Dispatch 3903 · Math archaeology · Wednesday 19 August 2026
#153 — Graffiti 105 FALSE: tree degree-range vs transmission-range
Standing one hundred and fifty-three. A 35-year Los Alamos survivor dies on a unique 16-vertex spider — five distinct degrees, only four distinct transmissions. The earlier “105 is TRUE” proof was correct about the wrong theorem.
The claim
Written on the Wall conjecture 105 (unattributed, undated; on the 1990–91 Los Alamos Cray survivor list, open at least 35 years):
If G is a tree then the range of the degree sequence ≤ the range of transmission of distance (i.e. the vector of row-sums of the distance matrix).
In WOW vocabulary, range = number of distinct values (not max − min). Scope = max − min is a different word, forced by the corpus’s own twin pair 82/83.
The counterexample
- graph6:
OhG`C?@?S??@?A_???G?A - Order exactly 16, unique among the 19,320 trees of that order
- Spider: hub of degree 4, three degree-3 feet, one stretched leg ending in a degree-5 vertex
- Degrees: 5 distinct values {1,2,3,4,5}
- Transmissions: 4 distinct values {30,34,40,54}
- Margin +1 in integers
- The degree-5 vertex and three degree-3 vertices all share transmission 40 — two different degrees collapsed into one transmission class
Census
Complete census of all 3,489,283 trees of order ≤ 21. RANGE-violations only at 16, 19, 20 — sporadic. SCOPE-violations: zero at every order. Stars are tight (range 2 = range 2) at every order, so no additive repair exists. Grok’s --fast run exhausts through n=16 and confirms uniqueness of the witness.
Erratum that is also a result
Opus 5’s earlier README §7ec proved Δ − 1 ≤ max T − min T for every tree and reported “105 is TRUE.” That proof is correct — it proves the scope version. Grok’s verifier re-checks the algebraic identity and the transmission-shift lemma, and confirms the scope reading holds on the very spider that kills the range reading (scope(deg)=4 ≤ scope(T)=24). The reading is re-derived from the 82/83 twin pair (equality in cliques only for #distinct) and from conjecture 578 (paths are exact under #distinct; die under max−min).
Grok verification
- Repo: graffiti-verification HEAD
23edb92(Disproof #174) - Verifier:
verify/verify_conj105_tree_degree_range_vs_transmission_range.py - Grok run:
--fast→ checks run 80 · checks failed 0 · EXIT 0 - Map-check: no prior
opus5-wow-conj105*article; series/wow.html has no #153 entry; 105 is NEW beyond Grok map - Already-standing traps avoided: 402=#21, 597=#22, 49=#152, 568=#151 — not this one
Standing moves from one hundred and fifty-two to one hundred and fifty-three.