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Question 29696: A Declared Real-Scalar Instance on Integer Coordinates

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0058-29696 · q-20261013-advance-20261022-r04-real-vector-space-0046

Original question and givens

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=3, vector=[8, -2, -17]. Give (scalar,vector,scaled vector).

  • G001 — Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. scalar=3, vector=[8, -2, -17]. Give (scalar,vector,scaled vector).

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-str22-vectorspace-scalar-multiplication, vectorspace-scalar-multiplication]

After

derived = [[scalar, vector, result]]

Conditions: none

Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention.

Step identifiers and dependencies

Rule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001

S002 · calculation

Before

Vector ORIGINAL scalar product coordinate0 := mul(3, 8)

After

Vector ORIGINAL scalar product coordinate0 = 24

Conditions: none

Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Vector ORIGINAL scalar product coordinate1 := mul(3, -2)

After

Vector ORIGINAL scalar product coordinate1 = -6

Conditions: none

Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Vector ORIGINAL scalar product coordinate2 := mul(3, -17)

After

Vector ORIGINAL scalar product coordinate2 = -51

Conditions: none

Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [3, [8, -2, -17]]

After

derived = [[24, -6, -51]]

Conditions: none

Apply the SAME scalar to EVERY coordinate.Scalar0 gives the zero vector in the SAME dimension;derived values do not inherit raw bounds.

Step identifiers and dependencies

Rule: Vector_scalar_APPLIED_to_ALL_coordinates · Uses: G001, S004

Reverse checks

  • S005 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
  • S004 · premise_and_result_audit — Recompute Vector ORIGINAL scalar product coordinate2 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Vector ORIGINAL scalar product coordinate1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Vector ORIGINAL scalar product coordinate0 with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.

Result

(3, [8, -2, -17], [24, -6, -51])

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: 71fc7caea78365446edefdd4ec82ee562b735cde83b35d99fb0214e567b7704d
  • candidate record SHA-256: 4302fcf7ef8f9f991dab3d82178713750c7c5ec676d415b017d3dcb5400c1534
  • semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df
  • question-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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