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R0058-29544 · q-20261013-advance-20261022-r02-present-value-discounting-0069
Use a fixed rate i=1/20 per declared period, with 1+i>0, fixed period conventions and no cash flows, fees, taxes or rounding. Amounts are nonnegative exact rationals. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. This is not investment advice, a quote or a market forecast. principal=86/1, periods=0. Give (principal,rate,periods,compounded future amount,recovered present amount,exact recovery?).
G001 — Use a fixed rate i=1/20 per declared period, with 1+i>0, fixed period conventions and no cash flows, fees, taxes or rounding. Amounts are nonnegative exact rationals. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. This is not investment advice, a quote or a market forecast. principal=86/1, periods=0. Give (principal,rate,periods,compounded future amount,recovered present amount,exact recovery?).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 = [qf-str22-presentvalue-forward-backward-audit, presentvalue-forward-backward-audit]
After
derived = [[principal, rate, periods, future, recovered, roundtrip]]
Conditions: none
Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention.
Rule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001
Before
source roles = [Present value rate, [1, 20]]
After
derived = [1/20]
Conditions: none
Check BOTH raw pair components within1e6 and nonzero denominator BEFORE sign/gcd normalization.This STR22 profile must not be imposed on legacy SAFE/BigInt tasks.
Rule: STR22_input_RAW_pair1e6_before_sign_gcd_normalization · Uses: G001, S001
Before
Present value ORIGINAL positive accumulation factor := sum(1, 1/20)
After
Present value ORIGINAL positive accumulation factor = 21/20
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [Present value ORIGINAL positive accumulation factor, sum, [1, 1/20]]
After
derived = [[1], [21, 20], 21/20]
Conditions: none
SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold.
Rule: STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction · Uses: G001, S003
Before
source roles = [1/20, 21/20]
After
derived = [Yes]
Conditions: none
Use the SAME fixed per-period rate/period convention without cash flows,fees,taxes or rounding.Require1+rate>0;the rate may be negative.This is not a market forecast or investment advice.
Rule: Present_value_FIXED_model_same_rate_period_NOcashflows · Uses: G001, S004
Before
source roles = [21/20, 0]
After
derived = [1, 21/20, 0, out*base^exp=factor^n]
Conditions: none
Initialize out=1,base=factor,exp=n with invariant out*base^exp=factor^n.At n=0 return the empty product1,but STILL qualify the original factor.
Rule: Present_value_BINARY_power_initial_emptyproduct_invariant · Uses: G001, S005
Before
source roles = [Present value principal, [86, 1]]
After
derived = [86]
Conditions: none
Check BOTH raw pair components within1e6 and nonzero denominator BEFORE sign/gcd normalization.This STR22 profile must not be imposed on legacy SAFE/BigInt tasks.
Rule: STR22_input_RAW_pair1e6_before_sign_gcd_normalization · Uses: G001, S006
Before
Present value ORIGINAL compound principal := mul(86, 1)
After
Present value ORIGINAL compound principal = 86
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
source roles = [Present value ORIGINAL compound principal, mul, [86, 1]]
After
derived = [[1, 1], [86, 1], 86]
Conditions: none
SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold.
Rule: STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction · Uses: G001, S008
Before
Present value ORIGINAL discount SAME compounded amount := div(86, 1)
After
Present value ORIGINAL discount SAME compounded amount = 86
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
source roles = [Present value ORIGINAL discount SAME compounded amount, div, [86, 1]]
After
derived = [[1, 1], [86, 1], 86]
Conditions: none
SOURCE addition first uses denominator gcd,and multiplication cross-cancels.Check BOTH resulting raw components within1e6 BEFORE final reduction.Division qualifies the nonzero reciprocal.Do not substitute another bank profile or loosen a source hold.
Rule: STR22_SOURCE_gcd_crosscancel_THEN_raw1e6_BEFORE_final_reduction · Uses: G001, S010
Before
source roles = [86, 1/20, 0, 1]
After
derived = [86, 86, Yes]
Conditions: none
The SAME positive accumulation gives(principal*acc)/acc=principal.Retain both actual calculations,including n=0.This exact-model algebraic roundtrip makes no claim about realized future returns.
Rule: Present_value_roundtrip_cancel_SAME_positive_power · Uses: G001, S011
S012 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S011 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S010 · premise_and_result_audit — Recompute Present value ORIGINAL discount SAME compounded amount with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S008 · premise_and_result_audit — Recompute Present value ORIGINAL compound principal with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S006 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S005 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S004 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S003 · premise_and_result_audit — Recompute Present value ORIGINAL positive accumulation factor with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.(86, 1/20, 0, 86, 86, Yes)
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.
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