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R0004-02549 · q-20261013-advance-20261021-r03-orbit-fixed-periodic-0074
Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=-1, b=15. 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=7. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).
G001 — Use the complete finite self-map on Fin(10)={0,…,9}, T(x)=(a*x+b) mod 10, a=-1, b=15. 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=7. Give (modulus,a,b,x,trace before first repeat,repeated state,zero-based cycle-start index,least positive period or nonperiodic-point).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=10; a=-1; b=15
After
S={0,...,9}; 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
start=7
After
record each state before its first successor calculation
Conditions: none
Start with the specified point itself,not its first successor.
Rule: orbit_initial_state · Uses: G001, S001
Before
least period step1 product := mul(-1, 7)
After
least period step1 product = -7
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
least period step1 affine value := sum(-7, 15)
After
least period step1 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, S003
Before
Euclidean division of 8 by 10
After
8=0·10+8
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S004
Before
least period step2 product := mul(-1, 8)
After
least period step2 product = -8
Conditions: none
Multiply the current state by the original coefficient a,retaining its sign.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
least period step2 affine value := sum(-8, 15)
After
least period step2 affine value = 7
Conditions: none
Then add the original constant b;this integer is not yet the canonical state.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Euclidean division of 7 by 10
After
7=0·10+7
Conditions: none
Check the transition using quotient and nonnegative remainder;n>0 and0≤remainder<n ensure uniqueness.
Rule: orbit_successor · Uses: G001, S007
Before
distinct trace=[7, 8]; first repeated state=7
After
cycle_start=0; starting-point least period=2
Conditions: none
The first repeated state locates the cycle entry.If it is the initial point,the preceding states are distinct,so the first return is the least positive period.If entry is later,the initial point lies outside the cycle and never returns.
Rule: finite_orbit_certificate · Uses: G001, S008
S009 · premise_and_result_audit — Replay the cycle of this fixed self-map;return or nonreturn is certified only for this complete finite model,not an unspecified system.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Recompute least period step2 affine value with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute least period step2 product with exact rational arithmetic, not rounded decimals.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 — Recompute least period step1 affine value with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute least period step1 product with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(10, -1, 15, 7, [7, 8], 7, 0, 2)
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: 25fcdf2102487e4a7a853456ee0da5279f1ae7888eff0044e35cc11099e8ebdecandidate record SHA-256: d18e56a7204ee4ca8e6647d66c3c7f40d611c0f3e68f63954030614c32d1e05bsemantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED