# Question 15827: A Failed Encoding Does Not Prove Uncountability

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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.

R0030-15827 · q-20261014-advance-20261031-r01-at-most-countable-set-0077



## Original question and givens

Declared evidence_mode=universal_affine_injection, certificate_parameter=2. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).


- G001 — Declared evidence_mode=universal_affine_injection, certificate_parameter=2. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).





## 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-at-most-countable-set](https://ocme.evemisslab.com/en/mko/mko-at-most-countable-set/)





## Step-by-step derivation

S001 · impliesBefore

registered DECLARED evidence mode=universal_affine_injection;numeric certificate parameter=2

After

qualifying declared certificate flag=Yes;classification=at_most_countable

Conditions: none

This audits the declared complete or universal certificate mode;the numeric parameter does not construct or verify an unspecified set.A failed encoding refutes that witness,not countability itself.At most countable does not mean countably infinite.

Step identifiers and dependenciesRule: countability_declared_evidence_mode_not_unspecified_set_construction · Uses: G001









## Reverse checks


- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(universal_affine_injection, 2, Yes, at_most_countable)

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: 7c8968f59397e62dea52139d5663c6a265002b33d4d1b9a8cf5877a265b310cf
- candidate record SHA-256: d63abbb6807350d60553bb8c74b4680a77a738c206779ed0985dd9b826745b44
- semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc
- question-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0077.json)