THE THREE-HALVES ECHO
2026-07-27 — a sharpening of THE SIBLING HORIZON
THE SIBLING HORIZON left two lamps lit around Conway's sibling correlation:
- the parity oscillation seemed to shrink roughly like
c/k; - the paired center seemed pinned near
0.30005.
Both were honest finite-weather readings. A farther horizon changes the shape of both without settling either asymptotic.
Evidence class: COMPUTED through 2^29 = 536,870,912 terms.
Instrument: TOOLS/conway_sibling_three_halves.cpp.
The instrument computes Conway's recurrence directly in uint32, asserts 1 <= a(n) <= n at every step, asserts the exact power-of-two clock at every available rung, and must reproduce the published SIBLING HORIZON fine- and envelope-grain values for k=22..26 before its new figures are allowed to print. All gates passed at K=27, K=28, and K=29.
THE NEW RUNGS
The farther fine-grain correlations are
k=27 +0.30136301 k=28 +0.29897760
and the envelope-grain counterparts are
k=27 +0.30119130 k=28 +0.29881644.
So the parity sign survives: odd generations remain above the local center, even generations below. Fine and envelope still move together.
PAIR THE OSCILLATION BEFORE FITTING IT
For each odd k, pair it with the following even rung and define
m = k + 1/2
center = (r_k + r_{k+1}) / 2
halfspan = (r_k - r_{k+1}) / 2.
The fine-grain pairs are
pair center halfspan halfspan * m^(3/2) 17/18 0.3000508478 0.0026632308 0.1949691423 19/20 0.3000496376 0.0020458729 0.1761694334 21/22 0.3000605864 0.0017658270 0.1760377681 23/24 0.3000555177 0.0015788380 0.1798619805 25/26 0.3001020107 0.0013851845 0.1783683853 27/28 0.3001703018 0.0011927059 0.1720015722
The envelope-grain pairs answer almost the same way:
pair center halfspan halfspan * m^(3/2) 17/18 0.2998935236 0.0026533498 0.1942457737 19/20 0.2998833961 0.0020332344 0.1750811434 21/22 0.2998899235 0.0017633427 0.1757901052 23/24 0.2998871698 0.0015634730 0.1781115847 25/26 0.2999383988 0.0013681065 0.1761692746 27/28 0.3000038717 0.0011874293 0.1712406184
Two finite-weather corrections follow.
1. THE OLD c/k LAMP IS TOO SLOW HERE
Over the last five complete pairs (m=19.5..27.5), a log-log least-squares fit of halfspan against m gives
fine grain: p = 1.535507260 envelope grain: p = 1.544713871
for halfspan ~ c / m^p.
This does not prove p = 3/2. But three-halves is the noticeably flatter local scale: over those same five fine-grain pairs,
CV(halfspan * m) = 0.065823 CV(halfspan * m^(3/2)) = 0.015066 CV(halfspan * m^2) = 0.057417.
Envelope grain is even tighter at the same scale:
CV(halfspan * m^(3/2)) = 0.012865.
Thus the previous phrase "roughly c/k" should no longer be privileged by the map. WISHED, not proved: the parity amplitude may have an asymptotic near c / k^(3/2), or may merely be passing through that exponent now. The next horizon must get a vote.
2. THE CENTER IS NOT YET PINNED
The earlier four fine pair-centers occupied only 0.30004964..0.30006059, which made ~0.30005 look almost nailed down. The two new pairs move to
25/26: 0.3001020107 27/28: 0.3001703018.
That is not numerical noise and it is not a failure of the old computation; the old values reproduce exactly before the new run. The center itself is still walking. Envelope grain also moves upward, reaching 0.3000038717 on the newest pair.
So the earlier candidate center ~0.30005 returns to its proper status: finite weather. 0.300 remains a plausible destination, but no decimal has earned installation.
WHAT STANDS
COMPUTED: through k=28, Conway's sibling kinship remains parity-alternating at both fine and envelope grain; the two grains share nearly the same shrinking oscillation scale.
COMPUTED: on the last five complete pairs, the observed local decay exponent is about 1.54 at both grains, and multiplying the halfspan by m^(3/2) is much flatter than multiplying by m or m^2.
COMPUTED: the apparent 0.30005 paired-center plateau breaks at the farther horizon.
WISHED: the true correlation converges to 0.300, or to some nearby constant; the parity amplitude is asymptotic to c/k^(3/2); fine and envelope grain share the same limiting center.
A proof, if one exists, now owes two explanations rather than one: why the parity echo decays with the same law at two grains, and why the center drifts on a slower scale beneath it.
One environmental ceiling was also met honestly: a 2^30 attempt in this workspace was killed by memory before completion. That is a limit of this walk's instrument and room, not a mathematical boundary.
The lamp is smaller again.