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Question 33290: A Selected Zero Vector Gives a Nontrivial Zero Relation

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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R0065-33290 · q-20261013-advance-20261025-r04-linear-independence-basis-0040

Original question and givens

Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 2], [2, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.

  • G001 — Deepen only the existing explicit finite-coordinate basis semantics, preserving vector and coefficient order. No new dimension or change-of-basis theorem is inferred. Coordinate uniqueness is used only after qualifying a nonzero-determinant ordered R² basis. vectors=[[1, 2], [2, 1], [0, 0]], selected zero-based index=2. Give (vectors,index,selected vector,is it zero?,coefficient list with 1 at this index and 0 elsewhere,valid dependence certificate?). If the selected vector is not zero, retain the proposed list but report No.

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-mod25-basisdeep-zero-vector-dependence-certificate, basisdeep-zero-vector-dependence-certificate, R2_basis_density_without_dimension_extension, []]

After

derived = [[vectors, index, selected, is_zero, nontrivial_coefficients, THIS_certificate]]

Conditions: none

MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status.

Step identifiers and dependencies

Rule: R65_SOURCE_bank_keys_operation_domain_ordered_output_roles · Uses: G001

S002 · implies

Before

source roles = [zero certificate ORIGINAL vector0, [1, 2]]

After

derived = [[1, 2]]

Conditions: none

RAW vectors are nonempty with1..6 components bounded±1000.ONLY the R² coordinate task adds dimension2;do not infer an extra2D gate elsewhere from a domain label.

Step identifiers and dependencies

Rule: Basis_RAW_vector_1to6_components_bound1000 · Uses: G001, S001

S003 · implies

Before

source roles = [zero certificate ORIGINAL vector1, [2, 1]]

After

derived = [[2, 1]]

Conditions: none

RAW vectors are nonempty with1..6 components bounded±1000.ONLY the R² coordinate task adds dimension2;do not infer an extra2D gate elsewhere from a domain label.

Step identifiers and dependencies

Rule: Basis_RAW_vector_1to6_components_bound1000 · Uses: G001, S002

S004 · implies

Before

source roles = [zero certificate ORIGINAL vector2, [0, 0]]

After

derived = [[0, 0]]

Conditions: none

RAW vectors are nonempty with1..6 components bounded±1000.ONLY the R² coordinate task adds dimension2;do not infer an extra2D gate elsewhere from a domain label.

Step identifiers and dependencies

Rule: Basis_RAW_vector_1to6_components_bound1000 · Uses: G001, S003

S005 · calculation

Before

zero proposed nontrivial relation ORIGINAL vector0 coordinate0 weighted term := mul(0, 1)

After

zero proposed nontrivial relation ORIGINAL vector0 coordinate0 weighted term = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

zero proposed nontrivial relation ORIGINAL vector1 coordinate0 weighted term := mul(0, 2)

After

zero proposed nontrivial relation ORIGINAL vector1 coordinate0 weighted term = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

zero proposed nontrivial relation ORIGINAL vector2 coordinate0 weighted term := mul(1, 0)

After

zero proposed nontrivial relation ORIGINAL vector2 coordinate0 weighted term = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

zero proposed nontrivial relation ORIGINAL coordinate0 all terms sum := sum(0, 0, 0)

After

zero proposed nontrivial relation ORIGINAL coordinate0 all terms sum = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

source roles = [zero proposed nontrivial relation, 0, [[1, 2], [2, 1], [0, 0]], [0, 0, 1]]

After

derived = [[0, 0, 0], 0]

Conditions: none

For EACH coordinate retain EVERY coefficient*vector component before summing.Do not recap derived reconstructions as RAW vector/coefficient inputs.

Step identifiers and dependencies

Rule: Basis_EACH_coordinate_complete_linear_combination_no_derived_raw_cap · Uses: G001, S008

S010 · calculation

Before

zero proposed nontrivial relation ORIGINAL vector0 coordinate1 weighted term := mul(0, 2)

After

zero proposed nontrivial relation ORIGINAL vector0 coordinate1 weighted term = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

zero proposed nontrivial relation ORIGINAL vector1 coordinate1 weighted term := mul(0, 1)

After

zero proposed nontrivial relation ORIGINAL vector1 coordinate1 weighted term = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

zero proposed nontrivial relation ORIGINAL vector2 coordinate1 weighted term := mul(1, 0)

After

zero proposed nontrivial relation ORIGINAL vector2 coordinate1 weighted term = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

zero proposed nontrivial relation ORIGINAL coordinate1 all terms sum := sum(0, 0, 0)

After

zero proposed nontrivial relation ORIGINAL coordinate1 all terms sum = 0

Conditions: none

Use ORIGINAL operands/order for exact arithmetic.Rational operations separately enforce this MOD25 SOURCE pre-reduction envelope.Do not globally cap every derived integer at one million or use floating approximations.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [zero proposed nontrivial relation, 1, [[1, 2], [2, 1], [0, 0]], [0, 0, 1]]

After

derived = [[0, 0, 0], 0]

Conditions: none

For EACH coordinate retain EVERY coefficient*vector component before summing.Do not recap derived reconstructions as RAW vector/coefficient inputs.

Step identifiers and dependencies

Rule: Basis_EACH_coordinate_complete_linear_combination_no_derived_raw_cap · Uses: G001, S013

S015 · implies

Before

source roles = [[[1, 2], [2, 1], [0, 0]], 2, [0, 0]]

After

derived = [[0, 0, 1], [0, 0], Yes]

Conditions: none

The coefficient is1 at THIS index and0 elsewhere,so the list is nontrivial and its combination equals the selected vector.ONLY a zero selection certifies dependence.No does not establish whole-family independence.

Step identifiers and dependencies

Rule: Basis_ZERO_selected_vector_nontrivial_e_index_relation_ONLY_THIS_certificate · Uses: G001, S014

Reverse checks

  • S015 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S014 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S013 · premise_and_result_audit — Recompute zero proposed nontrivial relation ORIGINAL coordinate1 all terms sum with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute zero proposed nontrivial relation ORIGINAL vector2 coordinate1 weighted term with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute zero proposed nontrivial relation ORIGINAL vector1 coordinate1 weighted term with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute zero proposed nontrivial relation ORIGINAL vector0 coordinate1 weighted term with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S008 · premise_and_result_audit — Recompute zero proposed nontrivial relation ORIGINAL coordinate0 all terms sum with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute zero proposed nontrivial relation ORIGINAL vector2 coordinate0 weighted term with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute zero proposed nontrivial relation ORIGINAL vector1 coordinate0 weighted term with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute zero proposed nontrivial relation ORIGINAL vector0 coordinate0 weighted term with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S003 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S002 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
  • S001 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.

Result

([[1, 2], [2, 1], [0, 0]], 2, [0, 0], Yes, [0, 0, 1], Yes)

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: 7b14ab217fa788bd021e847c2e00370806f16e2563b374687576c75ca2cd33be
  • candidate record SHA-256: a3ca06cd0b5aa2aa88f4b9a189b40c06d8fea17c0bf325cb5d3f2cc32327d9c2
  • semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6
  • question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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