Original question and givens
Use the declared rational deviation-probability tail p_n=1/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/20. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
G001 — Use the declared rational deviation-probability tail p_n=1/(n+5) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Use positive target delta=1/20. Give (numerator,shift,delta,minimal_nonnegative_N,bound_at_N,strict_tail?,forall_n_ge_N_tail_bound).
Step-by-step derivation
S001 · implies
Before
positive profile parameters c=1;s=5;raw fraction=c/(n+s)
After
nonnegative valid probability indices start at=0;only there0<profile<=1
Conditions: none
A positive denominator and value≤1 require n+s≥c.Keep the original index;do not clip values above1 or silently shift them.This checks arithmetic domain only.The infinite-process/common-space interpretation remains declared,and the prefix-only family gains no infinite continuation premise.
Step identifiers and dependencies
Rule: deviation_profile_probability_domain_not_clipping · Uses: G001
S002 · calculation
Before
strict profile cross-product numerator := mul(1, 20)
After
strict profile cross-product numerator = 20
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
strict profile reciprocal threshold := div(20, 1)
After
strict profile reciprocal threshold = 20
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
strict profile unclamped cutoff := sum(20, -5, 1)
After
strict profile unclamped cutoff = 16
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
raw target delta=[1, 20];normalized delta=1/20;positive denominator target numerator=1
After
floor(c*delta_denominator/delta_numerator)=20;threshold=20;raw least nonnegative strict N=16
Conditions: none
The strict inequality c/(N+s)<a/b is equivalent to N+s>cb/a,hence floor(cb/a)−s+1 clamped only at0.Equality still requires the next index.
Step identifiers and dependencies
Rule: strict_integer_cutoff_floor_and_nonnegative_boundary · Uses: G001, S004
S006 · calculation
Before
strict witness denominator := sum(16, 5)
After
strict witness denominator = 21
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
strict witness valid probability bound := div(1, 21)
After
strict witness valid probability bound = 1/21
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
previous raw reciprocal denominator := sum(15, 5)
After
previous raw reciprocal denominator = 20
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
previous raw reciprocal value := div(1, 20)
After
previous raw reciprocal value = 1/20
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · implies
Before
N=16;current valid probability=1/21;previous RAW reciprocal=1/20;delta=1/20
After
current strict bound holds,previous raw threshold fails;N is minimal
Conditions: none
The previous arithmetic reciprocal is still≥delta,so every smaller index fails the original strict threshold.If that earlier index is outside the probability domain,it is only a raw reciprocal,not a probability above1.The returned N must pass the original English domain guard;do not max-shift the source answer.
Step identifiers and dependencies
Rule: strict_profile_minimality_without_invalid_probability_claim · Uses: G001, S009
S011 · implies
Before
declared same-space deviation profile on valid tail;n>=N=16;positive denominators increase
After
every later profile<=bound=1/21<delta=1/20
Conditions: none
Conditionally on the declared deviation profile,increasing denominators bound every later index.This selected-delta witness neither constructs a common process from numbers nor proves all-epsilon convergence of an unknown sequence from finite data.
Step identifiers and dependencies
Rule: probability_tail_all_later_indices_not_samples · Uses: G001, S010