{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0001","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf8238176160e8de8cbfbe657001a898d23d298c9e6543e02be1d33d51ee6d91","family_id":"qf-app20-dynamics-state-update","seed":202610201001,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":-7,\"b\":-11,\"modulus\":5,\"x\":0}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=-7, b=-11. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=0. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(5, -7, -11, 0, 4)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, -7, -11, 0, 4)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0002","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a25d81f674c02cb63e6e84b97d510ea30f9738a8edeb82f4be778eb7e422d861","family_id":"qf-app20-dynamics-state-update","seed":202610201002,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":-6,\"b\":-9,\"modulus\":6,\"x\":1}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=-6, b=-9. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(6, -6, -9, 1, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, -6, -9, 1, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0003","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7264ed134d278c84ea78169a2673ff9bfae7577d8b670a043aa6cbe71584d7aa","family_id":"qf-app20-dynamics-state-update","seed":202610201003,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":-5,\"b\":-7,\"modulus\":7,\"x\":2}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=-5, b=-7. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=2. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(7, -5, -7, 2, 4)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, -5, -7, 2, 4)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0004","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2de1b5ee9300e8e0bb84f22aadad8f7a616949c07f60203b254d4ce78fa036cf","family_id":"qf-app20-dynamics-state-update","seed":202610201004,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":-4,\"b\":-5,\"modulus\":8,\"x\":3}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=-4, b=-5. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=3. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(8, -4, -5, 3, 7)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, -4, -5, 3, 7)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0005","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0704531be927f688bab2557a9e773ab9a469db7f79be0558a4b788bcdc3f7d04","family_id":"qf-app20-dynamics-state-update","seed":202610201005,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":-3,\"b\":-3,\"modulus\":9,\"x\":4}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=-3, b=-3. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=4. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(9, -3, -3, 4, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, -3, -3, 4, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0006","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be593e8d9cbb8d7e3c620ffd1a32d1cbb5d519fd82c4d41a1b87a1b93e490cf0","family_id":"qf-app20-dynamics-state-update","seed":202610201006,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":-2,\"b\":-1,\"modulus\":10,\"x\":5}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=-2, b=-1. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=5. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(10, -2, -1, 5, 9)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, -2, -1, 5, 9)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0007","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff6505df2b0880f3d5ac03bd496006f77ea1af024ab24b427de050d531cd5fd6","family_id":"qf-app20-dynamics-state-update","seed":202610201007,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":-1,\"b\":1,\"modulus\":5,\"x\":1}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=-1, b=1. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(5, -1, 1, 1, 0)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, -1, 1, 1, 0)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0008","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1b4a065674912ad307dddb948a8aeebb72e1bc532a553dee2904715bfa98e6f","family_id":"qf-app20-dynamics-state-update","seed":202610201008,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":0,\"b\":3,\"modulus\":6,\"x\":1}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=0, b=3. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(6, 0, 3, 1, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 0, 3, 1, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0009","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2269d0c93237721add61b16bebf39d3b951ff149c685e8cc995770b1b61f6aa4","family_id":"qf-app20-dynamics-state-update","seed":202610201009,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":1,\"b\":5,\"modulus\":7,\"x\":1}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=1, b=5. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(7, 1, 5, 1, 6)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 1, 5, 1, 6)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0010","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b04aeb5c288e5383c45ac1b57bddf300d5080e509a9a5ad72e070dcb1aab87b","family_id":"qf-app20-dynamics-state-update","seed":202610201010,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":2,\"b\":7,\"modulus\":8,\"x\":1}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=2, b=7. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(8, 2, 7, 1, 1)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 2, 7, 1, 1)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0011","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"607100ef3f018a9f5090548f4e0fbe657d97c82a02085899642249814dfb166a","family_id":"qf-app20-dynamics-state-update","seed":202610201011,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":3,\"b\":9,\"modulus\":9,\"x\":1}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=3, b=9. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(9, 3, 9, 1, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 3, 9, 1, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0012","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8146acc1fcbb9eb200b9ea0a0854f328feb5612c831f854f40fe03290ead43d2","family_id":"qf-app20-dynamics-state-update","seed":202610201012,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":4,\"b\":11,\"modulus\":10,\"x\":1}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=4, b=11. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(10, 4, 11, 1, 5)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 4, 11, 1, 5)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0013","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e36edd4357af808c19d2c8a28a9862b876695b46f6ed85868cfd877c6f2520fb","family_id":"qf-app20-dynamics-state-update","seed":202610201013,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":5,\"b\":13,\"modulus\":5,\"x\":2}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=5, b=13. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=2. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(5, 5, 13, 2, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 5, 13, 2, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0014","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"319db97130b68e1e00c6d90c0914eef51829860b3ea97208d35adb19ce1ff210","family_id":"qf-app20-dynamics-state-update","seed":202610201014,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":6,\"b\":15,\"modulus\":6,\"x\":1}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=6, b=15. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(6, 6, 15, 1, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 6, 15, 1, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0015","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebaf09f176ae39e22c213f459a3aa6a728dfd42a05f845caca8dca677922ad12","family_id":"qf-app20-dynamics-state-update","seed":202610201015,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":7,\"b\":17,\"modulus\":7,\"x\":0}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=7, b=17. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=0. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(7, 7, 17, 0, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 7, 17, 0, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0016","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9879f6e993d14fe81ba1707958ff783bc1bf86ceba355b7d088bd49b3ba99109","family_id":"qf-app20-dynamics-state-update","seed":202610201016,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":8,\"b\":19,\"modulus\":8,\"x\":7}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=8, b=19. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=7. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(8, 8, 19, 7, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 8, 19, 7, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0017","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15de4d79dd9ecfc9f4ce6532f921509e335fb78c510352b5f85ca50f99458751","family_id":"qf-app20-dynamics-state-update","seed":202610201017,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":9,\"b\":21,\"modulus\":9,\"x\":7}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=9, b=21. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=7. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(9, 9, 21, 7, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 9, 21, 7, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0018","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c77f26c2197afbbcae99b22005f7c7547dc93b23d6427d0908d6445ce019476","family_id":"qf-app20-dynamics-state-update","seed":202610201018,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":10,\"b\":23,\"modulus\":10,\"x\":7}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=10, b=23. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=7. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(10, 10, 23, 7, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 10, 23, 7, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0019","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5093a650101e406bb2296f94f7111ee3205313369d5dafa7c40d01c0fc0552a","family_id":"qf-app20-dynamics-state-update","seed":202610201019,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":11,\"b\":25,\"modulus\":5,\"x\":3}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=11, b=25. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=3. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(5, 11, 25, 3, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 11, 25, 3, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0020","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f7ea034225e094f33dbad14ca4eb183caa65ceb7b50af5d3deb54c3441109d3","family_id":"qf-app20-dynamics-state-update","seed":202610201020,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":12,\"b\":27,\"modulus\":6,\"x\":1}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=12, b=27. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=1. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(6, 12, 27, 1, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 12, 27, 1, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0021","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1566007d849df6af8c5b41b1f1b4bb9dc9ee001e252fd6c6463db6ff088e033","family_id":"qf-app20-dynamics-state-update","seed":202610201021,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":13,\"b\":29,\"modulus\":7,\"x\":6}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=13, b=29. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=6. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(7, 13, 29, 6, 2)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 13, 29, 6, 2)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0022","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6887725be00a113b72aed493b545b7c5225416fd0516e0c22423da92dabffc49","family_id":"qf-app20-dynamics-state-update","seed":202610201022,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":14,\"b\":31,\"modulus\":8,\"x\":5}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=14, b=31. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=5. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(8, 14, 31, 5, 5)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 14, 31, 5, 5)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0023","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b576b53b07f7a678cdab1f2cb4591ac65bfc61d8e4cf5b6ceed661e7bb2dce2","family_id":"qf-app20-dynamics-state-update","seed":202610201023,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":15,\"b\":33,\"modulus\":9,\"x\":4}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=15, b=33. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=4. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(9, 15, 33, 4, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 15, 33, 4, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0024","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"152a53e7833cf3895856831500e8d21d07cb9ec14a9f93e4fe54f9af341d4154","family_id":"qf-app20-dynamics-state-update","seed":202610201024,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":16,\"b\":35,\"modulus\":10,\"x\":3}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=16, b=35. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=3. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(10, 16, 35, 3, 3)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 16, 35, 3, 3)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0025","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d27642af5adeb6a63ce9235a94f078d718a11fb5413c769fe6ae8ce78a835a95","family_id":"qf-app20-dynamics-state-update","seed":202610201025,"semantic_template":"qf-app20-dynamics-state-update/{\"a\":17,\"b\":37,\"modulus\":5,\"x\":4}"},"fields":{"title":"A Typed Finite State and Its Canonical Residue Update","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=17, b=37. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x=4. Give (modulus, a, b, x, T(x)).","answer":{"kind":"expression","value":"(5, 17, 37, 4, 0)","explanation":"Apply the specified affine rule and normalize the residue rather than using a negative programming-language remainder. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 17, 37, 4, 0)."},"skills":["A Typed Finite State and Its Canonical Residue Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0026","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd842cc1849a3d83a60a464ef772cee877fbd83ed63657ecee1785b13d60752b","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201026,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":1,\"b\":-7,\"modulus\":5,\"steps\":0,\"x0\":0}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=1, b=-7. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=0. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(5, 1, -7, 0, 0, [0])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 1, -7, 0, 0, [0])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0027","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a37a55d91f3aca3be4686ab86e18dc5a4d838c5b622d73224e527d28138f3656","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201027,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":2,\"b\":-4,\"modulus\":6,\"steps\":1,\"x0\":1}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=2, b=-4. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=1, updates=1. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(6, 2, -4, 1, 1, [1, 4])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 2, -4, 1, 1, [1, 4])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0028","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73d4418f0dbbe48af735dd5ea5b396eb43c8a563e68d6bfac0014fbed62460c3","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201028,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":3,\"b\":-1,\"modulus\":7,\"steps\":2,\"x0\":2}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=3, b=-1. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=2, updates=2. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(7, 3, -1, 2, 2, [2, 5, 0])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 3, -1, 2, 2, [2, 5, 0])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0029","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3441a785f8c8f9c4fd112c4881242ba88239656b5dedef929be46a3e8030450","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201029,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":4,\"b\":2,\"modulus\":8,\"steps\":3,\"x0\":3}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=4, b=2. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=3, updates=3. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(8, 4, 2, 3, 3, [3, 6, 2, 2])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 4, 2, 3, 3, [3, 6, 2, 2])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0030","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3c5279f5c3763c2ccd50ff02517b61e202ae891751f18af18e79eadff9fea02","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201030,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":5,\"b\":5,\"modulus\":9,\"steps\":4,\"x0\":4}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=5, b=5. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=4, updates=4. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(9, 5, 5, 4, 4, [4, 7, 4, 7, 4])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 5, 5, 4, 4, [4, 7, 4, 7, 4])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0031","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb6a305e8287bddf5760ceb700cec102b794bc0fb7f65ae1b47493eff8aebf88","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201031,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":6,\"b\":8,\"modulus\":5,\"steps\":5,\"x0\":0}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=6, b=8. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=5. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(5, 6, 8, 0, 5, [0, 3, 1, 4, 2, 0])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 6, 8, 0, 5, [0, 3, 1, 4, 2, 0])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0032","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83ad3352700e47964d2b73e1f214d5f5f2a06f6a7e00b6d5a81efd5a13d370e2","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201032,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":7,\"b\":11,\"modulus\":6,\"steps\":6,\"x0\":0}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=7, b=11. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=6. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(6, 7, 11, 0, 6, [0, 5, 4, 3, 2, 1, 0])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 7, 11, 0, 6, [0, 5, 4, 3, 2, 1, 0])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0033","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ca6c765817a677003fa5314dbc714fca99d691963c45dd145009e81d7af9428","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201033,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":8,\"b\":14,\"modulus\":7,\"steps\":7,\"x0\":0}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=8, b=14. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=7. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(7, 8, 14, 0, 7, [0, 0, 0, 0, 0, 0, 0, 0])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 8, 14, 0, 7, [0, 0, 0, 0, 0, 0, 0, 0])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0034","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dfdfe94eab51a1a5555beed283bf31b7713f6a9b8c2c4ddd72d0814fc673fff","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201034,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":9,\"b\":17,\"modulus\":8,\"steps\":8,\"x0\":0}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=9, b=17. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=8. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(8, 9, 17, 0, 8, [0, 1, 2, 3, 4, 5, 6, 7, 0])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 9, 17, 0, 8, [0, 1, 2, 3, 4, 5, 6, 7, 0])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0035","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b00a5a4390d9b2dd9070ab063bd402d9b61f1918ecf4dc365bab98dcff50aa72","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201035,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":10,\"b\":20,\"modulus\":9,\"steps\":0,\"x0\":0}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=10, b=20. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=0. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(9, 10, 20, 0, 0, [0])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 10, 20, 0, 0, [0])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0036","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc36c1c048245674d0d855c02309ef54146b4e72065eb2336bee5c58dd9c4424","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201036,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":11,\"b\":23,\"modulus\":5,\"steps\":1,\"x0\":0}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=11, b=23. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=1. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(5, 11, 23, 0, 1, [0, 3])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 11, 23, 0, 1, [0, 3])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0037","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2986c75af15b51babe3eee781ff0f401773965469430564b1ace2c985fd3f1a1","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201037,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":12,\"b\":26,\"modulus\":6,\"steps\":2,\"x0\":5}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=12, b=26. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=5, updates=2. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(6, 12, 26, 5, 2, [5, 2, 2])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 12, 26, 5, 2, [5, 2, 2])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0038","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78e13b10bc68a13da1af59e82e20cc9dcf1a2d8f137319643f2230919191e423","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201038,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":13,\"b\":29,\"modulus\":7,\"steps\":3,\"x0\":5}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=13, b=29. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=5, updates=3. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(7, 13, 29, 5, 3, [5, 3, 5, 3])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 13, 29, 5, 3, [5, 3, 5, 3])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0039","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"314f10981e54924c7d216bfed38a0a4f152a680eb90c64e0b4314c1fb224a6b1","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201039,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":14,\"b\":32,\"modulus\":8,\"steps\":4,\"x0\":5}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=14, b=32. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=5, updates=4. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(8, 14, 32, 5, 4, [5, 6, 4, 0, 0])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 14, 32, 5, 4, [5, 6, 4, 0, 0])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0040","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6317aa022d1da34c9c2faefc7d04294f22c65584da5d59b6f4cea114a3c1229","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201040,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":15,\"b\":35,\"modulus\":9,\"steps\":5,\"x0\":5}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=15, b=35. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=5, updates=5. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(9, 15, 35, 5, 5, [5, 2, 2, 2, 2, 2])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 15, 35, 5, 5, [5, 2, 2, 2, 2, 2])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0041","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe9f053e526fbeb10642df80718892b49ec7d9ac7b60a48649ae80a4f8d14739","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201041,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":16,\"b\":38,\"modulus\":5,\"steps\":6,\"x0\":0}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=16, b=38. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=6. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(5, 16, 38, 0, 6, [0, 3, 1, 4, 2, 0, 3])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 16, 38, 0, 6, [0, 3, 1, 4, 2, 0, 3])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0042","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b401639d2fb0edb1279ee4140e7723faf8ba4066405068f74c9e49210413a98b","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201042,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":17,\"b\":41,\"modulus\":6,\"steps\":7,\"x0\":4}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=17, b=41. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=4, updates=7. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(6, 17, 41, 4, 7, [4, 1, 4, 1, 4, 1, 4, 1])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 17, 41, 4, 7, [4, 1, 4, 1, 4, 1, 4, 1])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0043","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bf7e10277d5be4f56acff2a939d99d25260efca8ddf9f1f2c8364db44ca1d59","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201043,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":18,\"b\":44,\"modulus\":7,\"steps\":8,\"x0\":3}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=18, b=44. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=3, updates=8. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(7, 18, 44, 3, 8, [3, 0, 2, 3, 0, 2, 3, 0, 2])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 18, 44, 3, 8, [3, 0, 2, 3, 0, 2, 3, 0, 2])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0044","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d157e2658dc1b43e8ee4b09c4d78fe3b9048b7fff0ef58be54fb443e33ec947","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201044,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":19,\"b\":47,\"modulus\":8,\"steps\":0,\"x0\":2}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=19, b=47. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=2, updates=0. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(8, 19, 47, 2, 0, [2])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 19, 47, 2, 0, [2])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0045","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"090816e5dcff14cf3f8514e171ae932cdf77feb9b4efed816d47b2360c8a3cac","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201045,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":20,\"b\":50,\"modulus\":9,\"steps\":1,\"x0\":1}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=20, b=50. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=1, updates=1. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(9, 20, 50, 1, 1, [1, 7])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 20, 50, 1, 1, [1, 7])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0046","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"656b806a33e21b9b86c2c437f6ede4ce4bf9f82e6c952671970171d9d8fe59e2","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201046,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":21,\"b\":53,\"modulus\":5,\"steps\":2,\"x0\":0}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=21, b=53. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=0, updates=2. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(5, 21, 53, 0, 2, [0, 3, 1])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 21, 53, 0, 2, [0, 3, 1])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0047","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e71ebcac1e29d769f55688a648ed6e6c5b2988901eaedc4ba2d41614629fd11","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201047,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":22,\"b\":56,\"modulus\":6,\"steps\":3,\"x0\":3}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=22, b=56. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=3, updates=3. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(6, 22, 56, 3, 3, [3, 2, 4, 0])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 22, 56, 3, 3, [3, 2, 4, 0])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0048","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78c17e407906b8fc2788fc53fb4a5da3b12846899ecf221f08a513df73719219","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201048,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":23,\"b\":59,\"modulus\":7,\"steps\":4,\"x0\":1}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=23, b=59. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=1, updates=4. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(7, 23, 59, 1, 4, [1, 5, 6, 1, 5])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 23, 59, 1, 4, [1, 5, 6, 1, 5])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0049","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09f87dac92bc92ff9dcbcea84b41c04f71101d8408e8f7e85abd284a528f4b9c","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201049,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":24,\"b\":62,\"modulus\":8,\"steps\":5,\"x0\":7}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=24, b=62. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=7, updates=5. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(8, 24, 62, 7, 5, [7, 6, 6, 6, 6, 6])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 24, 62, 7, 5, [7, 6, 6, 6, 6, 6])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0050","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23cc4118f038fdc8d53e66f058bc841586354c0f63ce7521dbb4d2e7e07a0cce","family_id":"qf-app20-dynamics-orbit-prefix","seed":202610201050,"semantic_template":"qf-app20-dynamics-orbit-prefix/{\"a\":25,\"b\":65,\"modulus\":9,\"steps\":6,\"x0\":6}"},"fields":{"title":"A Fixed-Length Discrete Orbit Prefix","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=25, b=65. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. x0=6, updates=6. Give (modulus, a, b, x0, update count, orbit including x0). Zero updates gives the one-element prefix [x0].","answer":{"kind":"expression","value":"(9, 25, 65, 6, 6, [6, 8, 4, 3, 5, 1, 0])","explanation":"Iterate the complete self-map exactly the declared number of times; do not treat the prefix as a long-term stability proof. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 25, 65, 6, 6, [6, 8, 4, 3, 5, 1, 0])."},"skills":["A Fixed-Length Discrete Orbit Prefix"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0051","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4fc3f54cf6605923cd97140940f4f4aa4c866bc58dd0d633e899a0a970ec7974","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201051,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":-4,\"b\":1,\"modulus\":5}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=-4, b=1. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(5, -4, 1, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, -4, 1, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0052","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6718132450009ffda1f74a4bcd00d797bd5c6eff6f303100c2288e1ffa9f6180","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201052,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":-3,\"b\":3,\"modulus\":6}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=-3, b=3. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(6, -3, 3, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, -3, 3, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0053","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e1d2766cf9e34d818d978f73667568b1c67b23df1f1d390231c0ab16742e84e","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201053,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":-2,\"b\":5,\"modulus\":7}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=-2, b=5. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(7, -2, 5, [4])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, -2, 5, [4])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0054","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9634ea41baa4cf773dd0affbf835083b12ce37ec95fb83038e54dc74d09fc6ec","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201054,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":-1,\"b\":7,\"modulus\":8}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=-1, b=7. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(8, -1, 7, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, -1, 7, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0055","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"667828c36d9c2fcbfd80b15aea827783463117d11dfd6506bfbbc8e07b072ab1","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201055,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":0,\"b\":9,\"modulus\":9}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=0, b=9. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(9, 0, 9, [0])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 0, 9, [0])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0056","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfb8f5a43d42e1a517365d2600857e108c975c335d54380c5b60b02be58f7556","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201056,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":1,\"b\":11,\"modulus\":10}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=1, b=11. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(10, 1, 11, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 1, 11, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0057","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a915c1be3f9a92afa2169ca1896910bc456ffd9aa324777dc398ffd6ff201d8","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201057,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":2,\"b\":13,\"modulus\":5}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=2, b=13. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(5, 2, 13, [2])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 2, 13, [2])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0058","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c56609f0ba313e0748af76f3103f65ee5ce1fe68c58ffd88601adc568a8edf7a","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201058,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":3,\"b\":15,\"modulus\":6}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=3, b=15. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(6, 3, 15, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 3, 15, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0059","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"927d53984080ad74374b04e0000848b8c33c7e20f07b7f34c2ea9e871188e89f","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201059,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":4,\"b\":17,\"modulus\":7}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=4, b=17. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(7, 4, 17, [6])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 4, 17, [6])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0060","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0afdd0100ec973b3891a28ea75c53553090c9e43a4c38d7c2b9707831dfefd98","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201060,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":5,\"b\":19,\"modulus\":8}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=5, b=19. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(8, 5, 19, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 5, 19, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0061","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f3d46a870a745afe18472218c52a1ae0920721d6d8970676f2e53c26b7973ba","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201061,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":6,\"b\":21,\"modulus\":9}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=6, b=21. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(9, 6, 21, [3])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 6, 21, [3])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0062","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"685f727f7e55bde3580697e83fbaa4378c2877303aae42a14e4ea9bdc3a2c323","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201062,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":7,\"b\":23,\"modulus\":10}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=7, b=23. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(10, 7, 23, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 7, 23, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0063","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b9e2fec6d377ee46a6a6a3258a8f4484d5943e945c015eb57595b629a9efbab","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201063,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":8,\"b\":25,\"modulus\":5}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=8, b=25. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(5, 8, 25, [0])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 8, 25, [0])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0064","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce74c23c3c139525dc6ecbd1b4c7d634b7c1f5bebd47505df5975fa48bd35dd3","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201064,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":9,\"b\":27,\"modulus\":6}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=9, b=27. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(6, 9, 27, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 9, 27, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0065","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8995f4fcc5465939f112c7d3626544e105d892642d7bc276af03eb5701b9270","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201065,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":10,\"b\":29,\"modulus\":7}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=10, b=29. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(7, 10, 29, [3])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 10, 29, [3])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0066","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f930d679732bb4695cb1ca64e2c3030d1fbf7537d6b5351d7eafad6523e513b6","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201066,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":11,\"b\":31,\"modulus\":8}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=11, b=31. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(8, 11, 31, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 11, 31, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0067","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3013845125ce1c5bda6949a390db35db2752b15db5c5ef6468f7790dd88a92c9","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201067,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":12,\"b\":33,\"modulus\":9}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=12, b=33. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(9, 12, 33, [6])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 12, 33, [6])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0068","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39c48d81f74ff13d8bcad733f3395e17efc4f5ef374a020a0c1f72a3c0cc473a","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201068,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":13,\"b\":35,\"modulus\":10}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=13, b=35. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(10, 13, 35, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 13, 35, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0069","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"629efab3968096a419c79f187be84ff4cf7934607a6592feac7ed1c7ae3892aa","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201069,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":14,\"b\":37,\"modulus\":5}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=14, b=37. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(5, 14, 37, [1])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 14, 37, [1])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0070","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dd7a6d2f0418bfe0f49718d4d9b3f3450275290ebba127dbedcf24a9856be6d","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201070,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":15,\"b\":39,\"modulus\":6}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=15, b=39. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(6, 15, 39, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 15, 39, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0071","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90fa1eaa2ff4d5d9bf9a35c6d73a65385ebfdc7cbb377663a28a2df7c23b0234","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201071,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":16,\"b\":41,\"modulus\":7}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=16, b=41. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(7, 16, 41, [1])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 16, 41, [1])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0072","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"582d9ee0c1a4d14050fa8dfc62e3076713c6bdf29bbf0e2a01790bdccc613e5e","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201072,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":17,\"b\":43,\"modulus\":8}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=17, b=43. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(8, 17, 43, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 17, 43, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0073","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52f227478e4f6b216816551d136c93562a4116933f2c85debd732b06f7af17f6","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201073,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":18,\"b\":45,\"modulus\":9}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=18, b=45. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(9, 18, 45, [0])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 18, 45, [0])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0074","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e3646f31da6bb4c4dfb9a96bc6ffe8d637719f66cae749c3650ea27a712d5f0","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201074,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":19,\"b\":47,\"modulus\":10}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(10)={0,…,9} given by T(x)=(a*x+b) mod 10, a=19, b=47. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(10, 19, 47, [])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (10, 19, 47, [])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0075","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51a9199f5379ff3b9d9a4c31f59b308736f8740f024675fc29da894d9bb7e7bc","family_id":"qf-app20-dynamics-fixed-point-inventory","seed":202610201075,"semantic_template":"qf-app20-dynamics-fixed-point-inventory/{\"a\":20,\"b\":49,\"modulus\":5}"},"fields":{"title":"Every Fixed Point of the Given Finite Self-Map","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=20, b=49. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Give (modulus, a, b, every x with T(x)=x in increasing order).","answer":{"kind":"expression","value":"(5, 20, 49, [4])","explanation":"Enumerate the complete finite state set, retaining an empty fixed-point inventory when appropriate. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 20, 49, [4])."},"skills":["Every Fixed Point of the Given Finite Self-Map"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0076","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21326605a7f870b59ba7a1e383136bdcecede38deaa4c71ce28adedffe1d569e","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201076,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":1,\"b\":-3,\"modulus\":5,\"subset\":[0,1]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=1, b=-3. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(5, 1, -3, [0, 1], [2, 3], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 1, -3, [0, 1], [2, 3], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0077","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d3469ff2a236990a4b00cf0de7a7c0f6e727a8854562fdd19699d2597db60bf","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201077,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":2,\"b\":-2,\"modulus\":6,\"subset\":[2]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=2, b=-2. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[2], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(6, 2, -2, [2], [2], Yes)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 2, -2, [2], [2], Yes)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0078","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e4eccedc9f318da8fdd52b0a3ed7734ee7c13346057d2158b07bcf0c303ceb4","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201078,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":3,\"b\":-1,\"modulus\":7,\"subset\":[0,2]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=3, b=-1. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 2], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(7, 3, -1, [0, 2], [5, 6], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 3, -1, [0, 2], [5, 6], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0079","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8381b043b0885d4150143277ce8659569833cbd35bbc73fffa472b4a9edee33a","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201079,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":4,\"b\":0,\"modulus\":8,\"subset\":[1,2]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=4, b=0. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[1, 2], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(8, 4, 0, [1, 2], [0, 4], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 4, 0, [1, 2], [0, 4], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0080","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4164b2edf1fbcb9cd9706dbb157725ee8de6aeda485b7ea8bbc20ca0b368fa8c","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201080,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":5,\"b\":1,\"modulus\":9,\"subset\":[0,1,2]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=5, b=1. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 2], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(9, 5, 1, [0, 1, 2], [1, 2, 6], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 5, 1, [0, 1, 2], [1, 2, 6], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0081","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"211bfa376022ab82746a6c4fa405220c8d563d93e8fe8fcd7bce0433c5996942","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201081,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":6,\"b\":2,\"modulus\":5,\"subset\":[3]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=6, b=2. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(5, 6, 2, [3], [0], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 6, 2, [3], [0], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0082","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"477af86e2a51c76076935649ed05a931d1e7267e25bbf42f4caf6f0edce7e799","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201082,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":7,\"b\":3,\"modulus\":6,\"subset\":[0,3]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=7, b=3. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(6, 7, 3, [0, 3], [0, 3], Yes)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 7, 3, [0, 3], [0, 3], Yes)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0083","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"810aae4ad1f3110ab5e47e68ad53320b3e3ed2c8786a7139a2fbd059f891fb95","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201083,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":8,\"b\":4,\"modulus\":7,\"subset\":[1,3]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=8, b=4. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[1, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(7, 8, 4, [1, 3], [0, 5], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 8, 4, [1, 3], [0, 5], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0084","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00e1954cc1d6a81e4af0b96bcc74619cac8a349df16e8068795596ffcb6d7b48","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201084,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":9,\"b\":5,\"modulus\":8,\"subset\":[0,1,3]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=9, b=5. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(8, 9, 5, [0, 1, 3], [0, 5, 6], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 9, 5, [0, 1, 3], [0, 5, 6], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0085","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac03ee20fc9ecf5ef3f4a2dc0c51f3efe8b0ac0f1b81765bb0723ae6360c069a","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201085,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":10,\"b\":6,\"modulus\":9,\"subset\":[2,3]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=10, b=6. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[2, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(9, 10, 6, [2, 3], [0, 8], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 10, 6, [2, 3], [0, 8], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0086","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d30e23c688acf0729f78f87155d88232686f9b600301c4cd2f1ce7a8678b92a3","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201086,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":11,\"b\":7,\"modulus\":5,\"subset\":[0,2,3]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=11, b=7. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 2, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(5, 11, 7, [0, 2, 3], [0, 2, 4], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 11, 7, [0, 2, 3], [0, 2, 4], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0087","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85983336ac93178b71a51e2ab82aff5e816dd2f7f73e6b4d69660fc75047a8fb","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201087,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":12,\"b\":8,\"modulus\":6,\"subset\":[1,2,3]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=12, b=8. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[1, 2, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(6, 12, 8, [1, 2, 3], [2], Yes)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 12, 8, [1, 2, 3], [2], Yes)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0088","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc7fad813a5165d5900df4425b9da0f39360208dd3b0359813d5f0830315bbd5","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201088,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":13,\"b\":9,\"modulus\":7,\"subset\":[0,1,2,3]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=13, b=9. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 2, 3], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(7, 13, 9, [0, 1, 2, 3], [0, 1, 2, 6], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 13, 9, [0, 1, 2, 3], [0, 1, 2, 6], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0089","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c543bdc08afb2981c8f83223112e0d21614f3740308855fb2e94185dd1647c3e","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201089,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":14,\"b\":10,\"modulus\":8,\"subset\":[4]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=14, b=10. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(8, 14, 10, [4], [2], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 14, 10, [4], [2], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0090","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8fc756c71650bd3099efe11dc5b515e78c134633019d9b7d34bad2c92ccf73c","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201090,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":15,\"b\":11,\"modulus\":9,\"subset\":[0,4]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=15, b=11. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(9, 15, 11, [0, 4], [2, 8], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 15, 11, [0, 4], [2, 8], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0091","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d728fae9dc9dfa72a843498f83a3e8e316194095a0957cd9d78950296295de51","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201091,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":16,\"b\":12,\"modulus\":5,\"subset\":[1,4]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=16, b=12. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[1, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(5, 16, 12, [1, 4], [1, 3], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 16, 12, [1, 4], [1, 3], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0092","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1dc522874c6c6db4f9cb29261c46031cc6317b7439471bdc6f2f60e3d517d7e","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201092,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":17,\"b\":13,\"modulus\":6,\"subset\":[0,1,4]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=17, b=13. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(6, 17, 13, [0, 1, 4], [0, 1, 3], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 17, 13, [0, 1, 4], [0, 1, 3], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0093","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9c9ad270fc862a713d65bb79199275438f4ac24b38235a4bba510bb219acba3","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201093,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":18,\"b\":14,\"modulus\":7,\"subset\":[2,4]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=18, b=14. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[2, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(7, 18, 14, [2, 4], [1, 2], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 18, 14, [2, 4], [1, 2], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0094","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56e73ef86208fad3de2cc79906f7d4c658c5bca90691d8415e58efc5ebce07ad","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201094,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":19,\"b\":15,\"modulus\":8,\"subset\":[0,2,4]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=19, b=15. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 2, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(8, 19, 15, [0, 2, 4], [3, 5, 7], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 19, 15, [0, 2, 4], [3, 5, 7], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0095","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9456f05a7ba4c1e881432b899b36762f638de65572bbfb24faabf3e4bde17613","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201095,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":20,\"b\":16,\"modulus\":9,\"subset\":[1,2,4]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=20, b=16. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[1, 2, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(9, 20, 16, [1, 2, 4], [0, 2, 6], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 20, 16, [1, 2, 4], [0, 2, 6], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0096","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"212982068ee9ab4ff5c0b658caab91b67751efe494e828ec8af951f7f4d0651b","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201096,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":21,\"b\":17,\"modulus\":5,\"subset\":[0,1,2,4]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(5)={0,…,4} given by T(x)=(a*x+b) mod 5, a=21, b=17. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 2, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(5, 21, 17, [0, 1, 2, 4], [1, 2, 3, 4], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, 21, 17, [0, 1, 2, 4], [1, 2, 3, 4], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0097","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88d727e755a9ccf3f0da69b3f4c99e35589af231cc8f14a54dee1fc1ff70f88a","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201097,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":22,\"b\":18,\"modulus\":6,\"subset\":[3,4]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(6)={0,…,5} given by T(x)=(a*x+b) mod 6, a=22, b=18. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[3, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(6, 22, 18, [3, 4], [0, 4], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6, 22, 18, [3, 4], [0, 4], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0098","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31dc3a40ba1c78b08a1080f1c80bb4139d0ec8dd8f0d84cb84693514af8a2966","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201098,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":23,\"b\":19,\"modulus\":7,\"subset\":[0,3,4]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(7)={0,…,6} given by T(x)=(a*x+b) mod 7, a=23, b=19. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 3, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(7, 23, 19, [0, 3, 4], [4, 5, 6], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 23, 19, [0, 3, 4], [4, 5, 6], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0099","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22112f4b87983b4ee13f98a9ed3f0865d81baf32fdd7701a6d0bf267848e69c3","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201099,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":24,\"b\":20,\"modulus\":8,\"subset\":[1,3,4]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(8)={0,…,7} given by T(x)=(a*x+b) mod 8, a=24, b=20. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[1, 3, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(8, 24, 20, [1, 3, 4], [4], Yes)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (8, 24, 20, [1, 3, 4], [4], Yes)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r01-discrete-dynamical-system-0100","batch_id":"batch-20261013-advance-20261020-r01-discrete-dynamical-system-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"587c1f4e9bb0c0add33857acc71f1932df7797ed16c4787f3811a3068e5163d0","family_id":"qf-app20-dynamics-invariant-subset-audit","seed":202610201100,"semantic_template":"qf-app20-dynamics-invariant-subset-audit/{\"a\":25,\"b\":21,\"modulus\":9,\"subset\":[0,1,3,4]}"},"fields":{"title":"Forward Invariance versus Pointwise Fixing","stem":"Use the complete finite self-map on Fin(9)={0,…,8} given by T(x)=(a*x+b) mod 9, a=25, b=21. Choose the least nonnegative residue, including for negative a*x+b. Time is discrete, indexed from 0; a finite orbit is not a general stability result. Candidate subset=[0, 1, 3, 4], normalized as a set. Give (modulus, a, b, normalized subset, sorted distinct image, is the image contained in the subset?).","answer":{"kind":"expression","value":"(9, 25, 21, [0, 1, 3, 4], [1, 3, 4, 6], No)","explanation":"Compute the whole image; the empty subset is forward invariant, and invariance does not require every point to be fixed. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (9, 25, 21, [0, 1, 3, 4], [1, 3, 4, 6], No)."},"skills":["Forward Invariance versus Pointwise Fixing"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}