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R0004-02509 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0009
Use the complete finite self-map on Fin(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=1, b=7. 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(7)={0,…,6}, T(x)=(a*x+b) mod 7, a=1, b=7. 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=7; a=1; b=7
After
S={0,...,6}; 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(1, 1)
After
fixed audit product = 1
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(1, 7)
After
fixed audit affine value = 8
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 8 by 7
After
8=1·7+1
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)=1
After
T(x)=x: Yes
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.(7, 1, 7, 1, 1, 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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 96ec84097dc9f34823156b1cd5cb4b59fd050c09219e0b5046e4ffcf9da9b818candidate record SHA-256: 2e3d8797bce87aafc9cb5e8f7bfd9e20c63bba5ab8748f04d5d60f1af0d12e52semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED