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+# A linear observable parameter is profilable only in the space its own noise family scores, so the log-family half is refused on the math and the linear-Gaussian half stays open on a measurement the corpus cannot make (issue #572)
+
+**Status: Narrowed, not decided (2026-08-21).** ADR-0066/0099 profile a declared column's
+optimal multiplicative **scale** out analytically, and ADR-0108 (#562) profiles out an
+estimated **noise scale**. #572 proposed the third member — an additive **offset**, and the
+coupled `(scale, offset)` pair — by variable projection, and asked for an evaluation before
+anything was built.
+
+The evaluation settles two things and leaves the main question open:
+
+* **Settled, on the mathematics.** Variable projection needs the family's residual to be
+  `d - y` with `y` affine in the profiled coefficients. PyBNF's **log** families
+  (`lognormal`, `lnnormal`) score `log d - log(a*s + b)`, which is affine in `log a` only when
+  `b == 0` and in `b` never. The coupled form has no closed form there and must be **refused**,
+  not approximated. The pure-scale form survives as ADR-0066's geometric-mean ratio.
+* **Settled, on the interface.** A parameter that any noise source also reads is not a free
+  linear coefficient and must be refused **by resolved name**, formula sources included —
+  because the obvious rule misses one of the two real instances.
+* **Open.** Whether profiling is worth having on a **linear-scale** family — the case #572 is
+  actually about — is *not* answered here, and this ADR is explicit that the evaluation did not
+  answer it. The two measurements that would are named at the end.
+
+No code changes.
+
+## This ADR replaces a wrong conclusion
+
+The first version of this ADR concluded "not adopted", on two arguments that do not hold:
+
+1. *"The corpus has no instances."* PyBNF is general-purpose. Our 23-slug corpus is PEtab
+   imports, which skew `lognormal`; a conf written by hand uses `sos` or `chi_sq`, which is a
+   **linear** scale — the case where the construction works. `Smith_BMCSystBiol2013`, the one
+   `objective = sos` slug, carries nine profilable scales. If anything the sample under-counts
+   the feature's domain relative to PyBNF's native idiom. Instance counts in this corpus are
+   evidence about Benchmark-Models-PEtab, not about the merit of a general facility.
+2. *"The landscape measurement kills it."* It does not, because of where it was taken. The
+   collapse documented below happens **because** the residual is on a log scale — and log
+   families are exactly what argument (1) above already excludes. The evaluation landscape-tested
+   the method only in the regime where it cannot be used. See "What was not measured".
+
+Both corrections are recorded rather than quietly applied, because the numbers underneath them
+are sound and reusable; it was the inference from them that was wrong.
+
+## How it was measured
+
+Two tools, both in the corpus (`Grein-2026-benchmark-subset-I/tools/`): `linear_scope.py`
+classifies every declared free parameter reaching an observable (affine? pure multiplicative
+factor? what residual space? tied across how many series? also read by a noise source?), and
+`linear_profile.py` simulates once per point and then minimizes **PyBNF's own
+`evaluate_multiple`** over the linear parameters with θ held fixed.
+
+The profile is taken *numerically* on purpose: it is the exact conditional optimum whatever the
+family is, so the landscape question separates from the closed-form question. ADR-0108 pinned
+its own σ closed form against a numeric minimization of the reported objective for the same
+reason. **Neither tool runs a fit** — a `Borghans` point is one simulation plus a few hundred
+re-scorings of it.
+
+Three self-checks, all passed: the independently computed flat-line reference reproduces
+`-165.982113` at every point with spread `0` (the attractor already on record for that slug);
+the reference optimum and PEtab nominal point re-score to their recorded values within the
+σ-profiling and build difference; and the profiled score never exceeded the searched score at
+any of the **279** points measured, which it cannot, since the searched value is in the set
+being minimized over.
+
+## Finding 1 — the residual's space decides what is profilable, and that is general
+
+`Phi = [s, 1]`, `c* = (Phi^T W Phi)^-1 Phi^T W d` is an identity about a residual `d - Phi c`.
+A PyBNF noise family declares the space its residual lives in
+(`family.additive_on.ln_base`: `0` linear, `ln 10` for `lognormal`, `1` for `lnnormal`). So:
+
+| family | pure scale `a*s` | offset `s + b` | coupled `a*s + b` |
+|---|---|---|---|
+| linear (`gaussian`, `laplace`, `sos`) | least squares | least squares | least squares |
+| log (`lognormal`, `lnnormal`) | **geometric-mean ratio** (ADR-0066) | none | none |
+
+This is the load-bearing result and it has nothing to do with which models we happen to own.
+A correct implementation refuses a log-family observable unless the formula is **homogeneous**
+in the parameter, and applies the geometric-mean form rather than least squares when it is.
+Note "homogeneous" is a property of the whole formula: `Borghans`'s `scale` in
+`Z_state*scale + offset` is *not* a pure scale, because setting it to zero leaves `offset`.
+
+## Finding 2 — the double-binding refusal is needed, and the obvious rule is wrong
+
+#572 asks for `Fiedler_BMCSystBiol2016` to be refused by name and asks whether anything else
+hides the pattern. Both check out, and the answer to the second is **yes**:
+
+* **`Fiedler`** — confirmed exactly as described: all eight `s_pErk_*` / `s_pMek_*` tokens are
+  bound to `observableParameter1_*` **and** `noiseParameter1_*` in the same per-measurement
+  tables.
+* **`Raia_CancerResearch2011`** — the same defect by a different route, and absent from the
+  issue. Five `scaling_*` parameters are the observable scale *and* appear inside a
+  `prediction_formula` σ
+  (`CD274mRNA*scaling_CD274mRNA*sd_CD274mRNA_rel + sd_CD274mRNA_abs`), so moving the scale moves
+  that observable's σ.
+
+A rule that detects the `noiseParameter` placeholder catches `Fiedler` and **misses `Raia`**.
+The test has to be on the resolved parameter *names* every noise source reads, formula sources
+included. Thirteen of the corpus's 47 affine observable parameters are double-bound this way —
+so this is not a corner, it is a quarter of the population, and it is a design constraint on
+any implementation regardless of what is decided below.
+
+## Finding 3 — the one in-domain measurement is positive
+
+`Laske`'s `Int_nuc_off` is the corpus's only additive offset on a **linear-scale** family: a
+Gaussian offset tied across nine observables. It is exactly #572's case, minus the coupling
+with a scale, and it behaves the way a redundant nuisance parameter should:
+
+* it profiles to ≈ `57.13` at essentially **every** box draw, regardless of θ — i.e. it carries
+  no information about the dynamics, which is the whole argument for removing it from a search;
+* it compresses the spread over the box by ~20× (IQR `1.01e9 -> 5.01e7`, 20 draws);
+* it moves the PEtab nominal point by `0.02`.
+
+One parameter in one slug is thin evidence, and σ is searched rather than profiled there
+(`Laske` refuses `noise_profiling`, so the absolute magnitudes are inflated). But it is the only
+measurement taken inside the construction's actual domain, and it points **for** the feature,
+not against it.
+
+## Finding 4 — the open question is redundancy with ADR-0108, not soundness
+
+`Weber_BMC2015`, five pure Gaussian scales, 20 box draws, measured twice:
+
+| | searched IQR | profiled IQR | rank corr |
+|---|---:|---:|---:|
+| σ **searched** (as the conf ships) | 1.96e18 | 3.38e5 | 0.43 |
+| σ **profiled** (`noise_profiling = 1`) | 1.67e3 | **1.72e3** | **0.87** |
+
+With ADR-0108 switched on, profiling all five observable scales changes the spread of the
+landscape by **nothing** — marginally worse — and leaves the ranking almost intact. The
+thirteen-orders-of-magnitude improvement in the first row is real and it is the **σ profile's**:
+a wrong scale inflates residuals, and a profiled σ absorbs the inflation.
+
+`Brannmark_JBC2010` points the other way, and the reason matters: `noise_profiling = 1` is
+**refused** there (`IRS1_P` takes σ from a `PerMeasurementFormulaSigma`), so the scale profile
+has the field to itself and does compress the landscape (IQR `1.01e17 -> 3.94e12`, rank corr
+0.62).
+
+So the real question this evaluation raises is not "is the construction sound" — on a linear
+family it plainly is — but **"how much of it is a re-delivery of `noise_profiling`, and does
+that depend on whether `noise_profiling` is available for the slug?"** Two slugs, n = 20 and
+n = 25, disagreeing for an explicable reason, is not an answer.
+
+This also disposes of #572's proposed ordering rule. "Linear-first, then σ in closed form"
+describes a situation that does not arise on `Brannmark`, the slug where the linear profile
+matters most, because σ is not profilable there at all. A linear-profiling switch has to work
+stand-alone.
+
+## Finding 5 — the scope, read for what it is
+
+Across all 23 subset-I slugs: **47** affine observable-layer parameters in **9** slugs, against
+the 22 in 6 the issue tabulates.
+
+| slug | k | affine obs params | linear-lsq | log-geomean | none | why not |
+|---|---:|---:|---:|---:|---:|---|
+| `Schwen_PONE2015` | 30 | 10 | 0 | 1 | 9 | log family, not homogeneous |
+| `Smith_BMCSystBiol2013` | 25 | 9 | 9 | 0 | 0 | — |
+| `Fiedler_BMCSystBiol2016` | 22 | 8 | 0 | 0 | 8 | double-bound as a noise parameter |
+| `Raia_CancerResearch2011` | 39 | 5 | 0 | 0 | 5 | double-bound as a noise parameter |
+| `Weber_BMC2015` | 36 | 5 | 5 | 0 | 0 | — |
+| `Brannmark_JBC2010` | 22 | 4 | 4 | 0 | 0 | — |
+| `Borghans_BiophysChem1997` | 23 | 2 | 0 | 0 | 2 | log family, not homogeneous |
+| `Elowitz_Nature2000` | 21 | 2 | 0 | 0 | 2 | log family, not homogeneous |
+| `Laske_PLOSComputBiol2019` | 13 | 2 | 1 | 0 | 1 | log family, not homogeneous |
+
+Two slugs the issue does not list: **`Smith`** (nine tied scales, 36 % of `k = 25`, all
+closed-form — the largest opportunity in the corpus) and **`Raia`** (finding 2). It also
+undercounts `Brannmark` (4, not 2 — `k_IRP_1Step` and `k_IRSiP_DosR` are scales too) and cites
+`Schwen`'s 42 % without noting that 9 of its 10 have no closed form.
+
+**Read this table as a fact about Benchmark-Models-PEtab, not about the feature.** What it does
+support is a *priority* claim about PEtab-imported work, and a warning that #572's motivating
+examples — `Borghans`, `Elowitz`, `Schwen` — are all in the refused half.
+
+## What was not measured, and why that matters
+
+**The coupled `(scale, offset)` pair on a linear-scale family was never tested, because the
+corpus has no instance of it.** That is the case #572 calls "not just one more scalar", and
+this evaluation has nothing to say about it. Absence from a 23-problem sample is not evidence
+against it.
+
+**Item 3 (head-to-head fits at matched budget) was not run.** It cannot be, without first
+building the feature that #572 exists to gate.
+
+**`Smith`'s box landscape was not measured** — 71 s per simulation, and a 9-parameter inner
+solve on top. Only its nominal point was scored (`888141 -> 114222`, a large gain, but its
+PEtab nominal `sc_*` are far from optimal so this says little).
+
+### The log-family collapse, demoted to what it actually shows
+
+For completeness, since the numbers are sound and were the previous version's headline: taking
+the profile **numerically** on a log family — the obvious "no closed form? do it numerically
+then" fallback — destroys the landscape.
+
+`Borghans_BiophysChem1997`, 76 box draws that integrate, σ profiled by ADR-0108, profiling
+`scale` and `offset`: the searched objective spans `[-160.88, +339.69]` with IQR `129.5`; the
+profiled objective spans `[-167.15, -165.98]` with IQR **`0.081`**. Forty-six of 76 draws land
+**exactly** on the no-dynamics score `-165.982113`, 73 of 76 within `1.0` of it, and 39 of 76
+profile the scale below `1e-6` — the inner solve discards the dynamics, because on a log
+residual a wrong-shaped trajectory fits worse than no trajectory at all. Along a section from
+the reference optimum out to a box draw, the profiled objective is bit-identical at six
+consecutive points where the searched objective rises monotonically. The reference optimum
+(`-248.054331 -> -248.054446`) and the nominal point (`-198.207 -> -198.262`) do not move, which
+is the envelope theorem. `Elowitz_Nature2000` reproduces it: IQR `70.21 -> 7.77`, 27/57 exactly
+at the flat line, 19/57 discarding the dynamics.
+
+**What this is evidence for:** do not offer a numeric profile as a fallback where the closed
+form is absent. **What it is not evidence for:** the linear-Gaussian construction, which has no
+such mechanism — there the constant column is genuinely in the span, the projection can only
+reduce the residual, and `a = 0` is optimal only in the degenerate case that `s` is orthogonal
+to the centred data. `Borghans` is also a landscape already on record as pathological (a
+wrong-period oscillator scores ~25 NLL units *worse* than a flat line), which is a further
+reason not to generalize from it.
+
+## What would decide it
+
+Two measurements, in this order. Neither needs the full feature.
+
+1. **A synthetic linear-Gaussian fixture carrying a real coupled `(scale, offset)` pair.** The
+   corpus cannot supply one; building one is a few lines in the tutorial-fixture style, and it
+   puts the question in its own domain on a landscape nobody thinks is pathological. Run
+   `linear_profile.py` against it exactly as above. This is the measurement the evaluation is
+   missing, and it is cheap.
+2. **The ADR-0108 redundancy question, on `Smith` and `Brannmark`** — the two slugs where the
+   scale profile is *not* redundant with a profiled σ (`Smith` has no σ at all, `Brannmark`
+   refuses `noise_profiling`). Not on `Weber`, where finding 4 shows there is nothing left for
+   it to collect. This is where #572's item 3 belongs.
+
+If (1) shows the coupled pair behaves like `Int_nuc_off` did and (2) shows a benefit that
+`noise_profiling` is not already delivering, the feature is worth building, scoped to:
+
+* linear-scale families, plus the ADR-0066 geometric-mean form for a homogeneous parameter on a
+  log family; every other log-family parameter refused with the reason;
+* refusal by resolved parameter name for anything a noise source also reads (finding 2);
+* a stacked solve over the series a parameter is tied across — **9** of the corpus's 19
+  closed-form-profilable parameters are tied across 2 to 11 experiments (`Smith`'s `sc_GLUT_2B`
+  and `sc_PI3K` across 11 each), so the per-series solve #572 writes down is not the right
+  object for them;
+* `k` still counting a profiled linear parameter, and a Bayesian-sampler refusal, both by the
+  same arguments ADR-0108 already makes.
+
+## Consequences
+
+* No behaviour change. No new config key. `noise_profiling` (ADR-0108) and the ADR-0066
+  `normalization = ..., scale` chain are unaffected.
+* #572 stays open, narrowed to the linear-scale case, with the two measurements above as its
+  gate.
+* Two reusable tools land in the corpus with their gotchas written down:
+  `Grein-2026-benchmark-subset-I/tools/linear_scope.py` and `.../linear_profile.py`.
+
+## Prior art
+
+Variable projection (Golub & Pereyra 1973); hierarchical optimization for ODE models (Loos et
+al. 2018); pyPESTO/AMICI, which profile scale, offset and σ analytically by default. All three
+are stated over an **additive-Gaussian** residual. That assumption is load-bearing, it is the
+one finding 1 turns into a refusal, and ADR-0108's citation of the same prior art should be
+read with it.