[aliromman]

This scoped proof raises the selected nested compiler's lower width coefficient from 2.943 to K/N > 3.054. Together with the inherited upper proof from PR #5109, the interval is 3.054 < K/N < 3.0694, with endpoint ratio below 1.0051. Here H = 64215200000000, N = 2^H and K = ceil(log2 B_N)/H. This tightens the estimate of the unchanged permitted width; it does not reduce the width or improve the complete kappa calculation. Selected kappa remains 60403/20000000.

Proof, exact scope and credit, deterministic receipt, source pins, and independent counter are at reviewed head cf7c21187a9d2d0ea689902b8727617affe70cde. Base and every consumed source pin: ff3cc58e370d0f678e529246f8c73e8094e18fcf.

The new all-height identity uses eligible ordered mixed products F_d(c), target count W and paid count Q:

S_n = W Q^g + (W-Q) sum_(d=0..g-1) F_d(c)
S_n - W^(g+1) = (Q-W) sum_(d=0..g-1) (Q^d W^(g-d)-F_d(c)).

For the default-height flow with every paid rank below every target rank and Q >= W, this proves S_n >= W^(g+1). It also gives the exact equality threshold and within-height monotonicity. Every missing-parent source remains charged. Applying the floor to the actual power-1024 completion/split rule yields the new lower coefficient without constructing N or its histogram. The upper coefficient remains a cited premise and is not independently reproved here.

From the repository root, Python 3.12 and standard library only:

python3 -I -W error -B research/separated-source-telescope/verify.py
python3 -I -O -W error -B research/separated-source-telescope/verify.py
python3 -B scripts/research_package.py --package research/separated-source-telescope

Expected direct result: PASS_SEPARATED_SOURCE_TELESCOPE K/N>3.054 no_new_kappa. Registered normal/optimized checks pass on Python 3.12.13; optimized 3.14.7 also passes. Evidence includes 864 small merged flows, 12 complete literal source/exit comparisons, 708 within-height checks, 16 distinct selected v=1 boundary cases and six discriminating controls. Committed-byte and official 11-file export checks pass; a fresh Git-free extraction passes under optimized Python and rejects source-byte and receipt corruption. The optional source-history preflight refused the shallow checkout; no full-history pass is claimed. All source pins use the initial contribution commit's direct parent, and the export includes every declared live input.

This is a written uniform count proof with finite corroboration. It does not admit physical stock, routing, owned storage, native operators, precision/tapes, whole-job adapters, outer recovery or the full multiplication theorem. No selected cold replay or formal proof-kernel run was performed. Hosted CI and external review remain separate.

The advance claim follows the source-count handoff at priority 7. Final overlap covered 363 open PRs, 4,582 Discussion records and all 7,091 tracked README/TASK notes. Canonical main still matched the base; unpublished work and arbitrary diffs are outside that coverage.

Credit @JoyboyBrian for the census in PR #5067, @thakrarsagar for the bracket and handoff in PR #5026, @petalas for the inherited upper proof, and @unpaidmoon for the height-one identity in PR #5273. The all-height result extends those premises. The flow from @zxv in PR #4436 retains Jacob Alan Sussman's lockstep credit; the selected invoice from @1337isnot1337 and @zxv in PR #4470 retains its full native/compiler lineage. Source notices are unchanged. Prepared by @aliromman with substantial OpenAI Codex assistance in proof, code, verification and review; Apache-2.0. No independent human review is claimed.