Phi-Genesis
Selection Protocol and Search Audit
Research question
Does any tested operator family on Sierpiński-gasket graph approximations select the target indices (T={1,12,17,40}) more often than expected by chance, after accounting for the number of configurations tried?
The target indices were identified from a mass fit; therefore a match is not independent evidence by itself. The selection search uses a look-elsewhere correction over the configurations within each family, null ensembles, target weights based on null base rates, and a predeclared decision threshold of (p\le0.01) in both null ensembles.
Preregistered results
| Protocol | Family / rule | Configurations | Result | Power status |
|---|---|---|---|---|
| v1.0 | F1 + F2 + F3; (k=12), window 60 | 75 | No evidence; p = 0.922 / 0.968 | Threshold unreachable; negative is uninformative about whether the family contains a rule. |
| v1.1 | Same family; (k=6), window 60 | 75 | No evidence; p = 0.966 / 0.997 | Threshold reachable; informative negative for these configurations. |
| v1.2.1 | SG × S¹ continuum circle, two spin structures; (k=8), window 60 | 10 | No evidence; p = 0.953 / 0.926 | Threshold reachable; informative negative for these configurations. |
The p-value pairs are ordered as documented in the source records (N-pot / N-edge for v1.0 and v1.1; N-circ / N-circpot for v1.2.1). Each pair fails the requirement that both nulls meet the threshold.
The cumulative ledger records 193 configuration evaluations from the documented main families, plus an earlier Sierpiński-carpet search of about 128 level-3 configurations and an unrecorded level-4 sweep. The honest cumulative statement is therefore at least about 321, not an exact exhaustive count.
Other relevant checks
- Generation twists: all 32 (2^5) configurations at level 5 were tested; maximum target overlap was 1 of 4, and index 40 was never selected.
- Base-rate audit: under random on-site potentials, the ratio-based rule selected index 1 about 95% of the time, while index 40 appeared about 3.4% of the time. Hits on the four indices are therefore not equally informative.
- Carpet branch: the early 4/4 match was reassessed against a null model; the record attributes it to a look-elsewhere effect and marks that branch abandoned.
- v1.0 power correction: even a configuration selecting all four targets would not have reached the required p-value. This was documented as a limitation of that protocol rather than treated as evidence against all possible mechanisms.
What the result means
The tested families did not satisfy the preregistered selection criterion. This is a bounded negative result: it does not prove that no operator family could select the indices. It does mean the current mass formula cannot be described as a spectral prediction.
Any new family must have an a-priori rationale and a new preregistration, with null-only power design completed before target evaluation. Re-running the same search on the same four targets would increase the look-elsewhere burden rather than resolve it.
Audit records
- Binding preregistration v1.2.1
- Cumulative protocol ledger
- Methodological write-up
- Protocol data and code
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