[unpaidmoon]

Change and mathematical scope

Negative result for one changed P104 premise. No new κ. The selected conditional result stays 0.001672660197076918.

The premise is the literal correction (Q=\Pi P_{\mathrm{enc}}\Pi), instead of the old correction (P). For a projection (H), invertible (T) and invertible (D), with no commutation hypothesis among (D), (T) and (H),

[
Q A_T=A_T P,\qquad A_T=\Pi L_T,\qquad P_{\mathrm{enc}}=L_T P L_T^{-1}.
]

Both (P) and (P_{\mathrm{enc}}) fail this identity on a separating state.

One helper coordinate from the PR CrocSwap#3954 identity extension is fixed by (Q). Every dirty value, including 0, is restored. (Q) does not move that helper's chart. The published entrance is the identity chart (0 of 81 progress units) and the published completed exit is full (F) (81 units), so the charts do not match and (Q) does not insert the return.

On the pinned mask (H=\mathrm{diag}(0_8,I_{96})), (\Pi) is 96 disjoint transpositions. The scratch-free exchange (x\mathrel{+}=y), (y\mathrel{-}=x), (x\mathrel{-}=y) costs 3 additions per port, 288 additions in total, and 0 helper additions. This is a value permutation, not a native-rank lower bound.

The retained scaffold arithmetic recomputes to lower mass (3009344), capacity (2965248), excess (44096). That reading already sets the (\Pi) occupancy term to 0. The 288 additions remove none of the counted mass. Counting them as rank-1 children raises the excess to (44384). The bill is not favorable.

Not established: a native helper schedule, an owner relabeling of tariff 0, the physical pivot (B_0), the 10,816-line census, or a selected-frontier change.

Task and overlap

Base 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7. Task p104-side-correction, scope refined.

Item taken: item 1 of the eligible list below. It is open question 1 of research/p104-literal-slot-defect: price (\Pi P_{\mathrm{enc}}\Pi) as a real slot exchange on one helper entrance and exit from the PR CrocSwap#3954 endpoint.

Eligible list, revision-7 priority-table order, after dropping a row whose stated next decision is already an open PR or already delivered by this account:

1. p104-side-correction: this changed premise.
2. Dropped before that cut: uniform-transfer-audit (Audit PR4100 arbitrary-source CRT repair with an explicit inverse CrocSwap/integer-mult-bounds#4169, Audit PR4100’s free-word tensor lift on all four actual programs CrocSwap/integer-mult-bounds#4184, Audit #4100's cached chirp odometer: O(p) head travel per record uniformly in d (no new κ) CrocSwap/integer-mult-bounds#4325, Audit literal head travel of #4100's sparse-union schedule: section 2.3 holds with constant 17 (no new κ) CrocSwap/integer-mult-bounds#4327, Certify the inverse 1-norm of #4319's normalized Gaussian Schur section: 2.8138, against the lemma's 3938 ceiling (no new κ) CrocSwap/integer-mult-bounds#4336 and the other open Propose unconditional κ = 0.00173097911567079247276057 via batched CRT CrocSwap/integer-mult-bounds#4100 audits), supplier-boundary-audit (Complete source-bound four-provider finite admission for PR4088 CrocSwap/integer-mult-bounds#4180), joint-query-controller (Share closed primitive engines to remove parent parking at child calls CrocSwap/integer-mult-bounds#4167, Prove native-history action coupling and reduce the QQ16 comparison CrocSwap/integer-mult-bounds#4172, Prove a complete terminal-band QQ16 cost exclusion CrocSwap/integer-mult-bounds#4177), r437-carrier-integration (Integrate the R437 carrier compiler, numerical invoice and paid local caller CrocSwap/integer-mult-bounds#4176, plus this account's Derive the reverse row coefficient of the two-window R437 carrier. CrocSwap/integer-mult-bounds#4147 and Price the R437 child-constructor tapes inside Pi_v. CrocSwap/integer-mult-bounds#4160), independent-moment (Independent transfer audit (no new κ): PR4091's pooled ceiling applies to the retained PR4088 head; remaining headroom 1.140318244487202e-9 CrocSwap/integer-mult-bounds#4170), and formal-interface-lemma (the compact-switch audit is Audit the compact-switch coefficient and guard contract. CrocSwap/integer-mult-bounds#4321).

Nearest predecessors are CrocSwap#3954 (the endpoint, the helper extension, and the gate) and CrocSwap#4133 (the scalar obstruction, which leaves this price open). Distinct from CrocSwap#4136, CrocSwap#4140, CrocSwap#4153, CrocSwap#4154, and CrocSwap#4159. Overlap search at 2026-10-21T13:28Z covered 458 open PR bodies and 3225 discussion records; the phrase hits are those predecessors or unrelated "Pi phase" / "helper chart" notes. No open PR prices this exchange.

Validation

Python 3.12, standard library only. From the repository root:

python3 -B research/p104-pi-helper-slot/verify.py
python3 -B research/p104-pi-helper-slot/test_verify.py
python3 -B scripts/research_package.py --package research/p104-pi-helper-slot

verify.py prints PASS_P104_PI_HELPER_SLOT swaps=96 additions=288 helper_fixed=1 chart_gap=81 excess=44096 favorable=0 no_new_kappa. The registered replay prints PASS_REGISTERED_RESEARCH_PACKAGE research/p104-pi-helper-slot outcome=negative-result. A local run of the registered replay finished in about 10 seconds. verify.py also passes under -O.

The 4-port witness uses a (D) and a (T) that commute with neither each other nor (H). The 104-port check uses (D=I) and a positive diagonal (T); it counts swaps and rechecks the identity extension. It is not a root census. test_verify.py rejects a two-step exchange, a mutated helper, and a bill that treats the 288 additions as rank-1 children.

Not run: Lean, Docker, the 10,816-line census, a native helper schedule, and the selected cold replay.

Credit

Chris R / Clear1nsight, PR CrocSwap#3954, for (\Pi), the 72-helper identity extension, the identity-entrance / full-(F) contract, and the scaffold gate, retained on main as research/p104-two-axis-compatibility. PR CrocSwap#4133 recorded the unpriced conjugation and asked this question. Rohan Arun, Chafik Boukhalfa, Adam Gall, and the native P104 lineage named in PR CrocSwap#3954 supply the endpoint. This package does not import those programs.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

See research/p104-pi-helper-slot/README.md.

1. Does one helper whose published entrance is already (F) remove the 81-unit chart gap without repaying it, and can that inventory beat excess 44096? First step: name the role and recompute (W). Blocker: the retained contract says every non-data entrance is the identity chart.
2. Can the 96 transpositions be an owner relabeling of tariff 0? First step: one physical owner map that identifies (X_N) with (Y_N) at a single port without a move. Blocker: no such map is pinned, and PR research: derive P104 two-axis endpoint intertwiners and cost gate CrocSwap/integer-mult-bounds#3954 refuses to grant (\Pi) as a free relabeling. A zero tariff would still leave excess 44096.