[DreamingOfClouds]

No new κ. Four exact statements about legal native words, replayed on the literal #6824 native word.

Hub normal form. Events at the zero frame on registers that start at {0} (K) commute to the start and events at the full frame on registers that end full (R) commute to the end. With T the invertible map of the rest and A = I + E, gauges G0 on K, G9 on R with G9·T·G0 = A exist iff T[R̄,K̄] = A[R̄,K̄], T[R,Hnn] = 0, T[Hnn,K] = 0 and rank T[R,K] = |H09| (equivalently ≤). The first condition with its dual T⁻¹[K̄,R̄] = A⁻¹[K̄,R̄] is not sufficient when Hnn ≠ ∅ (two-register control). There is no dyadic obstruction: a gauge over Z[1/2] exists iff those conditions hold and T ∈ GL(Z[1/2]), with an explicit completion. Feed-forward corollary: if all helpers start at {0} and end full, the mid map never writes X or reads Y and T[Y,X] = E, a dyadic gauge exists whenever T[H,H] ∈ GL(Z[1/2]). On the #6824 word (|K| = 556, |R| = 571, |H09| = 359, |Hnn| = 4) all conditions hold exactly, rank T[R,K] = 359, T⁻¹ is dyadic (denominators up to 64), and the gauge re-solved from T alone gives a legal word with identical MOVE events and the exact endpoint on all 772 rows (37,819 events against 18,622; 14 never-moving private lifetimes removed).

Descent-crossing bound. For an up-set U of frames, at least rank A[final ∉ U, init ∈ U] private descents lead from U to outside U. For pairwise odd ports this is the clique size: the clique number is 8 and each of the 759 maximum cliques is crossed by exactly 8 of the 24 descents (the bound is attained).

Scheduling. Every linear extension of the per-register event order is legal with the same endpoint, frame chains and invoice histograms; only the number of simultaneously live private registers changes (14 to 29 over ten random schedules; the word has 14). Scheduling alone cannot change the price of a fixed network.

Pair-exchange law. If the outputs of an exchange at frame G reach sinks s1, s2 along paths without a private descent, then G ⊆ final(s1) ∩ final(s2); two distinct 8-dimensional sink frames force dim G ≤ 7. Registers 679 and 686 exchange at a frame of dimension 7 that equals the intersection of the sink frames 175 and 226. Without the descent hypothesis the inclusion fails (1,559 events); all 645,102 (event, sink) pairs reached without descents satisfy it.

Overlap. The wall, the 5824 odd pairs and the E8 structure are #6831 section 6 (cited, not repeated); the up-set bound extends its argument. Ali Romman's dyadic helper-basis completion and #6827 concern bases at interior cuts, not zero-frame and full-frame events; #6345 shows that the price depends on the rank histograms only. Repository text search found no other statement of these items (prose may be missed).

Not established. The rewritten word is a witness of the freedom, has more events than the original (it exceeds the 26,000-event bound of the default profile; the package declares no event bound) and is not re-priced; the native chart checks and the five framework assumptions of #6824 are not re-run (its MOVE events and chart pairs are identical). The unit-pivot completion is a greedy rule that succeeded here; general existence is the theorem. The pair-exchange law gives no cost for alternatives. All arithmetic is exact rational (no modular rank or inverse).

Base: 7c49fe3. The one executable input is the #6824 native word (head 2a7ca44), rebuilt by that pull request's own construction and pinned by Git blob, SHA-256 and its published payload digest d3ccfce7bdbca3b79771844e6c9fee48f1a8066444a19218995648df96982821 in retained/SOURCE-PINS.json.

Reproduce (CPython 3.12, standard library, about 15 s per run):

python3 -B scripts/research_package.py --package research/p104-relay-hub-normal-form

Expected: PASS_RELAY_HUB_NORMAL_FORM (normal and -O) and PASS_REGISTERED_RESEARCH_PACKAGE.

Credit: CubeJerry (#6805), Ali Romman (dyadic helper-basis completion, verifier semantics), zxv (#6726), thu-nmrc (#6716), themilkman216 (#968), DreamingOfClouds (#6702, #6824, #6831, #6827). Apache-2.0.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.

🤖 Generated with Claude Code