Prove the exceptional single-hub literal charge is 48.

[unpaidmoon]

Change and mathematical scope

Scoped exact minimum. No new κ and no selected-frontier change. Selected κ stays 0.001672660197076918.

On the exceptional single-hub star of PR #3573 (marks E,F, queries g,d,t,h,x,y), the relaxed literal actual-reader model of PR #3698 has minimum charge 48 against baseline 40. The excess of 8 is exactly the four reader rows E dxy, E gtxy, F dxy and F gtxy. Each has a unique reader, and every admissible evaluator leaves at least two queries visible. The other sixteen extremal rows attain cost 1.

The coordinate-functional bearing count is not this minimum. dthx and dthy contain three mark-bearing queries, but the evaluators (1,0,-1,0) and (0,1,-1,0) see only d and pay 1. The two-bearing mechanism that survives the PR #3698 relaxation is the pairing on dxy and gtxy.

PR #3573's GF4 hub map still preserves all 128 oriented subset implications. The excess is a rational factorization bill, not a transfer obstruction. One attaining per-query split for g,d,t,h,x,y is [0, 20, 4, 8, 6, 10]. The total is the minimum. That split is not claimed unique.

Task and overlap

• Base: bb4bccbf42844d2801487450727c2f63e10d1d0f (canonical CrocSwap/integer-mult-bounds main, refreshed unchanged before submission; priority revision 6).
• Task: uniform-transfer-audit, scope changed. The card asks for one all-size, compiler, precision, hosting or restoration implication. This session takes discussion Handoffs and requests for independent checks #1166 comment 19021149 instead of redrawing a card. The finite charge is the comparison with the two-hub excess of 12.
• Ordered eligible handoffs used for this session, chronological on Handoffs and requests for independent checks #1166 after dropping delivered work, DONE handoffs, and subjects of open PRs:
  1. Typed actual-width physical H (comment 19016699), subsequently opened as PR Add a typed one-round actual-width H and rounded return. #4123 by a parallel session.
  2. Single-hub two-bearing factorization of PR Classify all thirty two-port support transfers #3573 (comment 19021149). This session.
  3. Integer-shear supplier/recurrence inequality left by Prove minimum binary carrier and native XOR bill for the integer-shear target #3947 (comment 19032401).
• Dropped from the earlier chronological prefix in PR Review the common-map tensor-pairing bound independently. #4113: the PR Reduce helpers 644/660 from width 8 to 7; conditional κ > 0.00173 #3605 full-wrapper replay (open PR Reduce helpers 644/660 from width 8 to 7; conditional κ > 0.00173 #3605), the QQ16 six-tape bind (comment 19030049, marked DONE/released), and the tensor-pairing review (delivered as PR Review the common-map tensor-pairing bound independently. #4113).
• Predecessor heads, not objects of this repository: PR Classify all thirty two-port support transfers #3573 f0cb59da21fbde87b4ca6a7480b0905d6f6fb9bc, PR The two-hub literal actual-reader tariff minimum is exactly 54 #3698 f8893fdfcb41a236ac085a92bca68ec6e5139cdc. The six columns are restated and the extremal table is rederived.
• Overlap at 2026-10-20T02:23Z: 814 open PR bodies and 2642 discussion records. No open PR performed this factorization.

Validation

CPython 3.12.3. Registered replay:

python3.12 -B scripts/research_package.py --package research/p112-single-hub-two-bearing

Passed locally as PASS_REGISTERED_RESEARCH_PACKAGE research/p112-single-hub-two-bearing outcome=completed.

• verify.py rederives the twenty extremal premises, checks the pairing, searches the row model, checks WITNESS.json, the 40×40 Gram identity and the GF4 transfer.
• independent.py does not import that file. Minors and Cramer's rule recompute the twenty minima; GF4 is {0,1,2,3}.
• test_controls.py rejects a broken dxy reader, a substituted y image, and equating the strict dthx count with the relaxed cost.

Not run: Lean, Docker, a selected cold replay, or hosted CI. The argument is the pairing in PROOF.md together with a finite enumeration of one row model. It is not a graph classification or a supplier admission.

Credit

The star, the extremal table and the GF4 map are Nick Petalas's PR #3573. The row model and the two-bearing mechanism are Nick Petalas's PR #3583 and PR #3698, retaining Philippe Van Velzen's expanded identity, ctoth's coverage question and the actual-reader lineage. New enumeration and pairing by unpaidmoon, with substantial AI assistance (Cursor Grok). No independent human or proof-kernel review is claimed.

Open next questions

1. Does one further independent hub add two-bearing rows beyond dxy and gtxy, or repeat the twelve-unit K3,2 excess? First step: write the third hub's queries the way PR Prove the arbitrary-map two-hub support-transfer obstruction #3583 added k,z,w and rerun this row model. Blocked by the absence of a charge-transfer lemma; PR Prove rooted-minor closure of marked support-transfer obstructions #3591 moves support obstructions, not this charge.
2. Can the working GF4 map lift to a characteristic-zero literal factorization with charge strictly below 48? First step: one portwise lift of map (5), or a proof that every lift pays at least 48. Blocked by the gap between a support transfer and an evaluated Gram factorization.

Workflow feedback (optional)

The handoff in comment 19021149 named the comparison and the row model, so the session could stop at one finite minimum. No separate workflow note.