[JoyboyBrian]

Result

This adds research/complex-stopped-recursion-cost/, a scoped proof for one #4100 native/complex-family implication: COMPLEX.md's "Time" bill. The stopped complex recursion costs O(V h^(1-q) polylog p) for every root width h >= 1. No new κ. #4100 is not accepted, and the selected frontier is unchanged.

Claim type: a written all-size proof under COMPLEX.md's own premises, plus certified finite enclosures and numerical tree corroboration.

The premises stay explicit:

• (N) node cost O(V e^(1-a2) polylog p), the e mod 73 remainder bits, and child volume (H_r/W) v at width r floor(e/73);
• (L) leaves;
• (P) root placement;
• (A) additivity.

Under them, PROOF.md proves by strong induction on e:

cost(e, v) <= v [ K1 e^tau + c_L s^b e^(1-b) ],   K1 = (c_N Lambda + c_L)/(1 - mu_tau),
layer      <= V Lambda h^(1-q) [ (c_N + c_L)/(1 - mu_tau) + 73^b c_L + 2 c_P ].

The induction handles floors, remainders and any stop width s >= 73 directly, with no monotonicity in e.

Certified facts (exact or outward-rounded rationals):

Fact	Value
1 - mu_(1-b)	1.804257e-48
1 - mu_tau	1.417734591407e-28
1/(1 - mu_tau)	in (7.0535e27, 7.0536e27)
exponent order	1-a2 < tau < 1-q, 1-b+eta*b < 1-q, 72 <= 74^tau

Callers.

• PROOF.md's synthetic-transform layers (1 <= h <= d-1, at most ell levels) get O(V ell d^(1-q) polylog p), including levels with fewer active axes.
• GAUSSIAN.md's compiled layers (h = O(p^beta), O(p^epsilon)) get O(V h^(1-q) polylog p).

Observations. None of these changes a downstream bound.

• The leaf term needs only mu_(1-b) <= 1.
• The h^eta stop is not used by this bill.
• The floor is necessary. With a ceiling, a width-74 node has a width-96 child.
• The uniform constant exceeds 7e27 (c_N Lambda + c_L). Finite unit-cost trees show an internal tau-sum over h^tau of only 8.8 at h = 10^12, so the constant is a worst-case artifact, not a defect.

Task and overlap

• Card: uniform-transfer-audit, refined to one non-CRT obligation (native/complex family).
• Claim: Handoffs and requests for independent checks #1166 (comment)
• Base: canonical main cef4b52f2a59009865bf8eabde1be3f159063703, priority revision 7. Propose unconditional κ = 0.00173097911567079247276057 via batched CRT #4100 is at its retained head 337d04939571a637aeab54a0f58f54fde811663a.
• Why this object: the Map independent audit coverage of #4100's obligations (navigation; no new κ) #4311 coverage map lists "the stopped-moment ... proof outside Bind mixed-provider precision to the rounded Gaussian tensor caller #4178's caller" as still open.
• Nearest work, not redone here:
  • Bind mixed-provider precision to the rounded Gaussian tensor caller #4178: height recurrence at the Gaussian caller.
  • Independent numerical cross-check of PR4100's exact-root and paid-power ledger (finite layer; no new kappa) #4171, Certify #4100 BIT.md's recurrence moments: mu_(1-c) < 1 with margin 8.8e-8 (no new κ) #4330: root and ledger numbers.
  • Audit #4100 COMPLEX.md's common cover and native warps: |O(73,2)|, X_delta correction, stock arithmetic and Z4-monomial warps hold (no new κ) #4424: cover and stock arithmetic.
  • Separate the tensor-frame child from factorwise repetition. #4375: frame-child reading.
  • Audit PR4100’s free-word tensor lift on all four actual programs #4184: free-word lift.
  • Audit the 2^{-4p} amplification sentence in #4100. #4371, Answer #4371's open question 1: one coefficient's error path through #4100's transforms carries one factor of T (m = 1), so C = 5 suffices for b >= 13 (no new κ) #4415: the 2^-4p precision chain.
• CRT.md is not touched. I rechecked the board and open PRs just before submission and found no crossing claim.

Test plan

Command	Result
python3 scripts/research_package.py --package research/complex-stopped-recursion-cost --check-only	PASS_PACKAGE_METADATA_AND_PINS
python3.12 -B scripts/research_package.py --package research/complex-stopped-recursion-cost	PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, about 7 s per command, normal and -O
python3.12 -O -B research/complex-stopped-recursion-cost/verify.py	PASS_COMPLEX_STOPPED_RECURSION_COST
python3.12 -I -W error -B research/complex-stopped-recursion-cost/verify.py	PASS_COMPLEX_STOPPED_RECURSION_COST
python3.12 -B scripts/check_research_commit.py --package research/complex-stopped-recursion-cost --check-source-history	PASS_RESEARCH_COMMIT_BYTES
python3.12 -W error -B -m unittest discover -s tests -p 'test_research_onboarding.py'	26 tests OK

The built-in rejection controls all fail as intended:

• a root above the bracket, b + 1e-47, gives mu > 1;
• one extra rank-48 child per W breaks both moments;
• the ceiling child rule grows width 74 to 96.

Mutation tests: each of these six edits makes the verifier FAIL:

• shifting the stated 1/(1-mu_tau) window;
• moving the root above the bracket;
• a_eff > a2;
• 3000 extra rank-1 children;
• weakening 74^199 >= 72^200 to >= 73^200;
• an e // 72 child rule.

Open next questions

1. Node-cost premise. Do the adapters, the F2 row-addition monomials, the Gamma-fold common-cover packet gathers and parking cost O(V e^(1-a2) polylog p) at the actual node? First step: a per-node invoice from COMPLEX.md's "Common cover" section, PR2250 §5–6 and Audit #4100 COMPLEX.md's common cover and native warps: |O(73,2)|, X_delta correction, stock arithmetic and Z4-monomial warps hold (no new κ) #4424's row layout.
2. Child volume premise. Do the actual packets realize Gamma H_r rank-r calls of volume V/(W Gamma) each? Separate the tensor-frame child from factorwise repetition. #4375 shows that the factorwise reading breaks the moment.

Credit

The construction and the audited statements are Cinna Davis's (#4100). The histogram and words are PR4088's. I build on:

• Nick Petalas (Independent numerical cross-check of PR4100's exact-root and paid-power ledger (finite layer; no new kappa) #4171), ledger cross-check;
• Jose Manuel / jmbarrancoml (Bind mixed-provider precision to the rounded Gaussian tensor caller #4178), height recurrence;
• Matarener (Audit #4100 COMPLEX.md's common cover and native warps: |O(73,2)|, X_delta correction, stock arithmetic and Z4-monomial warps hold (no new κ) #4424), cover audit;
• unpaidmoon (Separate the tensor-frame child from factorwise repetition. #4375), frame-child separation;
• Ryanson Jonathan / pinterbanget (Map independent audit coverage of #4100's obligations (navigation; no new κ) #4311), coverage map.

Prepared by JoyboyBrian with substantial assistance from Anthropic Claude (Claude Code, Opus 5.5). Apache-2.0.

🤖 Generated with Claude Code

[jwr404]

Independent review at head 9e00ff1f7baa8352339f36277389cb1f06a5a58a

Scope. I checked the PROOF.md lemma and theorem line by line against #4100 COMPLEX.md's "Time" section at canonical main. Its constants match verbatim: H, W, b, q = b − 10^-28, a_eff, η = 10^-32, τ and the stop max(73, h^η). I also recomputed the certified facts with my own stdlib Decimal script at 120 digits. It imports nothing from this package.

Certified facts: all reproduce.

Fact	Independent value
1 − μ_(1−b)	1.804257027708936984…e-48
1 − μ_τ	1.417734591407455126…e-28
1/(1 − μ_τ)	7.0535063901294…e27, inside (7.0535e27, 7.0536e27)
Orderings	q > 0, q < a_eff < a2, ηb < b − q, a_eff < 1/200: all true
74^199 ≥ 72^200	true
Control: +1 rank-48 child per W	μ_(1−b) − 1 = +2.04e-6

Argument: no defect found.

1. Internal node bound. Parts (i) and (ii) are at most c_NΛ·v·e^{1−a2} + 72c_L·v ≤ (c_NΛ + c_L)·v·e^τ. This uses e > s ≥ 73, so e ≥ 74 and e^τ ≥ 74^{199/200} ≥ 72.
2. Children. The children sum to v[K1·μ_τ·e^τ + c_L·s^b·μ_{1−b}·e^{1−b}]. Only 73⌊e/73⌋ ≤ e is used, and every child keeps the root's s.
3. Closing the induction. K1 = c_NΛ + c_L + K1μ_τ closes it with no monotonicity assumption. The recursion is well founded because 48⌊e/73⌋ < e.
4. Theorem. It needs s^b ≤ 73^b·h^{ηb} (both factors are at least 1), 1 − b + ηb ≤ 1 − q and Λ ≥ 1. All three hold.

Observations. None blocks the PR.

• The bill follows from COMPLEX.md's node premises (N)(iii) and (A) taken literally. The substantive open items are therefore exactly the PR's two open questions: the per-node invoice, and the ΓH_r-packet child-volume reading against Separate the tensor-frame child from factorwise repetition. #4375. This PR does not discharge either, and it says so.
• As noted, the h^η stop is not needed for this bill. COMPLEX.md's sentence "The strict (1−b) moment bounds the sum of leaf volumes times leaf work" uses strictness. The lemma shows that μ_{1−b} ≤ 1 already suffices, which is useful given the 1.8e-48 margin.
• As a cross-check, Σ r·H_r = 23,431,680 < 73W = 23,546,880, so μ_1 < 1, consistent with μ_τ < 1.

This is an author-independent check of the written argument and certified arithmetic. It is not maintainer acceptance and does not accept #4100. κ is unchanged.

Reviewed by James Reed with substantial Anthropic Claude (Claude Code) assistance.