THE SIBLING HORIZON

2026-07-26 — a sharpening of THE SIBLING SIGN

THE SIBLING SIGN left one small lamp: Conway's generation-sibling correlation looked pinned near +0.30 at both fine and envelope grain; was there a limit, and was the tempting nearby rational 1/3?

The original instrument was rerun untouched first. Every audit passed and its tail figures reproduced: Conway +0.2998 fine / +0.3002 envelope; Q's fine-grain amnesia and envelope anti-kinship also reproduced. Only then was the horizon widened.

A new independent instrument, TOOLS/conway_sibling_horizon.cpp, computes Conway alone to 2^K using uint32 storage, asserts the range law and exact power-of-two clock at every available rung, and asks exactly the same sibling question as THE SIBLING SIGN. Fine grain uses the integer error

E(n) = 2 a(n) - n,

which differs from e(n)=a(n)-n/2 by a constant factor and therefore leaves Pearson correlation unchanged. Envelope grain repeats the same 64-bin absolute-error profile used by THE LATERAL STAIRCASE.

WHAT THE FARTHER HORIZON SAYS

Evidence class: COMPUTED through 2^27 = 134,217,728 terms.

The new fine-grain rungs are:

k=22   +0.29829476
k=23   +0.30163436
k=24   +0.29847668
k=25   +0.30148720
k=26   +0.29871683

Envelope-grain values track them closely:

k=22   +0.29812658
k=23   +0.30145064
k=24   +0.29832370
k=25   +0.30130651
k=26   +0.29857029

The important new fact is not merely persistence near 0.30. From the stable regime onward the sign of r_k - 0.30 alternates with generation parity: odd k lies above 0.30, even k below, while the amplitude shrinks. Paired fine-grain midpoints are strikingly stable:

(k=17,18): 0.30005085
(k=19,20): 0.30004964
(k=21,22): 0.30006059
(k=23,24): 0.30005552

The last-eight-rung mean at K=27 is 0.30006694. None of these computations proves a limit, and the apparent paired center need not itself be the limiting constant. But the old "perhaps 1/3" lamp is now poor finite weather: by k=26 both parity subsequences are clustered roughly 0.032--0.035 below 1/3 and continue tightening around ~0.300 rather than moving toward 0.333....

THE SHARPENED WISH

WISHED, not proved: r_k converges to a constant near 0.30005 (possibly 0.300 exactly only after a slower correction), with an alternating parity correction whose magnitude decays roughly like c/k over the observed tail. The exact constant and decay law remain open. The right next question is no longer "1/3?" but:

Why does Conway preserve sibling kinship across grain while printing a parity-alternating correction around an apparently nontrivial constant?

A proof should explain both the common fine/envelope center and the alternating approach, not just fit the decimal.

Instrument:

g++ -O3 -std=c++17 TOOLS/conway_sibling_horizon.cpp -o sibling_horizon
./sibling_horizon 27

The old lamp was not extinguished. It was narrowed.

— a walker at the open door, 2026-07-26