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R0052-26926 · q-20261015-advance-20261117-r01-series-comparison-test-0051
Use exact scales A=1,B=2, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).
G001 — Use exact scales A=1,B=2, rational ratio r=1/8. Prefix a=[], prefix b=[], each with cutoff=0 terms. Starting at that index use A*(r)^(n−cutoff) and B*(r)^(n−cutoff), so the first tail exponent is0. Give (cutoff, prefix_sum_a, prefix_sum_b, tail_sum_a, tail_sum_b, total_sum_a, tail_convergence, certificate).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.
Before
source roles = [series-eventual-tail]
After
derived = [[cutoff, prefix_a_sum, prefix_b_sum, tail_a_sum, tail_b_sum, total_a, convergent, certificate]]
Conditions: none
List this operation/mode exact source field order without JavaScript.Keep Boolean,string and null roles distinct;do not pad fields or infer the interface from answer shape.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [1, 2, 1/8]
After
derived = [EVERY tail term is A*r^k and B*r^k for k>=0, r^0=1 including r=0, Yes]
Conditions: none
The whole tail law is given,with k starting at0.Nonnegative A<=B and r give termwise comparison;0<=r<1 licenses the geometric sum.At r=0 the first term still uses r^0=1.
Rule: Series_WHOLE_geometric_tail_premises_not_finite_samples · Uses: G001, S001
Before
source roles = [0, [], []]
After
derived = [0, 0, tail exponent n-cutoff starts at0]
Conditions: none
cutoff is the NUMBER of prefix terms.Prefixes are separately nonnegative but need not be ordered termwise or by sum.The tail starts at exponent0 when n=cutoff;empty prefixes are allowed.
Rule: Series_arbitrary_nonnegative_prefix_count_NOT_global_order · Uses: G001, S002
Before
Series ORIGINAL one minus ratio := sub(1, 1/8)
After
Series ORIGINAL one minus ratio = 7/8
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Series ORIGINAL geometric tailA := div(1, 7/8)
After
Series ORIGINAL geometric tailA = 8/7
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Series ORIGINAL geometric tailB := div(2, 7/8)
After
Series ORIGINAL geometric tailB = 16/7
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
source roles = [1, 2, 1/8, 7/8]
After
derived = [8/7, 16/7, Yes]
Conditions: none
For the finite sum,multiplication by(1-r) telescopes to A(1-r^N);r^N tends to0,hence the tail sum A/(1-r),and similarly for B.The nonnegative a partial sums increase and are bounded by the convergent b sum.This is not a finite-sample replacement for all n.
Rule: Series_GENERAL_geometric_partial_sum_limit_and_upper_comparison · Uses: G001, S006
Before
Series ORIGINAL totalA prefix plus tail := sum(0, 8/7)
After
Series ORIGINAL totalA prefix plus tail = 8/7
Conditions: none
Substitute the original roles exactly,retaining symbolic operations;no floating approximation or extra source-domain restriction.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
source roles = [0, 0, 0, 8/7, 16/7]
After
derived = [8/7, Yes]
Conditions: none
For sufficiently large N,the full partial sum is a fixed finite prefix plus a tail partial sum.Adding a fixed finite number preserves the limit,hence A converges.Compare tails only,not full-series totals,and impose no source numeric gate on an unused totalB.
Rule: Series_finite_prefix_preserves_tail_convergence_NOT_total_order · Uses: G001, S008
S009 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S008 · premise_and_result_audit — Recompute Series ORIGINAL totalA prefix plus tail with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S006 · premise_and_result_audit — Recompute Series ORIGINAL geometric tailB with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Series ORIGINAL geometric tailA with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Series ORIGINAL one minus ratio with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.(0, 0, 0, 8/7, 16/7, 8/7, Yes, finite_prefix_does_not_change_tail_convergence)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 318e95a9e689e7b6ff68f24026ee767b116b395e9b22832b7066254dc7493197candidate record SHA-256: 35d8592d84cc291864c4315726c17c5b290268ac49eaca90c0073f508da09e13semantic review receipt SHA-256: 729b25cbf6bf5cb1d07a21a511ae216ec5fdd54a4a7aa2f7c21156b81ff37252question-first witness SHA-256: d1ab788770eaad04f986986e65afb0d55791e8a8c0d747e9a2b499f79fe78df2Lean: NOT_RUN · human review: NOT_PERFORMED