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R0004-02519 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0019
Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=11, b=27. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,T(x),T(x)=x?).
G001 — Use the complete finite self-map on Fin(5)={0,…,4}, T(x)=(a*x+b) mod 5, a=11, b=27. Take least nonnegative residues even for negative a*x+b. Fixed points, return times and least positive periods are different notions; the conclusion concerns this finite model only. x=3. Give (modulus,a,b,x,T(x),T(x)=x?).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
n=5; a=11; b=27
After
S={0,...,4}; T(z)=(a·z+b) mod nConditions: none
Remainders lie in0,...,n−1,so every state has one successor in the same finite set;negative dividends do not produce negative canonical remainders.
Rule: finite_self_map · Uses: G001
Before
fixed audit product := mul(11, 3)
After
fixed audit product = 33
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
fixed audit affine value := sum(33, 27)
After
fixed audit affine value = 60
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Euclidean division of 60 by 5
After
60=12·5+0
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S003
Before
T(3)=0
After
T(x)=x: No
Conditions: none
A fixed point returns in one step;this is equivalent to least positive period1,not merely a later return.
Rule: fixed_point_definition · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute fixed audit affine value with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute fixed audit product with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(5, 11, 27, 3, 0, No)
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: d3adcd7fd2ee44e992d7e0274677e1a6acec4e4f29662a4d36099f3369edb933candidate record SHA-256: 4f16e205671c91eafa7f386a04f623ca552a25c427ade34d924aa0d20640136bsemantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED