Orientation, all degrees and a walk audit

Goal and scope

Audit a fully specified digraph under the declared loop-free simple convention.

n≥1; no undirected-degree or loop-model substitution.

Givens and symbols

• work

• G1 — V={0,1,2}, directed E={(0,1),(1,0),(1,2)}, in lexicographic order; candidate walk W=(0,1,2).

• V,E — complete vertex and edge sets
• d+,d- — out- and in-degree

Rules and sources

• mko_definition · cited — Each edge is (u,v) with u≠v; (u,v) and (v,u) are different edges. Edges form a set, so same-direction parallel duplicates are forbidden. In-degrees and out-degrees are separate, each summing over the whole graph to |E|. Successive walk steps must follow orientation. Explicit loop boundaries fail this declared model, not every possible digraph convention. · mko-finite-directed-graph
• work · prerequisite — Directed edges are ordered pairs; out-degree counts first coordinates and in-degree second coordinates; walks respect each orientation. Edges of a simple loop-free digraph are unique ordered pairs with distinct endpoints; a reverse pair is a distinct edge; every directed-walk step must belong to E in that orientation.

Main derivation

E=((0,1),(1,0),(1,2))

\text{unique ordered pairs, all endpoints in }V,\quad u\ne v

v=0

d^+(0)=1,\quad d^-(0)=1

v=1

d^+(1)=2,\quad d^-(1)=1

v=2

d^+(2)=0,\quad d^-(2)=1

\sum_vd^+(v),\quad\sum_vd^-(v)

1+2+0=3=|E|,\quad1+1+1=3=|E|

W=(0,1,2)

(0,1)\in E,\quad(1,2)\in E

W'=(2,1,0)

(2,1)\notin E\implies W'\text{ invalid}

(1,1)\text{ proposed addition}

u=v\implies\text{rejected under this convention}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — Adding a duplicate(0,1) also violates the simple-set model.
• S07 · premise_audit · G1, S06, work — Existence of(1,2) does not supply(2,1).
• S06 · premise_audit · G1, S05, work — Undirected adjacency would not suffice.
• S05 · premise_audit · G1, S04, work — One sum is not 2|E|; that combines both sums.
• S04 · premise_audit · G1, S03, work — All three vertices have been classified.
• S03 · premise_audit · G1, S02, work — Opposite edges do not cancel.
• S02 · premise_audit · G1, S01, work — Count incoming and outgoing edges separately.
• S01 · premise_audit · G1, work — (0,1) and(1,0) are not duplicates.

Error controls and negative controls

Orientation

Check each ordered pair itself, not its unordered version.

Complete degrees

List both degrees at all three vertices before summing separately.

• (0,1) and(1,0) are the same edge. → reject — Their ordered-pair orientations differ.
• The out-degree sum is 6. → reject — It is 3=|E|, not 2|E|.
• W' is valid because it reverses W. → reject — (2,1) is absent from E.

Result and limitations

Out-degrees are(1,2,0), in-degrees(1,1,1), each summing to 3; W is valid and W' is not.

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 29de68530ae033900877a5ed071984a0b1139166bdb2f8540730b83646126993
• original source SHA-256: e9c6f110a34b506927711f1056970b7cb9f9b0ad5005a7b6dd237f108c5bb012
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)