Part B - Compare, observe and act

PHY.1 - Construct Physical Similarity Across Changed Conditions

Type: Method Status: Stable Normativity: Normative within the stated use

PHY.1:1 - Problem frame

Use this pattern when you want to learn about a physical situation through another arrangement, but changing its size, material, speed or surroundings changes the effects that produce the answer. You may be designing a smaller experiment, comparing observations made under different conditions, or deciding which part of a previous physical result can be reused.

Start with the quantity or behavior you need to infer about the original situation. Then identify one physical effect whose relative importance could change in the proposed comparison. A smaller object has less weight, for example, but its ability to carry that weight changes by a different factor. That difference can determine how the comparison should be built and loaded.

The result is a set of physically compatible comparison conditions, a justified transfer of a selected result, or a conflict that directs the next construction. Here physical similarity means correspondence of the physical behavior needed by the question under stated changes of scale or conditions. Geometric similarity preserves shape under scaling; further conditions can be needed to preserve the relevant behavior.

You need the physical account used in the comparison and the mathematics needed to transform its relations. You can develop that account through B.5.FM and B.5.TU or obtain a missing physical contribution from a collaborator. Algebra and ratios suffice for many first comparisons; fields and time-dependent behavior can require differential equations. A ready comparison whose conditions fit the question can be used directly. If the problem is expressing an already understood account mathematically, MMP.10 supplies that formulation work.

PHY.1:2 - Problem

A physically different experiment can resemble the original while answering a different question. Keeping shape preserves length ratios. It can still change the competition between weight and strength, inertia and friction, propagation and absorption, or reaction and transport. An experiment that preserves a final value can also change the path by which that value is reached.

The difficulty is constructing the conditions of a useful comparison. A familiar dimensionless number can help, but selecting it requires knowing which physical effects it compares. Several relevant conditions can demand incompatible changes to the same material or control. The requested result determines whether that conflict must be removed or whether a more limited comparison will suffice.

PHY.1:3 - Forces

PHY.1:4 - Solution

Choose the result to transfer → recover its physical causes and conditions → express their relative contributions → construct compatible comparison settings → resolve consequential mismatches → use the result and revise it when the question changes.

PHY.1:4.1 - Choose the physical result and the proposed change

Name the original arrangement and the comparison arrangement. Say what each does and what the comparison is intended to reveal. Distinguish a proposed experiment from an observed one.

Specify the result at the detail needed for its use: total extension under a load, the location of a maximum, a response after a stated duration, or a distribution of repeated outcomes. Include the location, interval and preparation when they affect that result. A request for average deformation differs from a request for deformation at every point.

Identify what may change between the arrangements and what is fixed by the proposed work. The available fluid, gravitational acceleration, material, instrument range or support can constrain the construction. A control may be a time history rather than one setting. Preserve any known dependence between settings: changing temperature can alter both viscosity and density.

If the existing physical account already yields a sufficient answer, use it. Constructing another experiment is worthwhile when it supplies a contribution the work still needs. C.11.DUA helps choose between acting on a sufficient answer and obtaining more information.

PHY.1:4.2 - Recover the interactions that can change the requested result

Follow the physical route by which the proposed input affects the output. Identify the participants, interactions and constraints on that route. Include an exchange across the chosen boundary when it contributes to the result. In a deformation problem, distinguish the force applied at an end from weight distributed through the body. In a transport problem, distinguish material carried by motion from material spread by diffusion.

Use the applicable theory to express those contributions. B.5.TU supplies the passage from theory to the encountered case; A.3.3.TR helps when several interactions determine the same evolving state. State the physical premises still needed rather than filling an unknown interaction with a convenient equation.

Estimate the relative size or time scale of competing effects. Derive that comparison from their laws: which quantity multiplies each term, and what spatial or temporal variation makes the term large? In a regime where one effect is negligible for the output, retain the reason for that simplification and the conditions under which it can fail.

Bring the preparation into the same account. Geometry, contacts, constraints and initial conditions can alter the solution even when the material equations are unchanged. For a field, a boundary condition applies over a surface or interval; matching its value at one point may leave the intended problem different elsewhere.

PHY.1:4.3 - Express the comparison through meaningful scales

Choose reference quantities from the physical question. A reference length might be a gap rather than the total apparatus length; a response time might be compared with the duration of forcing. Explain that choice in the working terms.

For a quantity q, choose a positive reference magnitude Q with the same units and write q=Q*q_hat. The number q_hat is its value in that scale. When the law concerns a difference from a reference q0, use q=q0+Q*q_hat and carry q0 through the substitution. This matters, for example, when an absolute pressure and a pressure difference enter different relations.

Substitute the scaled quantities into the physical relations. Transform derivatives and integrals too: if x=L*x_hat and t=T*t_hat, then a time derivative contributes a factor 1/T and a spatial derivative a factor 1/L. Divide each equation by an appropriate nonzero reference contribution. The remaining coefficients show which relative effects must be compared.

For example, in a flow whose speed varies by order U over distance L, characteristic inertial acceleration is U^2/L, viscous acceleration is nu*U/L^2, and gravitational acceleration is g. Their ratios to the inertial contribution are nu/(U*L) and g*L/U^2. These ratios compare physical terms. A different physical account can introduce another contribution and another condition.

Scale the geometry and preparation as well as the equations. A prescribed forcing history is compared at corresponding scaled times; a boundary profile at corresponding scaled positions. Keep a separately imposed forcing duration as an independent condition unless the work makes it proportional to the chosen response time.

For many quantities, dimensional analysis can construct dimensionless combinations systematically. Choose a combination whose physical interpretation makes the comparison useful. MATH.11 supports the construction of quantities unchanged by stated transformations. Dimensionless dependence still needs the physical premises that selected the quantities and relations; it can leave an unknown function to be obtained through MMP.11 or other subject work.

PHY.1:4.4 - Construct settings that supply the needed correspondence

Write the original and proposed dimensionless relations together. Set equal the coefficients and preparation features needed by the intended transfer, then solve these conditions jointly with the available physical settings. MMP.10 supplies the constraint formulation when several choices interact.

The result may be a recipe: change length by one factor, forcing by another and reading time by a third. Check that the selected properties describe an available material or realizable arrangement. A material’s stiffness and density, for example, may not be independently adjustable. Changing a support or adding a mass changes the physical account as well as a number.

When the scaled equations, domains and preparation coincide, express how a solution in the comparison variables becomes a solution of the original model. Include the output scale. C.29.1 supplies that transfer argument. If the model permits several solutions, the correspondence relates the permitted solutions; identifying one realized history requires the relevant physical preparation or selection conditions. Statistical predictions require the corresponding statistical account.

For a question about one output, try a weaker construction when reproducing the whole problem is unnecessary. Derive how that output depends on the settings and match the dependence needed by the question. Additional loading can reproduce total extension without reproducing local strain, as :5.2 shows. Keep the resulting comparison tied to the output it determines.

PHY.1:4.5 - Resolve a mismatch by its effect on the answer

If the conditions conflict, identify which physical contribution changes and how that could change the requested result. The conflict can itself rule out the proposed experiment at the chosen settings.

Choose the next move from the remaining physical possibilities. You can change a free setting, construct a different arrangement, retain a simpler physical regime, or derive how the unmatched contribution modifies the output. If several mismatched experiments are to be combined, construct the relation that permits that combination; a fitted relation retains its assumptions and the range over which it can be used. MMP.9 and MMP.11 can supply reduction and constrained model construction for these returns.

A small coefficient can support an approximation when its influence on the output is controlled. Inspect where that argument could fail. Thin boundary layers, a threshold, resonance or a long observation interval can make a nominally small contribution consequential. Derive a useful bound, compare an applicable limiting solution, or obtain a discriminating observation when it can settle the use. C.29.1 carries a bound through the later inference.

Use an existing result when it resolves the mismatch at the required strength. A conditional result or a limit can be enough. Additional measurement is selected by what its possible answers would change, including its cost, through C.11.DUA.

PHY.1:4.6 - Use the comparison and follow a changed question

Translate the obtained result into the original quantity, position and time. State the physical conditions on which that translation depends. A short derivation can supply the whole explanation; another contributor needs only the settings, correspondence and unresolved premises that change their use.

If a material experiment performs the comparison, include the effects of preparing, driving and reading it where they affect the inference. C.29.3 supplies this connection when the physical evolution performs a computation. A numerical solution has its own discretization and calculation errors under C.29.2. Keep those errors separate from a changed physical mechanism.

For a changed question, revisit the result first. Asking about a different location, time, range or intervention can make a previously omitted contribution relevant. Retain the conditions that still apply and reconstruct the affected comparison. B.5.MPC.R helps locate a failure across the physical account, mathematical representation and computation.

The useful continuation can be a proposed experiment, a design choice, an interpreted observation, a narrower claim or a new physical question. When the comparison suggests a different way for a team to obtain its result, use B.1.5.CM to describe the proposed operations and their relations. A.3.1.CC checks the claims on which that composition and its description rely and returns a correction to the affected contribution.

PHY.1:5 - Archetypal Grounding

The following constructed cases derive conditional comparisons. The physical laws and idealizations used in each case are stated; no apparatus measurements are reported.

PHY.1:5.1 - Find conflicting requirements before building a smaller flow experiment

A team proposes a quarter-size free-surface water experiment to investigate a flow in which inertia, gravity and viscosity may affect the result. The proposed geometry is similar, gravitational acceleration is unchanged, and the same liquid is initially intended. Use an incompressible Newtonian-fluid account. A relevant surface-tension, compressibility or other omitted effect would add a condition to this account.

Let L be characteristic length, U speed, g gravitational acceleration and nu kinematic viscosity. Comparing the acceleration contributions from :4.3 gives the Froude and Reynolds numbers:

Fr = U/sqrt(g*L)
Re = U*L/nu.

Write L'=lambda*L for the model length. To preserve Fr at the same g, solve:

U'/sqrt(g*lambda*L) = U/sqrt(g*L)
U' = sqrt(lambda)*U.

The same liquid then gives Re'/Re=lambda^(3/2). For lambda=1/4, the required speed is U/2 and Re’=Re/8. A corresponding transit time L’/U’ is one-half of L/U. Those settings preserve the inertia/gravity ratio while changing the viscosity/inertia ratio.

If the desired output needs both ratios reproduced, changing speed alone cannot repair the comparison. At fixed g, solving both conditions requires nu'=lambda^(3/2)*nu, which is nu/8 in this case. Selecting a liquid with that viscosity remains a physical-material question, including its other properties. Alternatively, with the same liquid and variable effective gravity, the conditions give U'=U/lambda and g'=g/lambda^3. At quarter scale those values are 4*U and 64*g; the transit time is divided by sixteen. A rotating apparatus proposed to create that acceleration also introduces rotation and spatial variation that may affect the comparison.

The first result is a decision about the experiment. If viscosity is negligible for the intended output throughout the relevant regime, the Froude-scaled experiment may suffice under that approximation. If viscosity changes separation or another needed behavior, retain that dependence and change the arrangement or the method of obtaining the answer. Matching initial and boundary conditions remains part of either construction.

Now change the question from a large-scale surface response to a local viscous effect near a wall. The previous gravity-dominated approximation leaves the new output unsupported. The method returns to the relative contributions and near-wall scale; the old choice of speed remains useful only for the question it answered.

PHY.1:5.2 - Reproduce one deformation without claiming the whole field

Consider a straight uniform bar, fixed at the top, with an axial force F pulling down at the lower end. Let L be length, A cross-sectional area, Y Young’s modulus, rho density and g gravity. Use small-strain linear elasticity and uniform material properties. Let x measure height from the bottom. The part below that point contributes weight rho*A*g*x, so the tensile force there is F+rho*A*g*x. Hooke’s law gives local strain:

strain(x) = F/(Y*A) + rho*g*x/Y.

Integrating from 0 to L gives total extension delta and mean strain:

delta = F*L/(Y*A) + rho*g*L^2/(2*Y)
delta/L = F/(Y*A) + rho*g*L/(2*Y).

A geometrically similar bar made of the same material has L'=lambda*L and A'=lambda^2*A. To reproduce the applied-force contribution to strain, use F'=lambda^2*F. Its own weight instead changes by lambda^3. At unchanged g, the self-weight contribution to strain changes by lambda. Merely using a smaller copy with the same material does not reproduce the two load contributions together.

Suppose the original bar hangs under its own weight, with F=0, and the question concerns only total extension relative to length. An added lower-end force on the smaller bar can match that output. Solve the mean-strain equation for the new force:

F' = A'*rho*g*L*(1-lambda)/2.

At quarter scale, F’ is 3/128 of the original bar’s weight. Substitution into the smaller bar’s mean-strain equation gives rho*g*L/(2*Y), the original value. The comparison therefore reproduces the requested normalized extension within this model.

Now ask for the strain at the lower end. The original unloaded bar has zero strain there; the smaller bar with the added force has F'/(Y*A') greater than zero. The output-specific construction cannot answer this new local question. To reproduce the complete strain profile, revisit the distributed loading or the combination rho*g*L/Y. The successful first comparison is retained for total extension.

PHY.1:5.3 - Preserve competing physical time scales

A substance diffuses along an interval and is consumed by a first-order reaction. A smaller comparison is proposed using the same diffusivity D and reaction rate k. Let c(x,t) be concentration, with the idealized equation:

c_t = D*c_xx - k*c,   0 < x < L.

Both ends absorb the substance, so c(0,t)=c(L,t)=0. For a simple worked preparation take c(x,0)=c0*sin(pi*x/L). Substitution gives:

c(x,t) = c0*sin(pi*x/L)*exp(-(pi^2*D/L^2 + k)*t).

Use x=L*xi and the diffusion time T=L^2/D. At scaled time tau=t/T, the equation becomes partial c/partial tau = partial^2 c/partial xi^2 - Da*c on 0<xi<1, where Da=k*L^2/D compares reaction with diffusion. The midpoint concentration relative to c0 is exp(-(pi^2+Da)*tau).

After L’=L/4 at unchanged D and k, the diffusion time is T/16 but Da becomes Da/16. If the original Da is 1, then at tau=1 the smaller experiment has a midpoint concentration exp(15/16), approximately 2.55, times the original normalized value. At that corresponding time, the smaller experiment has lost less substance to reaction.

To reproduce the dimensionless evolution with the same D, the smaller model would require k'=16*k, together with the corresponding initial and boundary conditions. A physical change intended to obtain that rate may change D too; solve using the resulting pair of properties. If the question instead concerns diffusion during an interval when consumption has a negligible effect, derive and use that shorter-time approximation. The reaction can re-enter when the requested duration changes.

PHY.1:6 - Bias-Annotation

The explicit cases use classical continuum laws and positive scale factors because their derivations are easy to inspect. In a new regime, discreteness, quantum effects or a change in material behavior can invalidate such laws. Recover the applicable account before extending their scaling.

The worked algebra assumes access to the physical laws and a reader able to manipulate them. The prerequisites in :1 help locate a needed contribution. One participant can recover the interaction law while another derives the comparison settings; their shared explanation must preserve what the law describes and which settings the arrangement can supply.

PHY.1:7 - Conformance Checklist

• The original question identifies the result and the proposed physical change.
• The relations and preparation contain the effects that can alter that result, with unresolved physical premises visible where used.
• Reference scales have physical meanings and compatible units; substitution covers the relevant spatial and temporal conditions.
• The comparison settings satisfy the required conditions together and can be supplied by the proposed arrangement, or their unresolved feasibility is stated.
• A limited or distorted comparison has a derivation or bound for the output it is used to answer.
• The result is interpreted at the original scale, with any preparation, computation and readout losses that change its use.
• A changed question returns to the affected physical contribution; an already sufficient result is used without compulsory extra experiments.

PHY.1:8 - Common Anti-Patterns and How to Avoid Them

PHY.1:9 - Consequences

The construction can reveal an impossible experiment before equipment is built. It can also identify useful freedom: a different load, medium, time scale or restricted output may supply the answer with less work. A failed similarity condition becomes a physical design question rather than an unexplained disagreement between experiments.

The price is making the physical premises and coupled settings explicit. A complex account can require several comparisons or a numerical analysis. The method can reduce that work by selecting a sufficient output, but it cannot supply a missing physical law from dimensional consistency alone.

PHY.1:10 - Architectural Rationale

Physical similarity is constructed from the effects that produce the requested answer. Choosing those effects before solving scale equations makes it possible to explain why a setting matters and to revise the comparison when the question changes. Comparing geometry alone would omit the different scaling of physical contributions. Requiring complete behavioral similarity for every use would exclude cheaper comparisons that preserve one useful result.

The physical and mathematical work remain connected. Physical reasoning supplies the laws, their regime and the realizable changes; mathematical operations expose joint conditions and transfer the consequence. Computation can solve those conditions or examine a mismatch. Their common inference and result-use methods are provided by FPF, while this pattern develops the physical scaling construction.

PHY.1:11 - SoTA-Echoing

The working question is how to obtain a useful physical result after changing scale or conditions. The selected approach derives and reconciles the relevant physical ratios and preparation, then handles unmatched effects at the output that matters. It combines established similarity reasoning with explicit result-specific transfer and revision.

Mahajan, The Art of Insight in Science and Engineering (2014), sections 5.2-5.3, provides a methodological anchor: dimensionless dependence can expose omitted physics while leaving an unknown coefficient or function. This changes :4.2-4.3 by making physical selection and the remaining unknown explicit. Its examples supply historical grounding rather than a complete current physics inventory.

Price’s 2024 MIT course emphasizes selecting a physically useful dimensionless basis and relating coefficients to competing terms. Adopt that choice in :4.3. NASA’s explanation of similarity parameters makes viscosity and compressibility consequences tangible. The warning is applied to effects relevant to the requested output; differing parameters can still permit a useful limited approximation.

Li et al. (2021), Spatial and temporal scaled physical modeling of fluid convection using hypergravity, report a way to reconcile gravity and viscosity scaling and identify limitations introduced by the apparatus. Their abstract motivates the changed-gravity alternative in :5.1. That case derives the scale factors from its stated conditions; reproducing the reported experiments requires their full apparatus method.

An alternative to a single matched comparison is reconstructing a result from several scales. Davey and Ochoa-Cabrero (2023), section 2.2, derive finite-similitude combinations under additional assumptions about scale dependence. This keeps the multiple-experiment return in :4.5 available when simple matching fails. Their higher-order construction has its own assumptions; the present method does not supply its full calculus or infer them from several measured points.

The output-specific loading and changed-question cases are this publication’s conceptual synthesis. Revisit the selected account when a new regime, source result or failed comparison changes a physical premise, available setting or valid approximation. New material or computation can make a previously infeasible comparison useful.

PHY.1:12 - Relations

B.5.FM and B.5.TU supply first-model construction and theory use when the physical account is still being formed. A.3.3.TR supplies state and the joint representation of change. B.5.MPC connects the physical, mathematical and computational contributions; B.5.MPC.R locates a failed connection after a change.

MMP.10 formulates coupled conditions; MMP.11 constructs unknown dependence; MMP.9 derives reduced evolution where eliminating detail exposes an unresolved contribution. MATH.11 constructs invariants from transformation rules. C.29.1 supplies the mathematical transfer or bound, C.29.2 the computation, and C.29.3 its realization through physical action and readout.

C.11.DUA selects further inquiry by what it could change and what it costs. B.1.5.CM develops the proposed composition of the working method; A.3.1.CC checks its claims and description when the comparison changes how participants obtain a result.

PHY.1:End

PHY.2 - Construct a Physical Analogue from Interactions

Type: Method Status: Stable Normativity: Normative within the stated use

PHY.2:1 - Problem frame

Use this pattern when you want to understand or investigate a physical situation through another arrangement, and the needed analogue must be built or changed. You may know the target’s laws but need an arrangement that makes their consequences accessible. Or you may be exploring an unresolved mechanism and need to construct a physical possibility that produces a consequence worth pursuing.

Here the target is the situation about which you want to learn. The source is the physical arrangement whose behavior supplies the analogy. The source may first be a thought construction: interacting bodies, material, fields or devices whose behavior can be reasoned about. It can then be expressed mathematically, simulated or built, according to the contribution the work needs.

Start with one target behavior and ask what physical interaction could produce it in a source. If two target parts affect one another, for example, ask what connection in the proposed source transmits that influence. The first result is a source construction, a consequence obtained from it, and an account of what that consequence permits you to infer or try in the target.

You need the relevant physical laws or someone able to supply them, together with the mathematical or qualitative reasoning required by the proposed construction. A ready analogue with a sufficient explanation of its use can be used directly. PHY.1 helps when the remaining difficulty is constructing conditions for a change of scale. C.29.1 helps when the physical constructions are already available and the unresolved work is the correspondence between their mathematical descriptions.

PHY.2:2 - Problem

A resemblance can suggest an analogy while leaving the source unable to produce the behavior that matters. Two components that behave appropriately in isolation can behave differently when connected. A source can also reproduce one target observation through a mechanism that gives a different response to an intervention.

The difficulty is building the source itself. Its components, interactions, preparation and surroundings must jointly produce a useful consequence. When the target mechanism is unresolved, construction and criticism of the source also refine the hypothesis about the target. An unsuccessful source can expose the missing interaction or a physical restriction that the next attempt must address.

PHY.2:3 - Forces

PHY.2:4 - Solution

Choose the target consequence → propose physical mechanisms for it → connect their interactions → recover preparation and correspondence → derive or produce a consequence → use it and revise the construction.

PHY.2:4.1 - Choose what the source must help you obtain

State the target question in the terms of the work. You might need a response to an input, a possible mechanism for an observed change, a limit on a proposed behavior, or an arrangement that lets you manipulate an otherwise inaccessible relation. Say what is already known and what remains proposed about the target.

Follow the intended consequence to the relevant participants and interactions. A force on one body may change another through a stored deformation; a change at a boundary may propagate through transport. Recover the target relation that the source must help explain or use. B.5.FM and B.5.TU support this first account and its relation to a theory.

Keep the first construction as small as that question permits. For a question about coupling, two interacting parts can expose what a whole-device sketch leaves implicit. If the target mechanism is unknown, begin with an observed dependence and a proposed way it could arise. The proposal gives something to develop even before a complete target theory is available.

PHY.2:4.2 - Propose mechanisms by the physical effect they supply

Ask what the source must retain, transfer, resist, amplify, constrain or change. Then choose physical processes that can perform those contributions under known laws. These are prompts for selecting relevant effects, not a required inventory of every source.

For example, a mass retains motion while a force changes it; a deformable body stores energy under loading; a capacitor accumulates charge while current flows; a dissipative interaction converts organized motion into less accessible energy. A driven body can maintain motion by drawing on an energy supply. A stochastic transition account can express unresolved switching between physically different states. Choose among such mechanisms from the target relation and the proposed source conditions.

Write or explain the law of each chosen interaction, its participants and its regime. Include direction and sign: a coupling can restore a displacement or increase it. Identify what is being approximated when a law is linearized or treated as instantaneous. When no suitable law is available, retain the interaction as the physical contribution still to be found.

A source can combine contributions from several familiar arrangements. Derive the behavior of that combination. A chain of separately plausible resemblances can leave an unconstructed connection in the middle; locate the participants and interaction at that connection. Mathematical formulation through MMP.10 or MMP.11 makes the resulting conditions easier to solve.

PHY.2:4.3 - Construct the connection and its effect on the whole

For each connection that carries the target influence, identify what the connected parts share and what each can exchange. Write the interaction conditions together with the component laws. At an electrical junction, current conservation and common voltage connect the components; in a mechanical contact, the allowed motion and transmitted forces do that work. Other interactions need their own physical conditions.

Recover the state needed to continue the coupled behavior. The source may need a relative deformation, stored charge, field or direction of motion that neither isolated-component sketch retained. A.3.3.TR supplies the general state and joint-change construction. Here identify the physical means by which the additional state persists and affects the next change.

Check the physical restrictions that can defeat the construction. For an energy-carrying interaction, account for the relevant storage, exchange, dissipation and driving. A source that uses only passive dissipative elements cannot supply sustained amplification; an active source needs an identified supply. For a transport or switching mechanism, include the travel time, waiting law or constraint that changes the intended result. The useful restriction depends on the source, so follow it to the target consequence.

Use these restrictions to improve the construction. Add the missing coupling, replace an unsuitable element, change a boundary or keep a result limited to the regime the source can produce. If the target’s required relation conflicts with the source’s physical laws, choose another source or revise the hypothesis. This conflict is an informative result of construction.

PHY.2:4.4 - Recover preparation, interpretation and available range

Describe how the relevant source state starts and how it is driven. The initial deformation of a target spring, for example, can require stored current in an electrical source. Setting every source state to zero would solve a different initial-value problem. Likewise, the preparation of a distribution includes more than its mean when its later behavior depends on that distribution.

Connect each source quantity used in the result to what it represents about the target. Include units and scales: source voltage may represent target velocity through a dimensional conversion, and source time may run faster. Derive that correspondence by substituting the conversion into the laws and preparation. C.29.1 provides the general argument that a source result transfers to the intended mathematical target.

When proposing a material realization, determine whether its components can supply the required signs, ranges and rates together. Include the effect of observing or driving the source if it changes the relevant behavior. A nominally passive observation can load an electrical node; a finite switching time can matter when the requested event is short. C.29.3 supplies the preparation, action and readout relation for physical computation.

For a thought construction, state the physical idealizations on which its consequence depends. That conditional result can already direct a useful change or inquiry. Building an apparatus is a further choice, selected when its outcome would add a worthwhile contribution through C.11.DUA.

PHY.2:4.5 - Obtain a consequence and determine its reach

Manipulate the source according to its proposed laws. Start with a small input, a change of one interaction, or a limiting case that exposes how the construction works. Obtain the consequence by a derivation, a qualitative physical argument, a computation or an experiment. Use a known limiting case to find an omitted interaction or a sign error where it can discriminate between constructions.

Trace the obtained consequence back through its physical and mathematical premises. A computation can determine a property of the proposed source model. Interpreting an apparatus reading adds the account of the apparatus. Applying either result to a physical target adds the target correspondence and the physical premises used there. An existing explanation can supply those premises; no new evidence collection follows merely from enumerating them.

If the target mechanism remains proposed, use the result as a consequence of that hypothesis. Ask what change would make competing mechanisms disagree. This can reveal a useful intervention, a different observation interval or a parameter relation to examine. MMP.7 helps when the proposed distinction must be expressed through recorded data; C.11.DUA determines whether obtaining that distinction is worth the effort.

Different sources can contribute complementary results. One may expose an invariant or a bound, another the effect of noise or a difficult regime. Compare them for the usefulness, range and cost of the required contribution. G.5 distinguishes alternatives retained for later choice, non-dominated candidates and a set whose members contribute together. E.23 supports repeated improvement when a stated evaluation can judge the changed source.

PHY.2:4.6 - Return to the target and change the useful part

Use the result in the target work: choose a preparation, interpret an observation, propose a mechanism, reject a construction, or change the design of an experiment. Explain the physical route that makes the result relevant. EXD.3, in Explanation Design DPF, helps coordinate the words, diagram and equations another participant needs to recover that reasoning. C.2.8 characterizes the structure a recipient can extract with their available preparation and effort.

When a consequence fails or the question changes, locate the affected connection. Did the chosen source law fail, did connection change the source behavior, did computation or observation distort the result, or did the target correspondence omit an important difference? B.5.MPC.R supports this diagnosis. Revise the physical mechanism, state, preparation or correspondence that the failure implicates and retain the contributions that still work.

Keep a newly revealed question when pursuing it could open useful action. The next task can concern the target, the source or their connection. If the new construction changes what participants must produce or how their contributions join, B.1.5.CM helps develop that proposed composition. A.3.1.CC checks claims about the method and its description, and locates the contribution that needs correction.

PHY.2:5 - Archetypal Grounding

These constructed examples show two uses: obtaining consequences of a known physical account and developing a proposed physical mechanism. The calculations describe idealized sources; they report no apparatus measurements.

PHY.2:5.1 - Build the missing coupling in a mechanical/electrical analogue

Two bodies move along a line, joined by an ideal spring. Let their masses be m1 and m2, velocities v1 and v2, and the spring extension be d. External forces are f1 and f2; linear drag coefficients b1 and b2 are nonnegative. With spring stiffness k>0, the physical laws are:

m1*dv1/dt = f1 - b1*v1 - k*d
m2*dv2/dt = f2 - b2*v2 + k*d
dd/dt = v1 - v2.

The question is how a deformation transfers motion between the bodies. Two independent electrical elements for the masses would omit that interaction. Choose node voltages to represent velocities, injected currents to represent external forces, and charge-accumulating capacitors for the masses. A conductance from each node to the reference node can represent its drag. The missing coupling needs a state whose rate is proportional to the difference of the two voltages. An inductor connected between the nodes supplies that relation.

Let s=r*t be source time, V_i=a*v_i the voltages and I_i=h*f_i the injected currents, with positive dimensional conversion factors a and h and positive time ratio r. Let J be current through the inductor from node 1 to node 2. Current conservation and the ideal inductor law give:

C1*dV1/ds = I1 - G1*V1 - J
C2*dV2/ds = I2 - G2*V2 + J
L*dJ/ds = V1 - V2.

Substitute the conversions and J=h*k*d. All three relations match the mechanical account when:

C_i = r*h*m_i/a
G_i = h*b_i/a
L = r*a/(h*k).

The constructed inductor carries the coupling state. The initial source state must satisfy V_i(0)=a*v_i(0) and J(0)=h*k*d(0). For m1=2 kg, m2=1 kg, k=3 N/m, d(0)=1 m, zero velocities and zero applied forces, the first accelerations are -1.5 and +3 metres per second squared. The prepared inductor current produces the corresponding voltage changes. With J(0)=0, the source would instead remain at rest and miss the deformation-driven motion.

The stored mechanical energy is (m1*v1^2 + m2*v2^2 + k*d^2)/2. Differentiating gives power f1*v1 + f2*v2 - b1*v1^2 - b2*v2^2. The circuit’s stored energy is r*a*h times the mechanical energy; its supplied and dissipated powers have the compatible factor a*h. The correspondence therefore includes storage and transfer under the stated ideal laws.

The first result is a source design and interpreted initial response. Before building it, choose the conversion factors so capacitances, conductances, inductance, voltages and currents lie in the available ranges. Include losses or loading that would alter the requested response.

Now change the target: a controller supplies a force that increases with velocity, producing effective negative damping over a stated range. A nonnegative conductance cannot realize that contribution. The changed physical construction needs an active element and its energy supply, or another means of obtaining the result. The previously constructed passive analogue still answers the original dissipative question.

PHY.2:5.2 - Construct a mechanism whose short-time consequence differs from diffusion

A spreading population of moving particles is approximately diffusive over long observations. The target question is whether that approximation can describe the first short interval after a localized release. Consider a proposed mechanism in which motion persists for a while before its direction changes.

Build a simple source in thought: independently driven shuttles on a line move at speed u, reversing direction at random times. The waiting times are independent exponentials with reversal rate lambda. A drive maintains the speed, and a controller supplies the reversals. Treat reversal duration as negligible relative to the intervals being considered. This source combines sustained physical motion with stochastic switching. Using it for the particles proposes a relation between their direction persistence and their observed spreading.

The motion already supplies a useful qualitative consequence. Starting at x=0, a shuttle travels path length u*t in elapsed time t; reversals can only reduce its distance from the start. Thus abs(x(t)) <= u*t. Reaching a point at distance d requires at least time d/u. An ideal diffusion law with positive diffusivity instead assigns positive probability beyond every finite distance at every positive time. These are different predictions even when their long-time spreading agrees. An arrival before d/u would contradict the proposed bounded-speed target mechanism under its stated speed and preparation. Whether an earliest-arrival observation would discriminate the mechanisms also depends on its resolution and on the probability predicted for such arrivals. The bound is already a source result; the target correspondence remains a hypothesis.

For a small interval dt, reversal has probability lambda*dt to first order. If p_plus and p_minus are the position densities of right-moving and left-moving shuttles, transport and exchange between the two states give:

partial_t p_plus  = -u*partial_x p_plus  - lambda*p_plus + lambda*p_minus
partial_t p_minus = +u*partial_x p_minus + lambda*p_plus - lambda*p_minus.

The total density is p=p_plus+p_minus and the flux is j=u*(p_plus-p_minus). Adding and subtracting the equations yields:

partial_t p = -partial_x j
partial_t j = -u^2*partial_x p - 2*lambda*j.

Direction is the additional state that lets the source retain motion between changes. Omitting it too early would erase the short-time effect under investigation.

Release all shuttles at x=0 with equal probabilities of the two directions, on an unbounded line. The mean position stays zero. Multiplying the equations by x and x^2 and integrating, with vanishing boundary terms, gives for the mean-square displacement M(t):

M''(t) + 2*lambda*M'(t) = 2*u^2
M(0) = M'(0) = 0
M(t) = (u^2/lambda)*[t - (1-exp(-2*lambda*t))/(2*lambda)].

At short times M(t) is approximately u^2*t^2: particles mostly retain their direction. At long times its leading growth is (u^2/lambda)*t, corresponding to diffusion coefficient D=u^2/(2*lambda). The source thus produces a long-time diffusion law while giving a different short-time consequence. If u and lambda both have numerical value 1 in the chosen units, M(1) is approximately 0.568 square length units, while the leading diffusion expression gives 1.

The first result is a conditional explanation and a candidate change of observation interval. Long-time diffusive behavior alone leaves the proposed persistence mechanism unresolved. A useful return is to compare direction correlation or early spreading when either could distinguish it from a competing mechanism. An existing measurement can suffice. A physical shuttle apparatus adds no value if the derivation already supplies the consequence the work needs.

Now suppose target turns have an appreciable duration, or their occurrence depends on how long the present run has lasted. The source’s instantaneous, memoryless reversal rule no longer supplies that target behavior. Construct the missing turn state or waiting-time dependence, recover its physical interpretation, and derive its effect on the requested interval. The previous model remains a limiting construction where those effects are negligible. For a material shuttle implementation, finite acceleration also limits how short an interval it can reproduce.

PHY.2:6 - Bias-Annotation

The worked cases use classical interaction laws and stochastic switching that can be manipulated through ordinary differential reasoning. Other physical regimes can require different permitted states, transformations and observations. Reuse the construction method with the physical laws of that regime; the elementary sources here supply examples of it.

A familiar source can be easier to imagine than a suitable one. Follow the behavior required by the target question when choosing and revising mechanisms. Collaborators can supply unfamiliar physical contributions. Human mental-model studies motivate part of this approach; applying the method with AI contributors requires checking the construction they actually produce and the prerequisites its users possess.

PHY.2:7 - Conformance Checklist

• The target question and the source’s intended contribution can be stated in working terms.
• Proposed source mechanisms have physical laws and conditions sufficient for the consequence being obtained, or the unresolved contribution is identified.
• The connection includes the interaction, retained state and physical restrictions that can alter the result.
• Preparation, driving, interpretation and any scale conversions describe the same constructed case.
• A derived or produced consequence leads to an appropriate target inference, intervention, design choice or further question.
• A proposed target mechanism retains the premises still needed to use it; further inquiry is chosen by its possible contribution and cost.
• A changed question or failed consequence returns to the affected mechanism or correspondence while preserving useful existing results.

PHY.2:8 - Common Anti-Patterns and How to Avoid Them

PHY.2:9 - Consequences

The method can create a useful source where retrieval of a ready analogy fails. Construction exposes missing interactions, incompatible physical requirements and consequences that were difficult to see in the target. Those discoveries can refine both the source and the question.

Its cost is reasoning about two physical situations and their connection. An elaborate apparatus or simulation can be unnecessary when a short construction already answers the question. The source can also become misleading if its convenient mechanism is carried into the target beyond the inference it supports.

PHY.2:10 - Architectural Rationale

The operative contribution is constructing physical mechanisms that can work together. Correspondence becomes usable after there is enough source behavior to compare. Preparation, coupling and the means of producing an effect therefore belong inside construction, where they can change the selected source.

The method accommodates both a known target account and development of a target hypothesis. Their first results differ: a derived transfer can answer a target question under its physical premises; a proposed mechanism can reveal a consequence that directs inquiry. This keeps early physical imagination useful while making the strength of its result recoverable.

FPF supplies common first-model reasoning, state construction, correspondence, computation and inquiry choice. Mathematical Thinking and MMP supply the required mathematical operations and formulation methods. The physical contribution here selects interactions, their connections and their attainable behavior. A circuit or a moving-shuttle construction demonstrates that contribution.

PHY.2:11 - SoTA-Echoing

The working question is how to construct a useful physical source when no ready analogue supplies the required behavior. The approach combines iterative source construction with explicit physical interactions, preparation and result use.

Nersessian (2025), section 3, examines scientific cases in which the analogy source is itself developed through construction and manipulation. Adopt that contribution in :4.2-4.6: source revision can refine the target problem as well as its candidate solution. Her studies support a human scientific practice; the allocation of work among people and AI here is a methodological extension.

De Benedetto and Poth (2025, first online 2024), section 3.1, reconstruct how combined representational resources can create new inferential possibilities. This supports keeping mechanisms from different sources available during construction. The present pattern develops the physical work of connecting them rather than adopting their full theory of concept learning.

Hangleiter, Carolan and Thebault, Analog Quantum Simulation (2022), sections 8.3.3-8.4, distinguish the source apparatus, its mathematical description, the formal target and the physical target. Their analysis changes :4.4-4.5 by locating the premise needed for each inference. The general connection is reused from C.29.1 and C.29.3. Their validation discussion is applied according to the conclusion needed; a conditional hypothesis remains a useful result when a stronger target claim is unresolved.

Angelani (2016), section 2, supplies a transparent transport-and-switching construction for the second worked case. His tumble events choose a new direction, whereas lambda here counts actual reversals; the rate convention is stated in the case. Datta, Beta and Grossmann, version 3 of The random walk of intermittently self-propelled particles, makes richer run/turn dynamics and waiting-time distributions available. Its section II supports the return when the simple switching assumption fails. The one-dimensional shuttle example is a constructed illustration, not a reproduction of those authors’ apparatus or a complete active-matter model.

The dimensional circuit construction and paired changed-condition uses are this publication’s synthesis. Revisit a source choice when a new law, available mechanism or observed disagreement changes the correspondence or reveals a useful physical possibility. A new example alone leaves the general method unchanged.

PHY.2:12 - Relations

B.5.FM and B.5.TU connect the target question with an initial physical account and theory. A.3.3.TR supplies the general construction of state and combined change. PHY.1 develops physical similarity conditions when scaling or changing a material is the operative problem.

C.29.1 establishes the correspondence used in inference; C.29.2 formulates the computation; C.29.3 connects a physical realization with preparation and readout. MMP.10 and MMP.11 formulate and develop the mathematical relations. MMP.7 connects a proposed physical distinction with a model of recorded observations. MMP.9 supplies the mathematical reduction when removing a physical state changes the retained evolution.

G.5 distinguishes the results of choosing alternatives or complementary contributions; E.23 governs repeated improvement under a stated evaluation. C.11.DUA selects worthwhile inquiry and use of sufficient results. B.5.MPC and B.5.MPC.R connect the physical, mathematical and computational contributions and locate a failure between them. C.2.8 characterizes extractable structure, and EXD.3 helps express the explanation; B.1.5.CM develops the proposed composition account; A.3.1.CC checks its claims and locates a needed correction.

PHY.2:End

PHY.9 - Construct a Measuring Interaction for a Physical Distinction

Type: Method Status: Stable Normativity: Normative

PHY.9:1 - Problem frame

Use this pattern when a physical difference matters to the work, but the available arrangement does not make that difference observable. You may need to choose a sensing effect, prepare a reference, separate a wanted response from another influence, or change a probe that disturbs its subject.

The first result is a proposed measuring interaction with a usable response: what to prepare, what to couple, what to observe, and which physical differences the response can resolve. It can remain conditional on an identified coupling, calibration or operating range.

C.16.MR constructs the relation between a sought property and an indication. This method develops its physical realization when that realization is missing or inadequate. The work concerns designing an interaction, before or after a measurement exists. It need not produce a new instrument; changing a preparation or comparison can suffice.

You need to interpret the relevant physical theory and the conditions under which its effect occurs. Mathematical or experimental collaborators can supply calculations and implementation while you retain the wanted distinction. The examples supply their additional mechanical, fluid and quantum premises.

Use an existing arrangement and calibration when they already resolve the question. If the physical arrangement is suitable and only inference from its records remains, use C.16.IR or MMP.7. A more elaborate sensor is unnecessary when the present response already supports the decision.

PHY.9:2 - Problem

A property can affect a physical system without appearing in the chosen readout. Mass occurs in the force and inertia of a falling body but cancels from ideal gravitational acceleration. A field can change a quantum state’s phase while the selected measurement remains insensitive to it.

A responsive instrument can also confuse the sought property with its own interaction. A capillary responds to pressure through a liquid column, but surface forces contribute to the height. Repeating the same reading more precisely does not separate those contributions.

The difficulty is to construct a physical comparison that exposes the wanted difference under the available conditions.

PHY.9:3 - Forces

PHY.9:4 - Solution

Choose the wanted distinction → select a responsive physical effect → construct the preparation and comparison → derive the readout and its disturbance → determine what can be resolved → revise the arrangement where needed.

PHY.9:4.1 - Specify the physical distinction and conditions

Name two physical possibilities that the work needs to distinguish, or a quantity range within which an estimate is needed. State where and when the property is sought. A pressure before connecting a probe and the pressure maintained during readout can be different targets.

Choose the needed consequence. Detecting a change, estimating a value and deciding which side of a limit applies place different demands on the same arrangement. C.16 provides the characteristic and scale; C.16.IR determines what a given indication can resolve.

Include the conditions the work permits you to change. A prepared reference, imposed force, interrogation time or detector orientation is a potential experimental choice only when it can actually be controlled.

PHY.9:4.2 - Choose an effect that carries the distinction into a response

Find a physical interaction through which the sought difference changes a state, transition, force, flow or other observable response. Derive that dependence far enough to see whether the proposed readout retains it. PHY.4 can constrain an unknown law; PHY.6 or PHY.7 can construct the relevant physical evolution.

Compare a small number of plausible arrangements. For each, ask what differs in the predicted readout between the selected physical possibilities under the same preparation. Reject an insensitive arrangement for this question even if it uses a well-established effect. The falling-body case in :5.1 shows why the occurrence of a quantity in one equation is insufficient.

Identify what supplies a reference. A known imposed quantity, a second preparation or a stable response can make the comparison interpretable. State the physical relation that allows the reference to be transported into this measurement. Numerical labels on two devices do not establish that relation.

Retain an unknown coupling as unknown. An available bound or a calibration can be enough; a proposed response without grounds for its coupling remains conditional. Select new characterization only when it can change the intended use.

PHY.9:4.3 - Construct the preparation and comparison

Specify how the subject, probe and reference are prepared before coupling. Then choose what is held fixed, what is varied and what is read. Include the order when the first interaction changes the next state.

Use a differential, reversed, ratio or null comparison when its physical invariance removes a consequential influence. Derive the cancellation from the arrangement. For example, reversing a known applied force while preserving a common force offset changes the wanted response but leaves the offset shared. A reversal that also changes the offset does not support that subtraction.

For a quantum probe, choose a prepared state, the parameter-dependent interaction and the measurement together. The physical interaction may be described by a channel taking rho to rho_theta; measurement operators E_k give outcome probabilities P(k|theta)=Tr(rho_theta E_k). Here rho is the prepared state, theta the sought parameter, and the positive operators E_k sum to the identity. If those probabilities do not change with theta, that preparation and readout supply no information about it.

A reference can also select which part of a response is read. A phase-sensitive measurement needs a phase reference; a differential displacement needs a reference position. Preserve the conditions that make this reference stable during the comparison.

PHY.9:4.4 - Include the effect of measuring on the subject

Construct the coupled subject-probe account for the duration that matters. Determine whether connecting the probe changes the sought quantity, changes only another quantity, or leaves its relevant value effectively unchanged. Use the physical coupling rather than a universal claim that measurement must disturb the target.

If the wanted value precedes coupling, derive how it can be recovered from the disturbed response, or change the arrangement to reduce the consequential disturbance. Keep uncertainty in that recovery. When correction is poorly determined, a different interaction or a conditional range can be more useful than more decimal places.

Compare the retained effect with the allowed error or decision margin. Loading, finite response time and backaction matter through their consequences for this question. PHY.5 helps justify an approximation. A usable result does not require modeling every interaction.

A nondestructive or non-demolition measurement protects a specified property under particular conditions; it does not establish that every later use of the subject is unaffected. Preserve any disturbance that changes the receiving experiment or work.

PHY.9:4.5 - Determine the distinctions the readout supports

Compose the physical response with the actual recording procedure through C.16.MR. Use MMP.7 when the conclusion requires a probability law for those records. Retain common influences, saturation, timing and finite resolution when they affect the conclusion.

Determine whether different target values remain compatible with the same possible records. If they do, identify which interaction, preparation or readout could separate the relevant alternatives. A local slope measures a local response; it does not resolve a periodic ambiguity or a shared unknown parameter.

Use the uncertainty or error model appropriate to the inference. Strongly nonlinear conversion can change both an estimate and its uncertainty; do not assume that converting an average is equivalent to averaging the converted quantity. Apply the necessary mathematical propagation or inference method, rather than treating a sensitivity coefficient as a complete answer.

When several parameters or performance goals matter, compare achievable combinations under the available resources. Different measurements can favor different parameters. Use C.16 to characterize those combinations and C.11 when choosing between available arrangements.

PHY.9:4.6 - Obtain a first result and revise the physical arrangement

Work one admissible preparation through to a predicted readout and an interpreted result. This can reveal an unresolved physical relation before an instrument is built. It also gives collaborators a concrete construction to implement or compare.

Change a condition that could defeat the intended use. Check the shared premise of a subtraction, the range of an inverse, the time available for a response or the disturbance caused by the probe. Return to the smallest implicated part of the arrangement.

Use available observations where they can change confidence or selection. C.11.DUA helps choose between another measurement, a conditional result and proceeding with uncertainty. A statistical comparison or a calibration certificate does not substitute for a missing physical response.

Return the proposed arrangement, its preparation and readout, the result it supports and the conditions that would change that use. The next work may be implementation, calibration, inference or a different measuring interaction. PHY.10 constructs a physical test when the purpose is to separate rival accounts.

PHY.9:5 - Archetypal Grounding

PHY.9:5.1 - Make mass affect the observation

Two bodies fall in a sufficiently uniform gravitational field with negligible drag. The equation m a=m g gives a=g. Timing their fall cannot distinguish their masses in this idealization, even though gravity exerts different forces.

Choose instead a known horizontal applied force F whose generation does not depend on the unknown mass. Track the body’s acceleration under that force. With other horizontal forces negligible, m a=F gives m=F/a for nonzero a. The physical change is from a force proportional to the unknown mass to an independently supplied force.

Now a constant horizontal force offset b matters during the short observation. Use two otherwise identical preparations with applied forces +F and -F. Assume the same b and the same mass in both, and measure their initial accelerations:

m a_+=F+b, m a_-=-F+b.

Subtracting gives

m=2F/(a_+-a_-).

With F=6 N, a_+=4 m/s² and a_-=-2 m/s², the mass is 2 kg and b=2 N. Using only F/a_+ would give 1.5 kg. The controlled reversal made mass distinguishable from the shared force offset.

The two preparations must preserve that offset. Velocity-dependent drag need not do so after the trajectories diverge. Measuring comparable initial responses, including the changed drag law, or selecting another arrangement can repair that use. Taking more samples of two physically different offsets does not justify the shared-b subtraction.

The acceleration readout and the force reference retain their own calibration and uncertainty. A decision whose margin exceeds those effects can use the mass estimate; a tighter use returns to the consequential contribution.

PHY.9:5.2 - Separate pressure from the probe’s surface force

A large liquid reservoir maintains an unknown gauge pressure p during a small probe’s readout. A vertical circular capillary opens to the atmosphere. Let rho be liquid density, g gravitational acceleration, r the tube radius and h the meniscus height above the pressure reference. Assume hydrostatic equilibrium and a capillary regime in which the meniscus curvature is described by the wetting angle theta.

Hydrostatic pressure and the surface-pressure jump give

p=rho g h - s/r, where s=2 gamma cos(theta)

and gamma is surface tension. The height responds to the wanted pressure and to the capillary surface force.

If s is unknown but the same surface condition can be prepared in two narrow tubes of radii r_1 and r_2, use

rho g h_1=p+s/r_1, rho g h_2=p+s/r_2.

For distinct radii,

p=rho g (r_1 h_1-r_2 h_2)/(r_1-r_2).

Take rho g=10000 Pa/m, r_1=0.1 mm, r_2=0.2 mm, h_1=0.20 m and h_2=0.15 m. The inferred pressure is 1000 Pa and s=0.10 N/m. Interpreting the first height as p=rho g h_1 would give 2000 Pa. Both narrow radii must remain within the chosen capillary approximation.

Now suppose the two surfaces have different, unknown wetting conditions. The equations contain separate s_1 and s_2. The cancellation no longer determines p: two readings with three unknown quantities leave a consequential freedom. Restore the common surface condition, use useful bounds on the surface terms, or select another pressure-sensitive interaction.

The reservoir premise also matters. If filling the capillary changes the pressure whose earlier value was wanted, include the reservoir-probe volume and pressure relation through PHY.6 and C.16.MR. The maintained-pressure calculation cannot by itself recover that earlier value.

PHY.9:5.3 - Prepare a probe whose phase difference can be read

An ideal two-level probe couples to a constant classical field B through

H=(hbar gamma B/2) sigma_z,

where gamma is a known coupling coefficient and sigma_z has eigenvalues +1 and -1 for states |0> and |1>. Treat the field as unchanged by this probe during the interrogation time t. The wanted quantity is B within a supplied range.

Prepare |0>. Evolution multiplies it by a global phase; measurements of this state cannot reveal that phase. Increasing t alone does not create a readable field dependence.

Instead prepare |+>=(|0>+|1>)/sqrt(2). Evolution produces a relative phase phi=gamma B t. Reading sigma_z still gives equal probabilities and is uninformative about phi. A controlled rotation before detection permits a sigma_x or sigma_y measurement. Their means are cos(phi) and sin(phi), so the two preparations for readout can recover phase modulo 2 pi. Each mean requires the corresponding repeated preparation and measurement.

For gamma=1 rad/(s·field-unit), t=1 s and B=pi/3 field-units, the ideal means are 1/2 and sqrt(3)/2. They select phi=pi/3 modulo 2 pi. A supplied range -pi < gamma B t <= pi makes that field value unique.

If the field may instead lie between -4 pi and 4 pi field-units, the same records admit several values. Shorten the initial interrogation, for example to t=0.1 s with the same gamma: the whole supplied range then lies inside the unambiguous phase interval. A later longer interrogation can refine a value once the remaining range supports its interpretation.

Coherence loss, imperfect rotations or an uncertain gamma change the response law. They can be included or constrained where the required estimate needs them. The local phase response alone is not a claim that arbitrarily long interrogation or another quantum resource improves every sensing task.

PHY.9:6 - Bias-Annotation

The examples use controllable idealized arrangements to expose the construction. A field situation may limit access, preparation, reversibility or repeated sampling. Use those limits in choosing the interaction rather than treating the ideal case as an available apparatus.

Quantum sensing is one implementation of a wider physical move. Classical comparisons, chemical response and mechanical probes can serve the same construction when their physical relations and receiving use fit.

PHY.9:7 - Conformance Checklist

• The target distinction includes the physical conditions under which it is sought.
• The proposed interaction makes that distinction affect the selected readout.
• Preparation, reference and comparison operations are physically interpretable.
• A cancellation or transferred calibration retains its shared physical premises.
• Disturbance and finite response are included when they change the wanted result.
• Local sensitivity is not substituted for resolution over the required range.
• The first result, remaining ambiguity and useful return to the arrangement are clear.

PHY.9:8 - Common Anti-Patterns and How to Avoid Them

PHY.9:9 - Consequences

The measurement can be improved by changing how information is physically produced. Better inference remains useful, but its contribution is tied to an interpretable preparation and response.

Different contributors can construct the coupling, calculate the response, prepare the reference and infer the result without losing the common physical question. When an existing arrangement suffices, the method ends with its use.

PHY.9:10 - Architectural Rationale

The measurement relation and the measuring interaction solve connected difficulties. A relation can be derived for a fixed setup that is incapable of resolving the wanted difference. Constructing a different coupling, preparation or reference changes what the relation can supply.

This physical construction links theory to observation and intervention. It also links physical computing to readout: a physical state can carry a computed result that the chosen observation does not distinguish. The mathematical and statistical interpretation remains with the corresponding methods; it receives the interaction and its conditions.

PHY.9:11 - SoTA-Echoing

The VIM definitions of measurement principle and measurand distinguish the phenomenon supporting a measurement from the quantity sought under specified conditions. JCGM GUM-6:2020, especially sections 7.2 and 9, explains theoretical, empirical and hybrid response models and effects of realizing a measurement.

Compare the two mass procedures in :5.1. Timing ideal free fall is simpler but supplies no mass distinction. A known-force comparison supplies it at the cost of a force reference and acceleration readout. Reversal adds another preparation only when separating the force offset matters. Neither a more accurate clock nor more identical falls supplies the missing physical dependence.

Degen, Reinhard and Cappellaro, Quantum Sensing, Reviews of Modern Physics 89 (2017), is a foundational synthesis of preparation, coupling, control and readout. Pezzè and Smerzi, Advances in multiparameter quantum sensing and metrology, 2025, extends the selection question to several parameters, incompatible optimal measurements and finite resources. The adopted contribution is the joint choice of probe, encoding and readout; a quantum sensitivity bound needs its own estimation and attainability assumptions.

For the phase question in :5.3, the same Hamiltonian with an eigenstate probe gives no observable phase, while a superposition and suitable readout do. Short interrogation resolves the wider range; longer interrogation can improve local sensitivity while requiring range information and adequate coherence. More complex entangled or adaptive arrangements require a comparison under the actual resource and uncertainty conditions, not an advantage inferred from their name.

The 2026 amendment to JCGM 100 explicitly addresses significant nonlinearity in both a measurement estimate and its uncertainty. It supports returning from a local sensitivity approximation when the receiving calculation depends on curvature. The applicable mathematical inference remains separate from construction of the physical coupling.

The numerical cases are constructed demonstrations under stated laws and idealizations. A changed material, interaction, preparation or readout range reopens the corresponding physical premise.

PHY.9:12 - Relations

• C.16.MR: constructs and composes the relation from the sought property and measuring arrangement to an indication.
• C.16.IR: determines which target distinctions a supplied indication or response relation can resolve.
• PHY.4, PHY.6 and PHY.7: constrain a physical law or construct the evolution needed to derive a response.
• PHY.5: determines whether an interaction, disturbance or response time can be neglected for the intended consequence.
• PHY.8 and MMP.7: construct physical statistics and the probability law of recorded outcomes when variation and recording affect the result.
• PHY.10: uses a measuring interaction in a physical test separating rival accounts.
• C.29.2 formulates the needed computation; B.5.MPC connects what it produces with the mathematical claim and the physical question.
• C.16 and C.11: characterize attainable performance and resource combinations, then choose among available arrangements.
• C.11.DUA: selects further measurement or inquiry by what it can change and the effort it requires.

PHY.9:End

PHY.10 - Construct a Physical Test That Separates Rival Accounts

Type: Method Status: Stable Normativity: Normative

PHY.10:1 - Problem frame

Use this pattern when competing physical accounts remain compatible with what has been observed, and the work needs a distinction between them. The missing contribution is a physical test: a preparation, intervention or understood comparison whose recorded consequences can separate the relevant alternatives.

A resistor and a capacitor can produce identical current traces under one imposed voltage history. Changing that history can expose the difference between dissipation and storage. A decaying collective signal can also hide different individual dynamics; a suitable intervention can make those dynamics distinguishable.

The first result is a proposed physical comparison, with the response each account permits, the conditions needed to produce and read it, and what the possible outcomes would change. It can reveal that the available apparatus cannot make the wanted distinction. A performed test adds observations and an interpretation under those conditions.

You need the physical meaning and applicable laws of the accounts being compared. Mathematical, computational and experimental collaborators can supply particular derivations and implementations. B.5.TC helps align the accounts when their questions or predicted quantities still differ. PHY.9 constructs a missing measuring interaction.

Use an existing discriminating observation or sufficient comparison directly. Leave the difference unresolved when it does not change the intended use, or when acting with the remaining uncertainty is preferable to another test.

PHY.10:2 - Problem

Agreement with an observed response can have several physical explanations. A parameter can compensate for a missing interaction. A particular drive can make two response laws coincide. Different hidden dynamics can produce the same aggregate decay.

Collecting more data under the same uninformative conditions can leave the disagreement intact. Conversely, changing conditions to enlarge a predicted difference can introduce a new interaction, defeat the detector or leave the range in which either account was proposed.

A discrepancy also tests the combined account of the subject, preparation and observation. It need not identify the disputed physical premise by itself. The practitioner needs a comparison that exposes that premise while retaining the alternatives introduced by the apparatus and operating conditions.

PHY.10:3 - Forces

PHY.10:4 - Solution

Locate the consequential physical difference → find a condition that exposes it → construct the preparation and readout → compare the permitted responses → use the result and locate any defeated premise → develop the next useful test or stop.

PHY.10:4.1 - State which physical disagreement matters

Ask the accounts the same physical question under corresponding conditions. Recover what each treats as the subject, its state, its interactions and the measured quantity. Use B.5.TC if the apparent conflict instead concerns different questions, representations or approximations.

State the consequence the work needs to distinguish. It may concern a law, an omitted interaction, stored state, a response time or another physical dependence. A distinction matters through what it changes in explanation, prediction, construction or subsequent inquiry.

Retain the parameter freedom that each account still has. An account with an adjustable coefficient is a family of possible responses, not just the curve at one fitted value. Include the available information that restricts that freedom. Do not choose a new parameter value independently at every observation unless the account itself permits that dependence.

Identify the auxiliary premises on which the comparison turns. For a driven specimen, these can include its preparation, the applied input, its coupling to surroundings and the relation between its response and the recorded indication. A test of the specimen’s law can fail because one of these premises fails.

PHY.10:4.2 - Find a physical change that makes the accounts diverge

Trace the disputed difference forward to an observable consequence. Then ask which feasible change exposes it. For example:

• Change an amplitude, scale or operating regime when the accounts predict different dependence on it.
• Change the time course of a drive when the earlier input made different response laws coincide.
• Interrupt, reverse or refocus an interaction when the accounts retain different state or memory.
• Use a symmetry or balance when one account requires a response to vanish or remain within a bound.

Choose the change from the physical dependence at issue. PHY.4 supplies law constraints, PHY.5 examines a regime change and PHY.6-.8 derive the corresponding evolution or collective response.

Recompute both accounts under the proposed change. A prediction under the original preparation cannot be compared with a rival’s prediction under the new one. Retain any changed interaction that can imitate or obscure the intended difference.

If intervention on the subject is unavailable, look for an accessible physical contrast: for example, a naturally varying condition with an understood relation to the accounts. Identify other differences between the observed situations. Their effects remain part of the comparison; naming the contrast does not make it controlled.

PHY.10:4.3 - Construct the experiment that can realize the contrast

Translate the mathematical change into physical preparation and operation. Identify the means of imposing the input, the initial or boundary conditions it requires, and the observable response. Include timing when a transient, memory or change of state is decisive.

Determine which shared conditions must survive. Maintaining geometry, material state or a reference can be more consequential than increasing the number of readings. When the intervention itself changes one of these conditions, either include that change in both predictions or construct another contrast.

Derive the subject-apparatus response far enough to establish that the difference reaches the recorded indication. PHY.9 constructs a missing coupling, reference or readout; C.16.MR represents the resulting measurement relation. A detector that clips both predictions to the same indication cannot implement their proposed separation.

Account for feasibility through the parts that can change this design. Available force, pulse duration, preparation time or observation range can rule out a proposed contrast or leave it conditional. An established apparatus or supplied capability can settle those questions without rebuilding its justification. If the required operation remains unavailable, return that specific limit or choose another contrast.

Use a reference or control for a named alternative explanation. A no-drive comparison can reveal an offset; a reference pulse can bound a control error. Add it when the result can change the interpretation.

PHY.10:4.4 - Compare the responses that can actually be distinguished

For each candidate design, obtain the readouts allowed by each physical account, including consequential parameter freedom and uncertainty in preparation, conversion and recording. MMP.7 supplies a probability law when records are random; C.16.IR helps determine what an indication can resolve.

Disjoint bounded response ranges give a particularly simple separation under their premises. If the ranges overlap, some outcomes can still discriminate the accounts while others remain compatible with both. Equal expected values likewise need not mean equal distributions or equal time dependence.

When discrimination depends on statistical evidence, use the appropriate model-comparison method with its dependence, sampling and error assumptions. Repeated readings do not automatically supply independent trials. A mathematical design criterion is useful only after the physical design and the interpretation of its records are supplied.

Compare designs on the distinction needed by the work and their full resource demand. A more informative record about one parameter need not be more informative about the rival mechanisms. Use a cheap sufficient contrast before constructing a costly optimization. C.11 compares available options; C.11.DUA addresses the value of another observation or computation.

Before performing the comparison, make its outcome interpretation clear enough to resist changing the question after seeing the result. State what would remain compatible with each account and which apparatus or preparation failure could mimic the disputed effect. The existing explanation or protocol can carry this reasoning.

PHY.10:4.5 - Obtain the result and locate the failed premise

Work the proposed preparation through to the predicted indication before committing the apparatus. This can expose an uninformative design or missing physical contribution without a new experiment.

When observations are obtained, interpret them under the stated recording and physical conditions. A response compatible with one supplied account and incompatible with another narrows this comparison. It does not exclude physical accounts that were never represented in the alternative set.

If the result disagrees with both, inspect the consequential premise rather than forcing a winner. Separate a computational error, an incorrect readout relation, an unachieved preparation and a defeated physical account. Follow the dependence of the failed prediction to identify a useful return. A supplied bound on an apparatus effect can sometimes exclude that explanation; when it cannot, retain the competing interpretations.

An agreement can also reveal a design limit. Determine whether the test retained the distinction it was meant to expose. If both accounts predict the observed outcome within the relevant uncertainty, report that limited result and preserve any unaffected consequence.

Reuse a result for the preparation, response range and question it supports. A successful test of a small-signal account does not determine its response after a regime-changing drive.

PHY.10:4.6 - Revise the account or develop the next contrast

Return the interpreted result and the physical reason for the next move. This may be use of a sufficient account, a narrower parameter range, an added interaction, a different preparation or readout, or an unresolved difference that the work can tolerate.

Develop a subsequent test from what remains unresolved. One outcome can identify which regime, hidden state or control error matters next. Sequential design can exploit that information, but a complete sequence need not be invented when one comparison already supplies the useful result.

When a revision changes the physical account, propagate it to the mathematical formulation, computation and interpretation that depend on it. Use B.5.RR and B.5.MPC.R for the corresponding reasoning and cross-contribution revision.

Stop when the obtained distinction suffices, or when no worthwhile available test changes the present use. Preserve a promising unresolved physical question when it opens a useful later construction or investigation.

PHY.10:5 - Archetypal Grounding

PHY.10:5.1 - Change speed to separate two drag accounts

A specimen moves through the same fluid at an imposed positive speed v. In the proposed operating range, account L takes the opposing force magnitude to be F=b v; account Q takes it to be F=c v². The coefficients are positive and constant under their respective accounts.

At v=1 m/s, an earlier calibrated force comparison places the true drag between 0.95 and 1.05 N. Thus L permits b between 0.95 and 1.05 N·s/m, while Q permits c between 0.95 and 1.05 N·s²/m². Both explain that observation.

A repeat at the same speed leaves this disagreement intact. Instead impose v=2 m/s while retaining specimen geometry and the fluid conditions on which the coefficients depend. L predicts a force between 1.9 and 2.1 N; Q predicts between 3.8 and 4.2 N. With an additional bounded force-readout error of ±0.1 N, the possible indications lie in [1.8,2.2] N and [3.7,4.3] N. They are disjoint.

The drive and readout must sustain that comparison. Under Q the required mechanical power can reach 8.4 W, since P=Fv. A 5 W drive cannot establish the proposed steady speed for every admitted Q response. A 10 W drive at that speed can meet this particular power demand, while its other operating limits remain relevant.

Suppose the comparison returns 2.05 N. It is compatible with L and incompatible with Q under the stated ranges and conditions. A return of 2.9 N instead disagrees with both. It prompts examination of the retained physical regime, parameter constraints and readout; it does not justify selecting whichever nominal curve is closer.

Heating or a geometry change can make a coefficient vary between runs. If that effect can span the separation, the test no longer has the same interpretation. Restore the shared condition, include the changed dependence or choose another contrast. Increasing speed without this return can defeat the very accounts being compared.

PHY.10:5.2 - Change a drive that makes storage and conduction coincide

A two-terminal element is driven with V(t)=V_0 exp(t/tau). One proposed account is an ideal resistor, I=G V. Another is an ideal capacitor, I=C dV/dt, prepared with charge C V_0 at the start of the recorded ramp.

Since dV/dt=V/tau, choosing C=G tau makes the complete current traces identical during this ramp. More accurate recording of that same drive cannot distinguish the accounts.

Use V_0=1 V, tau=2 s, G=1 mS and C=2 mF. Stop increasing the voltage when it reaches 2 V and hold it there. After the drive and readout have settled, the resistor predicts 2 mA and the ideal capacitor predicts zero current. The change removed dV/dt while retaining V.

The waiting interval must be interpreted physically. A source with finite output resistance and a detector with finite response can create a transient after the change. Include it or choose a readout time after its consequential effect. The ideal predictions also exclude a significant leakage path.

Now suppose the held-voltage current is 0.6 mA, while the earlier ramp still has I/V=1 mS. The ideal pair is inadequate for these records. A parallel conductance and capacitance give

I=G V+C dV/dt.

The hold gives G=0.3 mS. Substitution into the ramp relation G+C/tau=1 mS gives C=1.4 mF. The test has opened a physically different account in which conduction and storage coexist. These two records determine its two parameters under the stated idealization; they do not establish that this account suffices at every frequency or voltage.

The resulting distinction changes use. A continuously held voltage dissipates power through the conductance, while the capacitance stores charge and supplies a transient response. Subsequent pulse or frequency use can therefore ask a question that the original exponential ramp could not resolve.

PHY.10:5.3 - Refocus a hidden physical difference

A prepared ensemble has a transverse phase signal. Consider two idealized accounts of its free decay. In account S, each member has a fixed frequency offset delta, drawn from the Lorentzian density

p(delta)=Gamma/[pi (delta²+Gamma²)], with Gamma>0.

Each phase advances by delta t. Averaging over the ensemble gives M_S(t)=exp(-Gamma t) for t≥0. The individual offsets remain fixed even though the mean signal decays.

In account D, the phase instead has independent Gaussian increments with variance 2 Gamma dt over an interval dt. This Markov dephasing gives the same free signal, M_D(t)=exp(-Gamma t). The two accounts agree on this free-decay observation.

Apply a refocusing rotation at time tau and read the signal at 2 tau. In the ideal pulse comparison, the sign of phase accumulation is reversed for the second interval. Under S, each accumulated phase becomes delta tau-delta tau=0; the ensemble signal returns to 1. Under D, the two intervals have independent phase increments. Subtracting them leaves variance 4 Gamma tau, so the signal remains exp(-2 Gamma tau).

For Gamma=10 s⁻¹ and tau=0.1 s, the predicted refocused signals are 1 and approximately 0.135. The intervention exposes a difference hidden by the equal free decays.

A physical pulse has finite duration, range and accuracy. Its response over the occupied frequency range, other relaxation during the sequence and readout error must be included where they can change this separation. For illustration, if their combined effect on each predicted normalized signal is bounded by 0.05, the predicted indication intervals around 1 and 0.135 remain disjoint. This bound is a condition of that proposed implementation, not supplied by the ideal calculation.

If the pulse cannot refocus a consequential part of the ensemble, a small return can have that cause as well as irreversible dephasing. Change the pulse or reference comparison, retain its bounded effect, or leave the interpretation conditional. A partial echo can also motivate an account with both static variation and changing noise. The performed comparison then guides which hidden dynamics to retain.

PHY.10:6 - Bias-Annotation

The cases use controlled laboratory comparisons and idealized response laws. Astronomical, geological and other field questions can require naturally available contrasts whose backgrounds cannot be independently fixed. The relevant physical premises must then support the observational comparison.

A finite candidate set can exclude a useful account before the test begins. A result inconsistent with every candidate, or a new physical dependence exposed by the experiment, can justify developing the set. It need not become a competition among the original labels.

PHY.10:7 - Conformance Checklist

• The accounts answer the same physical question under corresponding conditions.
• Their remaining parameter freedom and consequential auxiliary premises are retained.
• The proposed physical change makes a difference in an observable response.
• Preparation, drive and readout can realize that change, or their unresolved limit is stated.
• Discrimination is judged with the relevant physical and recording uncertainty.
• Possible results retain their different implications, including agreement with several accounts or disagreement with all supplied accounts.
• The first result and subsequent physical, mathematical or computational return are usable.

PHY.10:8 - Common Anti-Patterns and How to Avoid Them

PHY.10:9 - Consequences

Physical comparison becomes a construction that can change what is observable. Its result can be a discriminating experiment, a useful limit on current apparatus, an improved account or a justified stop.

The calculation, apparatus and interpretation can be developed by different contributors while retaining the common physical disagreement. The cost is concentrated on the dependencies that determine the comparison. A result may remain local even when it resolves the present work.

PHY.10:10 - Architectural Rationale

B.5.TC compares how theoretical accounts answer a working question. This method develops a physical consequence of that comparison: how a preparation or interaction can expose the disputed dependence. PHY.9 supplies an observable readout; the statistical or mathematical design uses that physically realizable relation.

Equal unperturbed responses can conceal different retained state. Changing a drive or reversing an interaction therefore supplies more than another sample of the old observation. It asks the accounts how the physical system responds to a different operation. That response can also reveal how the system might be controlled or used.

The test retains the apparatus and preparation because they participate in the consequence being compared. This makes an unsuccessful comparison informative: a return can change the subject account, the experimental construction or their mathematical description, according to which premise failed.

PHY.10:11 - SoTA-Echoing

Huan, Jagalur and Marzouk, Optimal experimental design: Formulations and computations, Acta Numerica (2024), with corrections and clarifications in the 2026 author revision, distinguishes design for parameter information, prediction and model discrimination. Its sections 2.2 and 6.1 retain uncertain influences and the possibility of model misspecification. These distinctions enter :4.1 and :4.4-.5. A design based on a supplied model set does not establish that the set contains an adequate account.

For the drag comparison, the disjoint force ranges give a sufficient design without a numerical optimizer. When feasible inputs are numerous and predicted records are uncertain, an experimental-design method can compare their expected contribution. Its advantage depends on a useful objective and credible response models. The cost of constructing and evaluating that optimization matters alongside the cost of the experiment.

Vezvaee et al., Fourier transform noise spectroscopy, npj Quantum Information (2024), develops noise reconstruction from free-induction and spin-echo measurements. The adopted physical contribution is changing how a prepared probe couples to temporal structure. The paper’s spectral reconstruction relies on its pure-dephasing and stationary Gaussian-noise setting; those assumptions are not imposed on the Lorentzian static-offset construction in :5.3.

For a noise question within that setting, the paper compares reconstruction from simple sequences with methods requiring many pulses. It also exposes different limitations: finite pulse timing, a restricted accessible frequency range and sensitivity of time derivatives to measurement error. Choosing more elaborate control is therefore not a general improvement. The simpler sequence can be sufficient for the needed distinction, while a different noise or control question calls for another method.

The three numerical cases are constructed comparisons under stated physical laws. Their purpose is to exhibit the changed preparation, response and interpretation; they are not reports of performed experiments. New physical effects or apparatus limits change the corresponding case premises.

PHY.10:12 - Relations

• B.5.TC and B.5.TU: align the theoretical comparison and construct a missing application of an account.
• PHY.4-.8: constrain a law and derive the physical response to a changed preparation or interaction.
• PHY.9 and C.16.MR: construct the measuring interaction and its relation to the recorded indication.
• C.16.IR and MMP.7: determine what the indication resolves and supply the applicable recording probability law.
• PHY.1-.3: receive a discriminating comparison when similarity, a physical analogue or a limiting transformation needs examination.
• C.11 and C.11.DUA: compare available designs and the value of further inquiry.
• B.5.RR and B.5.MPC.R: revise the affected reasoning and its physical, mathematical and computational contributions.

PHY.10:End

PHY.3 - Derive a Physical Limit from Permitted Transformations

Type: Method Status: Stable Normativity: Normative within the stated use

PHY.3:1 - Problem frame

Use this pattern when you need to determine what a physical process or device could accomplish before choosing its detailed mechanism, or when a familiar physical prohibition appears to rule out a proposed change. A candidate may have an acceptable energy balance yet require an unavailable preparation, consume a resource that was omitted, or perform differently on inputs that one device is supposed to handle.

The first useful result is a necessary condition on the requested performance, a conditional exclusion of a class of proposals, or a changed premise that opens a different construction. You obtain it by describing the complete physical transformation and constructing a comparison that the applicable physical laws constrain. “Permitted” here means allowed under those physical premises.

You need a working description of the requested effect, the resources and surrounding systems it may use, and the physical principles relevant to that effect. A qualitative argument can suffice. A quantitative result also needs the quantities and mathematical operations in its chosen comparison. The two worked cases require different preparation: energy exchanges and elementary ratios in the first, state vectors and inner products in the second. Obtain an unfamiliar physical premise or calculation from a suitable source or collaborator and retain the conditions on which its answer depends.

When an established bound already answers the same question under the same physical conditions, apply it directly. A measurement of a particular device, estimation of an unknown parameter and design of a mechanism have their own methods. Return here when their result changes the allowed transformation or the reach of a physical restriction.

PHY.3:2 - Problem

A physical proposal usually names the desired effect more readily than the whole change needed to produce it. “Return the machine to service,” “copy an input” and “extract more work” can leave different resources, input families and end conditions implicit. Those differences determine which physical restrictions apply.

Detailed calculation of one design can reveal its failure while leaving other designs available. Conversely, a quoted limit can be applied outside its premises and suppress a useful construction. The difficulty is to derive a restriction whose physical description covers the proposals being considered, and then use it to decide what to change or investigate.

PHY.3:3 - Forces

PHY.3:4 - Solution

Construct the complete proposed transformation, identify the physical restrictions shared by its candidate mechanisms, and use an admissible comparison to derive their consequence. Keep the requested performance and its physical conditions attached to the result. Revisit the premise that matters when the resources, input family or required performance changes.

PHY.3:4.1 - Specify the physical change and what must remain usable

Start with the proposed input and output. Include the range of inputs and the performance required on them. For a machine expected to handle an unknown input, ask whether one fixed arrangement must work throughout that range or whether the arrangement may be chosen after learning the input. For fluctuating processes, distinguish a guarantee on every run from an average or an allowed failure probability. MMP.8 supplies that formulation when the choice or information condition is difficult.

Follow the resources that can participate in the transformation: material, energy, prepared states, information about the input, external controls and connections to the surroundings. Describe their relevant initial and final conditions. A charged auxiliary, a memory that accumulates records and an initially correlated pair can be consumed resources even if the visible output is unchanged. Use the physical theory to decide which such differences matter to the proposed restriction.

For repeated operation, state what the device must remain able to do. Return to an identical state is needed only when the requested operation or the physical argument requires it. If the construction uses a reset, include its physical change and resource use. A device that still performs the next operation can have changed state; a device restored in one observed variable can have exhausted another needed resource. A.3.3.TR helps choose a sufficient state description where these cases are unresolved.

The result is a physical transformation with its allowed side effects and input conditions. Keep an unspecified preparation visible as an unresolved premise. It can be useful to derive a conditional bound before that premise is settled.

PHY.3:4.2 - Establish which physical restrictions cover the candidate mechanisms

Select the physical account that connects the requested change with a conserved quantity, an ordered change, a preserved relation or another necessary condition. State its regime and the systems to which it applies. A local balance law can require a larger boundary when something crosses it; a law for a closed process can require explicit surroundings when the proposed device exchanges matter or information.

Explain why every device in the proposed conclusion admits the chosen description. MMP.10:4.4 supplies this coverage requirement. Physically, it can mean including a reservoir and its controller, representing an unobserved auxiliary, or admitting all interactions allowed by the theory instead of only one circuit. Restrict the conclusion to the designs described if this coverage remains incomplete.

An enlargement of the allowed class can make exclusion easier: if even a class with additional resources cannot realize the requested transformation, its contained class cannot do so. Show that containment. An approximation chosen because it is easy to calculate does not automatically give such an enlargement; it can remove a coupling or noise source that changes what is possible. When only an approximate account is justified, propagate a suitable error allowance or retain a conditional result at that model’s stated scope.

Use B.5.TU to recover the physical theory’s premise and its consequence for the case. The output of this step is the applicable restriction and the reason it covers the proposed physical class. Further measurement is useful only if its possible result changes that premise or the next choice; C.11.DUA helps resolve a consequential uncertainty about that effort.

PHY.3:4.3 - Construct a comparison that exposes the restriction

Choose the operation that fits the available physical structure. The following two constructions address different situations and can be used independently.

Compose with a reference and cancel selected changes. Use this when an admitted reference process exchanges the relevant quantities or restores a needed condition. Choose the part of the proposal you want to eliminate from the combined account, then determine the reference’s direction, amount and preparation that match it. Establish that the reference can perform that operation under the same connecting conditions. For example, two processes may exchange equal energy but require incompatible temperatures, pressures, phases or forms of delivery. Numerical equality alone cannot establish the physical connection.

Combine the proposed and reference operations, including their auxiliaries. Cancel only the matched exchanges or restored states; keep all remaining changes. C.29.BB supplies the accounting for additive quantities. Apply the physical restriction to the composite. If it would have a prohibited net effect, derive the inequality or condition on the original proposal that avoids that effect. A reference may be an ideal process admitted by the theory; building it in a laboratory is unnecessary for this conditional argument. Using the inverse in an equation still requires a reason that the needed physical reverse process is admitted.

Compare inputs of the same operation. Use this when one device must perform the requested transformation on a family of inputs and a physical law relates its actions on different inputs. Select inputs whose relation can expose the limitation. Write their proposed outputs, including allowed auxiliary outputs, while holding fixed the device and its input-independent preparation. Derive the relation the law requires between those outputs. If the requested outputs violate it, the common device cannot provide them under the stated conditions.

An auxiliary output may depend on the input even when the initial preparation cannot. Allow that dependence in the comparison. Requiring the auxiliary to return to a fixed state when the proposal permits a changed state would prove a narrower restriction. Conversely, choosing a different apparatus separately for each input would answer a different question from the fixed-device requirement.

These constructions can reveal a remaining physical premise rather than a bound. Name that premise and what supplying it would enable. Forcing either comparison when its connection conditions or input relation are absent adds no physical result.

PHY.3:4.4 - Derive the consequence with every remaining physical change included

Carry the quantities, signs, state conditions and input relations through the chosen comparison. Interpret each term by the physical contribution it represents. Check that the canceled effects have compatible units and conditions, and that the uncanceled effects describe the complete remaining process.

Obtain the requested inequality, an excluded transformation, or the unresolved condition on which either result depends. When using a mathematical invariant, MATH.11 supplies the derivation from the admitted transformation rules. When a numerical tool performs the algebra, retain the symbolic relation or a sufficient explanation that lets another participant inspect what the computation establishes. C.29.2 and C.29.3 govern a computation whose formulation or realization needs further work.

Distinguish a necessary bound from attainability. A proposed device can satisfy one physical restriction and still fail another, or approach a theoretical limit only as time or another resource grows. The useful output at this step says which candidates have been excluded, which remain under consideration, and which premise or construction controls the next question.

PHY.3:4.5 - Change the proposal at the premise that controls the result

Return the physical consequence in the terms of the working question. For a proposal outside the bound, identify a consequential change: supply a previously forbidden resource, allow a side effect, narrow the input family, relax precision or reliability, or change the condition for reuse. Derive the affected comparison again for the chosen change. Keep the original result for its original premises.

When the obstruction disappears, continue with a mechanism, an approximation or another limiting principle. PHY.2 can help construct a physical analogue for that inquiry; PHY.1 supplies physical similarity if transferring between regimes is the next difficulty. If the new question concerns an unfamiliar mathematical construction, obtain that contribution from Mathematical Thinking. B.5.MPC helps divide and reconnect those contributions across people and AI agents.

Stop the present derivation when its result is sufficient for the next decision. Further information or a physical trial is a separate action whose value depends on what remains unresolved. A concise explanation of the compared transformation and controlling premise is enough when it allows the intended user to continue.

PHY.3:5 - Archetypal Grounding

PHY.3:5.1 - Bound a proposed engine without designing its mechanism

Question. A proposed cyclic machine takes 100 J from an ideal reservoir at 600 K and delivers 60 J as work. Its only other interaction is heat rejection to an ideal reservoir at 300 K. The machine returns to its initial state. Can a different mechanism make the proposal possible under these same conditions?

The physical preparation includes both reservoirs, the work store and restoration of the machine. It supplies no fuel, depleted auxiliary store or initially available nonthermal resource. The two temperatures are absolute thermodynamic temperatures. Energy balance gives 40 J rejected to the cold reservoir; that balance alone leaves the proposed output admissible.

Use the classical second-law account for these conditions. Its Clausius restriction excludes a composite process whose only net effect is transfer of heat from the colder reservoir to the hotter one. An admitted reversible reference between these reservoirs has eta_ref = 1 - 300/600 = 1/2 and can run backward. This is a theoretical reference; no claim of a finite-power reversible laboratory device is needed.

Choose cancellation of the work exchange. A backward reference consuming the proposed 60 J extracts 60 J from the cold reservoir and delivers 120 J to the hot reservoir. The compatible exchanges combine as follows; positive entries increase the named store.

The net process transfers 20 J from cold to hot with no other change. It violates the selected restriction. The argument uses only the proposed exchanges and cyclic condition, so changing the candidate’s internal mechanism cannot repair it within this class.

For a general positive requested work W, the backward reference requires Q_hot_ref=W/eta_ref. If W > eta_ref*Q_hot, then Q_hot_ref > Q_hot; canceling work again produces the prohibited cold-to-hot transfer. Therefore:

W <= eta_ref*Q_hot = (1 - T_cold/T_hot)*Q_hot.
For the stated inputs: W <= 50 J.

The practical result redirects the design question to work below this bound or to a change in the stated resources. It does not supply the design that attains a chosen value, its power or its operating cost.

Change the allowed physical operation. Suppose the proposed device may also receive 20 J of external work. A forward reversible reference producing 40 J from 80 J of hot-reservoir heat, together with routing those 20 J through the work store, can supply 60 J gross output. The net work produced is 40 J. The denominator and net exchange must now reflect the actual question; this construction supplies no engine producing 60 J net work from the original 100 J alone. If exactly 100 J must still be taken from the hot reservoir, an admitted additional transfer of 20 J from hot to cold accounts for the remainder. The total cold-reservoir gain is 60 J, and both laws permit the resulting exchanges. The changed resource condition opens a construction rather than altering the earlier bound.

For a microscopic proposal with initial correlations or a nonthermal auxiliary, use the physical account appropriate to those resources. A reservoir-only calculation leaves their contribution out. The source comparison in :11 identifies a current treatment; it does not prescribe the same generalized formula for every macroscopic device.

PHY.3:5.2 - Test copying across inputs while allowing auxiliary outputs

Question. Can one device take one input qubit in an unknown pure state and produce two perfect copies on every run? The device can use an auxiliary prepared independently of the input, and its final auxiliary state may depend on that input. Discarding an auxiliary is allowed.

Use the standard quantum description of a deterministic operation. Include the device’s environment and any measurement records in the description of the complete process. A fixed mixed auxiliary preparation can be purified by adding a reference system. The total evolution can then be represented by one isometry V, which preserves inner products. This includes deterministic operations obtained by interaction and later discarding part of the system; it does not impose that the visible two-qubit map itself be unitary. If both required output copies are pure, their joint output factors from the remaining pure total state.

Choose two distinct nonorthogonal input states |a> and |b>. Let |0> be the blank second qubit and |e> the fixed initial auxiliary. Perfect copying would require:

V(|a>|0>|e>) = |a>|a>|e_a>
V(|b>|0>|e>) = |b>|b>|e_b>.

The final auxiliary states are deliberately allowed to differ. Define s=abs(<a|b>), with 0<s<1, and r=abs(<e_a|e_b>), with 0<=r<=1. Taking inner-product magnitudes before and after the same isometry gives:

s = s*s*r.
Dividing by s>0 gives 1=s*r.
But s*r <= s < 1.

The requirements are inconsistent. For the concrete pair |a>=|0> and |b>=(|0>+|1>)/sqrt(2), s=1/sqrt(2) would require r=sqrt(2). That exceeds the allowed overlap of normalized auxiliary states. More unobserved auxiliary output cannot make this deterministic perfect copier possible under the stated account.

This is a physical restriction on one operation across its input family. The mathematical step is preservation of an inner product; the physical work is establishing why the candidate devices admit that common description with the stated preparation. Showing failure of one guessed gate arrangement would leave this broader question unanswered.

Change the input family. Restrict it to the computational-basis states |0> and |1>, with a blank second qubit |0>. Controlled-NOT gives |0>|0> -> |0>|0> and |1>|0> -> |1>|1>. The same arrangement therefore copies every input in that restricted family. A different known orthogonal pair can first be mapped to that basis, copied, and mapped back on both outputs.

Change the performance instead. If failed runs may be discarded, the accepted operation is conditioned on an outcome and the all-runs argument no longer directly characterizes its normalized successful output. One must specify the allowed input set, success probability and failure output, then derive their restrictions. Allowing imperfect copies likewise changes the output relation and requires an accuracy question. The present result identifies why either revised problem differs; it supplies no unexamined claim that a desired success rate or accuracy is achievable.

PHY.3:6 - Bias-Annotation

The method corrects two live errors: treating one device’s failure as a prohibition on every mechanism, and applying a quoted prohibition after its resource or input conditions have changed. Its own principal risk is selecting a convenient physical description that omits an allowed device or contribution. Keep the coverage argument next to the conclusion and use a changed-condition return to expose what the result actually depends on.

The worked proofs assume established classical thermodynamics and standard quantum operations in their stated regimes. Their conditional strength should remain visible when a research question concerns the adequacy of those theories themselves. A formal contradiction identifies incompatible premises; it does not choose by itself which empirical premise to revise.

PHY.3:7 - Conformance Checklist

• The requested physical effect includes its input family, performance criterion and relevant preparation.
• Consumed resources, surrounding systems and any required restoration or continued capability are included where they affect the restriction.
• The selected physical account has a stated regime and a reason to cover the device class in the conclusion.
• The comparison uses compatible admitted reference operations, or the same operation on inputs linked by the physical law.
• The derivation retains uncanceled changes and allowed auxiliary outputs.
• The result distinguishes the derived necessary condition or exclusion from the still-needed realization.
• The explanation shows which changed premise would require another derivation and what the current result enables next.

PHY.3:8 - Common Anti-Patterns and How to Avoid Them

PHY.3:9 - Consequences

A useful physical limit can be obtained before the internal mechanism is known. It can reduce unproductive design work, expose an omitted resource, or suggest a different input family or performance requirement. The derivation also gives collaborators a physical question to answer: a needed reference process, a covering law or an unsettled preparation.

The cost is obtaining a physical account whose reach matches the conclusion. Stronger exclusion usually needs broader coverage than failure of a candidate design. A conditional result is often sufficient to guide the next step while that coverage is developed. Near a limiting value, finite time, noise, size and implementation constraints can determine whether pursuing it is useful.

PHY.3:10 - Architectural Rationale

The complete transformation is the shared object of the two comparison constructions. It makes hidden resources and restoration conditions available to reasoning, while leaving the internal mechanism open. Reference cancellation reduces a proposal to a constrained composite effect. Input comparison constrains what one operation can do across alternatives. These differences explain the choice in :4.3; neither construction is a compulsory stage of the other.

The physical premise gives the mathematical operation its bearing on a device. A conserved sum or preserved inner product can be studied mathematically without that interpretation. Conversely, knowing the name of a physical principle does not yet specify the permitted exchanges, preparation or input family. The method supplies this physical construction and uses the existing mathematical and common inquiry methods for their contributions.

An ideal reference helps bound real proposals when its admitted operations and the coverage relation are explicit. It serves a different purpose from a realistic simulation of one device. Both kinds of account can remain useful in the same inquiry: the bound directs construction, and the constructed mechanism reveals further practical restrictions. The explanation of a failed proposal can also generate the next problem by identifying a consequential change of premise.

PHY.3:11 - SoTA-Echoing

Reversible-reference comparison. David Tong, Statistical Physics, §4.3.1-4.3.2 gives the established Carnot comparison and the thermodynamic temperature ratio. It is a historical methodological anchor for composing processes and canceling exchanges. The pattern uses the construction beyond a particular working substance.

Input comparison with auxiliaries. Peter Shor, Quantum Computation, lecture 5 presents the no-cloning comparison and explicitly includes an input-dependent ancillary output. The present worked case spells out the total-process interpretation, overlap bound and changed-input return. The classical thermal and quantum arguments exemplify different physical restrictions and different comparison operations.

Resource conditions in current thermodynamics. Aguilar and Lutz, Correlated quantum machines beyond the standard second law, version 2 (2025) analyzes initial correlations, nonthermal resources, interaction energy and changes remaining after a driving cycle. These contributions explain why a reservoir-only efficiency claim can need a different account. Its microscopic assumptions determine where its generalized expressions can be used.

One occurrence and repeatable performance. Marletto, Deutsch and Vedral, Tests of constructor theory (2026), §1.4 and §3.2 develops the distinction between an allowed evolution and a device able to repeat a specified transformation, including the difference between reversing dynamics and supplying a reusable inverse operation. The pattern adopts that problem distinction. The article’s new constructor-theoretic principles are research proposals with their own tests; they are unnecessary premises for the two established arguments above.

The synthesis selects operations for constructing a physical limitation and reopening it under changed conditions. It does not require one universal physical resource theory. A new account of the permitted operations, an omitted resource or a changed performance condition is a reason to revisit the affected argument and its source premises.

PHY.3:12 - Relations

• B.5.TU and B.5.MPC supply theory-to-case use and division and reconnection of mathematical, physical and computational work. B.5.QD helps develop the next question from an obtained limit or obstruction.
• A.3.3.TR supplies a sufficient state description and representation of interacting changes. C.29.BB supplies boundary accounting and cancellation for additive quantities.
• MMP.8 and MMP.10 supply information-conditioned choice, constraint formulation and the coverage requirement for a class-wide conclusion. MATH.11 supplies mathematical invariant construction and use.
• PHY.1 supplies physical similarity when the comparison must cross regimes; PHY.2 constructs a physical analogue for the continuing mechanism question.
• C.29.1-C.29.3 govern mathematical correspondence, computational formulation and realization where those are unresolved. B.1.5.CM develops a proposed composition of the work method using the physical result; A.3.1.CC checks the claims in that account and its description. C.11.DUA helps decide whether an additional inquiry can change the useful next action.

PHY.3:End