# Question 5757: Open-Sentence Binding and Truth Eligibility

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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.

R0010-05757 · q-20261008-advance-20261012-r03-proposition-0032



## Original question and givens

In the classical integer context, consider x≤-1 with assignment=-7. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).


- G001 — In the classical integer context, consider x≤-1 with assignment=-7. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).





## 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-proposition](https://ocme.evemisslab.com/en/mko/mko-proposition/)





## Step-by-step derivation

S001 · impliesBefore

classical integer formulas with explicit binding/data

After

a truth verdict is not a delivered general proof object

Conditions: none

A false closed statement is still a proposition.These finite substitutions and comparison witnesses do not promote a truth label to a general formal proof or completeness of arbitrary inference rules.

Step identifiers and dependenciesRule: proposition_truth_and_proof_scope · Uses: G001





S002 · calculationBefore

bound integer comparison difference := sub(-7, -1)

After

bound integer comparison difference = -6

Conditions: none

Substitute only after receiving an integer binding;the difference is x minus the threshold.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · impliesBefore

x=-7;threshold=-1;x−threshold=-6

After

bound statement truth=True

Conditions: none

A bound0 is a valid value too.The condition difference≤0 is exactly x≤threshold;programming-language truthiness is not mathematical binding.

Step identifiers and dependenciesRule: bound_integer_formula_verdict · Uses: G001, S002









## Reverse checks


- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Recompute bound integer comparison difference with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(Yes, True, -6)

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: 7bed20869eed2eab1722ca97d9b41b124f1aa671d299d3c7445a4dd3c7c684d7
- candidate record SHA-256: 8da38f4624c799c8e3fe5b2bf33375a89df520220fff9aa99f47001826a55f1c
- semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f
- question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r03-proposition-0032.json)