R0034-17520 · q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0045
Original question and givens
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[19, 20], coefficients=[20, -21], 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=1/29, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
G001 — Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[19, 20], coefficients=[20, -21], 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=1/29, point=1. Give (point,epsilon,N,error_at_N,strict_tail?,forall_n_ge_N), with error_at_N the absolute error bound.
Step-by-step derivation
S001 · implies
Before
same COMPLETE finite domain Fin(2);original aligned limits=[19, 20];signed coefficients=[20, -21];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
selected pointwise witness absolute coefficient cross product := mul(21, 29)
After
selected pointwise witness absolute coefficient cross product = 609
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
selected pointwise witness strict reciprocal threshold := div(609, 1)
After
selected pointwise witness strict reciprocal threshold = 609
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
selected pointwise witness raw strict integer cutoff := sum(609, -2, 1)
After
selected pointwise witness raw strict integer cutoff = 608
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
selected pointwise witness:point=1;SIGNED coefficient=-21;absolute coefficient=21;positive epsilon=1/29;shift=2
After
threshold=609;floor=609;least nonnegative strict cutoff N=608
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
selected pointwise witness cutoff denominator := sum(608, 2)
After
selected pointwise witness cutoff denominator = 610
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
selected pointwise witness absolute error at cutoff := div(21, 610)
After
selected pointwise witness absolute error at cutoff = 21/610
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
selected pointwise witness previous denominator := sum(607, 2)
After
selected pointwise witness previous denominator = 609
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
selected pointwise witness previous absolute error := div(21, 609)
After
selected pointwise witness previous absolute error = 1/29
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
selected pointwise witness:current N=608;current error=21/610;previous error=1/29;epsilon=1/29
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
selected pointwise witness:declared exact tail;positive denominators increase for all n>=N=608
After
for EVERY later n,absolute error<=bound=21/610<epsilon=1/29
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