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R0058-29553 · q-20261013-advance-20261022-r02-present-value-discounting-0028
Use a fixed rate i=0/20 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. future=106/3, periods=2≥0. Give (future,rate,periods,(1+rate)^periods,present value). Zero periods gives accumulation 1 and preserves the amount.
G001 — Use a fixed rate i=0/20 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. future=106/3, periods=2≥0. Give (future,rate,periods,(1+rate)^periods,present value). Zero periods gives accumulation 1 and preserves the amount.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-n-period-discount, presentvalue-n-period-discount]
After
derived = [[future, rate, periods, accumulation, present]]
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, [0, 20]]
After
derived = [0]
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, 0)
After
Present value ORIGINAL positive accumulation factor = 1
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, 0]]
After
derived = [[1], [1, 1], 1]
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 = [0, 1]
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 = [1, 2]
After
derived = [1, 1, 2, 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
Present value ORIGINAL binary necessary square0 := mul(1, 1)
After
Present value ORIGINAL binary necessary square0 = 1
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, S006
Before
source roles = [Present value ORIGINAL binary necessary square0, mul, [1, 1]]
After
derived = [[1, 1], [1, 1], 1]
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, S007
Before
source roles = [1, 2, 0, [1, 1, 2]]
After
derived = [No, 1, 1, 1, Yes]
Conditions: none
Write old exp=2q+r.Multiply by base when r=1,and square the base ONLY if q>0.This preserves the SAME exact power without an unused final square that could cause a spurious source hold.Do not replace the SOURCE binary sequence with linear multiplication and shrink its domain.
Rule: Present_value_EACH_binary_exponent_step_no_unused_final_square · Uses: G001, S008
Before
Present value ORIGINAL binary odd multiply1 := mul(1, 1)
After
Present value ORIGINAL binary odd multiply1 = 1
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 binary odd multiply1, mul, [1, 1]]
After
derived = [[1, 1], [1, 1], 1]
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 = [1, 2, 1, [1, 1, 1]]
After
derived = [Yes, 1, 1, 0, Yes]
Conditions: none
Write old exp=2q+r.Multiply by base when r=1,and square the base ONLY if q>0.This preserves the SAME exact power without an unused final square that could cause a spurious source hold.Do not replace the SOURCE binary sequence with linear multiplication and shrink its domain.
Rule: Present_value_EACH_binary_exponent_step_no_unused_final_square · Uses: G001, S011
Before
source roles = [Present value future, [106, 3]]
After
derived = [106/3]
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, S012
Before
Present value ORIGINAL future over actual binary accumulation := div(106/3, 1)
After
Present value ORIGINAL future over actual binary accumulation = 106/3
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, S013
Before
source roles = [Present value ORIGINAL future over actual binary accumulation, div, [106/3, 1]]
After
derived = [[1, 1], [106, 3], 106/3]
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, S014
Before
source roles = [106/3, 0, 2, 1]
After
derived = [106/3]
Conditions: none
The accumulation is n copies of the SAME factor;divide future by its positive value.At n=0 preserve the amount.Do not add unspecified cash flows or charges.
Rule: Present_value_Nperiod_exact_discount_NOT_unlisted_adjustments · Uses: G001, S015
S016 · 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.S015 · 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.S014 · premise_and_result_audit — Recompute Present value ORIGINAL future over actual binary accumulation with exact rational arithmetic, not rounded decimals.S013 · 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.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 binary odd multiply1 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 — 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.S007 · premise_and_result_audit — Recompute Present value ORIGINAL binary necessary square0 with exact rational arithmetic, not rounded decimals.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.(106/3, 0, 2, 1, 106/3)
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: 072f38fe801247e66ba17ce3d521ad8cbf25a921923149ec802cb2db5990c507candidate record SHA-256: 20033247c0933adf613d0a8682dc8810460c903fda48d5c2a81bafc8e5f3a0f9semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05dfquestion-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7Lean: NOT_RUN · human review: NOT_PERFORMED