Original question and givens
Use only the explicitly supplied finite list p_n=4/(n+5) for n=0 through prefix_length-1, with prefix_length=4. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
G001 — Use only the explicitly supplied finite list p_n=4/(n+5) for n=0 through prefix_length-1, with prefix_length=4. Every listed entry must be in [0,1]. The formula describes these finitely specified entries only: no continuation beyond this finite prefix, infinite-process construction or full forall-epsilon tail premise is supplied. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Give (numerator,shift,prefix_length,probability_values,sufficient?,finite_prefix_not_convergence_certificate).
Step-by-step derivation
S001 · implies
Before
positive profile parameters c=4;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
finite prefix index0 denominator := sum(0, 5)
After
finite prefix index0 denominator = 5
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
finite prefix index0 probability := div(4, 5)
After
finite prefix index0 probability = 4/5
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
finite prefix index1 denominator := sum(1, 5)
After
finite prefix index1 denominator = 6
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 · calculation
Before
finite prefix index1 probability := div(4, 6)
After
finite prefix index1 probability = 2/3
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, S004
S006 · calculation
Before
finite prefix index2 denominator := sum(2, 5)
After
finite prefix index2 denominator = 7
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
finite prefix index2 probability := div(4, 7)
After
finite prefix index2 probability = 4/7
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
finite prefix index3 denominator := sum(3, 5)
After
finite prefix index3 denominator = 8
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
finite prefix index3 probability := div(4, 8)
After
finite prefix index3 probability = 1/2
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
only indices0 through 3 supplied;finite probability entries=[4/5, 2/3, 4/7, 1/2]
After
general convergence established=No;no infinite continuation or every-epsilon tail supplied
Conditions: none
List the original zero-indexed prefix without dropping an invalid first entry.Even correct decreasing entries supply no every-epsilon/all-later obligation;an empty prefix is not a certificate either.
Step identifiers and dependencies
Rule: finite_probability_prefix_not_infinite_process_certificate · Uses: G001, S009
Result
(4, 5, 4, [4/5, 2/3, 4/7, 1/2], No, finite_prefix_not_convergence_certificate)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.