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Question 33266: Explicit Reconstruction without Conflating Span and Independence

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

R0065-33266 · q-20261013-advance-20261025-r04-linear-independence-basis-0016

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, 0], [0, 1]], coefficients=[3, -2], target=[4, -2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).

  • 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, 0], [0, 1]], coefficients=[3, -2], target=[4, -2]. Give (original vectors,coefficients,target,reconstructed sum,equal?).

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-span-witness-audit, basisdeep-span-witness-audit, R2_basis_density_without_dimension_extension, []]

After

derived = [[vectors, coefficients, target, reconstruction, THIS_witness]]

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 = [span ORIGINAL vector0, [1, 0]]

After

derived = [[1, 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, S001

S003 · implies

Before

source roles = [span ORIGINAL vector1, [0, 1]]

After

derived = [[0, 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 = [span target, [4, -2]]

After

derived = [[4, -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, S003

S005 · calculation

Before

span supplied coefficients ORIGINAL vector0 coordinate0 weighted term := mul(3, 1)

After

span supplied coefficients ORIGINAL vector0 coordinate0 weighted term = 3

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

span supplied coefficients ORIGINAL vector1 coordinate0 weighted term := mul(-2, 0)

After

span supplied coefficients 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

span supplied coefficients ORIGINAL coordinate0 all terms sum := sum(3, 0)

After

span supplied coefficients ORIGINAL coordinate0 all terms sum = 3

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 · implies

Before

source roles = [span supplied coefficients, 0, [[1, 0], [0, 1]], [3, -2]]

After

derived = [[3, 0], 3]

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, S007

S009 · calculation

Before

span supplied coefficients ORIGINAL vector0 coordinate1 weighted term := mul(3, 0)

After

span supplied coefficients 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, S008

S010 · calculation

Before

span supplied coefficients ORIGINAL vector1 coordinate1 weighted term := mul(-2, 1)

After

span supplied coefficients ORIGINAL vector1 coordinate1 weighted term = -2

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

span supplied coefficients ORIGINAL coordinate1 all terms sum := sum(0, -2)

After

span supplied coefficients ORIGINAL coordinate1 all terms sum = -2

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 · implies

Before

source roles = [span supplied coefficients, 1, [[1, 0], [0, 1]], [3, -2]]

After

derived = [[0, -2], -2]

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, S011

S013 · implies

Before

source roles = [[[1, 0], [0, 1]], [3, -2], [4, -2]]

After

derived = [[3, -2], No]

Conditions: none

Yes witnesses target membership in the span with THESE coefficients,not independence or a basis.No rejects THIS coefficient list,not target membership in the entire span.

Step identifiers and dependencies

Rule: Basis_SPAN_THIS_witness_not_independence_or_nonmembership · Uses: G001, S012

Reverse checks

  • S013 · 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.
  • S012 · 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.
  • S011 · premise_and_result_audit — Recompute span supplied coefficients ORIGINAL coordinate1 all terms sum with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute span supplied coefficients ORIGINAL vector1 coordinate1 weighted term with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute span supplied coefficients ORIGINAL vector0 coordinate1 weighted term with exact rational arithmetic, not rounded decimals.
  • S008 · 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.
  • S007 · premise_and_result_audit — Recompute span supplied coefficients ORIGINAL coordinate0 all terms sum with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute span supplied coefficients ORIGINAL vector1 coordinate0 weighted term with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute span supplied coefficients 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, 0], [0, 1]], [3, -2], [4, -2], [3, -2], No)

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: 7e6469e923df7447aa694453ed1d81e8c15ae1167d8c4f21cac0001f856d78e0
  • candidate record SHA-256: 7ea2940a64c6d9f466fa86edefccb395ce08cc3701246ab4b21d6a4518d0e212
  • semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6
  • question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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