R0034-17512 · q-20261014-advance-20261103-r02-pointwise-convergence-of-functions-0037
Original question and givens
Use the same complete finite domain Fin(2)={0,…,1}, with aligned limit values=[11, 12], coefficients=[12, -13], 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=1/21, 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=[11, 12], coefficients=[12, -13], 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=1/21, 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=[11, 12];signed coefficients=[12, -13];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(13, 21)
After
selected pointwise witness absolute coefficient cross product = 273
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(273, 1)
After
selected pointwise witness strict reciprocal threshold = 273
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(273, -2, 1)
After
selected pointwise witness raw strict integer cutoff = 272
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=-13;absolute coefficient=13;positive epsilon=1/21;shift=2
After
threshold=273;floor=273;least nonnegative strict cutoff N=272
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(272, 2)
After
selected pointwise witness cutoff denominator = 274
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(13, 274)
After
selected pointwise witness absolute error at cutoff = 13/274
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(271, 2)
After
selected pointwise witness previous denominator = 273
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(13, 273)
After
selected pointwise witness previous absolute error = 1/21
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=272;current error=13/274;previous error=1/21;epsilon=1/21
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=272
After
for EVERY later n,absolute error<=bound=13/274<epsilon=1/21
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