[unpaidmoon]

Change and mathematical scope

Author-checked identity for the default height-1 slab of the #4470 product/completion/split rule under #4436 nested widths. No new κ and no selected-frontier change. The selected conditional exponent remains 60403/20000000.

If target multiplicities lie in ranks [L, m], paid ranks satisfy rmax < L, and cap = n//m satisfies L <= cap < L^2, the merged source count is

S = W*A(cap) + P*(W - A(cap)),

where W is the total target multiplicity, A(cap) is the multiplicity of target ranks at most cap, and P is the total paid multiplicity. Equivalently S - W^2 = (W - A(cap))*(P - W). When cap >= m, S = W^2.

At the published v = 1 invoice this is 328391680 at cap 21 and 119421184 = 10928^2 for every cap from 145 through 440. The drop is exactly (77564-10928)*(10928-7792). Because P > W, S falls as cap crosses a target rank, so S_n is not monotone inside one height.

Substituting this count does not prove K/N >= 2944/1000 and does not prove K/N < 3069/1000. The certified bracket remains (2943/1000) N < K < (3070/1000) N, with H = 64215200000000 and N = 2^H. There are fewer than 63945^1024 height-1 ranks, and that block is disjoint from the top slab (hlo = 14276389504).

This is a scoped proof plus exact finite checks. It is not a new supplier, a physical word, or a multiplication theorem.

Task and overlap

Base ff3cc58e370d0f678e529246f8c73e8094e18fcf. Card selected-stock-envelope, priority revision 7, scope refined.

Handoff comment 19140126 asked for one uniform truncated-source estimate because S drops from 328391680 to 119421184 between ranks 3045 and 63944. On this base, research/selected-truncated-source-envelope already proves that uniform census and K/N < 3070/1000. This package does not restate that bound. It proves the height-1 collapse, which that census does not state, and the integer fact that the collapse misses both the next lower thousandth (2944/1000, from #5026's Xlo) and the next upper thousandth (3069/1000, from the recomputed moment charge X = 197115103572543).

Fork draft: unpaidmoon#86, head d7271b4c28911378680c8c1658678ddb4d9b17ae.

Nearest work: #5026 / research/selected-complex-width-envelope and research/selected-truncated-source-envelope. Different from unpaidmoon's stock notes on bootstrap floors, log_q parents, and joint leaf reserves.

Validation

Python 3.12, standard library only:

python3 -I -W error -B research/height1-truncated-source/verify.py
python3 -I -O -W error -B research/height1-truncated-source/verify.py
python3 -B scripts/research_package.py --package research/height1-truncated-source

Registered normal and optimized replay both print PASS_HEIGHT1_TRUNCATED_SOURCE. Metadata replay prints PASS_REGISTERED_RESEARCH_PACKAGE research/height1-truncated-source outcome=partial.

The verifier authenticates flow.py, the selected invoice, the #5026 receipt and the uniform-moment receipt. It checks 40 small caps, 4 literal replays, all 420 selected height-1 caps, the six published calibration rows, and four rejection controls (paid rank at least L, truncated term omitted, extension to n = 63945, and a square lower bound where P < W). The endpoint gaps are exact integer bit-length comparisons. N and the v = 1024 histogram are not constructed.

Finite checks corroborate the written identity. They are not an enumeration of every rank at v = 1024. Not run: an all-height base reduction, exact B_N, physical admission, or the selected cold replay.

Credit

Sagar Thakrar / thakrarsagar (#5026 and handoff 19140126); JoyboyBrian (research/selected-truncated-source-envelope); James Reed / jwr404 (#4972); zxv (#4436); Jacob Alan Sussman (Theorem 1G); Cinna Davis / 1337isnot1337 and zxv (#4440/#4470); Mark / Matarener (#4612); Ryanson Jonathan (#4476); mihir-s-05 (#4507). Retained source notices stay intact.

Prepared by unpaidmoon with substantial Cursor Grok assistance.

Open next questions

The package question is whether a uniform eligible-boundary bound can reduce log2(T1/L) by at least three bits on every rank through N, enough to replace 3070/1000 by 3069/1000. The height-1 window cannot do that. The first check is any proposed uniform factor against the v = 1 count at n = 63944. The blocker is a bound that remains valid when a source rank sits just above n/M.