Question 17551: Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers

Original question and givens

Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/10. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.

• G001 — Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[0, 1], coefficients=[1, -2], positive shift=2, and f_n(x)=limit[x]+coefficients[x]/(n+shift) for every n≥0. Values and tail bounds are exact rationals. Pointwise witnesses may depend on both x and epsilon. A finite maximum of all pointwise witnesses gives a common witness only on this finite domain; it is not a general infinite-domain implication. A finite observed prefix alone is not a convergence certificate. Positive epsilon=1/10. Give (epsilon,pointwise_witnesses,max_N,uniform_on_finite_domain?,finite_domain_maximum_only_not_infinite_domain_theorem), with rows [point,least_N] in domain order.

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-pointwise-convergence-of-functions

Step-by-step derivation

same COMPLETE finite domain Fin(2);original aligned limits=[0, 1];signed coefficients=[1, -2];positive shift=2

declared for EVERY n>=0:f_n(x)=limit[x]+coefficient[x]/(n+shift);signed error=f_n(x)-limit[x]=coefficient[x]/(n+shift)

finite-domain witness point0 absolute coefficient cross product := mul(1, 10)

finite-domain witness point0 absolute coefficient cross product = 10

finite-domain witness point0 strict reciprocal threshold := div(10, 1)

finite-domain witness point0 strict reciprocal threshold = 10

finite-domain witness point0 raw strict integer cutoff := sum(10, -2, 1)

finite-domain witness point0 raw strict integer cutoff = 9

finite-domain witness point0:point=0;SIGNED coefficient=1;absolute coefficient=1;positive epsilon=1/10;shift=2

threshold=10;floor=10;least nonnegative strict cutoff N=9

finite-domain witness point0 cutoff denominator := sum(9, 2)

finite-domain witness point0 cutoff denominator = 11

finite-domain witness point0 absolute error at cutoff := div(1, 11)

finite-domain witness point0 absolute error at cutoff = 1/11

finite-domain witness point0 previous denominator := sum(8, 2)

finite-domain witness point0 previous denominator = 10

finite-domain witness point0 previous absolute error := div(1, 10)

finite-domain witness point0 previous absolute error = 1/10

finite-domain witness point0:current N=9;current error=1/11;previous error=1/10;epsilon=1/10

current strict bound holds;previous fails;all smaller indices fail by monotonicity

finite-domain witness point0:declared exact tail;positive denominators increase for all n>=N=9

for EVERY later n,absolute error<=bound=1/11<epsilon=1/10

finite-domain witness point1 absolute coefficient cross product := mul(2, 10)

finite-domain witness point1 absolute coefficient cross product = 20

finite-domain witness point1 strict reciprocal threshold := div(20, 1)

finite-domain witness point1 strict reciprocal threshold = 20

finite-domain witness point1 raw strict integer cutoff := sum(20, -2, 1)

finite-domain witness point1 raw strict integer cutoff = 19

finite-domain witness point1:point=1;SIGNED coefficient=-2;absolute coefficient=2;positive epsilon=1/10;shift=2

threshold=20;floor=20;least nonnegative strict cutoff N=19

finite-domain witness point1 cutoff denominator := sum(19, 2)

finite-domain witness point1 cutoff denominator = 21

finite-domain witness point1 absolute error at cutoff := div(2, 21)

finite-domain witness point1 absolute error at cutoff = 2/21

finite-domain witness point1 previous denominator := sum(18, 2)

finite-domain witness point1 previous denominator = 20

finite-domain witness point1 previous absolute error := div(2, 20)

finite-domain witness point1 previous absolute error = 1/10

finite-domain witness point1:current N=19;current error=2/21;previous error=1/10;epsilon=1/10

current strict bound holds;previous fails;all smaller indices fail by monotonicity

finite-domain witness point1:declared exact tail;positive denominators increase for all n>=N=19

for EVERY later n,absolute error<=bound=2/21<epsilon=1/10

finite-domain maximum witness denominator := sum(19, 2)

finite-domain maximum witness denominator = 21

finite-domain maximum absolute error point0 := div(1, 21)

finite-domain maximum absolute error point0 = 1/21

finite-domain maximum absolute error point1 := div(2, 21)

finite-domain maximum absolute error point1 = 2/21

positive epsilon=1/10;ALL pointwise witness rows=[[0, 9], [1, 19]];finite domain size=2

finite maximum N=19;all point errors at common N=[1/21, 2/21];same N works for every point and every later n

Reverse checks

• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Recompute finite-domain maximum absolute error point1 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute finite-domain maximum absolute error point0 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute finite-domain maximum witness denominator with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Recompute finite-domain witness point1 previous absolute error with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute finite-domain witness point1 previous denominator with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute finite-domain witness point1 absolute error at cutoff with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute finite-domain witness point1 cutoff denominator with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute finite-domain witness point1 raw strict integer cutoff with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute finite-domain witness point1 strict reciprocal threshold with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute finite-domain witness point1 absolute coefficient cross product with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 finite-domain witness point0 previous absolute error with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute finite-domain witness point0 previous denominator with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute finite-domain witness point0 absolute error at cutoff with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute finite-domain witness point0 cutoff denominator with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute finite-domain witness point0 raw strict integer cutoff with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute finite-domain witness point0 strict reciprocal threshold with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute finite-domain witness point0 absolute coefficient cross product 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

(1/10, [[0, 9], [1, 19]], 19, Yes, finite_domain_maximum_only_not_infinite_domain_theorem)

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.