Question 17557: 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=[6, 7], coefficients=[7, -8], 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=1/16. 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=[6, 7], coefficients=[7, -8], 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=1/16. 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=[6, 7];signed coefficients=[7, -8];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(7, 16)

finite-domain witness point0 absolute coefficient cross product = 112

finite-domain witness point0 strict reciprocal threshold := div(112, 1)

finite-domain witness point0 strict reciprocal threshold = 112

finite-domain witness point0 raw strict integer cutoff := sum(112, -2, 1)

finite-domain witness point0 raw strict integer cutoff = 111

finite-domain witness point0:point=0;SIGNED coefficient=7;absolute coefficient=7;positive epsilon=1/16;shift=2

threshold=112;floor=112;least nonnegative strict cutoff N=111

finite-domain witness point0 cutoff denominator := sum(111, 2)

finite-domain witness point0 cutoff denominator = 113

finite-domain witness point0 absolute error at cutoff := div(7, 113)

finite-domain witness point0 absolute error at cutoff = 7/113

finite-domain witness point0 previous denominator := sum(110, 2)

finite-domain witness point0 previous denominator = 112

finite-domain witness point0 previous absolute error := div(7, 112)

finite-domain witness point0 previous absolute error = 1/16

finite-domain witness point0:current N=111;current error=7/113;previous error=1/16;epsilon=1/16

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=111

for EVERY later n,absolute error<=bound=7/113<epsilon=1/16

finite-domain witness point1 absolute coefficient cross product := mul(8, 16)

finite-domain witness point1 absolute coefficient cross product = 128

finite-domain witness point1 strict reciprocal threshold := div(128, 1)

finite-domain witness point1 strict reciprocal threshold = 128

finite-domain witness point1 raw strict integer cutoff := sum(128, -2, 1)

finite-domain witness point1 raw strict integer cutoff = 127

finite-domain witness point1:point=1;SIGNED coefficient=-8;absolute coefficient=8;positive epsilon=1/16;shift=2

threshold=128;floor=128;least nonnegative strict cutoff N=127

finite-domain witness point1 cutoff denominator := sum(127, 2)

finite-domain witness point1 cutoff denominator = 129

finite-domain witness point1 absolute error at cutoff := div(8, 129)

finite-domain witness point1 absolute error at cutoff = 8/129

finite-domain witness point1 previous denominator := sum(126, 2)

finite-domain witness point1 previous denominator = 128

finite-domain witness point1 previous absolute error := div(8, 128)

finite-domain witness point1 previous absolute error = 1/16

finite-domain witness point1:current N=127;current error=8/129;previous error=1/16;epsilon=1/16

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=127

for EVERY later n,absolute error<=bound=8/129<epsilon=1/16

finite-domain maximum witness denominator := sum(127, 2)

finite-domain maximum witness denominator = 129

finite-domain maximum absolute error point0 := div(7, 129)

finite-domain maximum absolute error point0 = 7/129

finite-domain maximum absolute error point1 := div(8, 129)

finite-domain maximum absolute error point1 = 8/129

positive epsilon=1/16;ALL pointwise witness rows=[[0, 111], [1, 127]];finite domain size=2

finite maximum N=127;all point errors at common N=[7/129, 8/129];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

(1/16, [[0, 111], [1, 127]], 127, 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.