# Question 26789: Nonrectangle Opens, Invalid Factors and Pair Direction Have Different Typed Outputs

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0052-26789 · q-20261015-advance-20261117-r04-finite-product-topology-0089



## Original question and givens

Use mode=invalid_factor with the complete declared model={"x_points":[0,1,2,3,4],"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":[0,1,2,3,4],"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](https://ocme.evemisslab.com/en/mko/mko-finite-product-topology/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [{'x_points': [0, 1, 2, 3, 4], '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 dependenciesRule: 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: 3454ac84fb709fc6a0885afab12963d286548234a345a9ef3b55a61e50ad1eab
- candidate record SHA-256: 87f1b62c8f2ff4dcc13758408aeb5d8535e0724894c885c4672781275f40202a
- semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252
- question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261117-r04-finite-product-topology-0089.json)