Research note: retained-array catch-up in PR1664's transport moment; #1166 INIT64+d_A final-record route rejected in the model (3,648 / ≥2,496 / ≥1,344 ≥ 896)

[thakrarsagar]

Label: research note (a scoped lemma plus a conditional negative result). No new κ, no supplier, and the selected frontier is unchanged.

This answers Philippe Van Velzen's retention-price request, #1166 c18956869, which had no reply at 07:50Z. Package: research/retained-array-catch-up/.

Statement

The request offers two prices for an array that is retained across calls but read by none of them: (S) 1 unit per array-call, or (C) one rank-8 copy plus 8 per array-call. Under (S), the INIT64 + d_A final-record route's price gate gives a complex κ ceiling of 0.00228; we reproduce that value exactly. The 0.00221 figure (with PR2468-style read invoices) is quoted, not reproduced. The bit endpoint caps both. Under (C) the route has no saving.

All results are in PR1664's scalar common-frame transport moment:

1. Lemma 2. Holding an array at a fixed frame adds no paid mass. So (S) is a physical-storage premise, not a price the moment charges.
2. Proposition 3. Take records R at height h_r and partners P at h_p, with R ∩ P = 0. Delivering O ⊆ R⊕P at consumer height h_c costs exactly min_L ρ_r|L−h_r| + ρ_p|L−h_p| + ρ_o|L−h_c|. The minimum is at the median level and equals the sum of PR1664's cut deficits. This is a both-direction min-cut with the consumer level explicit.
3. Corollary 4 (Fact T and P4 on the pinned Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 template). A record consumed at its pair's current frame pays 8 coordinates per level the pair advances, for each record row independent of what the consumer's contract may combine. A level is advanced by the pair's own call or by an idle sweep.
4. The law versus (S) and (C). The law matches neither.
   • It equals (C)'s mass when a pair advances one level per lifetime call.
   • (S) under-prices PR2540: 16, against the forced and attained 128.
   • In replica 0 (12 of 16 replicas) there is a zero-advance window across six calls: the law gives 0, (S) 6 and (C) 48.

Priced decisions (units per replica; exact no-saving threshold 896 at C = 0)

(a) INIT64 + d_A final-record contract. Rejected in the model under the premises below.

Variant	P_final (end decoder)	P_Y (as soon as Y_(k,U) is final)	P_early (after each bank's last local call)
INIT64 + d_A	3,648	≥ 2,496	≥ 1,344
All-initial variant (#2074 c18950926), reads re-timed in favour of the route	3,456	≥ 2,304	≥ 1,152
Bank 0's INIT via f0 (one level later)	3,520 / 3,328	≥ 2,368 / 2,176	≥ 1,216 / 1,024

• Off-by-one in the gate. Under the gate's own INIT convention bank 0's d_A lives 7 calls, not 8. That gives 456 array-calls rather than 464, and the law then equals (C): 3,648.
• Without P_rank. Theorem B (CLAIM: one paid pre-F0 B2 source observation (seed 3e62c004) #2074) applied per cut still rejects at P_final: 1,952 / 1,824. At P_Y and P_early it needs our own-bank premise, giving 1,344 / 1,216 and 800 / 608. Below 896, the caps for 800 / 608 are still under the selected κ. The count alone does not decide at 544, 112 or 0.
• P_dec is load-bearing. At decoders right after their own last captures, the moment bound drops to 512 (cap 0.0020286). No decoder is known to run there.
• Capture times also matter. An AI review pass checked captures one level later that add a read of the transient Y. They give P_early 832 / 640, below 896, though the caps stay under the selected κ. P_Y and P_final stay ≥ 896.

(b) Gap-1 shared-source centre sharing (f0→f2). No moment gain under the stated premises, by catch-up alone: at least 64 per shared set across B1, which equals the saving. The f0→f2 information question stays open (PR2199 open question 2). #2391/#2461 cover only f0→b1.

(c) QQ16 retained caller #2540/#2651. Clarification only. Its retention row is exactly its 128 = 16 H8, one charge rather than two, with ρ = 16 from PR2511 and an F_U1 consumer. The moment comparison with zero-retention #2707 is #2707's own result, already incorporated with the maintainers' review. Physical storage is UNKNOWN.

Break-evens. The largest catch-up mass at which a route still gains is 579/576 against the selected record (idle-priced / rank-8-priced), 573/568 against PR2474 and 568/560 against the bit endpoint. The #1166 break-evens of 573 (S) and 71 (C) reproduce.

Premises (each is stated in PROOF.md §7.2 and §8)

• P_obs: the stated observation contract, with final-value reads.
• P_rank: finite-model evidence only. Uncompressed delivery is our reading of the contract.
• The stated capture times.
• P4: no slack in the inherited words.
• P_dec: the derived decoders satisfy it.
• Current-frame consumers.
• PR2540's ρ = 16.

Physical storage, tape movement and hosting windows lie outside the moment and are UNKNOWN. They can only add cost.

Base, pins, reproduce

• Base. de880ca1fa4540472ee8461d400914a60fc9dd41. The stage base 116a714 lacks a pinned input, so the package was built on de880ca.
• Pins. All 18 SOURCES.json inputs have identical git blobs on current main 38746ad3f (checked 2026-10-16). main's updated scripts/research_package.py --check-only also passes on this branch.
• Branch head. 7324b2526182206904f75e4037014bcc2f0be4f5.

python3 -I -B research/retained-array-catch-up/verify.py && python3 -I -B research/retained-array-catch-up/independent_check.py

Expected output (about 55 s total on CPython 3.14.4):

PASS_RETAINED_ARRAY_CATCH_UP pr2540=128 init64_dA=3648/2496/1344 all_initial=3456/2304/1152 dA_only=960 thmB_one_level=1952/1824 thmB_own_bank=1344/1216/800/608 thmB_count_only=608/544/112/0 without_P_dec=512/640 init0_via_f0=3520/2368/1216 array_calls=464/456 no_saving=896 breakeven_S=573 breakeven_C=71 zero_advance_calls=6 hosted_instants=21 frame_instances=34454/4044 negative_controls=17 no_new_kappa
PASS_RETAINED_ARRAY_CATCH_UP_INDEPENDENT pr2540=128 init64_dA=3648/2496/1344 all_initial=3456/2304/1152 dA_only=960 thmB_one_level=1952/1824 thmB_own_bank=1344/1216/800/608 thmB_count_only=608/544/112/0 zero_advance_calls=6 zero_advance_replicas=12/16 hosted_route_instants=21x16 breakeven_S=573 breakeven_C=71 no_saving=896 echelon_dijkstra_instances=2432 controls_rejected=2

The registered Python 3.12 replay was not run, because only CPython 3.14.4 was available. Both scripts parse under 3.12 grammar.

Certificates and controls

• Positive controls. PR2540 gives 128 (16 H8), PR2197 gives 144, PR2678 gives 1,792 and 3,696, and the centre path gives 576 per replica. The Handoffs and requests for independent checks #1166 break-evens 573/71 and PRICE-GATE.json's storage ceiling reproduce exactly.
• Two checkers with no shared code.
  • verify.py uses the pinned ceiling.py, exact rationals, a BFS over span lattices and a two-sided hosting check at 21 instants in replica 0.
  • independent_check.py does not import ceiling.py. It uses a coordinate-set replay, a per-family hosted-height replay over all 16 replicas, Decimal-80 moments and Dijkstra over echelon bases.
• 17 negative controls, each required to fail with its stated message. Examples:
  • mutated idle levels and bridge directions;
  • a PR2540 control with 15 rows;
  • a moved capture;
  • a delayed residual (undershoot);
  • Theorem B without the own-bank premise;
  • P_early claimed as a floor for decoders without P_dec;
  • 895 units still saving;
  • upward-only cut bounds, which fail on 14,426 of 34,454 instances;
  • the tensor toy with catch-up omitted or on the wrong axis (25/25 failures each).
• Tamper check. A tampered RESULT.json makes verify.py fail.

Search record

• Frame-model domains. Every domain is exhaustive, which is the stopping rule:
  • GF(2)^2 at 4 levels;
  • GF(3)^2 at 3 levels;
  • GF(2)^3 at 3 levels.
  • Together these are 34,454 record/partner instances, and in all of them opt = the sum of cut deficits = the meeting formula.
  • There are also 4,044 general instances: the bound is sound in all of them and tight in 3,999.
• Independent run. Dijkstra over 2,432 instances.
• What was not searched. There was no per-record-class schedule search; the stop rule excludes it. The tensor toy uses seed random.Random(20261016).

Unresolved obligations

• Physical storage is UNKNOWN.
• P_rank has no proof on the actual cover.
• Theorem B is an unreviewed posted claim, and the own-bank premise is ours.
• P4 is not re-derived event by event.
• P_dec and the capture times are load-bearing premises.
• Only nested frames are modelled.
• Consumers whose outputs stay below the partners' frame are not priced.
• f0 and the forward histories f2–f8 are not in the 464 array-calls. They would only add cost.
• Values computed by AI review passes rather than by the shipped verifiers are labelled as such in README.md.

Credit and what this adds

Credit.

• Andrey Mas (an664, PR1664: the model and the cut lemma); ctoth (CLAIM: can the centre path be hosted on idle coordinates like helper residuals? (ctoth, expires 05:00Z) #1516, CLAIM: where does the one unpaid coordinate per live family come from, and is it forced? (ctoth, expires 07:30Z) #1587); PR1012; the idle-axis-hosting / Conditional kappa 0.001249035299298004: hosting-selected rank147 kernel #434 lineage.
• ywz65 (PR2511 as the source of ρ = 16, PR2540, Normalize the zero-retention QQ16 decoder with paid rank-one children #2707, Prove a sharp 104-coordinate owner-local centre repair floor #2461); Arham Tahir (PR2681); petalas (PR2678); 1337isnot1337 (PR2197); Ross Richardson (PR2480); alephnan (PR2346); commdata2338 (PR2174, PR2651); navinor (Native centre sharing: zero first transcript, later query 1/4 #2391); the QQ16 lineage; the maintainers' combined review of Normalize the zero-retention QQ16 decoder with paid rank-one children #2707.
• Philippe Van Velzen (conventions, final-record census, price gate, decoders, Theorem B; fork files read as data only and never executed).
• Our PR2199 and PR2470.

What this adds. Earlier native climbs (PR2540/2492/2512/2197/2678) are one-time concrete charges, and PR2470 assumed a centre-like convention. This package derives the forced price, states the observation premise it needs, and decides the #1166 route at three consumer timings.

Provenance

• Sprint track. 50–59 Recursive interfaces, OS-random seed 59; loop 9d24, iteration 16.
• Claim. #1163 c18963409.
• Review. Reviewed by 13 independent AI review passes over four rounds in the same loop, covering reproducibility, adversarial mathematics, provenance and referee lenses. There has been no human review and no formal-kernel review.
• Attribution. Prepared by @thakrarsagar with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5) via Claude Code, in an automated sprint loop.

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