[DreamingOfClouds]

No new κ. Exact accounting and structure facts, replayed on the literal #6702 native word, the #6726 caller, #6805 and #6824 histograms, and the #6348 caller with the #6349 word.

Surplus. For a legal native word, public registers only climb and private registers only descend, so the native margin per call is exactly −(16p + O) (p = 104 ports, O = total private-descent drop, 56 on the current word; paid mass 5924 = 1664 + 4204 + 56). On the literal callers every stage is covered by 2p more target than idle incidences and the outer margin is p(17k − 1) (31720 = 104·305 at k = 18, 33488 at k = 19; 305 = 16·18 + 32 − 15). Hence S = Σ(T−P)r = (k−1)p − kO = 48k − 104: 760 for #6702, #6805 and #6824 on the #6726 caller, 808 for the #6348 caller on the #6349 word. S depends on the native word only through O, so rank-conserving edits (the #6805 retime and split, the #6824 lift) leave it fixed. Exactly S = κ·D̄, and S/D(0) < κ < S/D(κ) within 0.7 % either side (κ(#6805) = 0.0033081451891428, κ(#6824) = 0.0033082345661432).

E8 structure. Q(e) = wt(e)/2 on 1^⊥ ⊂ F2^9 is an O+(8,2) form with 120 nonsingular points (odd vectors of weight 3 or 7). The 104 ports are 70 + 34 of them; the 16 missing lines are two disjoint orthoframes (2025 frames in all, 759 inside the ports). The port stabiliser has order 384 with orbits 48, 32, 24. Frames disjoint from one of them split 896/336/28 by incidence rows. The public block P vanishes on all 5824 odd pairs (proved from frame monotonicity); O = I − P equals the sign-gauged E8 Gram matrix except in exactly 40 entries, all in the rows of ports 69 and 276; rank 10 = 8 + 2.

Private-descent mass. The 24 descent hops form exactly 8 components, each 8 → … → 1 of mass 7, so 56 = 8·7. For all 43435 seven-dimensional subspaces of F2^9 the ports outside the subspace span R^8 (16 hyperplane exceptions, exactly the 16 missing lines). Consequence, conditional: a rank-8 Gram operator realised by 8 disjoint single-stage chains costs at least 56. The literal word has rank-10 O and attains 56, so it is not covered by the hypotheses.

Mixed ports (model). With an explicit bookkeeping hypothesis and the #968 floor max(|A|−48, 56): S ≤ 48k − |∪A_j| ≤ 48k − 104, equality only for the uniform 104-port set.

Overlap. #4697 (Matarener) derived surplus = 48·|thick| − ports for the kernel-era controllers: the shape is theirs; the split 48 = p − O, the outer law on the tree and fragmented callers, and the invariance statement are new here. #968 (themilkman216) has the E8 root-line model with F2^9 labels, P104 = P120 − F0 − F1, the 1260-frame census (896 admit decoders, 364 do not) and the floor cst ≥ max(72 − p, 56): cited, not claimed; the quadratic-form description, literal port identification, stabiliser and orbits, O-versus-Gram comparison and the hop and subspace certificates are new. #6717 and #6398 fix the query-count contracts and are not re-derived. Repository search found no other statement of these items (text search; prose may be missed).

Not established. The outer law and stage balance are checked on three literal callers, not proved for every caller. The mixed bound depends on its hypothesis; no mixed word exists.

Base 7c49fe3. Inputs: base-commit files (#6348 caller, #6349 word) and unmodified copies of #6702/#6726/#6805/#6824 files pinned by blob and SHA-256 in retained/SOURCE-PINS.json (#6726 at 435bbe9, #6805 at 0ecd4fd, #6824 at 2a7ca44).

Reproduce (CPython 3.12, standard library, about 25 s per run):

python3.12 -B scripts/research_package.py --package research/p104-surplus-and-e8-structure

Expected: PASS_P104_SURPLUS_AND_E8_STRUCTURE (normal and -O) and PASS_REGISTERED_RESEARCH_PACKAGE. Ten negative controls are included.

Credit: themilkman216 (#968), Matarener (#4697), aliromman (#6348, #6349), zxv (#6726), CubeJerry (#6805), thu-nmrc (#6716, #6717), DreamingOfClouds (#6702, #6824). Apache-2.0.

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

🤖 Generated with Claude Code