# Strict Affine Index Maps, Composition and Finite-Prefix Evidence

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261107-r07-subsequence-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=0≥0 and positive integer index_step=1. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(0, 1, k(j)=0+1j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (0, 1, k(j)=0+1j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0001/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 2. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=1≥0 and positive integer index_step=2. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(1, 2, k(j)=1+2j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (1, 2, k(j)=1+2j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0002/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 3. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=2≥0 and positive integer index_step=3. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(2, 3, k(j)=2+3j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (2, 3, k(j)=2+3j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0003/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 4. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=3≥0 and positive integer index_step=4. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(3, 4, k(j)=3+4j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (3, 4, k(j)=3+4j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0004/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 5. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=4≥0 and positive integer index_step=5. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(4, 5, k(j)=4+5j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (4, 5, k(j)=4+5j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0005/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 6. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=5≥0 and positive integer index_step=1. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(5, 1, k(j)=5+1j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (5, 1, k(j)=5+1j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0006/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 7. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=6≥0 and positive integer index_step=2. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(6, 2, k(j)=6+2j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (6, 2, k(j)=6+2j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0007/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 8. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=7≥0 and positive integer index_step=3. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(7, 3, k(j)=7+3j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (7, 3, k(j)=7+3j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0008/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 9. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=8≥0 and positive integer index_step=4. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(8, 4, k(j)=8+4j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (8, 4, k(j)=8+4j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0009/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 10. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=9≥0 and positive integer index_step=5. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(9, 5, k(j)=9+5j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (9, 5, k(j)=9+5j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0010/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 11. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=10≥0 and positive integer index_step=1. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(10, 1, k(j)=10+1j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (10, 1, k(j)=10+1j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0011/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 12. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=11≥0 and positive integer index_step=2. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(11, 2, k(j)=11+2j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (11, 2, k(j)=11+2j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0012/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 13. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=12≥0 and positive integer index_step=3. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(12, 3, k(j)=12+3j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (12, 3, k(j)=12+3j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0013/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 14. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=13≥0 and positive integer index_step=4. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(13, 4, k(j)=13+4j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (13, 4, k(j)=13+4j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0014/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 15. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=14≥0 and positive integer index_step=5. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(14, 5, k(j)=14+5j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (14, 5, k(j)=14+5j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0015/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 16. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=15≥0 and positive integer index_step=1. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(15, 1, k(j)=15+1j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (15, 1, k(j)=15+1j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0016/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 17. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=16≥0 and positive integer index_step=2. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(16, 2, k(j)=16+2j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (16, 2, k(j)=16+2j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0017/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 18. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=17≥0 and positive integer index_step=3. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(17, 3, k(j)=17+3j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (17, 3, k(j)=17+3j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0018/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 19. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=18≥0 and positive integer index_step=4. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(18, 4, k(j)=18+4j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (18, 4, k(j)=18+4j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0019/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 20. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=19≥0 and positive integer index_step=5. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(19, 5, k(j)=19+5j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (19, 5, k(j)=19+5j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0020/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 21. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=20≥0 and positive integer index_step=1. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(20, 1, k(j)=20+1j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (20, 1, k(j)=20+1j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0021/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 22. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=21≥0 and positive integer index_step=2. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(21, 2, k(j)=21+2j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (21, 2, k(j)=21+2j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0022/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 23. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=22≥0 and positive integer index_step=3. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(22, 3, k(j)=22+3j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (22, 3, k(j)=22+3j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0023/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 24. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=23≥0 and positive integer index_step=4. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(23, 4, k(j)=23+4j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (23, 4, k(j)=23+4j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0024/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 25. A Positive Affine Index Step Gives Strict Increase for Every Natural Index

Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=24≥0 and positive integer index_step=5. Give (index_start, index_step, index_formula, strictly_increasing, certificate).

View answer and explanation(24, 5, k(j)=24+5j, Yes, strictly_increasing_for_all_natural_j)

The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (24, 5, k(j)=24+5j, Yes, strictly_increasing_for_all_natural_j).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0025/)

Subsequence · qf-mod38-subseq-index-certificate · candidate





## 26. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=1+(1)n for n≥0 and index map k(j)=0+1j, with nonnegative start and positive index step. Evaluate j=2. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(2, 2, 3, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (2, 2, 3, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0026/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 27. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=2+(2)n for n≥0 and index map k(j)=1+2j, with nonnegative start and positive index step. Evaluate j=3. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(3, 7, 16, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (3, 7, 16, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0027/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 28. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=3+(3)n for n≥0 and index map k(j)=2+3j, with nonnegative start and positive index step. Evaluate j=4. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(4, 14, 45, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (4, 14, 45, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0028/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 29. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=4+(4)n for n≥0 and index map k(j)=3+4j, with nonnegative start and positive index step. Evaluate j=5. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(5, 23, 96, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (5, 23, 96, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0029/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 30. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=5+(5)n for n≥0 and index map k(j)=0+5j, with nonnegative start and positive index step. Evaluate j=6. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(6, 30, 155, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (6, 30, 155, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0030/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 31. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=6+(6)n for n≥0 and index map k(j)=1+1j, with nonnegative start and positive index step. Evaluate j=7. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(7, 8, 54, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (7, 8, 54, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0031/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 32. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=7+(7)n for n≥0 and index map k(j)=2+2j, with nonnegative start and positive index step. Evaluate j=8. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(8, 18, 133, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (8, 18, 133, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0032/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 33. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=8+(1)n for n≥0 and index map k(j)=3+3j, with nonnegative start and positive index step. Evaluate j=9. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(9, 30, 38, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (9, 30, 38, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0033/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 34. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=9+(2)n for n≥0 and index map k(j)=0+4j, with nonnegative start and positive index step. Evaluate j=10. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(10, 40, 89, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (10, 40, 89, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0034/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 35. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=10+(3)n for n≥0 and index map k(j)=1+5j, with nonnegative start and positive index step. Evaluate j=11. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(11, 56, 178, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (11, 56, 178, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0035/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 36. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=11+(4)n for n≥0 and index map k(j)=2+1j, with nonnegative start and positive index step. Evaluate j=12. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(12, 14, 67, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (12, 14, 67, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0036/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 37. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=12+(5)n for n≥0 and index map k(j)=3+2j, with nonnegative start and positive index step. Evaluate j=13. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(13, 29, 157, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (13, 29, 157, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0037/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 38. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=13+(6)n for n≥0 and index map k(j)=0+3j, with nonnegative start and positive index step. Evaluate j=14. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(14, 42, 265, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (14, 42, 265, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0038/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 39. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=14+(7)n for n≥0 and index map k(j)=1+4j, with nonnegative start and positive index step. Evaluate j=15. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(15, 61, 441, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (15, 61, 441, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0039/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 40. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=15+(1)n for n≥0 and index map k(j)=2+5j, with nonnegative start and positive index step. Evaluate j=16. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(16, 82, 97, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (16, 82, 97, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0040/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 41. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=16+(2)n for n≥0 and index map k(j)=3+1j, with nonnegative start and positive index step. Evaluate j=17. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(17, 20, 56, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (17, 20, 56, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0041/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 42. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=17+(3)n for n≥0 and index map k(j)=0+2j, with nonnegative start and positive index step. Evaluate j=18. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(18, 36, 125, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (18, 36, 125, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0042/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 43. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=18+(4)n for n≥0 and index map k(j)=1+3j, with nonnegative start and positive index step. Evaluate j=19. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(19, 58, 250, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (19, 58, 250, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0043/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 44. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=19+(5)n for n≥0 and index map k(j)=2+4j, with nonnegative start and positive index step. Evaluate j=20. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(20, 82, 429, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (20, 82, 429, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0044/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 45. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=20+(6)n for n≥0 and index map k(j)=3+5j, with nonnegative start and positive index step. Evaluate j=21. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(21, 108, 668, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (21, 108, 668, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0045/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 46. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=21+(7)n for n≥0 and index map k(j)=0+1j, with nonnegative start and positive index step. Evaluate j=22. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(22, 22, 175, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (22, 22, 175, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0046/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 47. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=22+(1)n for n≥0 and index map k(j)=1+2j, with nonnegative start and positive index step. Evaluate j=23. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(23, 47, 69, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (23, 47, 69, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0047/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 48. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=23+(2)n for n≥0 and index map k(j)=2+3j, with nonnegative start and positive index step. Evaluate j=24. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(24, 74, 171, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (24, 74, 171, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0048/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 49. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=24+(3)n for n≥0 and index map k(j)=3+4j, with nonnegative start and positive index step. Evaluate j=25. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(25, 103, 333, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (25, 103, 333, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0049/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 50. A Subsequence Term Uses the Source at Its Increasing Selected Index

Use the entire arithmetic source a_n=25+(4)n for n≥0 and index map k(j)=0+5j, with nonnegative start and positive index step. Evaluate j=26. Give (j, source_index, subsequence_term, certificate).

View answer and explanation(26, 130, 545, a_of_k_j_preserves_original_order)

Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (26, 130, 545, a_of_k_j_preserves_original_order).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0050/)

Subsequence · qf-mod38-subseq-arithmetic-term · candidate





## 51. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=0+1i and inner index l(j)=0+1j, with nonnegative starts and positive steps. Evaluate j=1. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(1, 1, 1, 1, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (1, 1, 1, 1, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0051/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 52. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=1+2i and inner index l(j)=1+2j, with nonnegative starts and positive steps. Evaluate j=2. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(2, 5, 11, 4, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (2, 5, 11, 4, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0052/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 53. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=2+3i and inner index l(j)=2+3j, with nonnegative starts and positive steps. Evaluate j=3. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(3, 11, 35, 9, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (3, 11, 35, 9, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0053/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 54. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=0+4i and inner index l(j)=3+4j, with nonnegative starts and positive steps. Evaluate j=4. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(4, 19, 76, 16, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (4, 19, 76, 16, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0054/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 55. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=1+5i and inner index l(j)=0+5j, with nonnegative starts and positive steps. Evaluate j=5. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(5, 25, 126, 25, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (5, 25, 126, 25, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0055/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 56. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=2+1i and inner index l(j)=1+6j, with nonnegative starts and positive steps. Evaluate j=6. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(6, 37, 39, 6, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (6, 37, 39, 6, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0056/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 57. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=0+2i and inner index l(j)=2+1j, with nonnegative starts and positive steps. Evaluate j=7. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(7, 9, 18, 2, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (7, 9, 18, 2, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0057/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 58. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=1+3i and inner index l(j)=3+2j, with nonnegative starts and positive steps. Evaluate j=8. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(8, 19, 58, 6, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (8, 19, 58, 6, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0058/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 59. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=2+4i and inner index l(j)=0+3j, with nonnegative starts and positive steps. Evaluate j=9. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(9, 27, 110, 12, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (9, 27, 110, 12, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0059/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 60. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=0+5i and inner index l(j)=1+4j, with nonnegative starts and positive steps. Evaluate j=10. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(10, 41, 205, 20, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (10, 41, 205, 20, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0060/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 61. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=1+1i and inner index l(j)=2+5j, with nonnegative starts and positive steps. Evaluate j=11. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(11, 57, 58, 5, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (11, 57, 58, 5, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0061/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 62. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=2+2i and inner index l(j)=3+6j, with nonnegative starts and positive steps. Evaluate j=12. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(12, 75, 152, 12, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (12, 75, 152, 12, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0062/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 63. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=0+3i and inner index l(j)=0+1j, with nonnegative starts and positive steps. Evaluate j=13. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(13, 13, 39, 3, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (13, 13, 39, 3, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0063/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 64. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=1+4i and inner index l(j)=1+2j, with nonnegative starts and positive steps. Evaluate j=14. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(14, 29, 117, 8, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (14, 29, 117, 8, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0064/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 65. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=2+5i and inner index l(j)=2+3j, with nonnegative starts and positive steps. Evaluate j=15. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(15, 47, 237, 15, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (15, 47, 237, 15, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0065/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 66. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=0+1i and inner index l(j)=3+4j, with nonnegative starts and positive steps. Evaluate j=16. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(16, 67, 67, 4, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (16, 67, 67, 4, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0066/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 67. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=1+2i and inner index l(j)=0+5j, with nonnegative starts and positive steps. Evaluate j=17. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(17, 85, 171, 10, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (17, 85, 171, 10, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0067/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 68. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=2+3i and inner index l(j)=1+6j, with nonnegative starts and positive steps. Evaluate j=18. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(18, 109, 329, 18, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (18, 109, 329, 18, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0068/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 69. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=0+4i and inner index l(j)=2+1j, with nonnegative starts and positive steps. Evaluate j=19. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(19, 21, 84, 4, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (19, 21, 84, 4, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0069/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 70. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=1+5i and inner index l(j)=3+2j, with nonnegative starts and positive steps. Evaluate j=20. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(20, 43, 216, 10, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (20, 43, 216, 10, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0070/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 71. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=2+1i and inner index l(j)=0+3j, with nonnegative starts and positive steps. Evaluate j=21. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(21, 63, 65, 3, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (21, 63, 65, 3, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0071/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 72. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=0+2i and inner index l(j)=1+4j, with nonnegative starts and positive steps. Evaluate j=22. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(22, 89, 178, 8, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (22, 89, 178, 8, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0072/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 73. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=1+3i and inner index l(j)=2+5j, with nonnegative starts and positive steps. Evaluate j=23. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(23, 117, 352, 15, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (23, 117, 352, 15, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0073/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 74. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=2+4i and inner index l(j)=3+6j, with nonnegative starts and positive steps. Evaluate j=24. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(24, 147, 590, 24, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (24, 147, 590, 24, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0074/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 75. Compose the Inner Index into the Outer Index in That Order

Use outer index k(i)=0+5i and inner index l(j)=0+1j, with nonnegative starts and positive steps. Evaluate j=25. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).

View answer and explanation(25, 25, 125, 5, Yes, composition_remains_strictly_increasing)

Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (25, 25, 125, 5, Yes, composition_remains_strictly_increasing).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0075/)

Subsequence · qf-mod38-subseq-composed-index · candidate





## 76. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[0, 1, 2, 3]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([0, 1, 2, 3], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([0, 1, 2, 3], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0076/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 77. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[1, 3, 5, 7, 9]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([1, 3, 5, 7, 9], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0077/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 78. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[2, 5, 8, 11, 14, 17]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([2, 5, 8, 11, 14, 17], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([2, 5, 8, 11, 14, 17], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0078/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 79. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[3, 7, 11, 15]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([3, 7, 11, 15], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([3, 7, 11, 15], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0079/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 80. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[4, 9, 14, 19, 24]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([4, 9, 14, 19, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([4, 9, 14, 19, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0080/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 81. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[5, 6, 7, 8, 9, 10]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([5, 6, 7, 8, 9, 10], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([5, 6, 7, 8, 9, 10], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0081/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 82. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[6, 8, 10, 12]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([6, 8, 10, 12], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([6, 8, 10, 12], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0082/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 83. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[7, 10, 13, 16, 19]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([7, 10, 13, 16, 19], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([7, 10, 13, 16, 19], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0083/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 84. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[8, 12, 16, 20, 24, 28]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([8, 12, 16, 20, 24, 28], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([8, 12, 16, 20, 24, 28], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0084/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 85. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[9, 14, 19, 24]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([9, 14, 19, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([9, 14, 19, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0085/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 86. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[10, 11, 12, 13, 14]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([10, 11, 12, 13, 14], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([10, 11, 12, 13, 14], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0086/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 87. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[11, 13, 15, 17, 19, 21]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([11, 13, 15, 17, 19, 21], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([11, 13, 15, 17, 19, 21], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0087/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 88. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[12, 15, 18, 21]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([12, 15, 18, 21], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([12, 15, 18, 21], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0088/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 89. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[13, 17, 21, 25, 29]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([13, 17, 21, 25, 29], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([13, 17, 21, 25, 29], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0089/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 90. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[14, 19, 24, 29, 34, 39]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([14, 19, 24, 29, 34, 39], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([14, 19, 24, 29, 34, 39], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0090/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 91. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[15, 16, 17, 18]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([15, 16, 17, 18], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([15, 16, 17, 18], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0091/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 92. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[16, 18, 20, 22, 24]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([16, 18, 20, 22, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([16, 18, 20, 22, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0092/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 93. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[17, 20, 23, 26, 29, 32]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([17, 20, 23, 26, 29, 32], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([17, 20, 23, 26, 29, 32], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0093/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 94. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[18, 22, 26, 30]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([18, 22, 26, 30], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([18, 22, 26, 30], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0094/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 95. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[19, 24, 29, 34, 39]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([19, 24, 29, 34, 39], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([19, 24, 29, 34, 39], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0095/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 96. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[20, 21, 22, 23, 24, 25]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([20, 21, 22, 23, 24, 25], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([20, 21, 22, 23, 24, 25], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0096/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 97. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[21, 23, 25, 27]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([21, 23, 25, 27], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([21, 23, 25, 27], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0097/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 98. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[22, 25, 28, 31, 34]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([22, 25, 28, 31, 34], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([22, 25, 28, 31, 34], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0098/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 99. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[23, 27, 31, 35, 39, 43]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([23, 27, 31, 35, 39, 43], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([23, 27, 31, 35, 39, 43], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0099/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate





## 100. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate

Supplied finite natural-index list=[24, 29, 34, 39]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).

No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.

View answer and explanation([24, 29, 34, 39], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)

Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([24, 29, 34, 39], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r07-subsequence-0100/)

Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261107-r07-subsequence-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261107-r07-subsequence-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261107-r07-subsequence-foundations/validation-report.json)