Z0 Pi-Return Sampler

This generated report tests a bounded version of the fundamentalPi.md claim: a finite running process can produce pi operationally through closure-return statistics, without storing decimal digits of pi or assuming circles, radians, or primitive geometry.

This is the readable HTML companion to the Markdown source. The Markdown report is kept as an evidence file, but the wide numeric tables need a styled, scrollable view or they look ugly in the browser.

It is not a proof that Z0 uniquely encodes pi. The report exists to separate the runnable construction from the control ensembles.

Method

Z0 seed
-> circular XOR orbit, S_next = S XOR RotL1(S)
-> one tap tape per bit position
-> non-overlapping bit pairs mapped to four relation-walk channels
-> cyclic return windows of length 2n
-> pi_n = 1 / (n * P_return(2n))

The four bit-pair channels are a diagnostic harness for return statistics, not a claim that the substrate is a square lattice.

Z0 Forward Progression

Seed bits:101011110110110111000000110001011110101
Seed length: 39
Hamming weight: 22
XOR steps sampled: 4095

n	walk steps	returns	windows	P_return	pi_hat	error
1	2	19759	79833	0.247504	4.040336	+0.898743
2	4	11068	79833	0.138639	3.606478	+0.464885
4	8	5953	79833	0.074568	3.352637	+0.211045
5	10	4895	79833	0.061315	3.261818	+0.120226
8	16	3152	79833	0.039482	3.165966	+0.024373
16	32	1520	79833	0.019040	3.282607	+0.141014
32	64	804	79833	0.010071	3.102962	-0.038631
64	128	409	79833	0.005123	3.049855	-0.091738
80	160	319	79833	0.003996	3.128252	-0.013340
128	256	203	79833	0.002543	3.072391	-0.069202

Weyl Orientation Audit

The four canonical Z0 descriptions are tested as descriptions, not silently promoted to ontology. The table reports selected estimator values from the same circular-XOR pi-return sampler.

Orientation	bits	weight	n=8	n=32	n=80	n=128	mean abs error
Z0 forward	39	22	3.165966	3.102962	3.128252	3.072391	0.063050
Z0 reverse	39	22	3.185166	3.489205	3.108762	3.231582	0.156190
Z0 inverse	39	17	3.165966	3.102962	3.128252	3.072391	0.063050
Z0 inverse-reverse	39	17	3.185166	3.489205	3.108762	3.231582	0.156190

Shuffled And Same-Density Controls

The empirical p column is the fraction of controls whose pi estimate is as close or closer to mathematical pi than the Z0 forward estimate at the same scale, with one-count smoothing.

Same-bit-population shuffled Z0

n	Z0 pi_hat	control mean	stdev	min	max	controls as close	empirical p
8	3.165966	3.243308	0.069684	3.058267	3.428075	9/128	0.078
16	3.282607	3.175878	0.118586	2.892500	3.536189	100/128	0.783
32	3.102962	3.091205	0.188627	2.543100	3.674199	17/128	0.140
64	3.049855	3.030478	0.275910	2.489802	3.947439	33/128	0.264
80	3.128252	3.030499	0.309448	2.370338	4.023841	3/128	0.031
128	3.072391	3.083982	0.440633	2.219556	4.130433	15/128	0.124

Same-density random 39-bit seeds

n	Z0 pi_hat	control mean	stdev	min	max	controls as close	empirical p
8	3.165966	3.244940	0.074346	3.064842	3.441078	18/128	0.147
16	3.282607	3.186944	0.132388	2.799979	3.636707	91/128	0.713
32	3.102962	3.100118	0.165564	2.668215	3.563973	17/128	0.140
64	3.049855	3.035704	0.308095	2.422118	3.998047	24/128	0.194
80	3.128252	3.000295	0.293314	2.422118	3.838125	8/128	0.070
128	3.072391	3.061416	0.442868	2.058400	4.910987	12/128	0.101

CODATA Catalog Scan

Profiles scanned: 336 official pre-2019 CODATA rows.

The ranking below uses mean absolute error across the control scales. This is a diagnostic ranking only. It does not prove physical meaning.

Z0 rank by this crude score: 2.

rank	CODATA quantity	bits	weight	mean abs error	n=8	n=32	n=80	n=128
1	neutron Compton wavelength	37	24	0.046644	3.145307	3.139050	3.187668	3.233393
2	characteristic impedance of vacuum	39	22	0.063050	3.165966	3.102962	3.128252	3.072391
3	helion-proton mass ratio	39	17	0.064953	3.227401	3.106826	3.147989	3.282607
4	alpha particle-proton mass ratio	39	19	0.072742	3.051720	3.182119	3.005761	3.149976
5	Compton wavelength	35	20	0.095767	3.164532	2.907670	3.035805	3.009283
6	triton-proton mass ratio	39	20	0.105359	3.185166	3.282607	3.293441	3.214924
7	atomic mass unit-hartree relationship	35	22	0.119048	3.199580	3.100978	2.755577	3.198438
8	electron mass in u	39	26	0.135342	3.396571	3.122380	3.138090	3.214924
9	electron molar mass	39	26	0.135342	3.396571	3.122380	3.138090	3.214924
10	neutron-proton mass ratio	37	19	0.140949	3.288425	3.211457	3.242252	3.003609
11	electron mag. mom. to Bohr magneton ratio	47	20	0.146033	3.247671	3.043048	3.099517	2.713476
12	deuteron-electron mass ratio	39	18	0.147156	3.292354	3.102962	2.978843	2.676804

Result

The Z0 circular-XOR orbit does produce a sane finite pi-return sampler: it creates a running closure process whose return statistics enter the pi_n = 1 / (n * P_return(2n)) estimator without storing pi.

The control tables are the brake pedal. Any sufficiently unbiased four-channel walk can produce a pi-return estimator, so the important question is not whether pi appears at all. The important question is whether Z0 separates from shuffled seeds, same-density seeds, and other CODATA constants by drift, variance, anisotropy, finite-period behavior, or some stronger invariant.

Declared Limits

Bit-pair to walk-channel mapping is a diagnostic harness, not substrate.
Controls use deterministic RNG seeds for reproducibility.
Same-density controls preserve length and Hamming weight, but not every linear cellular-automaton invariant.
CODATA rows have different bit lengths, so catalog ranking is a lead generator, not a normalized final verdict.
A stronger report should add channel-map permutations, drift/aniso audits, exact period handling, and unit/precision transforms.