# Question 6198: Endpoint, Net Displacement, and Step Magnitude

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0011-06198 · q-20261009-advance-20261013-r02-integer-0048



## Original question and givens

Integer start=10, signed steps s1=23, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).


- G001 — Integer start=10, signed steps s1=23, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-integer](https://ocme.evemisslab.com/en/mko/mko-integer/)





## Step-by-step derivation

S001 · impliesBefore

declared exact integer or rational-representation protocol

After

retain signs,ordered operands and original index order

Conditions: none

Check these finite data with exact integer arithmetic,without floating approximations or a claim to prove all integer-ring axioms.

Step identifiers and dependenciesRule: integer_arithmetic_scope · Uses: G001





S002 · calculationBefore

signed net displacement := sum(23, -3)

After

signed net displacement = 20

Conditions: none

Add the two signed steps to obtain net displacement.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

final position := sum(10, 20)

After

final position = 30

Conditions: none

Add net displacement to the start to obtain the endpoint;net displacement alone is not an absolute position.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

steps=(23,-3)

After

absolute step lengths=(23,3)

Conditions: none

Step signs encode direction;path length uses each step's absolute value.Do not cancel first and then take an absolute value.

Step identifiers and dependenciesRule: signed_step_vs_length · Uses: G001, S003





S005 · calculationBefore

sum of step lengths := sum(23, 3)

After

sum of step lengths = 26

Conditions: none

Add both step lengths;returning to the start may still involve positive travel.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004









## Reverse checks


- S005 · premise_and_result_audit — Recompute sum of step lengths with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute final position with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute signed net displacement with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(30, 20, 26)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: b12f982cf311c53a15ff86acf595b6cb701f8f3337d2e88786b2a066debf8ac5
- candidate record SHA-256: 40a4ee05d7988209f1b5102d6e08ac6a597a6f1edac356e64e566d339da41837
- semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80
- question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261009-advance-20261013-r02-integer-0048.json)