# GENERATORS — theory.md

**Theme.** Operations derived from theory and mathematics: recipes that turn a formal structure
(an equivalence, a limit, a sum rule, an inequality, a definition) into a question that can be
answered by measurement.

### G016 — Cross-formulation spread
**Operation.** Write the same prediction in two formulations the theory calls equivalent (two gauges, two bases, two truncation schemes, two regulators at the same order). Compute the difference. It is formally higher order, so it is normally discarded — instead treat its size as a physical scale and find the parameter region where it exceeds the resolution of an existing measurement. That region is where the choice of formulation stops being a convention and becomes a testable statement. Then ask which observable separates the two formulations most cleanly.
**Produces.** An experiment placed where convention becomes content, plus a bound on how far the truncated description can be trusted.
**Worked example.** Light–matter coupling in a cavity: dipole-gauge and velocity-gauge two-level models give the same Rabi frequency at weak coupling and drift apart as the coupling approaches the mode frequency. The drift is the two-level truncation failing, not a gauge artifact. Sweeping coupling at fixed detuning and watching the splitting depart from the two-level formula locates the failure without needing a new model.
**Fails when.** The formulations are related by an exact untruncated transformation, so the difference is identically zero, or the difference stays below resolution everywhere accessible.

### G017 — Exact-corner continuation
**Operation.** List the corners of parameter space where the model is exactly solvable (infinite connectivity, large N, one dimension, zero temperature, an integrable coupling). Compute the target quantity exactly at the nearest corner. Then move one controlled step toward the real system and compute the first correction in the small parameter that measures the distance. The sign and size of that correction say how far the corner's intuition survives; the smallest physical system measurably off the corner is the target.
**Produces.** A measurement in the crossover, where exact and real differ by a computed amount.
**Worked example.** Mean-field theory is exact on a fully connected lattice. Take a connectivity-tunable simulator (trapped ions or superconducting qubits with programmable couplers), compute the leading 1/z correction to the ordering temperature, and find the connectivity at which it reaches the experimental resolution. That connectivity — not the largest available — is the interesting experiment.
**Fails when.** No exact corner is within reach, or the correction series is uncontrolled so the "first correction" is not small.

### G018 — Sum-rule budget
**Operation.** Express the measured quantity as a total over a complete set of states (a sum rule). Measure the total. Any shortfall must be assigned to a named channel before it is called error. Enumerate every channel that can carry the missing weight, including ones outside the measured band, and rank them by the matrix element that couples them to the probe. The top of that list is the next experiment.
**Produces.** A named missing channel and the measurement that would see it.
**Worked example.** Optical conductivity: integrate σ(ω) over the measured band and compare with the charge density. A shortfall is routinely attributed to correlations; the operation instead demands the bookkeeping first — interband weight, a phonon-assisted channel, weight above the band. Only when every channel is bounded does the residual become evidence. The same audit applies to spin sum rules in neutron scattering.
**Fails when.** The sum rule needs weight at energies no source reaches, or matrix elements are unknown so channels cannot be ranked.

### G019 — Degeneracy fingerprinting
**Operation.** List every exact degeneracy the theory predicts, from symmetry, index, or fine-tuning. For each, enumerate the perturbations that can lift it and compute the splitting pattern each produces: order in the perturbation, anisotropy, sign, dependence on a tunable field. Measure the pattern and invert it. A splitting that matches no candidate is the interesting one — it is either an uncontrolled field or an assumption that does not hold.
**Produces.** An identification experiment in which the pattern itself, not the size, is the signal.
**Worked example.** Valley degeneracy in a two-dimensional crystal. Strain, a perpendicular magnetic field, and alignment to a substrate lift it with different anisotropies and different dependences on valley index. Mapping the splitting as a function of field angle produces a fingerprint; comparing it with the computed patterns identifies which perturbation dominates, or exposes one nobody modelled.
**Fails when.** The degeneracy is only approximate and its intrinsic width exceeds every candidate splitting, or several perturbations produce indistinguishable patterns.

### G020 — Analyticity audit
**Operation.** Take a measured response on the real axis and impose the analyticity the theory requires (causality: no poles in the upper half-plane; passivity: positive real part). Reconstruct one part from the other and subtract. Structure in the residual — a peak, a sign change, a slope — is a candidate for a missed resonance or a violated assumption. Then find the minimal pole that would account for it and ask where that pole must sit.
**Produces.** A predicted feature at a frequency or wavevector not yet probed, with an implied strength.
**Worked example.** A reflectivity spectrum measured over a limited band and extrapolated for Kramers–Kronig. The extrapolation, not the data, dominates the residual outside the band. Vary the extrapolation family over physically admissible forms; a residual peak that survives every family is a mode, one that moves with the extrapolation is an artifact. Applying this before announcing a hidden excitation costs one afternoon.
**Fails when.** The band is too narrow for the transform, or the system is not causal or passive in the regime probed.

### G021 — Slack of a rigorous bound
**Operation.** Take a rigorous inequality that pins a measurable quantity between two independently measurable ones. Measure all of them. Report the slack, not the satisfaction. Then ask which physical effect consumes the slack and what configuration would saturate the bound; saturation is a distinct, falsifiable target. A negative slack is not a small discrepancy — it falsifies one of the three inputs outright.
**Produces.** A saturation experiment plus an exact consistency test.
**Worked example.** A variational upper bound on a ground-state energy compared with the energy measured by spectroscopy. The gap is usually read as the ansatz's quality; the operation reads it as a budget. If the measured energy sits above the bound by an amount that tracks a known correlation effect, the ansatz is the limitation; if it falls below, the Hamiltonian or the state assignment is wrong, and no fitting can repair that.
**Fails when.** The bound is so loose that saturation is unreachable, or it assumes a limit (infinite size, exact equilibrium) the experiment cannot approach.

### G022 — Null-space read of an inverse problem
**Operation.** Write the map from model parameters to the data and compute its Jacobian. Diagonalise. Directions with near-zero singular value are parameter combinations the data cannot resolve; report those combinations rather than individual parameters. Then ask what additional measurement adds a row that lifts the degeneracy, and design that measurement instead of collecting more of the same data.
**Produces.** A precise statement of what the data cannot determine, and one new measurement that fixes it.
**Worked example.** A decay curve fitted with two exponentials. At a given noise level the amplitude and lifetime of the weak component trade off along a near-null direction: the data fix the integrated weight but not the lifetime. A longer acquisition shrinks the noise but barely rotates the null direction. A second delay, or a different probe, does. The operation converts "we need better data" into "we need a different row".
**Fails when.** The problem is already well conditioned, or model error dominates noise so the Jacobian is the wrong object.

### G023 — Two-way counting
**Operation.** Count something the theory says can be counted two ways: bulk states against boundary states, degrees of freedom against equations, poles against zeros, spectral weight against charge. The theorem asserts equality. Classify any mismatch into three cases — the measurement ignores a reservoir or boundary, the missing object has zero coupling to the probe, or an assumption behind the theorem fails — and design the test that separates them.
**Produces.** A search for a dark object or an unaccounted reservoir, with a discriminator attached.
**Worked example.** Edge modes in a finite topological photonic lattice: the bulk index predicts a number of edge modes, the measurement sees fewer. Either a mode has moved out of the gap or the termination differs from the modelled one. Bevel or re-terminate the lattice: if the count changes with termination, the surface was the reservoir; if it does not, a mode is present with no weight at the probe's frequency.
**Fails when.** The counting theorem needs a limit the sample does not reach, or it is defined only modulo an integer so small mismatches carry no information.

### G024 — Collapse residuals
**Operation.** Form every independent dimensionless group the model allows and plot existing data in those coordinates. A complete model collapses the data onto one curve. Where the data split, examine the shape of the residual rather than its size: a split organised by a quantity the model excludes names the missing variable. Add it as a new group, then predict where the collapse must fail.
**Produces.** An identified missing variable and a scaling prediction for where it matters.
**Worked example.** Thermal conductivity of samples of different purity plotted against reduced temperature. If the curves separate by purity, the model is missing a length scale — the mean free path. Add the ratio of mean free path to a geometric or phonon length and the split should organise; the prediction is that it grows as that ratio approaches one, which is testable by thinning the sample.
**Fails when.** There are more free groups than independent ranges in the data, so a collapse can always be manufactured, or the missing variable is not dimensional (topology, parity, a conserved charge).

### G025 — Duality translation
**Operation.** Find an exact map between two descriptions of the same system (strong against weak coupling, particle against vortex, position against momentum). Carry the quantity of interest through the dictionary. Ask which side is cheap to compute and which is cheap to measure; a measurement that is hard on one side is often routine on the other, and the dictionary turns it into a prediction rather than an analogy.
**Produces.** A hard measurement replaced by an easy dual one, with an exact translation and stated limits.
**Worked example.** A two-dimensional lattice gas of bosons near the superfluid–insulator transition. In the insulating phase, phase winding is the natural order parameter but is hard to access; the dual description makes it a density observable, which site-resolved imaging measures directly. The dictionary also states how far from the transition it remains valid, which sets the experiment's operating point.
**Fails when.** The map holds only at a self-dual point, or it sends the measured quantity to an operator with no experimental handle.

### G026 — Floor of an asymptotic series
**Operation.** For a quantity computed as an asymptotic series, find the order at which terms stop shrinking. The size of that smallest term is the ambiguity the theory generates by itself — a number, not an error bar. Ask what non-perturbative object sets the scale, whether the predicted floor can be measured, and whether any observed discrepancy exceeds it. Below the floor, disagreement is uninformative; above it, something is missing.
**Produces.** A predicted magnitude for non-perturbative effects and a sharp threshold for when a discrepancy means something.
**Worked example.** A coupling that runs: at short distance the series is well behaved, at long distance it diverges. The minimal term defines a scale and a residual size. Comparing a fixed-order prediction with data at that scale, the residual should be of the predicted size regardless of how many orders are added; adding orders and watching the residual stop shrinking is the check, and a residual that stays larger points to a sector the series never contained.
**Fails when.** The series converges, or only the first terms are available so no minimal term has appeared.

### G027 — Non-commuting limits
**Operation.** Identify two operations the derivation performs in a fixed order: average then evolve, coarse-grain then linearise, long-time limit then large-size limit, drive then relax. Swap them and compute the difference. The commutator is not an error — it is an effective term created by the ordering. Estimate its size, then place an experiment in the window where the two orders differ by more than the noise.
**Produces.** An effective interaction with a predicted magnitude and a signature that exists only in the crossover.
**Worked example.** A system driven periodically and relaxing on a comparable timescale. Averaging the drive first gives a static effective Hamiltonian with no dc response; letting it relax during the drive gives a rectified dc component. The two orders differ by a term that peaks when the drive period matches the relaxation time, so sweeping drive frequency at fixed amplitude should show a dc response rising and falling — a signature absent in both limits.
**Fails when.** The operations commute exactly, or the intermediate regime is inaccessible because the two timescales cannot be made comparable.

### G028 — Degree-of-freedom accounting
**Operation.** Count the propagating degrees of freedom the model contains, then count the independent spectral features the measurement resolves. Equal counts mean the probe is complete. Fewer measured features mean a mode with no coupling to that probe or a mode outside its range. For each missing mode compute its selection rules against every available probe and choose the one that should see it.
**Produces.** A targeted "look here with this instrument" experiment for a mode that is currently invisible.
**Worked example.** A polar crystal: infrared absorption sees the transverse optic mode, Raman sees a different combination, and a mode silent to both can still be seen by a probe that breaks the relevant parity, such as hyper-Raman scattering or inelastic neutron scattering at nonzero wavevector. Counting first shows a mode is missing; the selection-rule table says where to look.
**Fails when.** The mode is decoupled from every available probe, so the mismatch cannot be tested, or hybridization makes the count itself ambiguous.

### G029 — Definition ambiguity
**Operation.** Find quantities the theory defines only up to a total derivative, a surface term, or a subtraction. Write two admissible definitions explicitly and compute the difference in the observable that is actually measured. If the difference is measurable, the comparison between theory and experiment is testing a definition and must say which one; if it vanishes identically, the ambiguity is harmless and can be dropped from every future comparison.
**Produces.** A definition-dependent prediction and a flag on comparisons that silently mix definitions.
**Worked example.** Heat current in a lattice spin model: the energy current is defined up to a total time derivative, and two definitions give different thermal conductivities for the same model and the same parameters. Two calculations can therefore disagree without either being wrong. The operation says: compute both, see whether the difference survives the probe (a steady-state gradient versus a correlation function), and state the definition whenever a number is quoted.
**Fails when.** A continuity equation fixes the quantity uniquely, or the difference cancels in the observable of interest.

### G030 — Cancellation census
**Operation.** Write the quantity of interest as a sum of contributions. Attach a sign to each from its physical origin. If the total is small, do not celebrate agreement — identify the cancellation and find the knob that flips one term's sign or magnitude while leaving the others. Near the cancellation the response is maximally sensitive, and the sign change itself is a measurable event.
**Produces.** A high-sensitivity operating point, often where a response changes sign, plus a prediction for which knob moves it there.
**Worked example.** Thermopower in a doped semiconductor or a correlated metal: electron-like and hole-like contributions carry opposite signs, so the Seebeck coefficient passes through zero at a carrier density where the two balance. That zero is not "no effect" — it is the point where the two partial contributions are equal in magnitude, and a small field, strain, or temperature change moves the balance measurably. The operation converts a null result into a lever.
**Fails when.** The small total comes from a symmetry that forbids the terms rather than a cancellation among allowed ones, or the individual contributions cannot be separated experimentally.