[unpaidmoon]

Change and mathematical scope

Scoped space lemma for the depth-first reading of BIT.md §2 at c = 8667/5000000. No new κ. The selected conditional exponent stays 60403/20000000. The PR2809 simulator is not run.

For every integer e ≥ 2, the level-1 child payload V/(W N) exceeds the allowance V/(W N q^{4096(w-1)}) named by research/bit-stock-recurrence-audit by exactly q^{4096(w-1)} ≥ q^{4096}. On 2 ≤ e ≤ 2^576, w = ceil(e^c) = 2.

In the geometric model, ancestor payloads sum to V/(W N - 1) < V. Parameter bits, one cell each, are at most 26040 e. For every stream with volume V ≥ e,

parked payload + parameter bits ≤ 26041 V.

The constant depends on q and not on e or V. A reading that parks a fresh copy of the root volume at every cut is not O_q(V): D(2^26050) ≥ 26042.

Remaining obligations: the serializer that would decide which reading the machine implements, and the marker/template/erased-backing cells, are not charged here.

Task and overlap

Base commit ff3cc58. Priority revision 7. Task card uniform-transfer-audit, scope refined to the continuation-stack question left by research/bit-stock-recurrence-audit.

This is item 2 of a 3-item eligible list (session r30; (8-1) mod 3 = 1). The other items are the non-restoring (w, q, v_N) caller and a 14-point F_2^6 walk of step weight at most 3. The algebraic BIT.md §§1–4 audit, the sibling ledger-boundary exponent, and the older c = 17/10000 space proof in research/physical-ordinary-interchange are different results.

Validation

Python 3.12, standard library only:

python3 scripts/research_package.py --package research/pr2809-continuation-stack-space --check-only
python3.12 -B scripts/research_package.py --package research/pr2809-continuation-stack-space

verify.py prints PASS_PR2809_CONTINUATION_STACK_SPACE C=26040 witness_e=448 witness_bits=11665872 K2_bits=729117 level1_factor_bits=364544 payload_ratio=1/(WN-1) full_copy_exceeds_bitlen=26051 no_new_kappa=true.

test_verify.py rejects w = 1 at e = 2, a non-halving coefficient 64, and the constant 26039 at e = 448.

Not run: the PR2809 simulator, Lean, Docker, and the selected cold replay. The proof is the integral argument in PROOF.md, not a finite sample of the machine.

Credit

Cinna Davis, #4100, for BIT.md and the O_q(V) sentence. Mark (Matarener), with Claude Code assistance, for research/bit-stock-recurrence-audit, which checked §§1–4 and left this question open. Jose Manuel's research/physical-ordinary-interchange is the older c = 17/10000 space argument and is not a dependency.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human review and no formal-kernel acceptance are claimed.

Open next questions

1. Does the PR2809 serializer park the ancestor chain, or one root-sized stream per cut? The second reading is not O_q(V). First compare bytes written at one recursive cut with the child capsule. Blocked because this package does not extract or run that serializer.
2. Are marker, template and erased-backing cells inside the 26041 V budget? First charge one fixed template per live role against V/(G-1). Blocked because BIT.md §2 does not give the live role count.