Part I — Dependence, transmission and continuation

DTM.1 - Distinguish Dependence, Cooperation and Parasitic Use

Type: Method Status: Stable

DTM.1:1 - Problem frame

Use this when a spreading practice, component or tool relies on other participants’ resources and someone concludes that this dependence makes it harmful—or that its popularity makes it beneficial.

A team using an external reasoning service may deliver better results while losing an alternative way to obtain them. A new exchange standard may help its users while imposing conversion work elsewhere. A teaching technique may be copied widely without producing the ability its learners wanted. In each case, determine what continues or spreads and what changes for those who support and use it.

The first result is a small account of the dependence and its consequences under a stated comparison. It lets the participants distinguish useful support, cooperation, extraction at another’s expense and an unresolved relation before choosing a response.

For a routine purchase with known service conditions and no question about continuation or spread, use the applicable procurement or operational method. This method is useful when the relation itself is being misclassified.

DTM.1:2 - Problem

Dependence, replication advantage and useful work are different relations. A method description can be copied while competent performance disappears. A provider can gain more customers while those customers also gain. A variant can persist because it brings benefits, because alternatives are inaccessible, or because its continuation is rewarded even when the work suffers.

Classifying the whole situation as “parasitism” conceals which of these mechanisms would have to change. Conversely, calling it “cooperation” can conceal a cost transferred to someone outside the comparison.

DTM.1:3 - Forces

DTM.1:4 - Solution

Name what continues, recover what supports it, compare the consequences, then characterize the relation.

DTM.1:4.1 - Identify the variant and its continuation

Choose the unit whose continuation is in question. For a working practice, distinguish the method used, a description of it, its competent execution and the participants capable of teaching it. For a device or program, distinguish installed instances from the design and its revisions.

State the criterion by which a later occurrence counts as continuation or a changed variant. A short operational distinction may suffice: “a report prepared with this procedure and checked against independent measurements,” rather than “the same culture.” Where learning changes the method, retain that change for DTM.2.

Then identify the participants and means that enable this continuation: an executor, a provider, an organization allocating time, a communication channel, or an archive. These examples are different relations, not a compulsory list of parts.

DTM.1:4.2 - Recover the enabling relation

Ask what contribution would be absent if the supporting participant or resource were unavailable. Describe the resulting loss and whether another available contribution could replace it. Distinguish dependence on one supplier from dependence on the kind of contribution.

For example, the practice may need an external solver, without needing this particular solver. Alternatively, its records may be readable only by the present service. These relations call for different continuations: obtaining a substitute, making the representation portable, changing the practice, or accepting the dependence for the intended work.

Use a model or a safe bounded comparison when withdrawal would disrupt real work. An unperformed counterfactual remains a hypothesis. Loss under withdrawal establishes dependence; it does not yet establish harm during supported use.

DTM.1:4.3 - Compare results for the affected participants

Recover whose result matters, for which work and over what horizon. A.6.P.RI helps repair expressions such as “our benefit,” “the system is safe” or “they are dependent.”

Choose a feasible comparison: continuing the present arrangement, using an available alternative, or changing one contribution. For each materially affected participant, retain useful output, required expenditure, exposure to failure and capacity for later work where those differences change the choice. The relevant domain methods define and measure these quantities.

A difference is meaningful only against that comparison. If a provider bears a conversion cost in one arrangement and the customer bears it in another, keep both allocations visible. Do not place the cost in an unnamed total and then attribute the resulting advantage to each participant.

When consequences remain unknown, identify the missing relation that changes the conclusion. It may be sufficient to proceed under two conditional accounts. A new investigation is useful only if its result could improve the receiving decision enough to warrant its cost.

DTM.1:4.4 - Relate continuation to consequences

Now compare the variant’s continuation with the effects just recovered.

These are not permanent types of people or organizations. Several relations may coexist: cooperation between two participants can impose harm on a third. The description also does not settle what intervention is legitimate; the applicable agreements, rights and decision methods do that work.

The minimal output can be one sentence: “This procedure depends on the shared service, improves current output while the service is available, and leaves the team without a tested substitute.” That is already enough to consider a substitute or a different assurance arrangement without calling either participant a parasite.

DTM.1:4.5 - Choose the next modeling question

Use DTM.2 when the uncertainty concerns adoption or further transmission; DTM.3 when participants change through use and thereby change transmission. Use DTM.5 for a conflict between within-group and between-group change, and DTM.7 only when a shared mechanism may be serving a different result.

If the comparison removes the alleged harm, stop the harmful-variant branch. If only the choice among already understood arrangements remains, return to the existing selection and improvement methods. A new dynamic model is not required to preserve an already sufficient conclusion.

DTM.1:5 - Archetypal Grounding

DTM.1:5.1 - A service improves work but concentrates dependence

Consider a hypothetical team whose old process yields 10 acceptable analyses per week. With a service it yields 14 at an additional cost affordable for the work. Under an outage the supported process yields only 4; a tested substitute would yield 9.

The method produces three separate comparisons. Current supported output improved by 4. Current outage exposure is larger than before. Replaceability would reduce that exposure, but would not preserve all normal output. None of these differences alone proves exploitation. The next action depends on the required continuity, supplier conditions and cost of the substitute.

If the supplier benefits from continued use, that fact also does not settle parasitism: both parties may benefit. Evidence that a supplier deliberately obstructs portability would introduce a further causal claim, not retroactively change these numerical comparisons.

DTM.1:5.2 - The same spread permits opposite evaluations

Suppose a conditional population model gives fractions 0.30, 0.25 and 0.45 in three modes of tool use. Scores of 1, 0.5 and 0.1 for a separately specified task give a mean of 0.47. Scores of 1, 1.4 and 0.9 for another justified assessment give 1.055.

The arithmetic is a constructed example, not measurements of users. Its purpose is to expose the missing premise: state fractions do not determine task performance. The practitioner must supply the task, capability criterion and evidence for the scores. A model of spread cannot supply them merely by naming one state “dependent.”

DTM.1:5.3 - A remembered movement is not the promised capability

A learner can recall a turn sequence and depend on a teacher’s cues while still failing to coordinate balance and timing. Separate the transmitted description, the cue-supported execution and independent execution under the intended conditions. The dependence may be useful temporary support. Calling it harmful before comparing learning outcomes would obscure the missing coordination method.

The next inquiry concerns the relevant bodily, rhythmic and learning methods. DTM does not infer a physical cause from the learner’s difficulty.

DTM.1:6 - Bias-Annotation

The word “parasite” can turn a disputed comparison into an apparent natural classification. Keep the claim about a specified relation and its effects. An evaluator’s preferred independence is not automatically the participant’s objective.

A focus on immediate output can hide future exposure; a focus on unaided performance can hide the value of reliable tools. Compare both only when the receiving work requires them.

DTM.1:7 - Conformance Checklist

• The account distinguishes the continuing variant, its realization and the participants or means supporting it.
• Withdrawal or substitution bears on dependence; a separate comparison bears on usefulness or harm.
• Each material consequence has a recipient, horizon and feasible comparison.
• A parasitic-use claim names the enabling or diversion mechanism and the impaired work, rather than relying on unfamiliarity or popularity.
• Unknown effects and conditional changes remain visible; the next action does not require a settled moral label.

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

Dependence is damage. Replace the inference with the enabling relation and a comparison of supported, interrupted and alternative work.

Spread is approval. Obtain evidence of the work’s result separately from counts of adoption.

The organization benefits, so everyone benefits. Restore recipients and cost allocation before combining outcomes.

Classify first, explain later. Describe the enabling relation and effects first; retain an unresolved characterization when the evidence does not select one.

DTM.1:9 - Consequences

The participant can choose what to preserve, replace or investigate without treating every dependence as failure. The account can also expose a conflict hidden by an aggregate improvement. It costs more than assigning a label, but may finish with a few sentences instead of a population model.

DTM.1:10 - Rationale

Continuation and useful work answer different questions. Their separation makes it possible to understand cooperation, exploitation and changing relations with the same method while retaining the differing causal mechanisms. The comparison is supplied by the actual work, not by an assumption that autonomy or collective growth is always preferable.

DTM.1:11 - SoTA-Echoing

Solé et al., Large-Language Models as a Cognitive Virus (2026 preprint) distinguish dynamics of coupling states from an illustrative competence measure. This method adopts that separation, without taking their state names as diagnoses or their chosen scores as universal measurements.

Fields and Levin, Cognitive Offloading Is a Cognitive Universal (2026 preprint) provide a competing starting point: external support is not inherently a defect. The method uses that challenge to the independence default; it does not require the paper’s stronger universal physical claims.

Koonin, The first major transition (2026 preprint) distinguishes replication autonomy from contribution to a collective in its early-life model. This supports keeping the two questions separate; it does not establish the corresponding cultural relations.

The subject-level synthesis here compares enabling dependence, continuation and consequences. C.36 supplies cultural relations, A.6.P.RI the reference repair, and D.3/D.4 the treatment of conflicts across interests and levels. None alone establishes the subject-specific effects, which remain a modeling and evidence question.

DTM.1:12 - Relations

• DTM.2 constructs adoption and variant-change laws after the continuing unit is identified.
• DTM.3 connects participant change and further spread.
• DTM.5 distinguishes selection within and among collectives; DTM.7 investigates a particular diversion mechanism.
• DTM.8/.9 use the identified consequences to model protective action and its feedback.
• C.36, A.6.P.RI, D.3/D.4 retain their general cultural, reference and conflict contributions.

DTM.1:End

DTM.2 - Build a Transmission Law from Exposure and Uptake

Type: Method Status: Stable

DTM.2:1 - Problem frame

Use this when a model predicts the spread of a way of working from contacts, messages or downloads, yet these events do not reliably produce use.

A standard can reach a group that lacks permission to adopt it. A demonstration can reach a learner who cannot yet perform its coordinated movement. A prompt can be copied into an AI workflow and then altered before it is executed. These differences change both the amount transmitted and what continues.

Build a law connecting observable exposure, uptake and the resulting variant. The first result is a defensible expected flow or transition probability, with the unresolved factors retained. It can be qualitative when the next decision needs only the missing condition.

A count of views is already sufficient if the question is only whether a notice was seen. Do not expand it into a model of competence or use unless the receiving question needs that result.

DTM.2:2 - Problem

A single “transmission rate” can combine contact, exposure, opportunity, willingness, successful performance and later continuation. It may fit one data series while giving the wrong answer when teaching, permissions, compatibility or access changes.

Adoption and faithful copying also differ. Two practices can have equal uptake and very different future composition because recipients reconstruct or combine what they receive.

DTM.2:3 - Forces

DTM.2:4 - Solution

Define the receiving event, reconstruct the path to it, then compose uptake with the resulting variant.

DTM.2:4.1 - Name the event to be predicted

Choose an event that answers the work question: first trial, successful execution under named conditions, continued use over an interval, or ability to teach another participant. Distinguish these when one may occur without the next.

Specify who can undergo the event and the opportunity over which its probability or rate is measured. For a weekly adoption count, identify the groups not already using the practice and what counts as a first use during that week. For a transition between two working methods, identify both starting and receiving states.

Return to DTM.1 if “the variant” sometimes means the description and sometimes the performed method. MMP.7 and MMP.16 help model what records actually reveal and choose observations that distinguish competing event definitions.

DTM.2:4.2 - Recover exposure and the conditions of uptake

Trace how a recipient encounters an actionable variant. A contact matters only through what it makes available. Identify any condition whose absence prevents the receiving event: a needed capability, usable tool, compatible interface, time or authority, for example.

For one period, let E be relevant exposure, G the required opportunity or support, and U the receiving event. When U requires E and G, the chain rule gives:

P(U) = P(E) × P(G | E) × P(U | E,G).

The conditional terms retain dependence; multiplying three unrelated marginal probabilities would assume more. If use can begin through independent invention or another channel, add that event path rather than treating all use as transmission from the selected source.

A brief account such as “the group saw the method but cannot run the required tool” may finish this step. Estimate numerical terms only when they change the next use.

DTM.2:4.3 - Choose a contact law that matches the mechanism

For independent opportunities with per-contact success probability p and n contacts, the chance of at least one success is 1−(1−p)^n. This construction requires comparable independent trials. It is not justified merely by counting contacts.

When uptake needs reinforcement, use the relevant history: distinct supporting neighbours, repeated successful demonstrations, trust, resource availability or a shared commitment. Give repeated and independent sources the effects supported by the case. Repetition may cease to add information, and another contact can discourage uptake.

In continuous time, define the hazard for a specified recipient who is still eligible for the event. The conditional probability of its first event over a small interval Δt is approximately hΔt; h has units of inverse time. Under a constant hazard, the interval probability is 1−exp(−hΔt), not h itself.

To construct the population flow, count the eligible recipients once. In an illustrative fixed population of N participants, with fraction f already using the variant, homogeneous mixing may support an eligible recipient’s hazard h=βf. Then:

eligible recipients S = N(1−f)
new-uptake flow J = S h = N(1−f)βf          [events per unit time]
flow of population share j = J/N = βf(1−f) [share per unit time].

Here β combines the supported encounter and conditional-uptake rates. The factor 1−f belongs to the eligible population, not again inside its per-recipient hazard. This is one two-state construction, not a universal law. Network, institutional or broadcast exposure may require recipient-specific hazards; sum those over the actual eligible set. New participants, departures and return to eligibility require their own transitions.

If competing variants or mechanisms could explain the same uptake, retain those alternatives and use MMP.16 to find a distinguishing observation. A fitted contact coefficient does not establish the missing mechanism.

DTM.2:4.4 - Describe what the recipient obtains

After determining uptake, specify whether the recipient preserves, reconstructs or combines variants. Keep the change conditional on the source, receiving conditions and chosen event.

For discrete alternatives, K(i|j) can represent the probability that an uptake from source variant j yields recipient variant i. Each source column sums to one over the included possible outcomes. If a failed or unclassified outcome matters, include it or separately state the conditioning that excludes it.

With accepted source flows a_j, the receiving flow is:

b_i = Σ_j K(i|j) a_j.

This does not require copying fidelity or a fixed biological analogue. A revised procedure may be a useful result. When several sources jointly produce one method, replace the single-source column with a rule conditional on that combination. An unchanged title does not demonstrate an unchanged method.

DTM.2:4.5 - Connect observation and further transmission

State which observed records estimate each event or term. A download can identify access, a trial can identify attempted execution, and an independently checked result can support a performance claim. None automatically supplies the next event.

If uptake changes the recipient’s ability or incentive to transmit, hand that dependence to DTM.3. If recipients stop using the variant, represent that transition separately from failure to adopt. The output is the event account plus the resulting law and its source of uncertainty; it is not a requirement to estimate every possible parameter.

Stop when the law supports the intended comparison. Return to the event path if a proposed intervention changes a factor that the current rate had combined with others.

DTM.2:5 - Archetypal Grounding

DTM.2:5.1 - Transmission can reconstruct the practice

Suppose 100 eligible groups face a modeled one-period uptake probability of 0.3. Among their accepted source encounters, 60% carry A and 40% carry B. The accepted flows are therefore 18 and 12.

Assume the recipient retains A after an A encounter with probability 0.9, while a B encounter yields A with probability 0.2. The other outcomes yield B. The receiving flows are:

A: 0.9 × 18 + 0.2 × 12 = 18.6
B: 0.1 × 18 + 0.8 × 12 = 11.4.

These are expected counts, not fractions of a particular person. Thirty uptakes do not mean thirty faithful copies. If accepted encounters favour one source differently, use their composition rather than the source population’s composition.

An intervention that helps more recipients obtain the required tool changes uptake. An intervention that helps recipients preserve a needed operation changes K. They are different proposed effects and need different observations.

DTM.2:5.2 - Reinforcement is a different law

In a constructed three-neighbour case, each neighbour independently provides a relevant endorsement with probability 0.4. If one endorsement is sufficient, the chance of reaching that threshold is 1−0.6³=0.784. If two are required, it is 3×0.4²×0.6+0.4³=0.352.

Both numbers concern the specified endorsement condition, not actual adoption. A recipient may still lack resources or choose not to proceed. Dependence among neighbours invalidates this binomial calculation; three repetitions from one source are not necessarily three independent endorsements.

The useful result is a choice between mechanisms to investigate, rather than adjusting one transmission coefficient until both stories appear to fit.

DTM.2:5.3 - A demonstration reaches learners without supplying coordination

A dance workshop distributes a turn demonstration. A view count measures exposure. A learner’s attempt, successful execution with a cue, uncued execution and later teaching provide different events.

If the learner needs a coordination method not supplied by the demonstration, raising the number of views need not increase the target performance. HCD and the relevant movement methods supply the acquisition work. The transmission model records its effect on the chosen transition, rather than treating a missing capability as unwillingness or resistance.

DTM.2:5.4 - A recipient hazard becomes a population flow

Suppose N=100, f=0.2 and β=0.5 per week. There are 80 eligible recipients, each with current hazard 0.1 per week. The instantaneous new-uptake flow is therefore 8 events per week, or 0.08 of the population per week.

For a short interval of 0.1 week with hazards approximately unchanged, the expected first-uptake count is approximately 0.8. With constant hazard throughout that interval, it is 80[1−exp(−0.01)], approximately 0.796. Counting the remaining fraction again would incorrectly reduce the instantaneous flow to 6.4 events per week. If uptake materially changes f during the interval, evolve the changing state rather than holding its initial hazard fixed.

DTM.2:6 - Bias-Annotation

A convenient event can displace the intended result: copied prompts and watched demonstrations are easier to count than retained competence. Preserve the receiving event even when its measurement is harder.

Do not treat every reconstruction as decay. Compare what the new variant enables before assigning a copying-error interpretation.

DTM.2:7 - Conformance Checklist

• The predicted event, eligible population and interval are stated.
• Exposure, opportunity and uptake remain separable where an intervention can change them differently.
• Multiplicative factors have appropriate conditional meanings; contact independence is not silently assumed.
• The resulting variant is linked to accepted source flows, including reconstruction or combination when relevant.
• Observed records are connected to the events they actually establish.
• The output states the conditions under which the law can be reused and what change requires reconstruction.

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

One click equals adoption. Select the receiving event before choosing a convenient record.

Every repetition is an independent trial. Recover the source and history of the encounters.

Uptake means copying. Add the resulting-variant rule after uptake; do not hide it inside a contact count.

A fitted rate explains the mechanism. Retain rival mechanisms when they imply different effects of the proposed change.

DTM.2:9 - Consequences

The model can distinguish increasing access from enabling performance or changing what is learned. It also shows when an intervention cannot affect the bottleneck it claims to address. More event distinctions require evidence; keep only those that alter the receiving comparison.

DTM.2:10 - Rationale

A law of transmission connects a source opportunity to a recipient change. Exposing that construction makes it possible to revise the appropriate factor when conditions change. Composing uptake with reconstruction preserves the possibility of cultural development instead of reducing it to faithful replication.

DTM.2:11 - SoTA-Echoing

Centola, The spread of behavior in an online social network experiment (2010) is a historical experimental counterexample to universally treating social adoption as independent simple contagion. It motivates the reinforcement branch, not a universal threshold or the illustrative numbers above.

Zilio et al., Co-circulation and co-infection: parasite interactions across scales (2026) distinguish contacts, transmission and establishment and examine their coupling. The method transfers that separation of events; it does not transfer biological event probabilities into cultural practice.

C.36 supplies creation, transmission, reconstruction and selection as distinct cultural relations. MMP.7/.16 supply the observation-model work. The contribution here is to construct the subject-specific event path and compose uptake with the resulting variant, using those providers.

DTM.2:12 - Relations

• DTM.1 identifies the continuing variant and affected participants.
• DTM.3 connects the adopted practice with changes in participants and further spread.
• DTM.4 analyzes the resulting dynamic regimes.
• DTM.6 handles interaction that changes exposure, uptake or persistence.
• C.36, MMP.7/.16 and the applicable learning methods retain their distinct cultural, measurement and capability contributions.

DTM.2:End

DTM.3 - Couple Change Within Carriers to Spread Between Them

Type: Method Status: Stable

DTM.3:1 - Problem frame

Use this when a practice changes its participants, and those changes affect whether, what or how they pass it on—or when wider adoption changes the conditions under which participants can use it.

A learner becomes able to demonstrate a method only after practice. Growing adoption can overload shared teaching support. An installed standard changes which exchanges need conversion. A model that holds these conditions fixed can therefore predict the wrong continuation even if its separate local descriptions are reasonable.

The first result is a coupled account: which participant state changes which transmission or uptake event, and which population change feeds back to participants. It may be two connected ordinary sentences or a small mathematical model. Use MMP.18 for the general work of reconciling mathematical descriptions once these subject relations have been identified.

If participant change has no material effect on the receiving question over its horizon, retain the simpler transmission model.

DTM.3:2 - Problem

Two common reductions erase the relation that drives the result. One treats every carrier as an unchanged source of identical transmission. Another computes a detailed participant trajectory but assumes its population consequences are just that trajectory multiplied by the number of participants.

“Carrier” here names the person, group, device or other participant in which the relevant variant is realized or retained for this model. It does not imply disease. A carrier’s state, its resources, the method it uses and the contact network remain different objects.

DTM.3:3 - Forces

DTM.3:4 - Solution

Find the relation crossing each modeling boundary, construct its effect on an event, then test what the chosen aggregation loses.

DTM.3:4.1 - Identify the participant state that can change transmission

Start from a receiving event defined by DTM.2. Ask what changes its occurrence or resulting variant: the source’s capability, time available to teach, quality of a demonstrated result, retention, equipment compatibility, or the receiver’s current preparation, for example.

Keep a state only if it can change the receiving outcome. Recover how that state changes through use, learning, wear, recovery, support or another mechanism of the actual practice. Describe the process before choosing its equation.

A person’s ability to coordinate a turn and a robot controller’s learned parameters can each affect a demonstration. That does not give them the same learning law. Obtain the internal process from the relevant human-development, movement or engineering methods.

DTM.3:4.2 - Convert the state into a transmission contribution

Determine the quantity delivered to the next model. Examples include demonstrations per period that meet a performance criterion, an uptake probability conditional on readiness, the composition of reconstructed variants, or the cost of exchange with an adopter.

Preserve the event selected in DTM.2. If each encounter is eligible for a repeatable success, the expected success count is the sum of its event probabilities. With n comparable encounters and common probability p, that count is np; this expectation does not require independence.

First uptake by a recipient is different: after it occurs, later encounters cannot create that recipient’s first uptake again. For n independent comparable opportunities before such uptake, its probability is 1−(1−p)^n. More generally, let p_k be the chance of first uptake at opportunity k conditional on no earlier uptake and the stated history. Compose those conditional chances along that history; if histories vary, also account for their probabilities. Alternatively, update the eligible recipient set as events occur.

Aggregate first-uptake probabilities over distinct eligible recipients, not encounters. For a continuous-time model, sum their hazards to obtain the population flow as in DTM.2. This preserves the quantity passed to DTM.4.

Use DTM.2’s conditional factors and resulting-variant rule where needed. Do not multiply a capability score by a contact count without explaining how that score changes the selected event probability or rate.

Then name the observation that could establish or challenge the proposed state-to-event relation. If two mechanisms predict different outcomes for the same state, retain both until the receiving decision warrants discrimination by MMP.16.

DTM.3:4.3 - Recover the feedback from wider spread

Ask what changes for a participant when more, fewer or different others use the variant. Common possibilities include access to compatible partners, demand on shared support, availability of teachers, recognition rules and a protective response.

Build only the feedback supported by the case. For example, increasing adoption may create more teachers and also consume their time. The net effect is a question about those quantities, not an automatic positive feedback.

Where the influence is one-way over the chosen horizon, state that reduction and its reason. A diagram with arrows in both directions is not a requirement to invent the second mechanism. Where both directions matter, trace the loop far enough to see which state or event the returning effect changes.

C.32.MWA helps keep the structures separate: the vertical of methods enacted in one performance, resource provision, control, contacts and units of selection need not share the same boundaries.

DTM.3:4.4 - Choose an aggregation that preserves the needed effect

Choose among individual states, a few relevant classes, a distribution over state or age since adoption, and a justified common state. Use the least detail that preserves the comparison.

When event contribution is q(x), the relevant population quantity is the appropriately weighted average of q(x), not automatically q of the average x. Weights come from the modeled events: an individual making many demonstrations can contribute more exposure than an individual making none. Population shares alone need not be the weights.

A time-since-adoption description is useful when newcomers and experienced users produce materially different events. An average state is more defensible when the effect is approximately linear over the relevant range or when variation is small enough for the decision. Test that approximation, rather than choosing a fixed number of classes.

Return to MMP.18 if shared quantities, units, conservation, state boundaries or incompatible assumptions prevent composition.

DTM.3:4.5 - Decide whether a fast process may be reduced

If local adjustment is much faster than changes in adoption, a local equilibrium may provide the needed output. First identify the relaxation time and the population-change time over the conditions being compared. Examine startup and interventions, not only a final stationary point.

Retain the local dynamic state when learning delay, fatigue, memory, periodic loading or loss of support changes the prediction. A fast measured response does not establish the absence of slower memory.

The result states what is passed across the boundary, the conditions under which it can be summarized, and the change that would require restoring detail. DTM.4 then investigates persistence and return using that coupled model.

DTM.3:5 - Archetypal Grounding

DTM.3:5.1 - Equal average readiness can imply different spread

Consider an illustrative readiness x between 0 and 1. Suppose a relevant demonstration occurs once per period and produces the receiving event with probability q(x)=x². This law is a declared hypothesis for the example, not a general law of learning.

In population A every source has readiness 0.5, giving q=0.25. In population B half have readiness 0.1 and half 0.9. Both have mean readiness 0.5, but population B has mean contribution (0.01+0.81)/2=0.41.

A model using only mean readiness would treat both populations identically. If the comparison is about transmitted successful practice, that reduction loses the effect. Preserve the readiness distribution or sufficient classes. If the actual event law were linear and contact opportunities equal, the same particular loss would disappear.

This case also separates evidence obligations. Measurements of readiness do not establish q(x)=x²; that link requires observations of the chosen receiving event.

DTM.3:5.2 - Growth can reduce the support available to each learner

Suppose local readiness follows the conditional model:

x' = α(f)(1−x) − δx
α(f) = α₀/(1+kf).

Here f is the fraction adopting, α is the effective support rate per learner, δ a loss rate, and k specifies how adoption loads the support arrangement. The signs and functional form express the assumed sharing mechanism; they would have to change if adoption added support faster than demand.

For α₀=1, δ=0.25 and k=3 in the chosen time unit, the stationary readiness is α/(α+δ). It is about 0.714 at f=0.2 and 0.541 at f=0.8. If q=x² remains the receiving-event law, the corresponding contributions are about 0.510 and 0.292.

The loop is now visible: adoption changes support per participant, readiness changes the receiving-event rate, and that rate changes adoption. These stationary substitutions are useful only when readiness adjusts fast enough. Immediately after a large influx of beginners, replacing their states with these equilibria can be wrong.

A trial to increase adoption might therefore need additional teaching support. The equation does not prove that such support exists or how to teach the missing operation; those are provider questions.

DTM.3:5.3 - Spread changes compatibility without changing a learner

Consider a group comparing two exchange standards. Let f be the share of its relevant partners on the new standard. Over the same horizon, the group compares switching with continuing. Suppose it pays conversion costs while its old-standard partners keep their process unchanged. Its gain from switching is b−c−l(1−f): improvement b, switching cost c and conversion cost l for the remaining old-standard partner share.

Here the feedback is through compatible exchanges. No readiness state or training equation is needed. DTM.2 constructs the actual adoption event; the resulting adoption changes f and thus the gain available to later groups. A broadcast announcement is not the same as an authorized, affordable transition.

Use ECO.8 to retain the coordination and cost-bearing relations. Use the engineering compatibility method to establish that an adapter actually works. The spread model couples those supplied results rather than replacing them.

DTM.3:5.4 - Repeated opportunities do not create repeated first uptake

For one initially eligible recipient, suppose two independent comparable opportunities each succeed with probability 0.8. The probability of first uptake during those opportunities is 1−0.2²=0.96. It can also be obtained as 0.8+0.2×0.8: the second term includes only the recipients still eligible after the first opportunity.

The product 2×0.8=1.6 instead gives the expected number of successes when both attempts are actually performed and success is repeatable. It is not a first-uptake probability. Either quantity can be useful, but they feed different receiving models. The participant-state coupling must preserve that distinction even when the same internal readiness supplies both probabilities.

DTM.3:6 - Bias-Annotation

A biological source can make “within” suggest a single organism and “between” a population of organisms. Recover the actual participant boundaries instead. A shared provider or a network may cross organizational boundaries.

A detailed internal model can also attract attention away from the only output the receiving question needs. Detail is justified by a changed prediction or decision, not by anatomical or organizational completeness.

DTM.3:7 - Conformance Checklist

• The account names the event whose law changes and the participant state that changes it.
• Each boundary-crossing quantity has a subject interpretation, units where relevant, and an identified receiving use.
• Any feedback from spread is a stated mechanism; an omitted feedback has a bounded reason.
• Event-weighted contributions are distinguished from unweighted participant averages.
• A stationary reduction has a timescale and startup boundary; material delay or memory remains represented.
• A change in aggregation or participant boundary returns to the relevant modeling provider.

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

Everyone transmits like the average participant. Average the event contributions under the relevant contact weights and compare with the proposed reduction.

A vertical of methods is a contact hierarchy. Separate the simultaneously enacted operations from the network through which another participant encounters the practice.

Fast equilibrium from the first moment. Check newcomers and changed conditions before discarding internal time.

Always draw two arrows. Keep one-way coupling where justified; construct reverse influence only from an actual mechanism.

DTM.3:9 - Consequences

The model can explain why a locally successful method fails to spread, or why its spread changes the conditions that made it successful. The cost is additional state or data when simple aggregation fails. Some cases become simpler after the correct interface quantity is identified.

DTM.3:10 - Rationale

Cross-level modeling becomes useful when one process supplies a quantity that another process changes in return. The subject method must identify those quantities; general mathematical composition cannot choose them from a shared vocabulary alone. Explicit aggregation and time assumptions make the connection revisable.

DTM.3:11 - SoTA-Echoing

Zilio et al. (2026) motivate separating within-carrier interaction from co-circulation and the feedback through transmission conditions. Their biological mechanisms are not used as universal cultural laws. The readiness and compatibility examples above are separate conditional constructions.

The historical Katz–Shapiro model of compatibility and network effects (1985) supplies a contrasting mechanism in which others’ adoption changes a user’s outcome. Its equilibrium approach does not by itself provide the adoption or learning dynamics used here.

MMP.18 provides general coupling and reduction methods. C.32.MWA preserves distinctions between structures. DTM adds the construction connecting participant state, transmission contribution and conditions changed by spread.

DTM.3:12 - Relations

• DTM.2 identifies receiving events and variant changes; DTM.4 examines the resulting regimes.
• DTM.6 supplies interaction mechanisms that can change local or population conditions.
• DTM.8/.9 add protective interventions and response dynamics where they matter.
• MMP.18, MMP.16 and C.32.MWA supply mathematical composition, distinguishing observations and structural correspondence.
• HCD, relevant movement methods and engineering methods supply their own acquisition or physical mechanisms; DTM does not replace them.

DTM.3:End

DTM.4 - Find Conditions for Variant Invasion, Persistence and Return

Type: Method Status: Stable

DTM.4:1 - Problem frame

Use this when early growth, a fitted trend or a single threshold is being used to claim that a practice will establish itself, disappear, or return after an intervention.

A method may decline when few groups use it yet persist once compatible partners or supporting capabilities become common. A brief reduction can be followed by return to the same regime. A favorable gain from switching may still produce no uptake when no capable, authorized recipient receives the proposal.

The first result is a conditional account of possible continuation: whether a rare variant grows in a specified environment, what can maintain it, and what would change the reachable regime. Use the relevant mathematical methods to establish the properties of the chosen model; this pattern selects and connects the questions that a spread claim must answer.

For an immediate one-off choice with no reliance on persistence or later return, a comparison of the available actions may be sufficient.

DTM.4:2 - Problem

“Increases now,” “can establish,” “persists under present conditions” and “cannot be reversed” are different claims. Local growth does not settle long-term coexistence. A stable regime need not attract every admissible initial condition. A mathematical equilibrium at zero remains possible even when a nonzero introduction would grow.

A threshold without its governing event law, surrounding state and admissible initial conditions can therefore support the wrong intervention.

DTM.4:3 - Forces

DTM.4:4 - Solution

Specify the surrounding regime, test growth from rarity, find maintained regimes, then examine reachable change and return.

DTM.4:4.1 - Preserve the law and the surrounding conditions

Recover the transmission and uptake construction from DTM.2 and the relevant feedback from DTM.3. Identify the variables being varied and those held fixed, the feasible state region, units of time and the environmental conditions. Keep the counted event and eligible population unchanged across that handoff. A per-recipient probability or hazard, a repeatable-event count and a population flow are different inputs; convert them using the stated receiving event before analyzing regimes.

State the practical question before solving: establishment from a small introduction, persistence of an existing variant, coexistence, reduction to a desired range, or return after a temporary change. “What happens?” is too broad when those questions depend on different initial conditions.

If a capability, permission or resource required for uptake is absent, preserve that absence in the law. A favorable comparison is not itself an event.

DTM.4:4.2 - Test a rare variant in a specified resident regime

Place the proposed variant at a small positive amount in an otherwise specified regime. Determine its initial growth from the modeled receiving events and losses.

For one rare quantity y, a local form y’=r y plus smaller terms gives growth when r>0 and decline when r<0. r concerns this surrounding regime and these assumptions. If r=0 or the neglected terms can dominate at the relevant scale, the first-order test does not decide.

With several linked rare states, use the appropriate coupled linearization or generation-to-generation operator supplied by the mathematical modeling method. Do not assign one scalar “reproduction number” until its construction and threshold apply to that model.

A zero initial amount can remain zero in a deterministic model with no external introduction. The rare-variant test asks about a small positive amount; it does not predict that an introduction will occur.

DTM.4:4.3 - Find what can maintain the variant

Find admissible stationary or recurring regimes and determine their relevant stability. Retain coexistence, oscillation or continued replacement if the model supports those forms; persistence need not mean a constant population share.

Where reinforcement or compatibility matters, examine finite initial amounts as well as rarity. A rare variant can decline while a sufficiently established variant persists. In that case, identify what separates the continuations and whether the proposed action can cross that boundary.

Use analytical reasoning, validated numerical calculation or a justified qualitative argument at the formality needed by the decision. A solver’s last point alone is not evidence of an attracting regime; check the event law, feasible region and behavior under relevant nearby conditions.

DTM.4:4.4 - Distinguish a temporary displacement from a changed law

Model what the contemplated action changes. A one-time alteration of current use changes an initial condition. Continuing support, a compatibility adapter or a revised recognition rule can change a rate or a dependence. A newly available variant changes the candidate set.

Follow the trajectory during the action and after its removal. Ask whether the post-action state remains in the same region of attraction, enters another one, or encounters a different law because capabilities, resources or variants have changed.

For an intended return, identify the route and means needed to realize it. Lowering a parameter is insufficient if a necessary alternative has been lost or an endpoint is invariant under the model. Use C.36.RP and the relevant domain method to obtain an unavailable contribution; do not invent it as a free change in the state variable.

DTM.4:4.5 - Express the result as conditional continuations

Give the conditions and their consequences together:

• the surrounding regime in which a rare variant grows or declines;
• the maintained regimes relevant to the question;
• initial conditions or interventions that distinguish reachable continuations;
• the observations or changed assumptions that would defeat the account.

Use the existing comparison and portfolio methods when several actions remain viable. DTM.1 preserves whose consequences are being compared. DTM.6/.9 reopen the law when another variant or protective response changes the surrounding regime.

Stop with the weakest supported claim that answers the practical question. A useful conditional threshold need not become an unconditional prediction, and a qualitative return may be enough to reject an ineffective intervention.

DTM.4:5 - Archetypal Grounding

DTM.4:5.1 - Rare growth and persistence give different answers

Consider a constructed reversible uptake model for the share f using a practice. The modeled adoption flow is s f²(1−f): encounters must provide the reinforcement represented by f². The modeled abandonment flow is s θ f(1−f): access to alternative users contributes to departure. Thus:

f' = s f(1−f)(f−θ),    0≤f≤1,    s>0,    0<θ<1.

The two flows are nonnegative on the stated region. Their forms are hypotheses for this example, not universal properties of social learning. With θ=0.4 and s=1 per chosen time unit, the fixed points are 0, 0.4 and 1.

For 0<f<0.4 the derivative is negative; for 0.4<f<1 it is positive. The endpoint regimes attract interior states on their respective sides, while 0.4 separates them. At f=0.2 the rate is −0.032; at f=0.6 it is +0.048.

The rare-variant test reports decline near zero. It does not establish that an existing majority will disappear. The model instead gives persistence from the higher initial range. The first useful result is this distinction, before any choice of intervention.

DTM.4:5.2 - A temporary reduction can be followed by return

In the same model, suppose an admissible action temporarily changes f from 0.8 to 0.5. When the action ends and the old law resumes, f increases again because 0.5 remains above 0.4. A displacement to 0.3 would instead place the trajectory in the lower range.

These statements do not establish an available or legitimate way to change f. They tell the project what a claimed temporary intervention would need to achieve under the model. Changing ongoing support would require a revised law, not a fictitious one-time jump.

The endpoints in this model are invariant: departure is absent at f=1 and uptake absent at f=0. If the actual practice permits independent departure, outside introduction or invention, add those processes before using the endpoint predictions. A change in the underlying event account may remove the threshold or replace the two regimes.

DTM.4:5.3 - Compatibility changes a threshold but does not guarantee uptake

Suppose a switching group bears conversion costs while old-standard partners keep their process unchanged. Over one horizon, its gain relative to continuing is g(f)=2−0.5−3(1−f), where f is the adopter share. Assume equally weighted encounters in a homogeneous population, so f also gives the group’s relevant partner share on the new standard. The gain is positive only above f=0.5.

An available adapter costing 0.4 and reducing the conversion coefficient to 0.2 gives gₐ(f)=2−0.5−0.4−0.2(1−f). At f=0.2, g=−0.9 while gₐ=0.94.

This is a threshold of the group’s comparison, not yet a population stability result. Add the explicit event rule that a fraction e of the remaining groups receive a feasible, authorized offer and adopt when the gain is positive. Then one step gives:

f_next = f + (1−f)e × indicator(g>0).

For f=0.2 and e=0.25, the share remains 0.2 without the adapter and becomes 0.4 with it. For e=0, neither positive gain nor a favorable threshold produces uptake.

This constructed rule omits departure, unequal partners and changing expectations. To conclude persistence or reversibility, supply those mechanisms where they matter and repeat the corresponding question. ECO.8 retains the coordination and allocation of costs; the engineering method establishes compatibility.

DTM.4:6 - Bias-Annotation

The term “invasion” denotes growth from a small positive presence in a specified model. It assigns neither moral value nor an instruction to spread something.

An attractive tipping-point narrative can encourage modelers to choose reinforcement in advance. Compare a simpler uptake law when the evidence does not establish it, and preserve the resulting difference in continuation.

DTM.4:7 - Conformance Checklist

• The practical question distinguishes establishment, persistence and return.
• The resident state, feasible region and event law are identifiable.
• A rare-variant result is conditional on a specified surrounding regime and a nonzero introduction.
• Maintained regimes are distinguished from a trajectory’s transient values; finite-introduction effects are examined where relevant.
• The proposed action changes an initial condition, law, resource or variant set explicitly.
• The result after action removal and any unavailable return contribution remain visible.
• A comparison threshold is not presented as proof of uptake, stability or authority.

DTM.4:8 - Common Anti-Patterns and How to Avoid Them

A positive trend proves inevitability. Recover the surrounding regime and test the continuation that the claim actually needs.

Rare decline proves eventual disappearance. Examine persistence from established states when reinforcement or compatibility supplies a different regime.

One successful reduction proves durable change. Follow the post-action law and reachable state.

A profitable transition occurs automatically. Include exposure, means and authority in the event rule.

DTM.4:9 - Consequences

The result can reject an intervention that only displaces the state temporarily, or reveal that a failed small introduction does not rule out persistence under different conditions. Its value depends on the event law and feasible actions; precise calculation cannot repair a missing mechanism.

DTM.4:10 - Rationale

Establishment, persistence and return are related questions on one dynamic account, but they use different evidence. Keeping them together prevents a local calculation from answering a more expansive question by implication. Mathematical methods supply the analysis; the subject method keeps its conditions connected to the intended change in practice.

DTM.4:11 - SoTA-Echoing

Solé et al. (2026 preprint) provide a current example of reinforcement producing multiple stable coupling regimes. This motivates examining establishment and persistence separately; the one-variable construction above is a different illustrative model, not a reduction or empirical validation of that paper.

The historical Katz–Shapiro compatibility analysis (1985) shows why outcomes may depend on other adopters. A gain comparison does not supply an adoption process. The worked adapter construction adds its own explicit event rule and leaves stronger dynamic claims open.

The pattern uses existing mathematical stability and continuation methods. Its contribution is the linked subject inquiry from rare growth through maintained regimes to feasible return.

DTM.4:12 - Relations

• DTM.1 preserves the participants and consequence comparison.
• DTM.2/.3 supply the event law, internal change and feedback.
• DTM.5/.6 can change the resident composition or selection conditions.
• DTM.8/.9 supply protective actions and their dynamic effects.
• MMP, relevant mathematical and computational methods perform the selected analysis; C.36.RP helps recover a contribution needed for an otherwise unavailable return.

DTM.4:End