[Matarener]

Requested by Mark

Change and mathematical scope

Before: the retained compiler premise prescribes zxv's chronological product, paid completion of small targets and disjoint splitting (p7-TENSOR.md, higher-moment-strictification) and then Sussman's lockstep flow; the completion/splitting step exists only as moment arithmetic, and FINITE-PRECOMPILATION.md §6 allocates one owned-centre bank per compiled unit instance, noting that "a smaller reuse allocation requires its own lifetime proof".

After: research/lockstep-completion-pass executes the pass and proves the lifetime statement. Status: two written lemmas with event-level corroboration; no new κ, no selected-record change.

• Lemma A. Completion to the floor L on each small target's own owner, then splitting of every paid job at the cutoff J, is a transformation U -> U' of finite native units. U' satisfies the lockstep hierarchy; its compiled programs pass the ownership invariants I1–I6 of our research/lockstep-live-role-audit: live roles in the lockstep compiler and the #4470 outer constant C (no new κ) #4691 executor; every added job is a whole job on a field its unit owns; the rank-one surplus is preserved exactly; the net-tax identity (1) of net-tax-rank-compiler holds; the added records (pieces, completions, normalizer pairs, block relocations) have a closed per-unit count that the executor reproduces.
• Lemma B. Owned-bank lifetimes in the lockstep schedule are properly nested (the live units are the classes on the current path plus the nested programs), so a LIFO stack of bank slots suffices; its peak is an explicit recursion. The executor runs the stack allocator with a new invariant I7 and its measured peak equals the recursion on every case. The owned-field inventory O in Z = W + O can be the peak instead of the instance count.

Remaining obligations: the native content of completion jobs and normalizers, the all-column lift, the strict moment inequality, precision and analytic recovery are inherited. The physical half of native_all_columns (tape execution of g^{⊗f} at the retained tariff, see #5442) is not addressed; this package supplies exact record counts for that bill, not prices.

Task and overlap

Base: canonical main 32890bc5de034ab8fc93ac88ead7279a6a416d04; package commit 38097c7f1. Inputs pinned in SOURCES.json: FINITE-PRECOMPILATION.md, p7-TENSOR.md, higher-moment-strictification/README.md, net-tax-rank-compiler/PROOF.md, the #5413 invoice.json/funding.json, FINITE-DAG-AUDIT.json, and our lockstep-live-role-audit/live_roles.py. Object: the completion/splitting pass and the lockstep flow on the p7 and selected #5413 profiles (cutoffs c=10,b=15 and c=20,b=21), tensor powers 1..3 at histogram level, one generation for compiled counts.

Task card selected-stock-envelope, scope refined (TASK.json): it continues #4691 (same card) toward the compiler/owned_erasure premises named in target 10 of #5428. Nearest work: #4691 (executor, separated rules only), #4832 (sharp split charge; moment arithmetic only), #4586/#4638 (chronological batch widths), #5364/#5404 (continuation tapes), #4192 (Beneš arena scan), #5442 (algebraic half of native_all_columns). None executes the pass or proves bank lifetimes.

Validation

python3.12 -B scripts/research_package.py --package research/lockstep-completion-pass

PASS locally on CPython 3.12 (Windows), standard library only: verify 148 s, controls 168 s; check_research_commit.py passes. Evidence:

• counting parity with research/lockstep-live-role-audit: live roles in the lockstep compiler and the #4470 outer constant C (no new κ) #4691 on the toy rule (N = 4..64) and the BIT-shaped rule (N = 50..3200), and the FINITE-DAG-AUDIT receipt at toy rank 40;
• complete p7 and selected profiles at tensor powers 1, 2, 3: identity (1) to 40 digits, surplus unchanged (608 and 752 to the power v), equal added masses, separation; at v = 1 the selected unit has 77,204 rounds after the pass (64,260 original, 5,152 pieces, 7,792 completions), 15,584 normalizer records and 12,944 relocations;
• event-level execution with the stack allocator on the toy rule (full width to N = 28), an overlapping toy rule after the pass (full width, N = 40 and 64: 12,048 and 112,738 bank allocations, peaks 732 and 8,751), p7-shaped and selected-shaped rules with the caller's chronology shape and random, all-on-centre and one-owner owners: peak, total and all record counts equal the formulas;
• compiled counts of the complete profiles at one generation: selected profile N = 3045, 84,996 units per copy, 7,793 live at peak, total/peak banks = 346;
• twelve rejecting controls (test_verify.py): overlapping rule before the pass, cutoff not below floor, dropped remainder, short completion, foreign completion owner, release-early allocator, per-depth bank, eager parent, repeated exit field, peak recursion without its nested term, wrong tax, tampered RESULT.json.

Finite executions corroborate the written lemmas; they are not the all-size proof. Not run: the literal owner of each selected paid call (shaped chronologies instead), two-generation compiled counts for the selected profile, the v = 1024 template, Lean, hosted CI before this PR.

Credit

Jacob Alan Sussman (Theorem 1G lockstep flow), Cinna Davis (PR4412/4414/4416 product and completion), zxv (higher-moment-strictification, net-tax-rank-compiler, #5428), Nick Petalas (#4832), Chris R / Clear1nsight (#5413), Dillon (#5402), Ali Romman (#5383/#5398), Cao Yifan (#4610), and our #4691 executor. Apache-2.0. Prepared by Mark (Matarener) with substantial Claude Code assistance; no independent human or formal-kernel review.

Open next questions

1. Bind the actual owners of the selected caller's paid calls and execute the literal 77,204-round rule at two lockstep copies (first step: per-event owner fields from NEUTRAL-BACKWARD-v16c.json).
2. Price the added records with the retained native tariff (171-record monomial bound, block-permutation price) for a numeric movement bill.
3. Evaluate peak(N)/total(N) at two generations with memoisation over dilations.

🤖 Generated with Claude Code

[zxv]

Reviewed current head 1fb616aaa6f32fd5500ec58a407f0c46e32e7aeb. One correctness finding remains.

[P2] The peak recurrence combines maxima from different execution paths. counts.py:220–228 adds the largest live class-path inventory to the largest nested exit-program peak. These need not occur together: the exit rank depends on the paid ranks along that same path, which also determine its class sizes. This invalidates the equality asserted in Lemma B.

Full-width counterexample, run from this checkout:

import sys
sys.path.insert(0, 'research/lockstep-completion-pass')
import counts as C, executor as E
raw = E.RawRule('two ranks', (('a', 3), ('b', 5)),
                ((1, 'C0'), (2, 'C0')))
rule = E.complete_and_split(raw, J=2, L=3)
cc = C.Counts(rule)
run = E.audit(rule, 45, owned_policy='stack')
print(run['full_width_executed'], cc.flow(45).G,
      cc.banks(45)[1], run['peak_banks'])
# True 2 2050 2048

Here a=512, B=4096, and the executor completes all ownership checks. An independent path-conditioned recursion also gives 2048. The formula's 2048 live parent banks and its extra 2 nested banks occur on different paths. Please keep the maximization over compatible (path, exit) pairs, or state/prove this expression as an upper bound and change the exactness claims accordingly. Add a multi-generation regression where path sizes vary. This counterexample does not show that the LIFO allocation is unsafe; the current formula overestimates this peak.

The earlier JSON comparison failure at 38097c7f1 was fixed by the new head’s JSON normalization; it is not an outstanding finding. The complete registered replay, including all twelve controls, passes on the revised head. The counterexample above is a full-width two-generation execution that passes the executor and disagrees with the claimed exact peak.

Validation ran serially on CPython 3.12.8 / Darwin arm64 under a sampled 1 GiB descendant-RSS limit and explicit timeouts, with no resource stop. Literal selected-call owners, the 1024-fold template, physical tape prices, and the full retained compiler theorem were not independently validated here.

Review performed with OpenAI Codex assistance.

[Matarener]

Thanks, confirmed and fixed in 38119b3. Counts.banks now keeps, for each deepest dilation, the largest live inventory of a path reaching it, and maximises over compatible (path, exit) pairs: exit ranks r * dil(path) of that same path. Your two-generation rule is a verify.py case at N = 45 and 69 (executed peaks 2048 and 16,386, equal to the formula). It is also a control: the decoupled sum gives 2050. On the full rule the formula matches the executor at every N from 3 to 69. Lemma B in PROOF.md states the path-conditioned recursion. All earlier frozen results are unchanged, including total/peak 346 for the selected profile.