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R0004-02510 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0010
Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=2, b=9. 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=1. Give (modulus,a,b,x,T(x),T(x)=x?).
G001 — Use the complete finite self-map on Fin(8)={0,…,7}, T(x)=(a*x+b) mod 8, a=2, b=9. 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=1. 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=8; a=2; b=9
After
S={0,...,7}; 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(2, 1)
After
fixed audit product = 2
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(2, 9)
After
fixed audit affine value = 11
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 11 by 8
After
11=1·8+3
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(1)=3
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.(8, 2, 9, 1, 3, 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: 9bbaeb9bc99944e0de4e60cb038031197f4327ef3c30b19f205906a5221c3671candidate record SHA-256: b91d40ea9758a71e08955336845841e0b0f8e5d57f7ed5148c9add46f5cf4a14semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED