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Question 33244: Candidate Coordinate Lists and the Conditional Uniqueness Obligation

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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R0065-33244 · q-20261013-advance-20261025-r04-linear-independence-basis-0094

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. ordered basis=[[1, 1], [0, 1]], target=[6, 7], first integer coordinates=[6, 1], second=[6, 1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).

  • 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. ordered basis=[[1, 1], [0, 1]], target=[6, 7], first integer coordinates=[6, 1], second=[6, 1]. Give (basis,target,signed determinant,first coordinates,first reconstruction,first matches?,second coordinates,second reconstruction,second matches?,if both match then their coordinates agree?).

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-r2-coordinate-uniqueness-audit, basisdeep-r2-coordinate-uniqueness-audit, R2_basis_density_without_dimension_extension, []]

After

derived = [[basis, target, D, first, reconstruction1, match1, second, reconstruction2, match2, implication]]

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 = [ordered R2 basis COLUMNS vector0, [1, 1]]

After

derived = [[1, 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, S001

S003 · implies

Before

source roles = [ordered R2 basis COLUMNS 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 = [R2 target, [6, 7]]

After

derived = [[6, 7]]

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

R2 ORIGINAL b1x times b2y := mul(1, 1)

After

R2 ORIGINAL b1x times b2y = 1

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

R2 ORIGINAL b2x times b1y := mul(0, 1)

After

R2 ORIGINAL b2x times b1y = 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

R2 ORIGINAL signed column determinant := sub(1, 0)

After

R2 ORIGINAL signed column determinant = 1

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

R2 true coordinate ORIGINAL targetx times b2y := mul(6, 1)

After

R2 true coordinate ORIGINAL targetx times b2y = 6

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 · calculation

Before

R2 true coordinate ORIGINAL b2x times targety := mul(0, 7)

After

R2 true coordinate ORIGINAL b2x times targety = 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

R2 true coordinate ORIGINAL first numerator := sub(6, 0)

After

R2 true coordinate ORIGINAL first numerator = 6

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

R2 true coordinate ORIGINAL b1x times targety := mul(1, 7)

After

R2 true coordinate ORIGINAL b1x times targety = 7

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

R2 true coordinate ORIGINAL targetx times b1y := mul(6, 1)

After

R2 true coordinate ORIGINAL targetx times b1y = 6

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

R2 true coordinate ORIGINAL second numerator := sub(7, 6)

After

R2 true coordinate ORIGINAL second numerator = 1

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 = [R2 source helper TRUEcoordinate0, RAW-pair, [6, 1], [], 6, 1]

After

derived = [6, 1, 6]

Conditions: none

This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer.

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S013

S015 · calculation

Before

R2 source helper TRUEcoordinate0 ORIGINAL raw numerator over nonzero denominator := div(6, 1)

After

R2 source helper TRUEcoordinate0 ORIGINAL raw numerator over nonzero denominator = 6

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, S014

S016 · implies

Before

source roles = [R2 source helper TRUEcoordinate1, RAW-pair, [1, 1], [], 1, 1]

After

derived = [1, 1, 1]

Conditions: none

This is the SOURCE operational slice:perform the specified cancellations,THEN bound the PRE-reduction numerator/denominator by±one million and require nonzero denominator,BEFORE sign/gcd normalization.Out-of-envelope is a SOURCE hold,not mathematical falsity;do not impose it on every derived integer.

Step identifiers and dependencies

Rule: Wave_RATIONAL_SOURCE_PREreduction_1million_guard_and_cancellations · Uses: G001, S015

S017 · calculation

Before

R2 source helper TRUEcoordinate1 ORIGINAL raw numerator over nonzero denominator := div(1, 1)

After

R2 source helper TRUEcoordinate1 ORIGINAL raw numerator over nonzero denominator = 1

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, S016

S018 · calculation

Before

R2 FIRST integer coordinate list ORIGINAL vector0 coordinate0 weighted term := mul(6, 1)

After

R2 FIRST integer coordinate list ORIGINAL vector0 coordinate0 weighted term = 6

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, S017

S019 · calculation

Before

R2 FIRST integer coordinate list ORIGINAL vector1 coordinate0 weighted term := mul(1, 0)

After

R2 FIRST integer coordinate list 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, S018

S020 · calculation

Before

R2 FIRST integer coordinate list ORIGINAL coordinate0 all terms sum := sum(6, 0)

After

R2 FIRST integer coordinate list ORIGINAL coordinate0 all terms sum = 6

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, S019

S021 · implies

Before

source roles = [R2 FIRST integer coordinate list, 0, [[1, 1], [0, 1]], [6, 1]]

After

derived = [[6, 0], 6]

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, S020

S022 · calculation

Before

R2 FIRST integer coordinate list ORIGINAL vector0 coordinate1 weighted term := mul(6, 1)

After

R2 FIRST integer coordinate list ORIGINAL vector0 coordinate1 weighted term = 6

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, S021

S023 · calculation

Before

R2 FIRST integer coordinate list ORIGINAL vector1 coordinate1 weighted term := mul(1, 1)

After

R2 FIRST integer coordinate list ORIGINAL vector1 coordinate1 weighted term = 1

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, S022

S024 · calculation

Before

R2 FIRST integer coordinate list ORIGINAL coordinate1 all terms sum := sum(6, 1)

After

R2 FIRST integer coordinate list ORIGINAL coordinate1 all terms sum = 7

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, S023

S025 · implies

Before

source roles = [R2 FIRST integer coordinate list, 1, [[1, 1], [0, 1]], [6, 1]]

After

derived = [[6, 1], 7]

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, S024

S026 · calculation

Before

R2 SECOND integer coordinate list ORIGINAL vector0 coordinate0 weighted term := mul(6, 1)

After

R2 SECOND integer coordinate list ORIGINAL vector0 coordinate0 weighted term = 6

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, S025

S027 · calculation

Before

R2 SECOND integer coordinate list ORIGINAL vector1 coordinate0 weighted term := mul(1, 0)

After

R2 SECOND integer coordinate list 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, S026

S028 · calculation

Before

R2 SECOND integer coordinate list ORIGINAL coordinate0 all terms sum := sum(6, 0)

After

R2 SECOND integer coordinate list ORIGINAL coordinate0 all terms sum = 6

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, S027

S029 · implies

Before

source roles = [R2 SECOND integer coordinate list, 0, [[1, 1], [0, 1]], [6, 1]]

After

derived = [[6, 0], 6]

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, S028

S030 · calculation

Before

R2 SECOND integer coordinate list ORIGINAL vector0 coordinate1 weighted term := mul(6, 1)

After

R2 SECOND integer coordinate list ORIGINAL vector0 coordinate1 weighted term = 6

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, S029

S031 · calculation

Before

R2 SECOND integer coordinate list ORIGINAL vector1 coordinate1 weighted term := mul(1, 1)

After

R2 SECOND integer coordinate list ORIGINAL vector1 coordinate1 weighted term = 1

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, S030

S032 · calculation

Before

R2 SECOND integer coordinate list ORIGINAL coordinate1 all terms sum := sum(6, 1)

After

R2 SECOND integer coordinate list ORIGINAL coordinate1 all terms sum = 7

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, S031

S033 · implies

Before

source roles = [R2 SECOND integer coordinate list, 1, [[1, 1], [0, 1]], [6, 1]]

After

derived = [[6, 1], 7]

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, S032

S034 · implies

Before

source roles = [[[1, 1], [0, 1]], [6, 7], 1]

After

derived = [[6, 1], [6, 1], [6, 7], Yes, [6, 1], [6, 7], Yes, Yes]

Conditions: none

The ordered vectors are COLUMNS;D≠0 qualifies unique coordinates.The SOURCE true-coordinate helper must pass its rational envelope EVEN when not in the output.Submitted lists are INTEGER coordinates;reconstruct each.The final BOTH-valid⇒equal implication may be true with an invalid list and does not certify both.

Step identifiers and dependencies

Rule: Basis_R2_nonzero_column_basis_source_TRUEcoordinate_gate_and_conditional_uniqueness · Uses: G001, S033

Reverse checks

  • S034 · 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.
  • S033 · 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.
  • S032 · premise_and_result_audit — Recompute R2 SECOND integer coordinate list ORIGINAL coordinate1 all terms sum with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute R2 SECOND integer coordinate list ORIGINAL vector1 coordinate1 weighted term with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute R2 SECOND integer coordinate list ORIGINAL vector0 coordinate1 weighted term with exact rational arithmetic, not rounded decimals.
  • S029 · 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.
  • S028 · premise_and_result_audit — Recompute R2 SECOND integer coordinate list ORIGINAL coordinate0 all terms sum with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute R2 SECOND integer coordinate list ORIGINAL vector1 coordinate0 weighted term with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute R2 SECOND integer coordinate list ORIGINAL vector0 coordinate0 weighted term with exact rational arithmetic, not rounded decimals.
  • S025 · 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.
  • S024 · premise_and_result_audit — Recompute R2 FIRST integer coordinate list ORIGINAL coordinate1 all terms sum with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute R2 FIRST integer coordinate list ORIGINAL vector1 coordinate1 weighted term with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute R2 FIRST integer coordinate list ORIGINAL vector0 coordinate1 weighted term with exact rational arithmetic, not rounded decimals.
  • S021 · 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.
  • S020 · premise_and_result_audit — Recompute R2 FIRST integer coordinate list ORIGINAL coordinate0 all terms sum with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute R2 FIRST integer coordinate list ORIGINAL vector1 coordinate0 weighted term with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute R2 FIRST integer coordinate list ORIGINAL vector0 coordinate0 weighted term with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute R2 source helper TRUEcoordinate1 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
  • S016 · 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.
  • S015 · premise_and_result_audit — Recompute R2 source helper TRUEcoordinate0 ORIGINAL raw numerator over nonzero denominator with exact rational arithmetic, not rounded decimals.
  • 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 R2 true coordinate ORIGINAL second numerator with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute R2 true coordinate ORIGINAL targetx times b1y with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute R2 true coordinate ORIGINAL b1x times targety with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute R2 true coordinate ORIGINAL first numerator with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute R2 true coordinate ORIGINAL b2x times targety with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute R2 true coordinate ORIGINAL targetx times b2y with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute R2 ORIGINAL signed column determinant with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute R2 ORIGINAL b2x times b1y with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute R2 ORIGINAL b1x times b2y 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, 1], [0, 1]], [6, 7], 1, [6, 1], [6, 7], Yes, [6, 1], [6, 7], Yes, 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: eda98e9509f683bdf316b892144f633be2a4a0b8a01a08e9bbd373d7e273607b
  • candidate record SHA-256: 1ded2924a5b4fa5cf966af083a52212ac406ac78b41b4fa622645534df183bd1
  • semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6
  • question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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