Question 17575: Take a Finite Maximum without Exchanging Infinite-Domain Quantifiers

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0034-17575 · q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0075

Original question and givens

Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[24, 25], coefficients=[25, -26], 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=1/34. 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=[24, 25], coefficients=[25, -26], 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=1/34. 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

S001 · implies

Before

same COMPLETE finite domain Fin(2);original aligned limits=[24, 25];signed coefficients=[25, -26];positive shift=2

After

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)

Conditions: none

The complete common finite domain and all-n formula are declared premises,not extrapolated from samples.Function values and errors are not probabilities;they may be signed or exceed1.Do not clip or shift indices.

Step identifiers and dependencies

Rule: pointwise_explicit_common_domain_all_n_tail_not_probability · Uses: G001

S002 · calculation

Before

finite-domain witness point0 absolute coefficient cross product := mul(25, 34)

After

finite-domain witness point0 absolute coefficient cross product = 850

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

finite-domain witness point0 strict reciprocal threshold := div(850, 1)

After

finite-domain witness point0 strict reciprocal threshold = 850

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

finite-domain witness point0 raw strict integer cutoff := sum(850, -2, 1)

After

finite-domain witness point0 raw strict integer cutoff = 849

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

finite-domain witness point0:point=0;SIGNED coefficient=25;absolute coefficient=25;positive epsilon=1/34;shift=2

After

threshold=850;floor=850;least nonnegative strict cutoff N=849

Conditions: none

The strict inequality is N+s>|c|b/a,so take its floor plus1 minus shift and clamp only at0.Equality is insufficient and a zero coefficient permits N=0.No probability≤1 guard belongs here.

Step identifiers and dependencies

Rule: pointwise_least_strict_cutoff_floor_not_probability_guard · Uses: G001, S004

S006 · calculation

Before

finite-domain witness point0 cutoff denominator := sum(849, 2)

After

finite-domain witness point0 cutoff denominator = 851

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

finite-domain witness point0 absolute error at cutoff := div(25, 851)

After

finite-domain witness point0 absolute error at cutoff = 25/851

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

finite-domain witness point0 previous denominator := sum(848, 2)

After

finite-domain witness point0 previous denominator = 850

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

finite-domain witness point0 previous absolute error := div(25, 850)

After

finite-domain witness point0 previous absolute error = 1/34

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

finite-domain witness point0:current N=849;current error=25/851;previous error=1/34;epsilon=1/34

After

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

Conditions: none

The preceding error is still≥epsilon.As denominators increase with n,every smaller nonnegative index also fails the strict bound.

Step identifiers and dependencies

Rule: pointwise_minimality_previous_index_strict_threshold · Uses: G001, S009

S011 · implies

Before

finite-domain witness point0:declared exact tail;positive denominators increase for all n>=N=849

After

for EVERY later n,absolute error<=bound=25/851<epsilon=1/34

Conditions: none

The same declared formula bounds every later denominator,which supplies an all-tail bound rather than a conclusion about an unknown sequence from one successful value.

Step identifiers and dependencies

Rule: pointwise_all_later_tail_not_single_numeric_sample · Uses: G001, S010

S012 · calculation

Before

finite-domain witness point1 absolute coefficient cross product := mul(26, 34)

After

finite-domain witness point1 absolute coefficient cross product = 884

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

finite-domain witness point1 strict reciprocal threshold := div(884, 1)

After

finite-domain witness point1 strict reciprocal threshold = 884

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

finite-domain witness point1 raw strict integer cutoff := sum(884, -2, 1)

After

finite-domain witness point1 raw strict integer cutoff = 883

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

finite-domain witness point1:point=1;SIGNED coefficient=-26;absolute coefficient=26;positive epsilon=1/34;shift=2

After

threshold=884;floor=884;least nonnegative strict cutoff N=883

Conditions: none

The strict inequality is N+s>|c|b/a,so take its floor plus1 minus shift and clamp only at0.Equality is insufficient and a zero coefficient permits N=0.No probability≤1 guard belongs here.

Step identifiers and dependencies

Rule: pointwise_least_strict_cutoff_floor_not_probability_guard · Uses: G001, S014

S016 · calculation

Before

finite-domain witness point1 cutoff denominator := sum(883, 2)

After

finite-domain witness point1 cutoff denominator = 885

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

finite-domain witness point1 absolute error at cutoff := div(26, 885)

After

finite-domain witness point1 absolute error at cutoff = 26/885

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

finite-domain witness point1 previous denominator := sum(882, 2)

After

finite-domain witness point1 previous denominator = 884

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

finite-domain witness point1 previous absolute error := div(26, 884)

After

finite-domain witness point1 previous absolute error = 1/34

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

finite-domain witness point1:current N=883;current error=26/885;previous error=1/34;epsilon=1/34

After

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

Conditions: none

The preceding error is still≥epsilon.As denominators increase with n,every smaller nonnegative index also fails the strict bound.

Step identifiers and dependencies

Rule: pointwise_minimality_previous_index_strict_threshold · Uses: G001, S019

S021 · implies

Before

finite-domain witness point1:declared exact tail;positive denominators increase for all n>=N=883

After

for EVERY later n,absolute error<=bound=26/885<epsilon=1/34

Conditions: none

The same declared formula bounds every later denominator,which supplies an all-tail bound rather than a conclusion about an unknown sequence from one successful value.

Step identifiers and dependencies

Rule: pointwise_all_later_tail_not_single_numeric_sample · Uses: G001, S020

S022 · calculation

Before

finite-domain maximum witness denominator := sum(883, 2)

After

finite-domain maximum witness denominator = 885

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

finite-domain maximum absolute error point0 := div(25, 885)

After

finite-domain maximum absolute error point0 = 5/177

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

finite-domain maximum absolute error point1 := div(26, 885)

After

finite-domain maximum absolute error point1 = 26/885

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · implies

Before

positive epsilon=1/34;ALL pointwise witness rows=[[0, 849], [1, 883]];finite domain size=2

After

finite maximum N=883;all point errors at common N=[5/177, 26/885];same N works for every point and every later n

Conditions: none

Every point of the complete finite domain has a witness,and the maximum of finitely many cutoffs works simultaneously.This does not interchange∀x and∃N on an arbitrary infinite domain.

Step identifiers and dependencies

Rule: pointwise_finite_maximum_no_infinite_quantifier_exchange · Uses: G001, S024

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

(1/34, [[0, 849], [1, 883]], 883, 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.

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input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: ca4edf137de1c08a7f090e7e773fdd3c3f54db601b7acbcd40d797fba22f5bd5
candidate record SHA-256: 1027184b0c3b22c353d5d717de08f2d7bf0223bfc1bbb1218e0520c8adf820b4
semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336
question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699
Lean: NOT_RUN · human review: NOT_PERFORMED

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