[Matarener]

Requested by Mark · project thread

Change and mathematical scope

Proof plus scoped review, no new κ. The package research/p104-owned-lineage-floor-review contains two things.

1. Scoped review of Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170's owned-layer floor. Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170 (DreamingOfClouds, designer toolkit v2) is merged for archive only. Its intake record says "independent scientific review and source-bound replay remain open". The review reads Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170's own files as follows:

   • The "proved floor 794.53 R0 (~10%)" in P104-THEORY.md is withdrawn in the same file. Its CHARTORB addendum marks the inherited cap48 row invalid and corrects the figure to 754.14.
   • P104-LEMMAS.md corrects 754.14 to 751.46. Its sections 9-11 state that no floor above 0 rests on SPEC + paid + literals alone. Every positive floor carries measured scopes (NEED = 4972, own/owns/ownc = 22/18/64). With the proved bound NEED >= 90 the floor is 0.00, and it becomes positive only from NEED >= 3597, which is open.
   • The LP solver and certificate are not shipped, so the LP is not re-solved here.

   Own code (nothing imported from Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170) recounts Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170's P_M owned-delivery matrix C on Conditional κ .003125282 (+5.22e-7 over #5827): remove helper 468's frame-593 rung with rerouted reads #5907, Conditional κ .003125804 (+5.22e-7 over #5907): remove helper 470's frame-593 rung, the twin of #5907's removal #6030, Conditional κ .003128386 (+2.582e-6 over #6030): owned registers 770, 771 and 772 descend through the frames of helpers 705, 706 and 731, which no longer relay them; helper 732 reads 771 at frame 117 #6144 and Conditional κ = 0.003128645 (+2.59e-7 over #6144): retain an operand to remove a root stop #6167:

   word	owned symbols	owned paid	rank C	NEED	Z
   Conditional κ .003125282 (+5.22e-7 over #5827): remove helper 468's frame-593 rung with rerouted reads #5907	20	56	8	4992	0
   Conditional κ .003125804 (+5.22e-7 over #5907): remove helper 470's frame-593 rung, the twin of #5907's removal #6030	20	56	8	4992	0
   Conditional κ .003128386 (+2.582e-6 over #6030): owned registers 770, 771 and 772 descend through the frames of helpers 705, 706 and 731, which no longer relay them; helper 732 reads 771 at frame 117 #6144	26	56	10	4972	20
   Conditional κ = 0.003128645 (+2.59e-7 over #6144): retain an operand to remove a root stop #6167	25	56	10	4972	20

   PUB + C = I holds exactly. R1, R3 and P3a hold with 0 violations. There are 0 STRICT relay edges, 0 merges and 0 branchings. These numbers agree with Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170.

2. Owned-lineage factorization (new).

   • Lemma 1 (proved). A relay-free owned lineage contributes C_ji = alpha_j beta_i on every non-orthogonal port pair, and its owned paid is >= rank_F2 Gamma[supp alpha, supp beta]. So the relay-free owned layer is an instance of the repository's masked decoder problem C(P104) >= 56: lineages are the terms and lineage paid is the term cost.
   • Lemma 2 (proved). The same holds for branching lineages, with one rank-one term per leaf.
   • Theorem 4. A relay-free, merge-free owned layer with exactly 8 leaves has owned paid >= 56 even when lineages share a common descent. The proof uses finite facts: over all 440,880 one-dimensional flats of the E8 arrangement, only the 16 lines of the two removed frames have support span of dimension 8. From this, a branching landing has dim >= 8, and two cost-7 leaves cannot share. A shared step can at best tie.
   • Check on real words. All four words are tight: 8 lineages, each with paid = rank_F2 = 7.

   The floor for at most 8 terms without sharing is already known. Theorem 3 re-derives it by a different finite input. New here: the connection between Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170's accounting and the decoder floor, and the branching case (cost sharing absent from the plain decoder model).

Remaining obligations:

• Not covered: merges, STRICT relays, 9 or more terms (the open masked floor), and the orthogonal entries of C.
• Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170's open premise NEED >= 3597 stays open.

Task and overlap

The task card is complex-rank-shape-decoder (priority revision 8). Scope is refined: owned lineages as decoder terms, plus the #6170 review. Base: canonical main 2485805a6d47f36ea83ddde6327d453ac82d44dc. #6170 head 445fb9124.

Nearest work:

• Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170: framework, NEED/REL, lemmas R1, P3a, G1a, N1, N2, N2(e).
• research: P104 B-form nine-term decoder floor >= 56 (no κ claim) #6188: nine-term B-form decoder floor.
• research/p104-carrier-code-uniqueness: 8-dimensional codes.
• Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212: Theorem U.
• Prove confined two-clique P104 relays need eight paid units (no κ) #6229 (research/p104-two-clique-relay-floor, Nick Petalas) and research/p104-ambient-relay-floor (AAAmirinu): 8 owned paid units for one 8-port two-clique block in the confined and ambient REL models, merges included. Those are local 8-port statements; this package covers the whole 104-port relay-free owned layer of real words but not merges. PR6229's cumulative row-space invariant is the natural tool for the merge case.

Apart from #6229 and the ambient follow-up above, no PR after #6170 reviews its floor (checked main 2485805 and PR heads up to 6266).

Validation

python3.12 -B scripts/research_package.py --package research/p104-owned-lineage-floor-review

Python 3.12.3 with NumPy 2.5.4, hosted Linux:

• verify.py (layers 1-3, about 40 s) prints PASS_P104_OWNED_LINEAGE_FLOOR_REVIEW.
• verify.py --flats (layer 4, about 2 min); full run about 2 min on the rebased branch prints PASS_P104_OWNED_LINEAGE_FLOOR_REVIEW_WITH_FLATS.
• research_package.py prints PASS_REGISTERED_RESEARCH_PACKAGE.
• check_research_commit.py --check-modes passes.

Arithmetic: rationals for the replays, integers for the root data and W-orbits.

Not run:

• --check-source-history, because the clone is shallow.
• Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170's own toolkit.
• Any re-solve of Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170's LP (its solver and certificate are not shipped).

The proofs are in PROOF.md. The finite facts F1-F3 are computations, and the remaining steps are written arguments. No independent human or proof-kernel review is claimed.

Credit

DreamingOfClouds retains #6170, its P_M accounting, the lemmas used here and the REL framework. Ali Romman retains #6167. #5907, #6030 and #6144 and their predecessors retain their attribution. Theorem U (#1212) and the masked decoder floor are prior repository work. Lemmas 1-2, Theorems 3-4 and the code were prepared for Matarener with Claude (Anthropic) via Claude Code. Apache-2.0.

Open next questions

• Extend Lemma 1 to merges and STRICT relays: split a merged symbol's content per ancestor.
• Control the orthogonal entries of C lineage by lineage (pairs with e(i) > r(j)). This is the gap between X and C, and the missing link to Add P104 designer toolkit v2: exact pricing and legality, #6030 rebuild with fold/hoist/retain, generators, sub-circuit tools (no kappa claim) #6170's NEED premise.

Both are listed in TASK.json.

🤖 Generated with Claude Code

https://claude.ai/code/session_011VhYm9sv5h3SL8idn3hUem