Question 18567: A Finite Maximum Is Not an Infinite-Domain Quantifier Exchange

Original question and givens

Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[16, 17], coefficients=[17, -18], 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. Positive epsilon=1/26. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, boundary).

• G001 — Use the same complete finite domain Fin(2)={0,…,1}, aligned limit=[16, 17], coefficients=[17, -18], 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. Positive epsilon=1/26. Give (pointwise_witnesses, N, uniform_on_this_finite_domain, 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-uniform-convergence-of-functions

Step-by-step derivation

source operation=uniform-pointwise-max;original snapshot and mathematics retained

explicit ordered source output fields=[pointwise_witnesses, N, uniform_on_this_finite_domain, boundary]

original complete domain Fin(2);aligned limit=[16, 17];signed coefficients=[17, -18];shift=2

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

uniform point0 strict numerator threshold := div(17, 1/26)

uniform point0 strict numerator threshold = 442

uniform point0 floor threshold minus shift := sub(442, 2)

uniform point0 floor threshold minus shift = 440

uniform point0 first strict integer index := sum(440, 1)

uniform point0 first strict integer index = 441

uniform point0 cutoff denominator := sum(441, 2)

uniform point0 cutoff denominator = 443

uniform point0 cutoff gap := div(17, 443)

uniform point0 cutoff gap = 17/443

uniform point0 previous denominator := sub(443, 1)

uniform point0 previous denominator = 442

uniform point0 previous gap := div(17, 442)

uniform point0 previous gap = 1/26

original point=0;epsilon=1/26;abs coefficient=17;exact threshold=442;floor=442;unclamped index=441

least allowed N_x=441;gap_at_Nx=17/443<epsilon;previous gap=1/26;all later denominators only increase

uniform point1 strict numerator threshold := div(18, 1/26)

uniform point1 strict numerator threshold = 468

uniform point1 floor threshold minus shift := sub(468, 2)

uniform point1 floor threshold minus shift = 466

uniform point1 first strict integer index := sum(466, 1)

uniform point1 first strict integer index = 467

uniform point1 cutoff denominator := sum(467, 2)

uniform point1 cutoff denominator = 469

uniform point1 cutoff gap := div(18, 469)

uniform point1 cutoff gap = 18/469

uniform point1 previous denominator := sub(469, 1)

uniform point1 previous denominator = 468

uniform point1 previous gap := div(18, 468)

uniform point1 previous gap = 1/26

original point=1;epsilon=1/26;abs coefficient=18;exact threshold=468;floor=468;unclamped index=467

least allowed N_x=467;gap_at_Nx=18/469<epsilon;previous gap=1/26;all later denominators only increase

ALL original ordered point witnesses=[[0, 441], [1, 467]];finite domain size=2;maximum attained at point=1

common N=467;for EVERY x and EVERY n>=N,n>=N_x and gap_n(x)<=gap_Nx(x)<epsilon

Reverse checks

• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute uniform point1 previous gap with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute uniform point1 previous denominator with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute uniform point1 cutoff gap with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute uniform point1 cutoff denominator with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute uniform point1 first strict integer index with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute uniform point1 floor threshold minus shift with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute uniform point1 strict numerator threshold with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute uniform point0 previous gap with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute uniform point0 previous denominator with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute uniform point0 cutoff gap with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute uniform point0 cutoff denominator with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute uniform point0 first strict integer index with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute uniform point0 floor threshold minus shift with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute uniform point0 strict numerator threshold 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

([[0, 441], [1, 467]], 467, Yes, finite_maximum_only_not_infinite_domain_implication)

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.