# Question 18587: Compute the Exact Finite-Domain Supremum without Reformatting the Source Ratio

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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.

R0036-18587 · q-20261014-advance-20261104-r02-uniform-convergence-of-functions-0037



## Original question and givens

Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=12. Give (n, sup_gap, certificate).


- G001 — Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[11, 12], coefficients=[12, -13], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Uniform convergence requires one N(epsilon) for every point and every n≥N. The finite maximum of pointwise witnesses works here because the domain is finite; do not extend that argument to arbitrary infinite domains. A finite observed prefix alone does not prove uniform convergence. n=12. Give (n, sup_gap, certificate).





## 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-uniform-convergence-of-functions](https://ocme.evemisslab.com/en/mko/mko-uniform-convergence-of-functions/)





## Step-by-step derivation

S001 · impliesBefore

source operation=uniform-sup-gap;original snapshot and mathematics retained

After

explicit ordered source output fields=[n, sup_gap, certificate]

Conditions: none

Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

original complete domain Fin(2);aligned limit=[11, 12];signed coefficients=[12, -13];shift=2

After

for EVERY n>=0 and EVERY original point x,f_n(x)-limit[x]=coefficient[x]/(n+shift);same positive denominator on the whole domain

Conditions: none

Cancel the same limit[x] from the declared whole-sequence formula,not merely observed samples.Retain original point order;these coefficients and limits are not probabilities.

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





S003 · calculationBefore

uniform selected denominator := sum(12, 2)

After

uniform selected denominator = 14

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

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





S004 · calculationBefore

uniform point0 absolute gap := div(12, 14)

After

uniform point0 absolute gap = 6/7

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

uniform point1 absolute gap := div(13, 14)

After

uniform point1 absolute gap = 13/14

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

uniform finite maximum gap := div(13, 14)

After

uniform finite maximum gap = 13/14

Conditions: none

Substitute original data and established intermediates exactly,retaining indices,signs and the relevant space.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

original n=12;all exact point gaps=[6/7, 13/14];maximum absolute coefficient=13;attaining original point=1

After

EVERY gap<=maximum and selected point attains it;source direct ratio text=13/14;exact reduced value=13/14

Conditions: none

The nonempty finite domain supplies both an upper bound and an attaining point,hence the supremum rather than merely an upper bound.Keep potentially unreduced source ratio bytes while checking their exact value separately.

Step identifiers and dependenciesRule: uniform_finite_sup_attained_source_ratio_bytes_retained · Uses: G001, S006









## Reverse checks


- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Recompute uniform finite maximum gap with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute uniform point1 absolute gap with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute uniform point0 absolute gap with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute uniform selected denominator with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(12, 13/14, exact_finite_domain_supremum)

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: 4a1ec131d3e366a7054df984dd51a0b1a28e2c8ddad377d5f8d133c01d0f0ae1
- candidate record SHA-256: 1570f9ae3b3c9fc11d874a7adc8fbaf99828f7a72a2ba955678f93065b37dc83
- semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301
- question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2
- Lean: NOT_RUN · human review: NOT_PERFORMED

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