Research: port count / puncture depth priced with the exact packing ceiling — P104 caps the lever at ×1.054, 1/99 unreachable (no new kappa)

[jackmgaller]

Status: planning bound under a favorable relaxation. No new κ; no word, decoder realization or supplier. The selected record is unchanged. Adds only research/complex-puncture-depth-ceiling/.

Task card

• Card and base. complex-new-helper-inventory / complex-mechanisms. Chooser nonce f27886ce38911b0496e374cd6c06b666, priority revision 3, base 81e81af67714bdc8d1693b0bf5c29468b92f62ff.
• Scope: changed. This answers the second open question in Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240's complex-local-word-ceiling README: how does the target budget scale with port count V? No helper interface is constructed, so the card's acceptance gate is not met. The reason for the change is recorded in TASK.json.
• Claim. CLAIM: port-count / puncture-depth lever priced with the exact packing ceiling (p = 0..24) #1606

Question

On the 120-line E8 alphabet, V = 120 − p. The decoder constant is tied to p by the attained floor cst ≥ max(72 − p, 56), which complex-frame-pair-decoders proves for exact eight-term decoders with p ≤ 24. Apply PR717's per-port knob (one source climb (1,1,1,5) and one target climb (1,1,4,1,1) per call) to the selected #1094 local word, and price every p with #1240's exact ceiling.py. The questions are:

• Which p is best, and does a wider word ever help?
• Can any V reach a = 1/99?
• What cst must a decoder at V ports reach to break even?

Result (certified exact-rational brackets)

p	V	cst	cst/V	× selected-word ceiling (knob)	κ cap	× (proportional sensitivity)
0	120	72	3/5	0.9486	0.0016006	0.9103
8	112	64	4/7	1	0.0016872	1
16	104	56	7/13	1.0544	0.0017788	1.1035
22	98	56	4/7	0.9200	0.0015525	0.9996
24	96	56	7/12	0.8739	0.0014748	0.9621

1. P104 is the argmax on the attained floor, in both models. Its ceiling exceeds the bit primitive (0.0017334), so this lever can at most unbind the complex side. A floor decoder beats the selected word's ceiling only for p = 9..18.
2. The two components are large but nearly cancel.
   • Ports alone (cst 64): about +2.0% per port. V = 120 gives ×1.167.
   • cst alone (V = 112): about +2.6% per unit. cst 56 gives ×1.225.
   • The floor couples them one to one, so each puncture nets only about +0.66%.
3. This answers Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240's question: a port pays for itself, but E8 caps V.
   • A port's excess spend is below its budget increment even at a = 1/99 (0.00696 < 0.00899).
   • Extrapolating the knob at cst 64, a = 1/99 needs V ≥ 1,328.
   • No (V ≤ 120, cst ≥ 0) reaches 1/99, even with no centre at all.
4. cst/V is not a monotone proxy for the moment. P112 (cst 64) and P98 (cst 56) share ratio 4/7, but P98's ceiling is certified 8% lower. The cause is a large V-independent part of the word: about 1,628 rank-1 children per call remain at V = 0.
5. Break-even decoder constants. The package gives a sharp table of the largest cst per V that beats (i) the selected word's ceiling, (ii) the bit primitive and (iii) 1/99, for V ∈ {82, 96..120}.
   • Examples: V = 104 needs cst ≤ 57; V = 96 needs ≤ 51; V = 120 needs ≤ 70.
   • The Research: sharp preserved-frame P104 COPY-drop threshold at 38 punctures (no new kappa) #1187/Research: sharpen the fixed P104 first-drop boundary to 54 copy cost #1455 82-port witnesses (cst 55 and 54) price at ×0.56–0.59. Matching the selected word at V = 82 would need cst ≤ 41.

Not covered / limitations

• Favorable relaxation only. Idle counts are fractional at rank ≤ 51. The bound omits the costs of realizing asymmetric centres, frame legality, restoration and PR717's unpaired-centre parity costs.
• The per-port knob is a marginal model, not a derived word. Rows with V > 112 extrapolate it. The proportional-scaling sensitivity changes the magnitudes but not conclusions 1 and 3.
• The centre split is balanced and reproduces every split on the attained chain. The floor covers exact eight-term decoders with p ≤ 24 on E8 only.
• Planning conversion. κ < a/(1+a) is the retained planning conversion, not a moment certificate or full replay.
• Inherited assumptions remain: all-size/compiler, common-label/mutable-source, whole-call hosting, precision/tape, analytic transfer and bit H1–H5.
• Predecessor check note. At this base, complex-frame-pair-decoders/verify.py fails only its G.selected-record gate, because the selected record moved from PR942 to PR1094 after the package was frozen. Its decoder census and chain sections pass, and only the chain values are used here.

Overlap

• Scoped thresholds: what a coverage-law change must deliver for the selected complex local word (no new kappa) #1348 (Navinor). Prices families per replica at a fixed word. Here the word scales with V and cst is tied to p.
• Research notes (no κ claim): E8 decoder floors are tight; extension-ring, triple and Lorentzian units lose after helpers; cold-end, dispatch and fusion-search negatives #620 and No new κ: BW16 as a unit — window c ≥ 72 puts its ceiling at 0.649× E8's (checked) #895. These screen other units with the proxy x = c·h/v. Finding 4 suggests re-pricing them with this ceiling, which is left as an open next question.
• The P104/P111/82-port decoder PRs (Research: sharp preserved-frame P104 COPY-drop threshold at 38 punctures (no new kappa) #1187, Research: sharpen the fixed P104 first-drop boundary to 54 copy cost #1455, Research: masked rank-8 complex decoders are locally rigid on all P104 frame-pair punctures (no new kappa) #1292, Research: rank-8 masked decoders are gauge-rigid on P111 and type-(4,4) P104 units (no new kappa) #1389, Research: rank-8 masked decoders are gauge-rigid on the 896 admissible P104 units (no new kappa) #1399) concern decoder structure. This package supplies the moment-side screen for them.
• Refresh. The final overlap refresh (732 open PRs, 253 discussions, complete pagination) found no other ceiling pricing of puncture depth.

Reproduce

python3 -I -B research/complex-puncture-depth-ceiling/verify.py          # ~80 s, PASS_COMPLEX_PUNCTURE_DEPTH_CEILING (22 checks)
python3.12 scripts/research_package.py --package research/complex-puncture-depth-ceiling
python3 -B -m unittest discover -s tests -p 'test_research_*.py'

All three pass locally on macOS: CPython 3.14.7, plus the registered replay on CPython 3.12. The unit tests ran 20, all OK with 1 skipped.

The checks include:

• Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240's ceiling reproduced exactly from the word re-derived from the pinned assembly;
• sharp break-even thresholds;
• four failure controls: the coverage law rejecting V = 77, the floor refused beyond p = 24, monotonicity in cst, and the knob mattering;
• a byte-for-byte match of RESULT.json.

Credit

• Douglas Colkitt: the ceiling and its open question (Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240).
• DreamingOfClouds: the B14 model and P1 knob (PR717).
• The complex-frame-pair-decoders authors: the floor chain and knob transport.
• Navinor: Scoped thresholds: what a coverage-law change must deliver for the selected complex local word (no new kappa) #1348.
• Cinna Davis and woklerx: Research: sharp preserved-frame P104 COPY-drop threshold at 38 punctures (no new kappa) #1187 and Research: sharpen the fixed P104 first-drop boundary to 54 copy cost #1455.
• Joel Pulikkan and DaysSky: the unit screens (Research notes (no κ claim): E8 decoder floors are tight; extension-ring, triple and Lorentzian units lose after helpers; cold-end, dispatch and fusion-search negatives #620, No new κ: BW16 as a unit — window c ≥ 72 puts its ceiling at 0.649× E8's (checked) #895).
• The selected Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 authors: as listed in its review.

Prepared by Jack Galler (@jackmgaller) with substantial Claude Opus 5.5 (Anthropic, Claude Code) assistance with the model choice, code and text. No independent human review is claimed. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at 6c96bc1a0e374a713e9ae493228247f49be49dd5.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Exact packing-ceiling arithmetic prices two explicit port-count models and their balanced centre splits. P104 maximizes these models, not all possible architectures; integration should correct the phrase actual moment to admissible saving in the relaxation caveat.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.