Scientific math workbench

The Flagellar Motor · docs/MATH-WORKBENCH.md @ b909801f3db4 (hierarchical-aif/motor-stack) — opens the published snapshot 8b4b5935bcba

Three ways to read this page. Precise is the document itself, exactly as it is written in the repository. Plain and Clear were written for this website to help you meet that document — they are about it. They are not it, and they are not evidence.

A laboratory built around the bacterial flagellar motor. It holds a deterministic reduced model of the motor, analysis of recorded single-motor events, and a cross-study parity programme. Alongside those sit the scientific gates the work has to clear, and independent audits of both the model and the repository around it. The framing throughout is hierarchical active inference.

It is for a reader with a scientific interest, and especially for one who has come to check whether a model fit has quietly become a claim about biology. The laboratory's central discipline is a labelling one: every visible layer carries exactly one class — recorded observation, structural reconstruction, reduced model, or physical teaching analogue — and those classes may not be blended. Behavioural observations of one species are held apart from structural work on another, so that nothing on the page can read as a single measured specimen.

Start with the Living Science Walkthrough, which sets out those classes and the truth contract they belong to. Then the scientific and mathematical contract, then the parity gates, which state what would have to hold before a parity claim could stand.

What it is not: a claim of biological parity. The walkthrough is explicit that the release does not turn a model fit into a biological identity claim, and full biological parity is recorded as false and printed as false. Passing this repository's software tests is necessary here and is not the same thing as agreement with a living motor.

PlainClearPrecise

Your browser cannot switch reading levels, so the document itself is shown.

This is the document. Rendered from the repository at the commit above, with nothing rewritten for the web. A gate re-renders it on every deploy and fails the build if a single byte differs.

Route: /math-workbench. Source: app/math-workbench/page.tsx (server, loads the frozen reports) and app/math-workbench/scientific-math-workbench.tsx (client, 6 views).

This document exists because CLAUDE.md requires every surface to declare its evidence, assumptions, units, falsifier, gate ledger, reproduction command and limitations. Nothing here is new science; the workbench renders quantities computed elsewhere and executes the repository's own model functions.

1. What the workbench is

A repository-native instrument that makes the committed mathematics executable and inspectable in the browser without re-implementing any of it. Every number it displays comes from one of two places:

live execution of lib/uni-motor.js, lib/duration-models.js or lib/source-first-passage.js in the browser, on CPU, deterministically; or
frozen JSON reports loaded server-side from experiments/results/ and audits/.

It computes no quantity of its own. tests/math-workbench.test.mjs pins the calculators to the library functions and guards against shadow re-implementations.

2. What the workbench is not

It is not a new experiment. It runs no fit, produces no result artifact, and writes nothing.
It is not evidence that a bacterial motor performs Bayesian inference. The synthetic agent loop is a reduced teaching model over a reduced synthetic world.
It is not a claim of biological parity. fullBiologicalParityAchieved is false and is printed on the page.
The Execute the math view runs a synthetic world, not recorded data. Its "observed ligand"/"observed speed" tiles are observations of the simulation.

3. Views

View	Content	Evidence class
Model stack	the nine-step world → observation → prior → likelihood → posterior → policy → action → prediction sequence, and four status cards	modelled + report-derived
Execute the math	live stepping of `stepSyntheticSystem`, four controls, prior/likelihood/posterior/free-energy panels	modelled, synthetic
Equation library	11 catalogued modules with equation, plain meaning, inputs, outputs, truth class and source binding; plus the M0–M3 dwell calculator and the DLT first-passage calculator	mixed; each card declares its own class
Measured evidence	the observed cohort, held-out model scores, and both gate ledgers	observed + report-derived
Planned, not run	B3's nine frozen competitors and B4 robustness, all marked pending/not-run	frozen prediction
Print worksheets	twelve hand-calculation worksheets	teaching material

4. Evidence, units and assumptions

4.1 Observed dwell analysis

Protocol: UNI-FLAGELLUM-OBS-001. Source: Wadhwa et al. 2022 single-motor stator-remodeling dataset, pinned by identities.rawSourceSha256 and identities.rawSourceCommit.
Species: Escherichia coli. This is behavioural/kinetic evidence and is kept separate from the Salmonella and Bacillus structural evidence used elsewhere in the repository. The two are never combined into one specimen claim.
Experimental unit: the motor. Events are not independent replicates; all uncertainty is motor-cluster bootstrap.

Cohort reconciliation (the page shows the first number; this is the full chain):

quantity	value
source motors	129
source events	1349
excluded: left-truncated dwells	129
excluded: right-censored dwells	109
excluded: out-of-range dwells	3
training motors / events	80 / 793
holdout motors / events	19 / 233

80 + 19 = 99 motors carry scored events; the remaining 30 of the 129 source motors contribute no eligible event after exclusions.

Split rule: sha256(motorName) mod 5, at motor granularity, recomputed independently by tests/semantic/train-holdout-leakage.semantic.test.mjs.
Fitting: all parameters are estimated on training events only. One declared nuance: the set of eligible states (N = 1…8) was chosen using all data, not training only.
Units: durations in seconds; log scores in nats per event; μ_N in seconds.

4.2 Duration models

Scored on the normalized duration y = t/μ_N, then mapped to seconds with the Jacobian log f_seconds(t) = log f_normalized(y) − log μ_N. Each normalized model is constrained to unit mean, so μ_N carries the scale and the shapes are compared on equal footing.

model	form	frozen training parameters
M0 memoryless	`f(y) = e^{−y}`	none
M1 Weibull	scale `= 1/Γ(1+1/k)`	`k = 0.625088844276203`, `scale = 0.6996038164387606`
M2 lognormal	`μ_log = −σ²/2`	`σ = 1.5783076021679734`, `μ = −1.245527443530609`
M3 two-timescale	`w λ_f e^{−λ_f y} + (1−w) λ_s e^{−λ_s y}`	`w = 0.6066448974609373`, `λ_f = 5.239865393555934`, `λ_s = 0.44485855808292624`

4.3 Held-out result, including the adverse one

Mean log score, nats per event (higher is better):

model	score
M0 exponential	−3.259938023731547
M1 Weibull	−3.0962841768378104
M2 lognormal	−3.012890170946554
M3 UNI two-timescale	−3.0497565695441344

Paired advantages of M3, with motor-cluster bootstrap 95 % intervals:

contrast	point	95 % interval	reading
M3 − M0	+0.210181454187411	[0.06918462236407662, 0.3247183938187771]	resolved: M3 beats the memoryless null
M3 − M1	+0.046527607293676804	[−0.01796916498254155, 0.084962018775231]	crosses zero — not resolved
M3 − M2	−0.03686639859758028	[−0.06803225348050351, 0.014817954207609459]	crosses zero — not resolved

The adverse result is retained: a lognormal baseline out-scores the UNI two-timescale model on held-out point score. The M3 − M2 interval crosses zero, so the correct statement is "not resolved at this sample size", not "no difference" and not "M2 wins". With 19 holdout motors most contrasts are underpowered.

Mean CV² across states: 3.149531339591441, 95 % interval [1.5141240937044127, 3.5675042610575742] — above 1, so held-out dwell timing rejects the homogeneous memoryless prediction within the frozen population.

4.4 Fences that apply to every reading of this page

Carried verbatim from observedReport.claims:

Overdispersion does not uniquely identify a molecular hidden state; heterogeneity and nonstationarity remain alternatives.
Predictive superiority does not prove that the fitted timescales are molecular states, that the organism performs the inference, or that any mechanism is identified.

4.5 Variational and expected free energy

F[q] is in nats. It is model evidence, not mechanical work, not joules. Thermodynamic work τ·Δθ is a separate quantity with separate units and is never added to, converted from, or compared with F.
The likelihood is normalized to a categorical evidence vector over the three declared hidden states before the update, so the reported "surprise" is the surprisal under a renormalized categorical channel, not under a continuous density.

Known defect, open at the time of writing. The shipped expected free energy at lib/uni-motor.js:328 evaluates to KL[q(o)‖C] + 2·ambiguity + effort, not the canonical KL + ambiguity + effort, because risk is a cross-entropy and the epistemic term is counted twice. On the real model this inflates G by ~61 % and inflates G(RUN) − G(TUMBLE) by 6.5×, which changes action selection. See docs/audit/PHASE-E-WORKBENCH-AUDIT.md finding E-M01. Until it is corrected, the EFE equation card and worksheet 5 describe a quantity the tool does not produce.

4.6 DLT first-passage module

S_N(t) = Σ_j a_j e^{−r_j t} with r_j = k₊(N) + N σ₋ + j(σ₊ − σ₋); P₊(t|N) = k₊(N) S_N(t); P₋(t|N) = −dS_N/dt − P₊(t|N). A censored dwell contributes only its survival probability. Verified numerically in this repository: Σ_j a_j = 1, S monotone non-increasing, P₊ + P₋ = −dS/dt to <1e-4 relative, and mean dwell = ∫S dt to <1e-3.

The DLT calculator executes parameters from a fit whose parameter-recovery (G05) and held-out mechanistic prediction (G06) gates FAILED, and whose public-artifact parity gate (G03) also FAILED. The numbers are reproducible; they are not validated mechanism.

5. Gate ledgers as rendered

Single-study mechanistic gates (14 total): 4 PASS, 3 FAIL, 1 SOURCE_ONLY, 1 NOT_ESTABLISHED, 5 BLOCKED_EXTERNAL. Of the seven computational gates G00–G06: 4 pass (G00, G01, G02, G04), 3 fail (G03, G05, G06).

Cross-study parity gates (16 total): 8 PASS, 3 FAIL, 2 NOT_ESTABLISHED, 3 BLOCKED_EXTERNAL. Overall PARTIAL_PARITY_ONLY; fullBiologicalParityAchieved: false; 11 attributed studies; conservative lower bound of 409 independent motors/cells.

Known defect, open. X01_SOURCE_INTEGRITY renders PASS while zero of its twelve declared cached artifacts are present on disk. Per CLAUDE.md it must be BLOCKED_EXTERNAL in a clone without experiments/upstream-cache/. See audit finding E-B01. Do not read that green badge as verified source integrity.

6. Falsifiers

The workbench itself is falsified if any of the following is shown:

A displayed value differs from the value produced by the named source function for the same inputs. Test: tests/math-workbench.test.mjs.
A truth badge names an evidence class the underlying quantity does not have. Currently unprotected — see audit finding E-M04.
A gate with status FAIL, NOT_RUN, BLOCKED_EXTERNAL or NOT_ESTABLISHED renders with the pass affordance. Currently latent — see TC-06.
The rendered fullBiologicalParityAchieved is anything but false while any required gate is unmet. Currently guarded by a vacuous assertion — see E-M03.
Two identical runs from the same controls produce different trajectories. Verified deterministic: 16,200-step sweep, bit-identical replay.

7. Reproduction

npm ci
npm test                      # 103 tests + build + 2 rendered-route tests
npm run lint
npx tsc --noEmit
npm run science:verify
npm run cross-study:verify
npm run cross-study:verify-raw   # BLOCKED without experiments/upstream-cache/ (4.09 GB)
npm audit --omit=dev --audit-level=moderate
npm audit
npm run dev -- --port 3107

Then open http://localhost:3107/math-workbench.

Note. vinext dev binds [::1] (IPv6 loopback) only and does not honour --host. If your browser resolves localhost to 127.0.0.1 first, use http://[::1]:3107/math-workbench. The route is not reachable from another device.

The page footer's reproduction strip lists only four of these commands. npm run lint, npx tsc --noEmit, npm audit and cross-study:verify-raw are also required by CLAUDE.md and are omitted from that strip.

8. Limitations

No live browser verification exists in the audit record. All product findings to date are static-analysis or build-output based.
The truth-status colour channel is currently non-functional: app/globals.css references four CSS custom properties that are never declared, so the probability bars render invisible and PASS/FAIL/UNRESOLVED badges render identically (audit E-M02).
No species label appears anywhere on the served page (audit TC-04), although the underlying report carries the E. coli attribution.
The four overclaim fences in §4.4 are not rendered by the workbench (audit TC-05).
Only the default flow view is server-rendered; the other five views require JavaScript and have no rendered-HTML test coverage (audit E-M05).
B3 shows PENDING on this branch. Executed B3 and B4 results exist on phase-2/b3-model-competition. This branch's badge is correct for its own commit graph and stale with respect to the science. See audit §5.
npm test covers 103 tests; tests/red/* is excluded by design and several of its cells are red at HEAD.

9. Related documents

docs/audit/PHASE-E-WORKBENCH-AUDIT.md — the audit this document's warnings come from
docs/UNI-STACK-BUILDER-PLAN.md — the hierarchical builder that supersedes this surface
docs/OBSERVED-EXPERIMENT.md · docs/SCIENCE-GATES.md · docs/CROSS-STUDY-PARITY.md · docs/VERUM.md

sha256 392b7cc6e8359d5b — of the original file, so what was ingested stays checkable.

Written for this website — not the document. This is a plain-language retelling, written to help you meet the document. It is not the source, and it is not evidence. It has not yet been checked by a person. Read the document itself → (or choose Precise in the reading-level control above)

This page documents one screen in the project: a workbench that lets you run the project's own mathematics in a browser and look inside it. Nothing on that screen is new science. It either executes code that already exists in the repository, or displays numbers from frozen reports.

The page is unusually honest about what the workbench is not. It is not an experiment, it writes nothing, and it is not evidence that a real motor performs the calculation. One of its views runs a synthetic world, so the tiles labelled as observations there are observations of a simulation.

Two open defects are declared in the middle of the document rather than buried. One equation as shipped counts a term twice, which changes which action gets chosen, so an equation card describes a quantity the tool does not produce. And one gate shows a pass while none of its declared files are present on disk; the page tells you not to read that badge as verified.

Plain · written 2026-08-01 by claude-opus-5 · not yet checked by a person · about the document whose sha256 is 392b7cc6e8359d5b

Written for this website — not the document. This is a clearer retelling, written to help you meet the document. It is not the source, and it is not evidence. It has not yet been checked by a person. Read the document itself → (or choose Precise in the reading-level control above)

The document exists because the project's contract requires every surface to declare its evidence, assumptions, units, what would show it wrong, its gate record, reproduction command and limitations. This is that declaration, for one screen.

It begins by defining the workbench as an instrument that makes committed mathematics executable and inspectable without re-implementing it. Every number it shows comes either from live execution of existing library code, on the processor and repeatable, or from frozen report files. It computes no quantity of its own, and a test pins the calculators to the library functions so a shadow re-implementation cannot creep in.

Then a section on what it is not. Not a new experiment: it runs no fit and writes nothing. Not evidence that a bacterial motor performs the inference: the agent loop is a reduced teaching model over a reduced synthetic world. Not a claim of biological parity: the flag for that is false and is printed on the page. The view that steps through the mathematics runs a synthetic world, so its observation tiles are observations of the simulation.

A table lists the views: a model stack, a live stepper, a library of equation cards each declaring its own class, a measured-evidence view, a panel of work planned but not run, and printable worksheets.

The long middle section covers evidence, units and assumptions. The dwell analysis names its protocol and its pinned source. The species is stated, and this behavioural evidence is kept separate from the structural evidence elsewhere. The experimental unit is the motor, events are not treated as independent replicates, and all uncertainty comes from resampling motors. A reconciliation table shows how the source cohort shrinks at each declared exclusion. One nuance is declared rather than hidden: parameters are fitted on training events only, but the set of eligible states was chosen using all data.

The result on data kept back is given with the adverse part in bold. A lognormal baseline out-scores the project's two-timescale model on point score, and two of the three paired intervals cross zero. So the page insists the correct wording is that those comparisons are not resolved at this sample size — not that there is no difference, and not that the baseline wins. Limits carried over verbatim from the report say that overdispersion does not uniquely identify a molecular hidden state, and that better prediction does not show the fitted timescales are molecular states or that the organism performs the inference.

Two known defects are declared inline. The shipped expected free energy adds an epistemic term twice, because one component is a cross-entropy rather than a divergence. The page gives the algebra, says the inflation changes action selection, and states that until it is corrected one equation card and one worksheet describe a quantity the tool does not produce. Separately, a source-integrity gate renders a pass while none of its declared cached artifacts are on disk; the page tells the reader not to read that green badge as verified source integrity.

A list follows of conditions under which the workbench itself would be shown wrong, each annotated with whether a test protects it. Several are marked unprotected or latent.

The page ends with reproduction commands and a list of limitations. No live browser verification exists in the audit record. The colour channel distinguishing pass from fail is non-functional, because four style properties are used and never declared. No species label appears on the served page. And the overclaim limits are not rendered.

Clear · written 2026-08-01 by claude-opus-5 · not yet checked by a person · about the document whose sha256 is 392b7cc6e8359d5b