[unpaidmoon]

Change and mathematical scope

Partial audit. On a rewound parent program, children.extend has the exact program-tape meters

parent reads = 2 R0 + 3 h c + 3, writes = 0, moves = 2 R0 + 2 h c + 4
child  reads = 2 S + 1,          writes = S, moves = 2 S + 2
R0 = 2I + L + 5,  c = Q + 2I + 90,  S = R0 + (h + 1) c

with pinned FINITE_BITS = 16. The S child writes are strictly positive and belong to Pi_v in PR #3983 equation (6), not to the later capsule copy in equation (5). SelectedBinding.check_stock is a single pass, so this constructor does not admit original-width stock. No new κ. Selected κ stays 0.001672660197076918.

This is not a stock envelope, not the rest of Pi_v, and not a multiplication theorem.

Task and overlap

Base bb4bccbf42844d2801487450727c2f63e10d1d0f, priority revision 6.

Item 6 of the chronological open-#1166 list in the session report: instantiate producer Pi under original-width stock (discussion comment 19035639 / PR #3983 open question 2). Catalogue card selected-stock-envelope, scope changed: the card asks for a selected bootstrap stock coefficient; this prices one constructor term and records that the selected stock gate is empty.

Nearest work: PR #3983 states the uninstantiated Pi_v and the length P_h but does not count these tapes. PR #3967 is the R437 source pin used there. This round's #4130 and #4141 price other stock quantities and do not instantiate Pi. No open PR prices children.extend.

Validation

python3.12 -B scripts/research_package.py --package research/r437-pi-child-constructor

Observed PASS_R437_PI_CHILD_CONSTRUCTOR cases=40 finite_bits=16 stock_gate=pass prefix_in_extend=false index_reads_seen=6706 no_new_kappa, then PASS_R437_PI_CHILD_INDEPENDENT cases=3, then PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=partial. CPython 3.12.3. The forty fixtures cover good, pair, unit, and half-width children. A parent head moved one cell off the rewind postcondition does not match. A one-read-per-copied-symbol invoice does not match. Index-arena reads are nonzero and are not given a closed form.

Not run: selected cold replay, Lean, Docker, Callback._validate_program, directory production, and any stock-coefficient comparison.

Credit

Chris R / Clear1nsight, PR #3983 and PR #3967, for the Pi_v ledger and P_h. The constructor and tapes are the retained physical-caller sources (Jose Manuel / jmbarrancoml, ywz65, David Arias, Nick Petalas, PR #2753 / #2769). Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. Price Callback._validate_program on the same rewound programs. The prime-power division depends on the modulus.
2. Name a selected original-width stock word that check_stock should compare, or show that no selected caller reaches it.