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R0004-02505 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0005
Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-3, b=-1. 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=4. Give (modulus,a,b,x,T(x),T(x)=x?).
G001 — Use the complete finite self-map on Fin(9)={0,…,8}, T(x)=(a*x+b) mod 9, a=-3, b=-1. 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=4. 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=9; a=-3; b=-1
After
S={0,...,8}; 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(-3, 4)
After
fixed audit product = -12
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(-12, -1)
After
fixed audit affine value = -13
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 -13 by 9
After
-13=-2·9+5
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(4)=5
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.(9, -3, -1, 4, 5, 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: f31dcd0f965bcfabc4976c600b7c97c6564c5f9e88c191b67552ed2fa596bea4candidate record SHA-256: e1d12b1ba77731bb28de56d8b9cc5dae1687a0abd5fc6c8c00409925c874dd38semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED