The map of ideas

The lab in one picture: the proton's unknown structure (left) is probed by experiments, turned into predictions and a misfit number (top), inferred by Bayesian machinery and its rivals (centre) — and nothing is believed until it survives the trust cluster (bottom-left). Every node links to its entry below.

Parton

Anything inside the proton: quarks, antiquarks, gluons. At collider energies the proton is not three quarks but a seething many-body state — quantum fluctuations constantly split gluons into quark–antiquark pairs and back, so the "sea" is as real as the valence quarks. What actually collides at the LHC is one parton from each proton; the rest of both protons is spectator debris. "Parton" predates QCD (Feynman coined it in 1969 precisely to stay agnostic about what the constituents were). IN THIS LAB — the species we fit: up/down valence, the sea, strangeness, gluon — eight basis shapes in the MSHT20 model, four combinations in the toy model.

Parton Distribution Function (PDF)

The proton's ingredient list: f(x, Q) is the number density of a parton species carrying momentum fraction x when the proton is probed with resolution Q. Two deep facts make the whole field: (1) PDFs are universal — the same functions appear in every process, so fitting them from one experiment predicts another (factorization); (2) their Q-dependence is fully calculable (DGLAP evolution) — only the x-shape at one starting scale Q₀ is unknown. That x-shape at Q₀ is the entire object of this lab: everything we fit lives at Q₀ = 1.65 GeV, and evolution does the rest. IN THIS LAB — every result card is ultimately a statement about x·f(x, Q₀ = 1.65 GeV) in the evolution basis.

Momentum fraction x

The field's iconic variable: the share of the proton's momentum a parton carries, 0 < x < 1. The proton's anatomy by region: valence quarks peak around x ~ 0.1–0.3 (they carry the identity "two ups, one down"); the gluon explodes toward small x (at x = 10⁻³ there are hundreds of gluons per unit rapidity); the x → 1 tail is exponentially rare and hard to measure. Data constrains PDFs only inside the x-window experiments actually probed — our SLAC+BCDMS sets cover roughly 0.01 < x < 0.75 — outside it, any fit is extrapolating on shape assumptions alone. IN THIS LAB — the grey bands in the PPDF-2 plots mark exactly where our data goes blind (x < 0.01).

Evolution basis (Σ, g, V, V3, T3, T8…)

Linear combinations of quark flavours chosen so each evolves independently under QCD: the singlet Σ (sum of all quarks and antiquarks, which mixes with the gluon), the total valence V (= q − q̄ summed), and non-singlets like V3 = u⁻ − d⁻ and T3 = u⁺ − d⁺ that evolve alone. Physics lives in specific members: T3 is what distinguishes a proton from a neutron/deuteron target — mask it and proton/deuteron data become indistinguishable; V8 carries the strange-valence difference. NNPDF's FK tables speak this basis, so every model must translate into it. IN THIS LAB — the flavour-masking disaster of PPDF-8 was precisely a zeroed T3; the MSHT20 wrapper's final act is rotating MSHT's u,d,s,c basis into this one.

Sum rules

Exact integrals any proton PDF must satisfy, no matter the parametrization: the momentum sum (all x·f integrate to 1 — the proton's momentum is all accounted for), valence counting (∫u_V = 2, ∫d_V = 1 — a proton is two ups and one down at heart), and zero net strangeness (∫(s − s̄) = 0). Imposed either analytically — solving for normalization parameters, as MSHT20 does (that's why 60 raw parameters become 52 free) — or as penalty terms. They're also a free correctness test: a model that can't hit them exactly is mis-assembled. IN THIS LAB — the MSHT20 port carries four analytic sum rules; our smoke test confirmed momentum = 1.000025 (numerical) and valence counting exact. The grid model ran with none — a PPDF-9 caveat.

Positivity

Physical cross-sections can't be negative, which constrains PDFs (in suitable schemes) even where no data exists — a rare piece of free information at extreme x. NNPDF and Colibri impose it through dedicated pseudo-observables (like the F2 of a fictitious up-only target) added to the loss with a penalty weight. Subtle in the MS̄ scheme: individual PDFs may dip slightly negative; it's observables that must not. IN THIS LAB — present in every runcard (NNPDF_POS_2P24GEV_F2U), penalty currently off for comparability with the paper's example; revisit when we widen datasets.

LHAPDF

The community-standard grid format and access library — the "PDF file format of particle physics". Every prediction code (MadGraph, POWHEG, experimental frameworks) reads PDFs through it; every fitting group publishes to it. A "PDF set" ships a central member plus error members (replicas or eigenvectors), so downstream users propagate PDF uncertainty by looping over members. IN THIS LAB — truth PDFs are read through LHAPDF (PPDF-2); every Colibri fit exports 100 posterior members to it, making our Bayesian fits consumable by any collider tool.

Deep-Inelastic Scattering (DIS)

Fire a lepton (electron, muon, neutrino) at a proton and measure how it scatters: the exchanged photon/W/Z acts as a microscope with resolution Q, and the scattering rate at (x, Q) reads off the parton densities almost directly. It's the cleanest proton probe — no second proton to confuse the interpretation — and historically where quarks were discovered (SLAC, 1969). Theory prediction is linear in the PDF, which keeps fits fast and the statistics benign. Structure functions like F2 package what's measured. IN THIS LAB — our current 648 points are DIS F2 from SLAC and BCDMS on proton and deuteron targets; the p–d pairing is what gives (weak) flavour separation.

Global fit

Fitting all PDF flavours simultaneously to every usable experiment — DIS, Drell-Yan, W/Z production, jets, top pairs — because each process reads a different flavour combination in a different x-range, and only the union over-constrains the proton. NNPDF4.0: ~4,600 points, 80+ datasets, 40 years of experiments, χ²/N ≈ 1.16 globally. The dark side: datasets can tension against each other, and a subset can behave very differently from the whole. IN THIS LAB — we deliberately fit a small DIS subset first; the χ²/N ≈ 5.6 floor both our models hit there, where the global fit reports ≈1, is our sharpest open question (PPDF-9).

Factorization

The theorem licensing everything here: to controllable accuracy, hadronic cross-sections split as (short-distance physics, calculable in perturbation theory) ⊗ (long-distance PDFs, universal). The scale Q separating the two is arbitrary — physical answers don't depend on it order by order — and PDFs absorb the long-distance part identically in every process. Universality is the payoff: DIS-fitted PDFs predict LHC Higgs production. It's proven rigorously only for some processes and assumed (with evidence) for others. IN THIS LAB — factorization is why our DIS-only posteriors are even meaningful for anything else, and why FK tables can precompute the perturbative half once.

FK table (fast kernel)

Precomputed interpolation grids that collapse an entire QCD calculation into one linear map: prediction = FK ⊗ PDF(Q₀). The FK table absorbs both the hard matrix element and DGLAP evolution from Q₀ to the data's scale, so evaluating a new candidate proton costs a matrix product — microseconds — instead of a fresh QCD computation. Built by NNPDF's PineAPPL → EKO pipeline under pinned "theory IDs" (ours: 40000000 = NNLO, Q₀ = 1.65 GeV conventions). Linear for DIS; quadratic (two PDFs) for hadronic data. IN THIS LAB — the speed that lets a laptop run Bayesian fits at all; also the scene of the PPDF-8 lesson: flavour masks zero out FK columns silently.

t0 method

A fix for the d'Agostini bias: experiments quote normalization (multiplicative) uncertainties, and if you build the covariance matrix from the measured values, fluctuations downward get artificially small errors — the fit learns to prefer low predictions everywhere. The t0 prescription computes multiplicative uncertainties from a fixed reference PDF's predictions instead, decoupling the bias; iterate the reference if needed. Standard across the field since 2009 (arXiv:0912.2276). IN THIS LAB — use_t0_covmat: True with NNPDF4.0 as reference in every single runcard; identical t0 across models is part of what makes our evidence comparisons apples-to-apples.

χ²/N

Misfit per data point, weighted by the full experimental covariance — correlated systematics and all. Calibration: ≈1 means the model matches data as well as the noise allows (the statistical ideal); ≪1 on noiseless closure data means correct recovery; ≪1 on real data means over-fitting; ≫1 means the model, the theory settings, or the data's internal consistency is strained. Its expected scatter is √(2/N) — for N=648, ±0.056 — which is what "consistent with 1" means quantitatively. IN THIS LAB — the spine of every card: 2×10⁻⁴ (clean drill), 0.977 (noisy drill), 5.60/5.84 (real data — the unexplained floor).

Parametrization

The functional form assumed for the PDFs at Q₀ — the single biggest modelling choice in the field, because data can only push against the shapes you allow. The spectrum: rigid polynomials (MSHT20: 52 params via Chebyshev expansions; our toy: 13), flexible grids (values at x-nodes), neural networks (NNPDF: ~800 weights). More flexibility ≠ better: unconstrained freedom inflates uncertainty where data is silent and invites over-fitting where it isn't. Judging this trade-off objectively is exactly what the evidence was invented for. IN THIS LAB — our model ladder: 13p toy → 36p grid → 52p MSHT20, all judged on identical data (PPDF-9 and the fits running tonight).

Prior (and prior sensitivity)

What the model was allowed to try before seeing data: uniform boxes on parameters, or a band around a reference PDF. In parameter estimation the prior washes out as data accumulates; in model comparison it never does — the evidence integrates over everything the prior permits, so a model confined to a narrow band can't be caught wasting space, and one roaming free pays rent on every acre. Fair contests therefore need deliberately designed, pre-declared priors — and the verdict should be quoted with its prior, not as an absolute. IN THIS LAB — the central caveat on PPDF-9 (narrow NNPDF-band prior vs wide boxes), and why the MSHT20 priors were declared in the log before any fit ran.

Posterior

The complete Bayesian answer: a probability distribution over every candidate proton, each weighted by likelihood × prior. Everything else is a summary of it — central values (means), error bands (quantiles), correlations (the corner plot), derived quantities (evaluate on samples). Its honesty property is the whole sales pitch: where data constrains, it narrows; where data is silent, it faithfully returns the prior — no bowl-shaped assumptions, no linearity requirements, valid for any model. IN THIS LAB — 1,715 posterior samples verified against planted truth in PPDF-2; the honest wide valence bands there are the posterior "returning the prior" exactly as designed.

Nested sampling (UltraNest)

The algorithm behind every Bayesian fit here: maintain N "live points" spread over the prior, repeatedly replace the worst one with a better one, and the shrinking volume traces likelihood contours from prior to peak. Its unique bonus: the same walk computes the total integral — the evidence — with an error budget, which MCMC methods can't do directly. The operational trap we learned: above ~10 dimensions the default region sampler collapses; a slice (step) sampler must be configured, with enough steps to decorrelate (we use ~3× the dimension). IN THIS LAB — UltraNest runs every fit; the 290× stall of PPDF-8 was this trap; tonight's 52-dim MSHT20 closure runs at healthy 0.6% efficiency because of that lesson.

Evidence (logZ)

One number scoring a whole model: Z = ∫ likelihood × prior over all parameters — "how probable is the data under this model, all things considered". It rewards fit quality but charges rent for wasted parameter space: automatic Occam's razor, no parameter-counting heuristics needed. Differences matter, not absolutes: ΔlogZ ≈ 5 is already strong preference (odds ~150:1); our ΔlogZ ≈ −109 is overwhelming. Two honest footnotes: verdicts inherit the priors, and logZ says which model explains this data best — not which is true. IN THIS LAB — priced missing flavour physics at ΔlogZ ≈ −800 (PPDF-8); refereed rigid-vs-flexible at −109 (PPDF-9); will referee MSHT20 next.

Bayesian complexity

The effective number of parameters the data actually determines, measured from the posterior (roughly: how much the posterior-average χ² exceeds the best-fit χ²). It's an honesty meter for parametrization design: a 36-parameter model with complexity 17 is telling you the data uses less than half its knobs — the rest are decoration the evidence will charge for. On noiseless closure data, posterior-avg χ² ≈ complexity is also a health check of the sampling itself. IN THIS LAB — 4.12 (toy, closure PPDF-1), 16.95 of 36 (grid, PPDF-9) — watch what MSHT20's 52 parameters report tonight.

Monte-Carlo replicas

NNPDF's uncertainty method: generate many pseudo-datasets by re-fluctuating the real data within its covariance, fit each independently, and quote the ensemble spread as the uncertainty — a bootstrap in spirit. Fast to reason about, embarrassingly parallel, and provably equivalent to the Bayesian posterior only for linear models with Gaussian likelihoods (arXiv:2404.10056). Neural networks and Chebyshev exponentials are not linear — hence this whole research program. IN THIS LAB — benchmarked to three decimals in PPDF-5 (loss/pt 1.009 and 2.000 vs theoretical ≈1 and ≈2); the MSHT20 replica ensemble is fit PPDF-13 in Maria's three-method comparison.

Hessian method

The MSHT/CT approach: find the single best fit, expand χ² quadratically around it, and quote uncertainties along the curvature eigenvectors (often inflated by a "tolerance" Δχ² = T² with T ~ 3 to absorb dataset tensions). Cheap, interpretable, and exactly right when the likelihood really is a bowl. Its failure mode is silent: along flat directions there is no bowl, the quadratic term vanishes, and the quoted band becomes an artifact of wherever the minimizer happened to stop. IN THIS LAB — found the same minimum as the Bayesian sampler to 0.04% yet under-covered the gluon by ~6σ on flat directions (PPDF-4); the MSHT20 Hessian fit (PPDF-14) closes Maria's three-method comparison.

Flat direction

A parameter combination the data cannot see: moving along it changes the PDFs but not any prediction, so χ² stays constant — a canyon floor, not a bowl. Generic causes: observables that only measure flavour sums (F2 barely resolves the valence split), x-regions with no data, and parameter degeneracies within a flexible shape. Each inference method shows its character here: Bayesian bands widen to the prior (honest), Hessian bands become meaningless (no curvature), replica bands reflect initialization luck. IN THIS LAB — watched the Hessian's valence parameters run to ~10⁹ with χ² unchanged (PPDF-4); the teal-vs-others contrast in the PPDF-6 plot is this concept in one picture.

Closure test

Fit fake data generated from a known "truth" — if you can't recover what you planted, fix the machine before touching real data. Level 0 (noiseless): a correct pipeline must drive χ² to ~0; any residual is a bug. Level 1 (noise drawn from the real experimental covariance): now χ²/N must land at ≈1 — perfection would mean fitting noise. The subtlety that bit us: the truth should come from the same model family being fitted, or model mismatch masquerades as machinery error. IN THIS LAB — PPDF-1/PPDF-3 for the toy; tonight's MSHT20 closure uses in-family truth at publication parameters for exactly that reason.

Pull

A deviation measured in units of the quoted uncertainty: (value − truth)/σ. The unit that makes "wrong" quantitative: |pull| ≲ 1 is healthy, and across many independent checks pulls should be standard-normal — systematically large ones mean the error bars lie, systematically tiny ones mean they're padded. Coverage (the fraction of checks with |pull| ≤ 1) is the ensemble version: ~68% is textbook. IN THIS LAB — max |pull| 0.42 across every flavour and x in the truth-recovery check (PPDF-2); the Hessian's ~6σ gluon pull (PPDF-4) is what a lying error bar looks like.

Parity gate

Our house rule for ported physics code: before any fit runs, the port must reproduce the reference implementation point-for-point — same functions, same inputs, agreement at numerical precision — over both published parameter values and random perturbations. The random draws matter: reference code often hides behaviour that's invisible at default values (normalization slots set to 1.0, branches that don't fire). A port that only matches at the publication point has been tested on one point of a 52-dimensional space. IN THIS LAB — the MSHT20 port passed 96 parity checks (worst gating deviation ≲10⁻⁹); the jittered draws caught a hidden ×A_g multiplication that publication values could never expose (PPDF-10).

Blind analysis

Decide the analysis before seeing the answer: priors, cuts, thresholds, and pass criteria get written down (and committed) before the result exists, so the result can't quietly shape them. The failure it prevents is goal-seeking — nudging a threshold here, a prior there, until the number looks like what you expected — which produces confident, irreproducible nonsense. Post-hoc changes are allowed only as new, documented decisions with the old result kept. IN THIS LAB — the MSHT20 prior boxes, charm convention, and closure criteria were logged and committed before the first fit launched (analysis/msht20/LOG.md); the one threshold we changed (PPDF-5's post-fit cut) is documented with its noise-model justification.

MSHT20 (our next rung)

One of the big-three global PDF sets (with NNPDF and CT), used across LHC phenomenology. Its parametrization: x·f = A(1−x)^η x^δ (1 + Σᵢ aᵢ Tᵢ(y)) with Chebyshev polynomials in y = 1 − 2√x, in the basis u_V, d_V, S, s₊, s₋, d̄/ū, g — 52 free parameters after analytic sum rules, uncertainties from Hessian eigenvectors with dynamic tolerance. Nobody has published a Bayesian posterior for it; the recent FPPDF release made its exact implementation public, which is what made our port possible. IN THIS LAB — ported and parity-verified tonight (PPDF-10); closure running; the three-method uncertainty comparison on it is Maria's stated final goal.