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Question 26785: Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs

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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R0052-26785 · q-20261015-advance-20261117-r04-finite-product-topology-0085

Original question and givens

Use mode=invalid_factor with the complete declared model={"x_points":[0,1,2,3],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"}. Give (mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary).

  • G001 — Use mode=invalid_factor with the complete declared model={"x_points":[0,1,2,3],"x_topology":[[]],"y_points":[100,101],"y_topology":[[],[100,101]],"invalid_side":"x"}. Give (mode, invalid_side, first_factor_valid, first_factor_error_or_null, second_factor_valid, second_factor_error_or_null, product_claim, boundary).

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 = [product-topology-boundary]

After

derived = [[mode, invalid_side, x_valid, x_error, y_valid, y_error, product_claim, certificate]]

Conditions: none

List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [{'x_points': [0, 1, 2, 3], 'x_topology': [[]], 'y_points': [100, 101], 'y_topology': [[], [100, 101]], 'invalid_side': 'x'}]

After

derived = [No, empty and full opens required, Yes, null, No]

Conditions: none

Audit each factor in source order,retaining its first actual error and null for the valid side.Exactly the declared side must fail;one False cannot replace the distinct failure reasons.

Step identifiers and dependencies

Rule: Product_EXACTLY_declared_invalid_factor_actual_FIRST_error · Uses: G001, S001

Reverse checks

  • S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
  • S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.

Result

(invalid_factor, x, No, empty and full opens required, Yes, null, No, product_topology_requires_valid_factor_topologies)

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: df634be90a17c5d47539e5cd95a0d81df99c1ef6528c55ea3a4a513eac9dd9bf
  • candidate record SHA-256: f0e5ddb7f64b9c105eb7183ff64eeb85849657484c4aa29d9b3f062a96f2b067
  • semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
  • question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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