# Question 1493: Equal Dimension Vectors Are Not a Physical Proof

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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.

R0001-01493 · q-20261013-advance-20261020-r05-dimensional-analysis-0093



## Original question and givens

Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 1, 3], right=[0, 1, 3]. Give (left vector, right vector, identical?).


- G001 — Use basic dimension exponent vectors [L,M,T] in that order. Input exponents and the integer power are bounded, but a computed result need not stay inside the input bound. Dimension agreement is necessary but not sufficient for physical correctness; unit scale conversions are separate. left=[0, 1, 3], right=[0, 1, 3]. Give (left vector, right vector, identical?).





## 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-dimensional-analysis](https://ocme.evemisslab.com/en/mko/mko-dimensional-analysis/)





## Step-by-step derivation

S001 · impliesBefore

dimension coordinate order=[L,M,T]

After

left/base=[0, 1, 3]; right=[0, 1, 3]

Conditions: none

These are dimension exponents,not measured magnitudes;the order is length,mass,time.

Step identifiers and dependenciesRule: dimension_order · Uses: G001





S002 · calculationBefore

L exponent difference := sub(0, 0)

After

L exponent difference = 0

Conditions: none

The exponent of every base dimension must agree across an equation.

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





S003 · calculationBefore

M exponent difference := sub(1, 1)

After

M exponent difference = 0

Conditions: none

The exponent of every base dimension must agree across an equation.

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





S004 · calculationBefore

T exponent difference := sub(3, 3)

After

T exponent difference = 0

Conditions: none

The exponent of every base dimension must agree across an equation.

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





S005 · impliesBefore

[Yes, Yes, Yes]

After

dimensionally consistent: Yes

Conditions: none

This checks a necessary dimensional condition;consistency does not prove numerical equality.

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









## Reverse checks


- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute T exponent difference with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute M exponent difference with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute L exponent difference 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

([0, 1, 3], [0, 1, 3], Yes)

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: 2e49d26eff71959d2f17f76b8e917e858443dd53d52f43394656760385e4c643
- candidate record SHA-256: 223752a2fbe836a93d8fdc745aec987991db944329708f7e4c02140209af009c
- semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc
- question-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261020-r05-dimensional-analysis-0093.json)