)^2+(41/21)*x1^2;zero only at x=0

Hilbert cross inner product coefficient(0,0) times left coordinate := mul(21, 1)

Hilbert cross inner product coefficient(0,0) times left coordinate = 21

Hilbert cross inner product term(0,0) := mul(21, 1)

Hilbert cross inner product term(0,0) = 21

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

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

Hilbert cross inner product term(0,1) := mul(1, -21)

Hilbert cross inner product term(0,1) = -21

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

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

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

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

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

Hilbert cross inner product coefficient(1,1) times left coordinate = 4

Hilbert cross inner product term(1,1) := mul(4, -21)

Hilbert cross inner product term(1,1) = -84

Hilbert cross inner product complete bilinear sum := sum(21, -21, 2, -84)

Hilbert cross inner product complete bilinear sum = -82

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

Hilbert vector sum coordinate0 = 2

Hilbert vector sum coordinate1 := sum(2, -21)

Hilbert vector sum coordinate1 = -19

Hilbert norm square of sum coefficient(0,0) times left coordinate := mul(21, 2)

Hilbert norm square of sum coefficient(0,0) times left coordinate = 42

Hilbert norm square of sum term(0,0) := mul(42, 2)

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

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

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

Hilbert norm square of sum term(0,1) := mul(2, -19)

Hilbert norm square of sum term(0,1) = -38

Hilbert norm square of sum coefficient(1,0) times left coordinate := mul(1, -19)

Hilbert norm square of sum coefficient(1,0) times left coordinate = -19

Hilbert norm square of sum term(1,0) := mul(-19, 2)

Hilbert norm square of sum term(1,0) = -38

Hilbert norm square of sum coefficient(1,1) times left coordinate := mul(2, -19)

Hilbert norm square of sum coefficient(1,1) times left coordinate = -38

Hilbert norm square of sum term(1,1) := mul(-38, -19)

Hilbert norm square of sum term(1,1) = 722

Hilbert norm square of sum complete bilinear sum := sum(84, -38, -38, 722)

Hilbert norm square of sum complete bilinear sum = 730

Hilbert first norm square coefficient(0,0) times left coordinate := mul(21, 1)

Hilbert first norm square coefficient(0,0) times left coordinate = 21

Hilbert first norm square term(0,0) := mul(21, 1)

Hilbert first norm square term(0,0) = 21

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

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

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

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

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

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(2, 2)

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

Hilbert first norm square term(1,1) := mul(4, 2)

Hilbert first norm square term(1,1) = 8

Hilbert first norm square complete bilinear sum := sum(21, 2, 2, 8)

Hilbert first norm square complete bilinear sum = 33

Hilbert second norm square coefficient(0,0) times left coordinate := mul(21, 1)

Hilbert second norm square coefficient(0,0) times left coordinate = 21

Hilbert second norm square term(0,0) := mul(21, 1)

Hilbert second norm square term(0,0) = 21

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

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

Hilbert second norm square term(0,1) := mul(1, -21)

Hilbert second norm square term(0,1) = -21

Hilbert second norm square coefficient(1,0) times left coordinate := mul(1, -21)

Hilbert second norm square coefficient(1,0) times left coordinate = -21

Hilbert second norm square term(1,0) := mul(-21, 1)

Hilbert second norm square term(1,0) = -21

Hilbert second norm square coefficient(1,1) times left coordinate := mul(2, -21)

Hilbert second norm square coefficient(1,1) times left coordinate = -42

Hilbert second norm square term(1,1) := mul(-42, -21)

Hilbert second norm square term(1,1) = 882

Hilbert second norm square complete bilinear sum := sum(21, -21, -21, 882)

Hilbert second norm square complete bilinear sum = 861

Hilbert sum of separate norm squares := sum(33, 861)

Hilbert sum of separate norm squares = 894

Hilbert cross term twice inner product := mul(2, -82)

Hilbert cross term twice inner product = -164

Hilbert expanded square including cross term := sum(894, -164)

Hilbert expanded square including cross term = 730

same inner product=-82;norm square of sum=730;separate squares sum=894;twice cross term=-164

bilinear expansion=730;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, 2], [1, -21], -82, 730, 894, 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.