Question 26781: Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs

Original question and givens

Use mode=invalid_factor with the complete declared model={"x_points":[0,1,2],"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],"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.

• mko-finite-product-topology

Step-by-step derivation

source roles = [product-topology-boundary]

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

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

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

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.