ESTIMATOR BIAS

The product of this skill is a gap: between what a fit returns and what is already provable without one.

The tribonacci constant has a nearest integer sequence that decays. The decay base is 0.7373527057603276, and it is not a fitted number — it is the modulus of the non-dominant conjugate pair, so deflating the minimal polynomial proves it before anybody regresses anything. The skill then estimates the same base from the data, by regressing ln|round(lambda**n) - lambda**n| on n, and reports the difference:

Three things follow, and they are the reason the skill exists.

• the fit is biased, and the bias is not rounding. Sixty more working digits move the n <= 400 answer by exactly zero. The gap belongs to the sequence. The mechanism is that the Pisot error is 2|alpha|**n|cos(n theta), so the cosine is a factor of the signal and the log-linear fit leaves a residual RMS of 0.74 nats or more at every range. A straight line in log space is the wrong model.
• the fit cannot see its own bias. The gap at n <= 40 is 1.26 standard errors of the fit's own slope, and at every longer range it is inside the error bar. A reader holding only the fit cannot detect any of this. Only the provable rate exposes it.
• "biased" is a property of the range, not of the estimator. At n <= 1000 the bias is 38 times below the 1e-03 threshold the verdict uses, and scanning the end of the range from 3 to 400 finds 44 of the 398 ranges where the fitted rate falls below the exact one. The sign is a property of where the fit stopped.

The instrument also measures one thing that refutes itself, and reports it as such. round(lambda**n) equals the trace of the three roots only from n = 4; at n = 1 and n = 3 the conjugate term is still above one half, so those two terms are not Pisot errors at all. About 63 percent of the short-range bias came from carrying them. See references/bias.md.

Run the gate

From the repository root:

python3 skills/elohim-harness/scripts/harness_run.py --skill-dir skills/estimator-bias

Exit 0 when all five gates hold, 1 when one fails, 2 when the skill or its instrument cannot be located. To run the instrument or either script on its own, cd nowhere: each resolves its own paths relative to its own file.

The instrument also honours an override, derived from the directory name the way the contract specifies:

ELOHIM_ESTIMATOR_BIAS_SCRIPT=/path/to/estimator_bias.py \
  python3 skills/elohim-harness/scripts/harness_run.py --skill-dir skills/estimator-bias

An override that points at a missing file is an error, exit 2, never a silent fallback to the bundled copy.

The five gates

pin — the ledger pins this instrument's sha256 and byte count. Holding the instrument beside its ledger fixes only which code runs; the pin is what makes an edit to that code visible instead of silent. Drift prints both hashes and can never resolve to PASS. Expect DRIFT the first time after any change to the instrument: that is the gate working.

ledger — each recorded fact is re-measured against a fresh shard and reported with its residual. A fact whose path has vanished reports missing rather than quietly passing. These fifteen facts started empty: nothing was written into the ledger that the instrument had not measured in a run of its own, and each one was re-derived by discover.py through a different route before it was promoted.

traps — seven traps in scripts/check_traps.py, each re-deriving its property from the minimal polynomial with code of its own. The sharpest one never forms lambda**n at all: it rebuilds every error term as 2 Re(alpha**n) from a conjugate pair derived from Vieta's sum and product, so a sequence the instrument invented would not survive it. The regression trap is exact — each natural logarithm is taken in Decimal, scaled by 10**60 and truncated, so every sum in the fit is an integer sum and the slope carries no rounding of its own. That is why the two slopes can be compared at 1e-12 and still mean something.

hygiene — scripts/check_hygiene.py lints these sources for network imports and for identifiers a shell filter can rewrite. A gate that reaches the network is not reproducible, and a gate whose source can be silently edited is not a gate. The natural logarithm here is spelled through a one-argument wrapper for the same reason.

claim binding — elohim-harness/scripts/claim_binding.py pulls every number out of every claim sentence and requires each one to be a rendering of a value something pins, part of a formula or a scan window, or declared in claim_binding_exemptions.json with a written reason. The four gates above measure the ledger; this one reads the sentence beside the number, which is where the worst defect this project shipped lived: prose asserting the opposite of its own pins while every gate reported green, and found by a human reading the output. A number nothing pins is therefore a failure and not a warning.

Promote on purpose

--discover re-derives each claim from scratch, with a different root finder, a different conjugate pair, a different logarithm and the other form of the least-squares identity, then writes it to backlog.json with a contrast: the distance the claim travels from the nearest other reading of its own sentence. The two routes agree to 1.1e-16 absolute on every entry, and the ledger pins with a 1e-09 tolerance rather than digit for digit, because agreement to the last double is not evidence of anything.

--promote ID:PATH[:TOL] then pins one measurement as a ledger fact, and from then on the harness re-measures it on every run.

Read every backlog entry before promoting it. Six of the twenty-one are left unpromoted on purpose, each with its reason written beside it in backlog.json: one restates Vieta's theorem, one pins a floating-point equality, one evaluates a formula on its own inputs, one is an extreme over a scan window, and two only re-express facts the ledger already pins. An unpromoted measurement is not a defect, and a measurement nobody can explain has no business in the ledger.

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