How a spectral clue became a reproducible research story.

The working idea was to ask whether operator spectra on a self-similar graph might pick out the integer set {1, 12, 17, 40}, which had been associated with four heavy-fermion masses through a one-scale relation, m = A·n.

That is a legitimate exploratory question. But because the indices were identified with the masses already in view, any later match needed to be tested against the number of operator families, configurations, rules, and targets explored.

Historical note: this page does not invent a precise start date where the committed record does not establish one.

An earlier Sierpiński-carpet oscillator search reported a 4-of-4 match. The later null analysis found the four hits statistically independent; three or more hits occurred in roughly 11% of random trials, while 4-of-4 occurred in about 0.1% under that diagnostic. Searching many configurations changes how a striking result should be interpreted.

The carpet branch was abandoned as evidence for a selection rule. Its approximate level-3 count (about 128 configurations) and an unrecorded level-4 sweep are disclosed in the ledger; they are not silently folded into the 193 main-ledger evaluations, so the all-in count remains a lower bound.

Lesson: a result that looks rare for one preselected configuration can be much less persuasive after many configurations have been tried.

Read the search ledger ↗

Version 1.0 tested 75 configurations using the rule with k = 12 selected positions and a window of 60. It returned no evidence (reported p-values 0.922 and 0.968 in the two null ensembles).

A crucial post-hoc power audit then showed that even a configuration hitting all four targets could not pass the required p ≤ 0.01 threshold: its p-values would still be about 0.136 and 0.216. So the negative outcome was correct but uninformative about whether that family contained a selection mechanism.

This was a methodological failure, not a reason to hide the result. It motivated a null-only power check before subsequent evaluations.

On the actual gasket graph, random on-site potential tests found index 1 in the ratio-based gap-selected set about 95% of the time, compared with about 37% for index 12, 22% for index 17, and 3.4% for index 40 in the reported audit.

That means a hit on index 1 is close to a structural default of the selection rule, not a distinctive fingerprint of the proposed physical mechanism. Index 40 would be much more discriminating—but the searches did not establish a rule that selects the full target set.

Inspect the base-rate test ↗

Version 1.1 changed the rule to k = 6, chosen using null-only draws so the threshold could be reached. It again found no evidence (p = 0.966 and 0.997). No configuration selected more than two of the four targets, and index 40 was selected by none of the 75 configurations.

Unlike v1.0, this was an informative negative for the tested 75-configuration family: the test had enough power to pass under some outcomes, but the observed outcomes did not.

Read the v1.1 protocol record ↗

The first 60 nonzero eigenvalue positions showed the same degenerate-cluster structure at levels 4, 5, and 6. A rule selecting cluster starts would pick two targets ({1, 12}), but 27 of 59 positions were cluster starts. That is not an unusual concentration.

The check is explicitly labelled exploratory and is counted in the ledger. It was not retroactively presented as a preregistered success.

The v1.2 document was a design-only draft. The binding v1.2.1 amendment specified a Sierpiński gasket × continuum circle family, two spin structures, five radius scalings, fixed seeds, null pools, and a null-only choice of k = 8. Only after that design was recorded were the 10 configurations evaluated.

Result: no evidence (p = 0.953 and 0.926). The best configuration selected only index 12. The test could have reached the threshold, so this is an informative negative for those configurations.

The continuum-circle family is the counterpart of an earlier lattice-circle family, not an independent line of evidence. The ledger says so explicitly.

Read the binding v1.2.1 protocol ↗

Recent commits made the Gate A decimation diagnostic distinguish mean, median, machine-exact matches, and near misses. The exact-subset mean was clarified as an arithmetic mean, and the machine-exact assertion was tightened to a maximum residual below 10⁻¹¹. The inertia module’s documentation also records that numerical counts can silently be wrong when a shift lies within about 10⁻⁹ of eigenvalues 3 and 4.

The estimator write-up also states that regression-window spread is intrinsic to the chosen three-period windows, while the window-free chord estimator reproduces the theoretical exponent to better than 0.1% at level 9 for N ≥ 1000. These are numerical results, not machine-checked theorems.

Inspect the commit history ↗

The main protocol ledger records 193 configuration evaluations across v1.0, v1.1, an exploratory rule check, and v1.2.1. Adding the earlier carpet branch yields at least about 321 attempts, with the true count unknown because the level-4 carpet sweep was not recorded.

The mass relation m = A·n still matches four masses with one fitted scale and four integer indices selected with the masses already in view. No independent spectral rule has derived those indices. The top-quark index remains inconsistent across documented routes (1654, 1664, or 1680); the repository does not choose one as correct.

The project is therefore closed in its current form as a search programme, not as a proof that no future mechanism can exist. Reopening it responsibly requires an operator family with an a-priori rationale, a new preregistered protocol, and a null-only power check.

Read the full current status ↗