[DreamingOfClouds]

Fragmentation-loss ceiling: the exact Λ ship threshold for the #4610 family, with Lean (no κ change)

Summary

No new word and no κ change. This adds research/fragmentation-ceiling/, which shows that for words in the #4610 family one per-call quantity decides everything: the fragmentation loss Λ(t) = Σ over fields of (Σ climbs c^t − R_f^t) ≥ 0.

• The bar κ ≥ 0.0030233 holds only if Λ(t*) ≤ 23.8536932758 per call. Conditional κ = 0.0030217758: joint helper interfaces and Lean compiler tightening #4610 has 23.8746264727, i.e. 0.0209331969 per call above. Five 1+1 → 2 merges per call (0.0041868 each) would close it.
• No helper is net-productive (Theorem C), and a descent never lowers the paid moment (Lemma 5). A rank-1 descent costs at least 1.99022, so at most 11 fit under the threshold.
• Under stated contact hypotheses the owned centre is moment-optimal at 8^t* = 7.9499 per use (A30-CUT).
• Applied to the centre-free coded constructions (Construct a center-free native producer with an explicit rank-8 port code (no new κ) #4929, Construct a reversible 640-field center interface and exact raw dual (no new K) #4694/Normalize eight center chains and construct a conditional two-call consumer (no new K) #4698, Emit a phase-correct coded p16 caller removing 144 center calls (no κ) #4933, Prove a scoped one-bank ceiling for the #4929 coded-port adapter (no new κ) #4932, Expose a clean native prefix and construct one homogeneous U(q) offer (no new κ) #4931, Consume the #4929 center-free word with coded reverse stages and A⁻¹-mixed query shears (no κ) #4940): the coded endpoint is Conditional κ = 0.0030217758: joint helper interfaces and Lean compiler tightening #4610 minus its centre, with exact budget 63.577975271157 per call. Within the A30-CUT scope (CONDITIONAL on P-ATOM, DELIV and mains net-up), every repair costs at least 8·8^t* per stage, short by exactly the Conditional κ = 0.0030217758: joint helper interfaces and Lean compiler tightening #4610 gap (0.376798 over the invoice).

What is proved, and how to check it

Every claim is tagged in PROOF.md:

• PROVED-LEAN (lean/Skel.lean, Lean 4 + Mathlib, 37 declarations, no sorry): subadditivity and exchange of x^t; the rank formula and parity for subspaces; path telescoping; the decomposition identity; Λ_f ≥ 0 and strictness; Theorem C; the descent price; the level-crossing count and dual feasibility of A30-CUT (its parity summation along a path is on paper). Build: cd research/fragmentation-ceiling/lean && lake exe cache get && lake build Skel (Mathlib pinned); #print axioms gives only propext, Classical.choice, Quot.sound.
• CHECKED-EXACT: verify.py rebuilds Λ (both orientations, 632 fields), the identity to 10⁻⁵⁷, the threshold, the best controller over all p (exact for p ≤ 400, proved bound beyond), the descent prices and the coded budget from Conditional κ = 0.0030217758: joint helper interfaces and Lean compiler tightening #4610's public package, 60 digits, stdlib only. Output is in data/verify_output.txt.
• PROVED-PAPER / CONDITIONAL: the cut lemma of A30-CUT (needs P-ATOM, DELIV, mains net-up); sufficiency of the threshold (needs a single crossing of f); the coded-family kills (inside the A30-CUT scope).

python3 -B research/fragmentation-ceiling/verify.py     # about 10–15 s, prints ALL CHECKS PASSED

Caveats

• The threshold is for the kap-family unit (112, 8) with the 8-job centre. It does not cover other units or controllers.
• A30-CUT leaves open cross-use sharing at a common frame, deliveries at frames meeting the port, decoders that break P-ATOM, warp-riding carriers and query shears.
• Statements about other PRs' constructions were priced from their published data; their claims were not re-run.

Credit

Cao Yifan (intqwq, #4610); Dugongue (#4929); Cinna Davis (1337isnot1337; #4694, #4698, #4470); dillona (#4933); Clear1nsight (#4931, #4932); SethBurkart123 (#4940).

Claude Opus 5.5 (Anthropic), via Claude Code.