Signal Decay and Fadeout

one_liner: Signals, influence, or effects systematically weaken over time or space.

origin_domain: physics

core_idea: Signal decay and fadeout is the structural pattern whereby a signal, influence, or effect systematically weakens or diminishes over time or distance, following predictable decay laws, as Rutherford (1900) first established empirically for radioactive emanation by demonstrating geometric-progression intensity decline. <!-- FACT-D53-496 -->[^rutherford-1900] The magnitude of effect decreases according to characteristic rates—exponential, power-law, logarithmic, or geometric—independent of domain. This pattern emerges wherever a quantity propagates, diffuses, or exerts influence across a system, and environmental or inherent factors attenuate that quantity predictably, a generality Friis (1946) captured for electromagnetic propagation in his canonical transmission formula relating received signal power to distance and aperture. <!-- FACT-D53-497 -->[^friis-1946] The prime focuses on the *structural regularity* of weakening, not on the mechanisms causing it; whether decay arises from absorption, dissipation, geometrical spreading, or loss of potency, the mathematical form often remains similar.

structural_signature:
Signal decay and fadeout encodes a pattern: **diminishing-magnitude → predictable-rate → asymptotic-or-threshold-boundary**. It separates an initial, strong state (high signal) from a final, weak state (negligible signal) and names the temporal or spatial progression between them, as Bateman (1910) formalized for serial decay chains by deriving closed-form expressions for time-dependent abundances under constant decay rates. <!-- FACT-D53-498 -->[^bateman-1910]

**Recurring features:**

- Weakening of effect over time or distance
- Exponential, power-law, or geometric diminishment
- Half-life and decay constants
- Signal-to-baseline ratio approaching zero
- Attenuation factor per unit distance or time
- Maintenance burden in systems with decay

The structural insight is robust: radioactive isotopes, drug concentrations, social influence, light intensity, acoustic energy, and organizational memory all exhibit the same weakening dynamics. The same mathematical structure governs decay across physics, chemistry, biology, social dynamics, and information systems, an empirical regularity Wixted and Ebbesen (1991) demonstrated by showing that retention curves across humans and pigeons across recall, recognition, and matching-to-sample tasks all conform to a common power-law form. <!-- FACT-D53-499 -->[^wixted-ebbesen-1991]

what_it_is_not:
Signal decay is not the same as loss or disappearance. When a car key is lost, it is simply gone, without pattern or predictability. When a drug concentration decays in the bloodstream, it follows a measurable half-life; the rate is predictable and lawful. The key distinction is that signal decay specifies *regular, characterizable weakening*, governed by decay constants or rates. Loss is sudden or random; decay is gradual and structured. This matters because decay enables prediction and modeling: we can forecast when a signal will become negligible if we know its decay law. Random loss offers no such predictive power. The prime's value lies in naming the *regularity* of weakening, not just its occurrence.

Nor is signal decay the same as damping or friction in physical systems. Damping describes the *dissipation of oscillatory energy* in wave or harmonic systems—how a pendulum's swing diminishes over time, how a tuning fork's sound fades. Damping assumes oscillatory behavior and describes the loss of energy per cycle or per unit of oscillation. Signal decay is broader, encompassing non-oscillatory weakening: the exponential decay of a radioactive isotope, the geometric attenuation of light intensity with distance, the logarithmic fading of a rumor from collective memory. These are all signal decay, but none involve oscillation or damping in the technical physics sense. Damping is a subset mechanism of signal decay in systems that exhibit oscillation, but damping language is not appropriate for non-oscillatory weakening.

The prime is also not identical to signal degradation or noise accumulation. A signal can degrade in quality (becoming noisier, more distorted) while maintaining the same or even increasing magnitude. A photograph scanned repeatedly loses fidelity without loss of brightness. Signal decay, by contrast, concerns the weakening of the signal's magnitude or strength itself, independent of quality degradation. Noise accumulation is typically due to environmental interference or measurement error; signal decay is the intrinsic weakening of the signal as it propagates or persists. These are separate phenomena, though they often co-occur in real systems.

Finally, signal decay is not a fundamental law of nature that applies everywhere uniformly. Decay rates and laws vary wildly across domains and systems. Some signals decay exponentially (radioactive decay, drug clearance), others follow power laws (social influence through networks), others are logarithmic (human memory without reinforcement). Some systems maintain signals with high fidelity across time or distance with minimal decay (quantum coherence in certain materials, fiber-optic transmission with amplifiers). The prime names the *structural phenomenon*—systematic weakening governed by rates and laws—not a claim that decay is inevitable. Systems can be designed to slow decay, extend the threshold of practical utility, or maintain signal strength through active reinforcement. The prime is about recognizing and managing decay where it occurs, not assuming it is universal.

broad_use:
**Physics and material science:** Radioactive decay (exponential half-life laws, nuclear transitions), light attenuation through absorbing media (Lambert-Beer law), acoustic dampening and sound intensity loss with distance (inverse-square law), thermal diffusion and heat dissipation, electromagnetic field strength decline with distance.

**Pharmacology:** Drug concentration decay following first-order or zero-order elimination kinetics, where plasma concentration diminishes as the body metabolizes and eliminates the compound. Half-life determines dosing intervals; multiple doses must account for decay of prior doses, as Gibaldi and Perrier (1982) detail in the canonical pharmacokinetics treatment of multi-dose plasma-level superposition. <!-- FACT-D53-506 -->[^gibaldi-perrier-1982]

**Social networks and information diffusion:** Influence decay over network distance (weaker persuasive effect at each hop); virality decay (a social media post's engagement diminishes exponentially with time); information decay (salience of a message fades from collective memory in weeks or months); reputation decay (an individual's social capital erodes as past actions fade from attention).

**Memory and cognition:** Forgetting curves (Ebbinghaus) where recall strength decays over time without reinforcement; memory consolidation depends on repetition to counteract natural decay; attention span for a topic decays as it recedes into the past.

**Political and organizational influence:** Temporal decay of policy effects as institutional structures and personnel shift; leadership directives decay in force when not actively reinforced (a mandate issued from the top decays toward prior norms in middle management and frontline operations), a phenomenon Lewin (1947) framed as the tendency of social systems to revert toward quasi-stationary equilibrium absent sustained driving forces. <!-- FACT-D53-507 -->[^lewin-1947] Institutional memory decays as institutional knowledge holders depart.

**Economics:** Market penetration curves (adoption rate follows an S-curve where initial growth accelerates then decays as saturation approaches), customer retention decay (churn increases as contact touchpoints decay), brand recall decay without advertisement reinforcement.

clarity:
A core function of "signal decay" is to distinguish between *transmission* (can the signal reach the target?) and *persistence* (does the signal retain sufficient strength when it arrives?), a separation Pozar (2011) operationalizes in the standard link-budget framework, which decomposes received signal magnitude into transmitter power, propagation loss, and receiver-threshold margin. <!-- FACT-D53-508 -->[^pozar-2011] A radio tower propagates a signal thousands of kilometers, but the signal decays; a phone call propagates across oceans via fiber optics, but the signal is amplified to counteract decay; a rumor spreads through a social network, but its credibility decays with each retelling.

Naming signal decay as a prime makes visible that not all effects persist uniformly; it enables practitioners to ask: What is the decay law governing this system? Is it exponential, power-law, or asymptotic? What is the characteristic time or distance scale (half-life, decay length)? Where does the signal become negligible (below noise floor, below detection threshold)? At what point does the system cease to respond? These are precisely the parameter-extraction questions Crank (1975) treats systematically in the canonical mathematics-of-diffusion framework, where boundary conditions and characteristic timescales are read directly from the governing decay equations. <!-- FACT-D53-509 -->[^crank-1975] This language creates a conceptual bridge between domains that otherwise appear unrelated: a pharmacist managing dosing schedules operates in the same decay-law universe as a social media strategist managing content freshness, as a leader reinforcing organizational norms.

manages_complexity:
Signal decay simplifies prediction by providing closed-form or parameterized models of effect diminishment. Rather than track individual instances or detailed mechanisms, practitioners can estimate remaining signal strength from decay rate and elapsed time, a reliability-engineering approach Barlow and Proschan (1965) develop into a formal mathematical theory of failure-rate–driven maintenance policy. <!-- FACT-D53-510 -->[^barlow-proschan-1965] This bounded representation enables both forecasting (when will this effect become irrelevant, undetectable, or negligible?) and resource allocation (where must we invest maintenance, reinforcement, or re-transmission to keep the signal above a critical threshold?).

For example, a therapist treats a patient with exposure therapy for anxiety. The anxious response decays with each repeated exposure according to a habituation curve. Knowing the decay rate allows the therapist to schedule exposures optimally: too infrequent, and the patient's habituation decays back toward baseline (reconditioning); too frequent, and the patient is overwhelmed. Understanding signal decay informs the treatment protocol.

In organizational settings, a change initiative's momentum decays without sustained attention. Leaders can model this decay and plan reinforcement: communicate progress at intervals that align with the decay rate, celebrate early wins before momentum fades, and design systems that embed the change (reducing reliance on constant reinforcement). This transforms a vague sense that "momentum is lost" into a concrete management problem with a structural solution.

abstract_reasoning:
Signal decay enables reasoning about *resilience* and *maintenance*: systems without decay (or with negligible decay) are simpler because they approach steady-state; systems with significant decay require ongoing inputs to maintain steady-state effect. It also enables reasoning about *temporal horizons*: a problem visible at short timescales (a news cycle, a policy announcement) may vanish at long timescales if the signal decays faster than the problem resolution timescale. Conversely, a signal that decays very slowly (CO₂ in the atmosphere, plutonium half-life, institutional inertia) creates mismatch between decision horizons (policy cycles measured in years) and effect horizons (environmental consequences over centuries).

Signal decay also licenses counterfactual reasoning: "What if we slowed the decay rate?" (add a catalyst, reinforce more frequently, use a different medium with lower attenuation). "What if the signal decayed faster?" (a norm that evaporates quickly is easier to reverse than one that persists). "At what decay rate does this system become unstable?" These questions are answerable in frameworks where decay is named and quantified.

knowledge_transfer: The pattern—systematic weakening following a characteristic rate—transfers cleanly across domains. A radioactive isotope decays exponentially; a drug concentration in the bloodstream decays exponentially; a social media post's engagement decays exponentially; a forgetting curve decays logarithmically. The mathematical structures are often identical or homologous, allowing transfer of reasoning and modeling techniques across domains. A pharmacologist familiar with first-order kinetics might recognize the same logarithmic relationship in the diffusion of innovation through a population; a physicist familiar with inverse-square laws might see the parallel in the attenuation of influence through a social network. The vocabulary and reasoning of signal decay help practitioners in one domain recognize and apply insights from another.

formal_abstract:
**Radioactive decay:** Cobalt-60 undergoes beta decay with a half-life of 5.27 years. A source with initial activity of 1,000 Bq (decays per second) will have activity of 500 Bq after 5.27 years, 250 Bq after 10.54 years, and so on. The decay follows N(t) = N₀ × e^(−λt), where λ is the decay constant. After 10 half-lives (52.7 years), the activity is approximately 0.1% of the initial value, and the source is negligible for most purposes. **Mapped back:** This exemplifies the core structural pattern: predictable, exponential weakening toward practical negligibility. The mathematical form enables precise forecasting (when will this source become safe to handle?) and resource management (when should we decommission this facility?).

**Pharmacokinetics:** A patient takes a 500 mg dose of amoxicillin with an elimination half-life of 1 hour. After 1 hour, the plasma concentration is 250 mg; after 2 hours, 125 mg; after 4 hours, 31.25 mg (below therapeutic effectiveness). The antibiotic must be dosed every 8 hours (approximately 2 half-lives) to maintain therapeutic plasma levels above a minimum threshold. Missing a dose allows the signal (drug concentration) to decay below the threshold, and the infection may resurge. **Mapped back:** This shows signal decay in action: the structural pattern of exponential weakening determines clinical protocol. Dosing schedules are *consequences* of decay law, not arbitrary choices.

applied_industry:
**Social media content decay:** A viral video receives 10 million views on day 1, 1 million on day 2, 100,000 on day 3, following roughly a power-law decay (each day captures ~10% of the prior day's views). By day 7, views have decayed to near-zero. Content creators manage this decay through: (1) creating follow-up content to restart the decay curve, (2) using algorithmic amplification (paid promotion that delays decay), (3) embedding the content in longer narratives (books, courses) that decay more slowly. **Mapped back:** Platform dynamics are governed by signal decay; successful strategies acknowledge and manipulate the decay rate rather than fighting it.

**Organizational norm change:** A manufacturing firm implements a new safety protocol. Day 1: executives emphasize the change, all supervisors cascade the message, compliance is high. Week 2: supervisors are less focused, workers revert to familiar patterns, compliance decays to 70%. Month 2: without reinforcement, workers have largely returned to prior norms (compliance back to 40–60%). Leaders counteract this decay through: (1) regular audits and feedback (reinforcement on a faster timescale than decay), (2) embedding the norm in systems (checklists, automated alerts) that do not decay, (3) celebrating compliance stories to reset the social norm baseline. **Mapped back:** Organizational culture change is an application of signal decay: norms without maintenance decay toward prior attractor states. Understanding this structure informs intervention design.

title: Decay is predictable in aggregate but opaque in mechanisms

body: In radioactive decay, the rate constant λ is precisely measurable and reproducible across identical nuclei; we cannot predict *which* nucleus will decay next, but we can predict the *total number* decaying per unit time. In social systems, we observe that influence decays with network distance, but the mechanisms are heterogeneous (some people forget, some people distrust the source, some people prioritize other information). Practitioners must often model decay empirically (fit a curve to observed data) without understanding the underlying mechanisms. This gap creates uncertainty: does the fitted model capture the essential structure, or are we overfitting noise? Will a model trained on past decay rates predict future decay accurately if conditions change?

title: Counteracting decay requires energy, but the energy itself decays in effectiveness over time

body: Adding reinforcement, reminders, or fresh signal input can slow decay (or reset it), but the reinforcement itself is subject to decay. A teacher reinforces a lesson; the reinforcement effect decays. A leader repeatedly communicates a change initiative; repetition fatigue sets in (the message becomes background noise, its effectiveness decays). In biological systems, repeated doses of medication maintain efficacy, but tolerance can develop (the signal decay rate itself increases). Practitioners often underestimate the *cumulative cost* of maintenance: sustained decay requires sustained energy input, and the marginal effectiveness of that input may decline over time.

title: Decay rates vary wildly across domains and systems, creating coordination problems

body: A drug decays with a half-life of 6 hours; leadership must reinforce organizational changes weekly or monthly to counteract norm decay; a social media post decays to irrelevance within days; institutional memory decays over years; climate forcing (CO₂) decays over centuries. When multiple signals with different decay rates interact in the same system, mismatches emerge. A policy may be designed (timescale: months) to address a social problem (decay timescale: days) or an environmental problem (decay timescale: decades). The designer must account for decay rate mismatch or the policy will be ineffective. This is difficult because practitioners are often trained in domain-specific timescales and unaware of decay rates outside their domain.

title: Slowing decay in one part of a system can accelerate decay elsewhere

body: In population dynamics, if a predator (signal) decays quickly due to starvation, the prey population (which would otherwise decay under predation) instead surges. In organizational systems, protecting a valued norm from decay (through heavy reinforcement or rules) can displace decay onto other norms or onto employee morale and agility. In pharmacology, if a drug is metabolized slowly (low decay rate), it accumulates in tissue and becomes toxic (a different kind of degradation). Focusing solely on slowing a particular signal's decay can create system-level instability. The tension is between *local decay control* and *system equilibrium*; optimal decay rates exist, and both too-fast and too-slow decay can be pathological.

title: Decay can be a feature or a bug depending on context and timescale

body: High activation energy (slow decay of barriers) protects systems from reactive, destabilizing changes; low decay of memories preserves historical knowledge but can trap systems in harmful past patterns. In chemistry, some reactions are stable precisely *because* they have high activation energy and proceed imperceptibly slowly—this is protective. In organizations, high decay of fads prevents overcommitment to transient trends; but high decay of core values can erode the organization's identity. The question "Should we slow or accelerate this decay?" requires understanding what stability or change we want at which timescale. Reflexively opposing all decay (trying to make everything permanent) or accepting all decay (assuming nothing persists) both fail. The tension is between preservation and adaptation.

title: Observing decay is not the same as understanding it

body: A practitioner can observe that patient compliance decays over a medication regimen (they see an S-curve of adherence declining over weeks), but this observation conflates multiple mechanisms: forgetting, side effects, improved health (no longer symptomatic), inconvenience, competing priorities, and loss of belief in the medication's value. Each mechanism may have a different decay rate and different interventions. Designing effective interventions (improve adherence) requires distinguishing mechanisms, but the aggregate decay curve obscures them. This is the classic problem of black-box modeling: we fit a decay law to the macro-pattern without understanding the micro-causes, and our predictions fail when the micro-causes shift. The tension is between practical prediction (use the best-fit curve) and mechanistic understanding (invest in understanding the causes).

prose:
1. **Reinforcement scheduling:** Apply maintenance inputs at intervals shorter than the decay timescale to keep the signal above a functional threshold. Examples: repeated doses of medication, periodic reminder messages, refresher training courses, regular communication of organizational values.

2. **Decay-rate reduction:** Identify and modify mechanisms causing decay to slow the rate. Examples: extended-release drug formulations (slower decay of plasma concentration), organizational structures that embed norms (slower decay of culture), writing systems that preserve knowledge (slower decay of information).

3. **Signal amplification:** Increase initial signal strength to extend the period before decay makes it negligible. Examples: higher drug doses (longer therapeutic window despite decay), louder announcements (wider propagation before attenuation), founding stories told repeatedly (greater cultural persistence).

4. **Substrate isolation:** Protect the signal from the environment that causes decay. Examples: sealed containers for radioactive sources (slower background decay), confidential handling of sensitive information (slower spread and decay of rumors), closed feedback loops in organizations (preservation of institutional knowledge).

5. **Threshold adjustment:** Instead of fighting decay, adjust the *acceptable threshold* of signal strength. Examples: lower clinical targets for chronic disease management (accept partial efficacy), lower standards for organizational compliance during transition periods, accept reduced engagement metrics for legacy content.

6. **Cascade restarts:** Design systems where one decaying signal triggers renewal of a fresh signal before crossing the negligible threshold. Examples: automated medication reminders trigger adherence before drug concentration becomes sub-therapeutic, scheduled organizational communications reset norm awareness, social media algorithms promote trending content before it decays to zero visibility.

notes:
Signal decay operates across multiple timescales and substrates. At the molecular scale, decay rates are determined by chemical and physical constants (half-lives, rate constants). At the organizational scale, decay timescales depend on human memory, attention, and institutional mechanisms. At the societal scale, decay of norms, values, and knowledge depends on generations, institutional continuity, and cultural transmission mechanisms. Understanding which scale applies in a given context is crucial. A social psychologist modeling norm decay might confuse individual-level decay (a person forgets a social norm) with group-level decay (a norm evaporates from the group if no mechanism reinforces it).

Signal decay also interacts with *bifurcation* and *critical transitions*. If a signal decays slowly, it may cross a critical threshold and trigger a sharp phase transition (a norm that gradually decays finally reaches a tipping point where it reverses suddenly; a confidence level that slowly erodes suddenly collapses). Understanding decay rates helps predict whether systems will undergo smooth transitions or sudden shifts.

The concept carries implicit assumptions: that decay is *regular and predictable*, that we can measure and model it, that decay can be managed (slowed, prevented, or exploited). When these assumptions fail—when decay is random or chaotic, when it is unmeasurable, when it is actively resisted by powerful actors—the signal decay framework becomes less useful and must be supplemented with other frameworks.


Signal Detection Theory

type: prime

one_liner: Every decision under noise factorizes into a sensitivity that fixes the achievable error trade-off and a freely-chosen criterion that distributes errors along it.

status: draft

core_idea:
In any setting where an observer must decide whether a particular state
of the world — the "signal" — is present against a noisy background,
every decision factorizes into two independent components. The first is
*sensitivity*: how well the observer's internal evidence separates
signal-present from signal-absent worlds. The second is a *criterion*:
how much evidence the observer requires before responding "present." The
choice of criterion is a free policy variable. It shifts the trade among
hits, misses, false alarms, and correct rejections, moving the operating
point along a receiver-operating-characteristic (ROC) curve whose *shape*
is fixed by sensitivity.<!-- FACT-D71-701 -->[^green-swets-1966] Confusing the two — reading high false-alarm
rates as low sensitivity, or vice versa — produces persistent diagnostic
errors.

The load-bearing structural commitment is that *sensitivity and
criterion are orthogonal*, and that this orthogonality bounds what any
decision policy can achieve. No criterion can transcend the underlying
sensitivity; it can only redistribute that sensitivity's errors. To
lower false alarms by tightening the criterion is to raise misses by a
quantifiable amount, sliding along the existing ROC; to lower both at
once requires a *different and better ROC*, which means improving
sensitivity — better instruments, better features, more evidence per
decision — not adjusting the cutoff. The framework reduces any binary
decision under uncertainty to a common 2×2 outcome matrix and two scalar
summaries, one for sensitivity and one for criterion, and it cleanly
separates three quantities that ordinary language fuses into "how good is
this test?": the discrimination capacity of the evidence, the decision
rule applied to it, and the cost-of-error structure that, together with
base rates, picks the optimal operating point. It is, in effect, a
coordinate system for decisions under noise.

structural_signature:
*the latent binary state* — *the noisy evidence variable* — *the two evidence distributions (signal-absent, signal-present)* — *the sensitivity that separates them* — *the freely-chosen criterion partitioning the evidence axis* — *the ROC relating the two error rates as the criterion sweeps* — *the cost-and-base-rate structure selecting the operating point*

The pattern is present when each of the following holds:

- **A latent binary state.** A signal is either present or absent —
  disease, target, perpetrator, guilt — and is not directly observed.
- **A noisy evidence variable.** The decider observes only a graded
  evidence signal correlated with, but not determined by, the latent
  state.
- **Two evidence distributions.** The evidence has one distribution
  when signal is absent and another when present; these overlap.
- **Sensitivity.** A scalar captures how far apart the two
  distributions sit — the discrimination capacity of the evidence,
  fixed by instruments, features, and information per decision.
- **A criterion.** A freely-chosen threshold partitions the evidence
  axis into "report present" and "report absent"; it is a policy
  variable independent of sensitivity.
- **An ROC and a 2x2 outcome.** Sweeping the criterion traces a
  receiver-operating-characteristic curve whose shape is fixed by
  sensitivity, mapping every decision into hit / miss / false alarm /
  correct rejection.
- **A cost-and-base-rate structure.** Error costs together with the
  population base rate select the optimal operating point on the
  fixed ROC.

These compose around one orthogonality: sensitivity fixes the achievable
error trade-off (the ROC), and criterion only redistributes errors along
it. Lowering both error rates at once requires a better ROC — improve
the evidence — not a moved cutoff, so the central diagnostic is whether
a performance complaint is a sensitivity problem or a criterion problem.

what_it_is_not:
- **Not `type_i_type_ii_errors`.** That names the two error *kinds* (false
  alarm, miss). Signal detection theory adds the *generative model* — two
  overlapping evidence distributions — that factorizes the error trade-off
  into a sensitivity that fixes the ROC and a criterion that distributes
  errors along it (see `type_i_type_ii_errors`).
- **Not `hypothesis_testing_null_vs_alternative`.** Null-hypothesis testing
  fixes a significance level and asks whether to reject. SDT treats the cutoff
  as a *free policy variable* selected by error costs and base rates, and
  separately characterizes the evidence's discrimination capacity across the
  whole ROC (see `hypothesis_testing_null_vs_alternative`).
- **Not `signaling`.** Despite embedding-nearness, signaling concerns a sender
  strategically emitting a costly cue to convey type. SDT concerns a *receiver*
  discriminating a latent state from noisy evidence. One is about emission;
  the other about detection.
- **Not `calibration`.** Calibration is whether stated probabilities match
  observed frequencies. SDT is about *discrimination* (separating the two
  states) and *criterion placement* — a perfectly discriminating detector can
  be miscalibrated, and vice versa.
- **Not `selection_bias`.** Selection bias distorts which data enter the
  sample. SDT presumes the evidence distributions and asks how to decide
  against them; a biased sample would corrupt the estimated distributions but
  is a different defect.
- **Common misclassification.** Reading a high false-alarm rate as a bad
  instrument (low sensitivity) when it is a lenient criterion, or the reverse.
  The two are orthogonal: moving the cutoff slides along one ROC and cannot
  lower both error rates. The tell: would the complaint be fixed by moving the
  cutoff (criterion) or does it require a better ROC (sensitivity)?

broad_use:
The same sensitivity-criterion factorization recurs across substrates
that share nothing but the structure of a decision against noise. In
**psychophysics**, it is the detection of faint stimuli against
perceptual noise, the original setting. In **radar and sonar**, it is
distinguishing real targets from clutter under jamming, the field where
the theory developed alongside its psychological form.<!-- FACT-D71-702 -->[^marcum-1948] In **medical
screening**, the entire ROC-and-AUC vocabulary of mammography, lab
assays, and rapid tests is signal-detection theory applied.<!-- FACT-D71-703 -->[^metz-1978] In
**machine-learning classification**, it is precision-recall and ROC
curves, decision thresholds on probability outputs, and cost-sensitive
threshold tuning. In **eyewitness identification**, it separates a
witness's sensitivity to recognize a perpetrator from the witness's
willingness to identify someone.<!-- FACT-D71-704 -->[^wixted-mickes-2014] In **security screening** — airport
scanners, content moderation, fraud detection — it is the choice of
criterion given the costs of false alarms versus misses.<!-- FACT-D71-705 -->[^swets-dawes-monahan-2000] In **judicial
decision-making**, the sensitivity of evidence to guilt and the criterion
of "reasonable doubt" are explicitly named in legal doctrine.<!-- FACT-D71-706 -->[^dekay-1996] In
**astronomy and gravitational-wave detection**, it is matched-filter
threshold-setting against detector noise.<!-- FACT-D71-707 -->[^abbott-etal-2016] And in **memory research**,
recognition memory is analyzed as old-versus-new signal detection.<!-- FACT-D71-708 -->[^egan-1958] The
framework is the same in each; the substrate-specific work is locating
the two evidence distributions, the criterion, and the cost structure.

clarity:
The theory clarifies by separating two questions that ordinary language
fuses into "how good is this test?": *how informative is the underlying
evidence?* — sensitivity — and *what decision rule are we applying to
it?* — criterion. Once separated, many disputes resolve. "The test has
too many false alarms" can mean either that the underlying sensitivity is
poor, a measurement problem, or that the criterion is set too lenient for
the cost structure, a policy problem — and the corrective actions are
entirely different. The frame likewise distinguishes the cost-of-error
structure from the discrimination capacity, two things that together but
separately determine the right operating point. The clarifying force is
to make visible that an error rate is a *joint* product of evidence
quality and decision policy, so that the right fix can be aimed at the
component actually responsible rather than at the conflated whole.

manages_complexity:
The theory collapses any binary decision under uncertainty into a common
2×2 confusion matrix — hit, miss, false alarm, correct rejection — with
two scalar summaries: a sensitivity index for the evidence and a
criterion for the policy. The full noise-plus-signal apparatus reduces to
one curve, the ROC, and one operating point on it. Disputes about test
performance, screen calibration, and decision thresholds all proceed
inside this shared frame across domains, so an argument about
mammography, a spam filter, and a legal standard can be conducted in the
same terms. The compression is sharp because it isolates exactly two
degrees of freedom: everything about a binary decision under noise that
matters is captured by where the ROC sits, set by sensitivity, and where
on it the operating point lies, set by criterion. A potentially
bewildering performance debate thereby reduces to two questions — is the
ROC good enough, and is the operating point in the right place — each
with a distinct remedy.

abstract_reasoning:
The framework supports a family of inferences that pre-theoretic talk
about "good tests" cannot pose cleanly. At a given criterion, what are
the costs of the unavoidable error mix given the population's base rate?
How would the optimal criterion change if base rates shifted? Could the
false-alarm rate be lowered without raising the miss rate — that is, is
there room on the sensitivity dimension, or only on the criterion
dimension? What is the cost of treating sensitivity and criterion as
linked when they are independent? These questions concern the relationship
among *discrimination capacity, decision policy, and error cost under
noise*, a relationship indifferent to whether the decider is a biological
perceiver, a physical detector, a statistical procedure, or a judicial
body. To reason with the theory is to reason about which of two
orthogonal levers — improve the evidence or move the cutoff — a given
performance complaint actually calls for, a distinction that holds in
every substrate where decisions are made against noise.

knowledge_transfer:
The portable interventions follow directly from the orthogonality. When
the cost of misses rises — a deadly disease, a security threat — lower the
criterion, accepting more false alarms. When the cost of false alarms
rises — expensive follow-up, low base rate of true positives — raise it.
To improve on the *sensitivity* dimension requires entirely different
work: better instruments, better features, more training data, more
evidence per decision. The framework forces this distinction explicitly,
so that effort is never wasted adjusting a cutoff when the real deficit
is in the evidence, or rebuilding an instrument when the real problem is a
mis-set criterion.

The structural roles map across substrates. The *latent binary state* is
the disease, target, perpetrator, threat, or guilt that may or may not be
present; the *noisy evidence variable* is the image, reading, memory
trace, or signal observed by the decider; the *two evidence
distributions* are the signal-absent and signal-present densities whose
separation is sensitivity; the *criterion* is the threshold partitioning
the evidence axis into "report present" and "report absent"; the *ROC
curve* relates the two error rates as the criterion sweeps; and the
*cost-of-error structure*, with base rates, selects the optimal operating
point. A radiologist deciding whether a higher false-positive rate
warrants a stricter cutoff or a better imaging modality, a machine-
learning engineer tuning a classification threshold against
misclassification costs, and a court setting an evidentiary standard are
performing the same structural act: choosing an operating point on a
fixed ROC, and recognizing that only a better ROC can lower both error
rates at once. The diagnostic — *is this a sensitivity problem or a
criterion problem?* — travels unchanged across psychophysics, radar,
medical screening, machine learning, eyewitness procedure, security,
law, and astronomy. Because the intervention vocabulary is identical
across these media, a practitioner who has separated sensitivity from
criterion in one domain can import the whole apparatus — characterize the
ROC, locate the operating point, price the error costs — into any domain
that makes binary decisions against noise.

formal_abstract:
Take the Gaussian equal-variance model, the theory's analytic core. The *latent
binary state* is signal-absent versus signal-present. The *noisy evidence
variable* $x$ is drawn from $N(0, 1)$ when the signal is absent and from
$N(d', 1)$ when present, so the *two evidence distributions* are unit-variance
normals separated by $d'$ — the scalar *sensitivity*.<!-- FACT-D71-709 -->[^green-swets-1966] The observer reports
"present" whenever $x$ exceeds a *criterion* $c$. From these two parameters every
outcome follows by integration: the hit rate is the area of the signal-present
density above $c$, the false-alarm rate is the area of the signal-absent density
above $c$, and *sweeping $c$ traces the ROC curve* whose bow toward the top-left
corner is fixed entirely by $d'$. The orthogonality is exact and visible: move
$c$ left and you raise both hits and false alarms, sliding *along* one ROC; you
cannot lower both error rates without increasing $d'$, which means a *different,
better ROC* — better evidence, not a moved cutoff. The *cost-and-base-rate
structure* selects the optimal $c$: the likelihood-ratio criterion that minimizes
expected cost is $c^* = \ln[(p_{\text{absent}} \cdot C_{FA}) /
(p_{\text{present}} \cdot C_{miss})]$, so a rarer signal or a costlier false
alarm pushes the criterion higher.<!-- FACT-D71-710 -->[^green-swets-1966] The diagnosis this licenses is the prime's
central question made quantitative: a performance complaint is a *sensitivity*
problem if it requires raising $d'$ and a *criterion* problem if it only requires
moving $c$.

**Mapped back:** the two normals are the evidence distributions, $d'$ is the
sensitivity fixing the ROC, $c$ is the freely-chosen criterion redistributing
errors along it, and the likelihood-ratio cutoff is the cost-and-base-rate
operating point — the framework's roles in closed form.

applied_industry:
Two applied substrates carry the identical structure. First, mammographic
screening. The *latent state* is cancer present or absent; the *evidence* is the
radiologist's read of the image; the *two distributions* are the appearance
densities of malignant versus benign tissue, whose overlap fixes *sensitivity*
(set by imaging modality and reader skill). The *criterion* is how suspicious a
finding must look before the radiologist recalls the patient. The clarifying
payoff is decisive: a program with "too many false positives" faces two entirely
different fixes. If the underlying read is poorly discriminating, that is a
*sensitivity* problem — buy better imaging (tomosynthesis), train readers, add
evidence — which moves to a better ROC.<!-- FACT-D71-711 -->[^pisano-etal-2005] If the read is fine but the recall
threshold is set too lenient for the cost structure and the low base rate of
cancer, that is a *criterion* problem — raise the cutoff, trading a few more
missed cancers for far fewer needless biopsies. Confusing the two wastes effort
rebuilding an instrument when the cutoff was the issue, or vice versa. Second, a
machine-learning fraud classifier outputs a probability score; the *evidence
distributions* are the score densities for legitimate versus fraudulent
transactions, their separation summarized by AUC (sensitivity), and the *decision
threshold* on the score is the criterion.<!-- FACT-D71-712 -->[^fawcett-2006] When fraud losses (cost of a miss) rise
relative to the friction of a false decline (cost of a false alarm), the engineer
*lowers the threshold* — the same move a court makes when it sets "beyond
reasonable doubt" high because the cost of a false conviction is judged to exceed
the cost of a false acquittal.<!-- FACT-D71-713 -->[^swets-dawes-monahan-2000] To lower both error types at once, only a better
model — more features, more data, higher AUC — will do.

**Mapped back:** tumor appearance and transaction-score densities are the evidence
distributions; imaging modality and model AUC are the sensitivity; recall
threshold and decision threshold are the criteria; and biopsy-versus-miss and
fraud-loss-versus-friction are the cost structures selecting the operating point —
the same sensitivity/criterion factorization across medicine, machine learning,
and law.


Signal Devaluation

one_liner: A signal erodes in informativeness as issuance expands and readers adapt.

core_idea:
Signal devaluation is the structural pattern in which a signal — a
grade, credential, title, rating, badge, certification, review star,
endorsement, or display feature — erodes in *informativeness over time*
through a self-reinforcing dynamic of issuer expansion, demand-side
pressure, and adaptive recipient expectations. The signal's *nominal
value* may be unchanged — "magna cum laude," "Senior Vice President,"
"five stars," "AAA" — but the *information content* it carries about the
underlying type falls toward zero as the signal saturates. The
characteristic endpoint is *separating-equilibrium collapse*: the signal
that once distinguished types no longer does, and recipients must turn to
finer-grained or alternative signals to recover the discrimination they
have lost.

Four commitments are load-bearing. There is a *signal* with an issuance
process — someone awards grades, confers titles, certifies products,
issues ratings. There is a *supply elasticity* — the issuer faces weak
or no enforceable constraint on how many strong-value units to award, so
expanding issuance is locally costless or even beneficial. There is
*demand-side pressure* — recipients want strong values for downstream
gating (careers, visibility, market access), pushing issuers toward
generosity. And there are *adaptive recipient expectations* — consumers
of the signal update their reference standards on what they observe, so
yesterday's strong signal becomes today's baseline, and the threshold at
which the signal discriminates drifts with the equilibrium.

Together these produce a ratchet: issuers expand the strong-signal
category to meet demand, recipients adapt their reading, issuers expand
again to maintain perceived value, until the signal saturates at the top
of its scale and its discrimination function collapses. The dynamic is
*not* monetary inflation in the strict sense — there is no unit of
account, no price index, no purchasing-power calculation — but the
structural arithmetic of expansion-plus-adaptation is the same, which is
why "inflation" is the natural metaphor and also why mistaking the
metaphor for identity leads to importing the wrong intervention levers.

structural_signature:
*the costly signal with an issuance process* — *the issuer with elastic supply* — *the demand-side pressure for strong values* — *the adaptive recipient updating reference standards* — *the expansion-plus-adaptation ratchet* — *the separating-equilibrium-collapse invariant*

The pattern is present whenever these components are configured together:

- **The signal (role).** A discriminating marker — grade, credential,
  title, rating, badge, star — issued to indicate an underlying type,
  whose *nominal* value is fixed but whose *information content* can vary.
- **The issuer with supply elasticity (role).** A party that controls how
  many strong-value units to award and faces weak or no enforceable
  constraint on issuance, so expanding the strong category is locally
  costless.
- **The demand-side pressure (relation).** Recipients want strong values
  for downstream gating, pushing the issuer toward generosity.
- **The adaptive recipient (relation).** Consumers of the signal update
  their reference standards on what they observe, so yesterday's strong
  value becomes today's baseline.
- **The ratchet (relation).** Expansion and adaptation feed each other:
  issuers expand to meet demand, recipients re-anchor, issuers expand
  again, driving the signal toward the top of its scale.
- **The collapse invariant.** The endpoint is separating-equilibrium
  collapse — the posterior that a strong signal indicates a high type
  decays toward the base rate, and discrimination is lost. Escape requires
  rising signal cost, a binding supply constraint, or non-adaptive
  anchored standards.

The components compose into the signature: an endogenously-costed signal
under elastic supply and adaptive reading erodes from a discriminator to
a saturated, uninformative marker.

what_it_is_not:
- **Not `signaling`.** Signaling is the static separating equilibrium in
  which a costly signal distinguishes types; signal devaluation is its
  *long-run breakdown* when the issuer controls the cost and relaxes it
  under demand. Signaling is the working mechanism; devaluation is its
  decay.
- **Not monetary inflation.** "Inflation" is the natural metaphor, but
  signal devaluation has no unit of account, price index, or
  purchasing-power calculation — its levers (supply gating, signal
  redesign) differ from central-bank tools, and importing monetary
  remedies presupposes an accounting structure the signal lacks.
- **Not Goodhart gaming (`performativity`).** Goodhart corruption is
  *agents optimising a still-scarce proxy*; devaluation is *more strong
  signals issued* under elastic supply. Anti-gaming defences do nothing
  against pure supply-elasticity erosion, and supply caps do nothing
  against gaming.
- **Not `signal_decay_and_fadeout`.** Signal decay is a signal *fading
  with distance or time* in a transmission channel; devaluation is loss
  of *informativeness* through issuance expansion and adaptive reading,
  with the nominal value perfectly intact.
- **Not `regression_to_the_mean`.** Regression is a statistical pull of
  extreme measurements toward the average; devaluation is a directed
  ratchet of the strong-category fraction toward the scale ceiling,
  driven by incentives, not sampling.
- **Not `adverse_selection`.** Adverse selection is hidden-type sorting
  under information asymmetry at a point in time; devaluation is the
  *temporal erosion* of a signal that once enabled sorting, as the
  separating equilibrium collapses.
- **Common misclassification.** Blaming an individual issuer for "lowering
  standards." Each issuer faces a local-optimum pressure to inflate while
  the equilibrium of universal inflation destroys the signal for all — a
  collective-action structure, not an individual lapse.

broad_use:
- **Grade inflation.** Average GPAs have risen steadily with
  concentration at the top of the scale; graduate schools and employers
  adapt by discounting high grades and shifting weight to alternative
  signals.<!-- FACT-D71-726 -->[^rojstaczer-healy-2012]
- **Credential inflation.** Roles that once required no degree now
  require one, and roles that required a bachelor's now require a
  master's, without corresponding skill changes — a positional arms
  race in credential acquisition.<!-- FACT-D71-727 -->[^collins-1979]
- **Title inflation.** Vice-President proliferation and C-suite
  bloat erode titles' information about seniority and responsibility;
  external readers adapt by reading titles alongside firm context.<!-- FACT-D71-728 -->[^cohen-gurun-ozel-2023]
- **Ratings inflation.** The pre-2008 expansion of AAA ratings on
  structured products eroded AAA's informativeness about default risk,
  with recipients adapting only after the failure.<!-- FACT-D71-729 -->[^white-2010]
- **Review-star and engagement inflation.** Platform star ratings drift
  toward the top of the scale, and follower or like counts drift upward,
  with recipients adapting by reading ratios and review text rather than
  scalar scores.
- **Certification and biological display.** Consumer certifications
  proliferate and weaken as low-bar bodies compete for issuance, while
  runaway sexual selection inflates costly displays — a structural,
  not merely metaphorical, parallel.<!-- FACT-D71-730 -->[^spence-1973]

clarity:
Naming signal devaluation as its own pattern separates two things
surface vocabulary persistently conflates: *monetary inflation*, a
specific case with money, prices, and purchasing-power calculation, and
*signal devaluation*, the broader informational-erosion pattern in any
costly-signal system. The two share a metaphorical surface — erosion
over time — but have different apparatus. Monetary inflation has a unit
of account, a price index, and central-bank levers; signal devaluation
has none of these, and its intervention levers are different:
supply-side gating, signal redesign, alternative signals, and recipient
education.<!-- FACT-D71-731 -->[^spence-1973] Treating signal devaluation as if it were monetary inflation
imports tools that do not fit.

The clarity move also exposes a chronic policy bind. Each individual
issuer faces a local-optimum pressure to inflate — a university that
holds the line on grades loses applicants, a firm that holds the line on
titles loses recruits — yet the equilibrium of universal inflation
destroys the signal's value for everyone. This is a classic
collective-action problem, and naming the pattern makes the structure
visible while identifying the levers that operate at the equilibrium
rather than at the individual issuer: industry-level coordination,
third-party audit, regulatory floors, and recipient-side adaptation. The
practitioner can then stop blaming individual issuers for a dynamic that
is structurally collective.

manages_complexity:
The pattern compresses a sprawling family of "the X used to mean
something" complaints into a single diagnosis with four levers. *Supply
elasticity*: who can constrain issuance, through accreditation, audit,
regulation, self-imposed limits, or market discipline. *Demand-side
pressure*: why recipients want strong values, and whether the downstream
gating that creates the demand can be reformed. *Adaptive expectations*:
how the recipient-side reference standard updates, and whether the
update can be slowed or anchored. *Signal design*: whether the signal can
be re-engineered for harder-to-inflate properties — cost, distribution,
verifiability, granularity.

The four levers are the same across substrates, with different practical
accessibility, and the pattern also makes a class of signal-redesign
moves diagnosable. *Granularisation* — moving from coarse to fine-grained
scales — buys temporary discrimination but is itself subject to
inflation. *Cost-imposition* — making the signal costlier to acquire —
is the strongest fix but politically expensive. *Contextualisation* —
publishing the distribution so individual signals are read against it —
is the lowest-cost partial fix. *Alternative signals* — letting the
market evolve replacements — is what happens by default and explains the
endless cycle of new signals emerging as old ones inflate. Naming the
moves lets a practitioner choose deliberately among them rather than
drifting into the default.

abstract_reasoning:
Signal devaluation is the long-run failure mode of signalling under
unconstrained supply. The classical separating-equilibrium analysis
assumes a fixed cost structure that prevents low types from imitating
high types; signal devaluation describes what happens when the issuer
controls that cost — and thus the supply — and faces local incentives to
relax it. The pattern is the long-run breakdown of the separating
equilibrium when the cost structure is endogenous to the issuer's own
incentives rather than fixed by nature.

The dynamic admits clean equilibrium analysis. In a static separating
equilibrium the cost structure keeps low types from acquiring the high
signal, so the posterior that a high signal indicates a high type sits
near certainty. In the dynamic devaluation process the issuer relaxes the
threshold under demand, more types acquire the high signal, recipients
update their prior over type given signal, and the equilibrium drifts
until the high signal is acquired by all types and the posterior
collapses back to the base rate — the signal has lost its discriminating
function. The escape conditions are precise: signal cost must rise to
match expanded issuance, the issuer must face a binding supply
constraint, or recipients must commit to non-adaptive anchored
standards. The first two are structural; the third is behavioural and
hard to sustain. Costly-signalling constraints in biology, supply caps
in some professional examinations, and quota-based grade reform are the
respective realisations of these escape routes.<!-- FACT-D71-732 -->[^zahavi-1975]

knowledge_transfer:
Because the underlying arithmetic — supply elasticity plus demand
pressure plus adaptive expectations — is substrate-blind, both the
theory and the interventions move cleanly across the signalling
substrates. The separating-equilibrium analysis transfers directly to
credential inflation: credentials function only when their cost is
binding on lower types and slack on higher types, so credentials with
diminishing acquisition cost inflate predictably, and the repair is to
restore the cost gradient or accept the inflation. The post-2008
rating-agency lesson — that issuer-pays compensation creates inflation
pressure governance reform alone cannot fix — transfers to platform
rating design, where the platform's incentive to keep the rated
population happy pushes ratings up, suggesting recipient-pays or
third-party-pays alternatives.<!-- FACT-D71-733 -->[^white-2010]

Two diagnostics transfer as portable checks. The title-inflation
diagnostic — "your structure has compressed at the top because
recruiting pressure exceeded cost discipline" — generalises to any
signal whose issuer faces demand without a binding constraint, and
points to decoupling the consequential reward from the signal so the
signal's information can be preserved. And the adaptive-expectations
finding — that recipient adaptation is what completes the inflation
cycle — transfers as a recipient-education intervention: publish the
distribution alongside the individual signal so readers anchor their
reference standard. Underlying all of these is the cycle-of-signals
lesson: old inflated signals are continually replaced by new ones —
credentials by portfolios, ratings by review text — so signal redesign
is iterative and ongoing rather than one-shot. A practitioner who has
watched one signal inflate carries the whole expectation, and the whole
lever set, into the next.

formal_abstract:
Grade inflation is the cleanest worked instance because the separating-
equilibrium machinery applies almost literally. The *signal* is the course
grade, whose *nominal* values (A, B, "magna cum laude") are fixed but whose
*information content* about student type can vary. The *issuer with supply
elasticity* is the instructor or institution, which controls how many top
grades to award and faces no binding external constraint on issuance — awarding
an extra A is locally costless or even rewarded through better evaluations.<!-- FACT-D71-734 -->[^rojstaczer-healy-2012] The
*demand-side pressure* is students wanting top grades for graduate admission
and employment, the downstream gating that makes the signal consequential. The
*adaptive recipient* is the admissions committee or employer, which re-anchors:
once most applicants present A-heavy transcripts, an A becomes the baseline
rather than a discriminator. The *ratchet* then runs — issuers expand the top
category to meet demand, recipients re-anchor, issuers expand again — driving
the distribution toward the scale ceiling. The *collapse invariant* is reached
when nearly all students earn the top grade: the posterior that an A indicates
a high-type student decays toward the base rate, and the signal's discriminating
function is gone, forcing recipients onto finer or alternative signals (class
rank, course difficulty, standardised tests).<!-- FACT-D71-735 -->[^rojstaczer-healy-2012] The escape conditions are
precise and structural: signal cost must rise to match expanded issuance, the
issuer must face a binding supply constraint (an enforced grade quota), or
recipients must commit to non-adaptive anchored standards — the third being
behavioural and hard to sustain. **Mapped back:** the grade is the signal,
the instructor is the elastic-supply issuer, admissions re-anchoring is the
adaptive recipient, and near-universal A's are the separating-equilibrium
collapse the invariant names.

applied_industry:
The pre-2008 inflation of AAA credit ratings is the applied worked case,
exercising a financial-institutions domain. The *signal* is the rating (AAA,
AA), nominally fixed but informationally variable about default risk. The
*issuer with supply elasticity* is the rating agency operating under
issuer-pays compensation — it is paid by the very bond issuers seeking high
ratings, so expanding the AAA category is not merely costless but directly
revenue-positive, the strongest possible supply elasticity.<!-- FACT-D71-736 -->[^white-2010] The *demand-side
pressure* is structured-product issuers needing AAA tranches to sell to
mandate-constrained buyers (pension funds permitted to hold only top-rated
paper).<!-- FACT-D71-737 -->[^coval-jurek-stafford-2009] The *adaptive recipient* is the investor who, over time, treats AAA as
routine rather than exceptional. The *ratchet* ran through the structured-
finance boom: more AAA tranches were minted, the category swelled far beyond
its historical meaning, and the posterior that "AAA implies negligible default
risk" decayed — the *collapse invariant* realised abruptly when AAA-rated
structured products defaulted en masse.<!-- FACT-D71-738 -->[^benmelech-dlugosz-2009] The diagnostic lever the pattern
isolates is the issuer-pays incentive: because the issuer controls the cost
structure and is paid to relax it, governance reform alone cannot fix the
inflation, which points to recipient-pays or third-party-pays alternatives —
the same structural repair that platform star-ratings need, where the
platform's incentive to keep rated sellers happy pushes ratings up. A third
genuine domain is corporate title inflation: Vice-President proliferation
erodes a title's information about seniority, and external readers adapt by
reading titles alongside firm context, the contextualisation repair. **Mapped
back:** the rating is the signal, the issuer-pays agency is the elastic-supply
issuer, investor re-anchoring is the adaptive recipient, and the en-masse
default of AAA paper is the separating-equilibrium collapse.


Signal Extraction

one_liner: Quantitatively recovering a target component from an entangled observation using a signal model, a noise model, and a discriminator.

origin_domain: philosophy

core_idea:
Signal extraction is the structural pattern of separating a target component (the *signal*) from a co-present non-target component (the *noise*), under the constraint that observations contain both as a sum, product, or otherwise entangled superposition. The pattern requires three structural ingredients: a *model of the signal* (what shape the target component is expected to take), a *model of the noise* (what shape the unwanted component is expected to take), and a *discriminator* (a rule, filter, or estimator that uses the difference between the two models to assign each piece of the observation to one side or the other). The output is a recovered estimate of the signal plus a residual taken to be the noise.

The structural commitment is the recognition that extraction is *not* the same as detection (presence/absence) and *not* the same as classification (sorting into categories): extraction is the *quantitative recovery* of one entangled component from another, with the discriminator's quality measured by how cleanly it separates them. The cross-domain unifying object is the signal-to-noise ratio: the fraction of recovered variance attributable to the modeled signal rather than the noise. The discriminator is the load-bearing piece; without it, the observation cannot be decomposed. Three structural failure modes follow from it: it can be too aggressive (suppressing real signal as noise, overfitting the noise model), too lenient (admitting noise as signal, overfitting the signal model), or biased (consistently mistaking one for the other when the assumed distinguishing feature fails, model misspecification). The pattern is bare separation-by-model-difference, recognized identically across substrates, with no imported home context.

structural_signature:
*the entangled observation containing both components* — *the model of the target signal* — *the model of the non-target noise* — *the discriminator exploiting the model difference* — *the recovered signal estimate* — *the residual taken as noise* — *the signal-to-noise ratio measuring separation quality*

A process is signal extraction when each of the following holds:

- **An entangled observation.** The observable contains a target and a non-target component superposed — summed, multiplied, or otherwise mixed — so neither is directly readable.
- **A signal model.** An explicit account of the shape the target component is expected to take.
- **A noise model.** An explicit account of the shape the unwanted component is expected to take; what counts as "noise" is a role assigned relative to a question, not an intrinsic property.
- **A discriminator.** A rule, filter, or estimator that uses the *difference* between the two models to assign each piece of the observation to one side; it is the load-bearing element, since without it the observation cannot be decomposed.
- **A recovered estimate plus residual.** The output is a quantitative reconstruction of the signal and a leftover residual treated as noise — quantitative recovery, not mere detection or classification.
- **A signal-to-noise ratio.** A scalar measuring the fraction of recovered variance attributable to the modeled signal versus the noise.

The components compose a contest between two models mediated by a discriminator, whose failure modes follow directly: too aggressive (overfit noise model, suppressing signal), too lenient (overfit signal model, admitting noise), or biased (misspecified distinguishing feature). The residual is where evidence of misspecification appears.

what_it_is_not:
- **Not signaling.** `signaling` is a *communication* pattern: a sender deliberately emits a costly cue to inform a receiver. Signal extraction is a *recovery* operation performed by an observer on an entangled observation; there is no intentional sender, and the "signal" is a target component, not a transmitted message.
- **Not signal detection.** `signal_detection_theory` answers *presence/absence* against a criterion (is there a signal?). Extraction answers *magnitude* (how big is it?) — a discriminator will dutifully fit a signal model to pure noise, so detection must be established before an extracted value is trusted.
- **Not classification.** `classification` sorts whole observations into discrete categories. Extraction *quantitatively recovers* a continuous component entangled within an observation, leaving a residual; it decomposes rather than labels.
- **Not pattern recognition.** `pattern_recognition` identifies that a known structure is present. Extraction goes further to *recover the structure's magnitude* against a modeled noise floor, and its central object — the signal-to-noise ratio — has no counterpart in mere recognition.
- **Not dimensionality reduction.** `dimensionality_reduction` compresses observations to fewer coordinates preserving variance. Extraction separates a *target* component from a *noise* component using an explicit noise model; the discarded residual is defined by the question, not by low variance.
- **Common misclassification.** Reading a clean extracted value as proof the signal is real. Because the discriminator fits a signal model even to noise, a precise estimate of something that is not there (a spurious peak, a coefficient on a non-effect) is the characteristic failure — test presence before trusting magnitude.

broad_use: Signal extraction, read as model-the-signal, model-the-noise, discriminate, recurs across substrates that must recover a quantity from an entangled observation. In engineering signal processing, low-pass, band-pass, and Kalman filters separate a signal of known shape from broadband noise, and matched filters recover a known waveform from a noisy channel.<!-- FACT-D71-751 -->[^kay-1993] In statistics and machine learning, regression decomposes outcomes into a fitted signal and a residual, PCA and ICA decompose a high-dimensional observation into target components and discarded variance, and denoising autoencoders learn the signal model implicitly.<!-- FACT-D71-752 -->[^hyvarinen-oja-2000] In astronomy and physics, pulsar signals are extracted from background by folding, gravitational-wave detection matched-filters a chirp template against detector noise, and the Higgs discovery was a multi-channel extraction of a mass peak against expected backgrounds.<!-- FACT-D71-753 -->[^atlas-cms-2012] In sensory neuroscience, receptive fields extract spatial- or temporal-frequency components from input, and cocktail-party processing separates a target voice from background.<!-- FACT-D71-754 -->[^cherry-1953] In economics and finance, price signal is extracted from market microstructure noise, long-term trend is separated from cyclical and idiosyncratic components, and factor models extract systematic returns from idiosyncratic noise. And in epidemiology, disease-incidence signal is separated from reporting noise and seasonal variation, and confounder control extracts the causal-effect signal from observational noise.

clarity: Naming signal extraction separates four questions that intuitively collapse but call for different machinery: *is there a signal at all?* — a detection question answered by hypothesis testing; *what is the signal's shape?* — a model-specification question; *what is the signal's magnitude?* — an extraction question, the one this prime names; and *how much noise was discarded along with the signal?* — a signal-loss or fidelity question, the second face of extraction. The third and fourth are the structural face of the prime, and the extraction perspective forces a designer to articulate the signal model *and* the noise model *and* the discriminator *and* the residual budget, since many practical failures of measurement and inference reduce to one or more of these being implicit when they need to be explicit. The frame also surfaces the *signal/noise duality*: what is noise to one question is signal to another. The shot noise that defeats a faint-image extraction is itself the signal for an experiment measuring photon statistics; the macroeconomic cycle discarded as "noise" relative to the trend is the signal for business-cycle research. Naming the prime makes the duality structurally explicit rather than rhetorical, and tells the analyst that "noise" is a role assigned relative to a question, not an intrinsic property.

manages_complexity: A signal-extraction frame compresses a large family of measurement, inference, perception, and control problems to a small canonical diagram: observation leading to signal model plus noise model, leading through a discriminator to an estimated signal plus residual. The same diagnostic questions can be asked in the same order regardless of substrate — what is the assumed signal shape? what is the assumed noise shape? what is the discriminator? what is its bias? what are the type-I and type-II costs? what is the residual distribution, and is it consistent with the model assumptions? This compression lets engineers, statisticians, neuroscientists, and economists trade vocabularies on the same problem: a Kalman filter, a generalized-least-squares regression, a center-surround receptive field, and a structural-break test are four substrates of the same diagram. The complexity managed is the surface diversity of these methods; the diagram reveals that they share a single structure, so an analyst who has mastered the residual diagnostics of one can apply them, suitably re-instantiated, to all the others, and a stuck extraction in any field can be debugged by walking the same four model components.

abstract_reasoning: Signal extraction supports a tight family of inferences. *Signal-to-noise scaling*: extraction quality scales with the square root of integration time under independent noise, regardless of substrate — the result behind astronomical exposure, repeat-measure averaging, replication in experimentation, and ensemble methods.<!-- FACT-D71-755 -->[^kay-1993] *Matched-filter optimality*: when both signal and noise models are correct, the matched filter (cross-correlation with the signal template) is the optimal discriminator, a result that holds identically in radar, NMR, and template-based outbreak detection.<!-- FACT-D71-756 -->[^vantrees-2001] *Misspecification penalty*: when the noise model is wrong, the extracted signal acquires bias whose magnitude is bounded by the signal-noise overlap on the misspecified dimension, a result that predicts confounding bias in observational studies and trend contamination in macroeconomics from the same arithmetic. And the *aggressiveness tradeoff*: tightening the noise model increases signal loss while loosening it increases noise leakage, so there is no free lunch unless additional information enters the model. The reasoner who holds the prime therefore reads any recovery problem as a contest between two models mediated by a discriminator, expects the square-root law to govern how much averaging buys, and treats the residual as the place to look for evidence that the noise model is wrong.

knowledge_transfer: Because signal extraction is bare separation-by-model-difference — signal model, noise model, discriminator, residual, signal-to-noise ratio — a technique built in one field transfers to another by re-identifying those roles, and the prime's reach is the reach of that re-identification. Matched-filter logic from signal processing ports directly to maximum-likelihood estimation under a Gaussian noise model, the formal identity behind both being the same.<!-- FACT-D71-757 -->[^vantrees-2001] The astronomical practice of averaging over independent observations to lift a faint signal above noise is the same arithmetic as pooling across cohorts to extract a small effect size in epidemiology, both governed by the square-root-of-N rule, so an exposure-time intuition becomes a sample-size intuition without changing the underlying calculation. Neuroscience transfers to engineering: center-surround spatial filtering ports to edge-detection algorithms, and cocktail-party blind-source separation ports to independent component analysis.<!-- FACT-D71-758 -->[^hyvarinen-oja-2000] Macroeconomics transfers to climate science: trend-cycle decomposition with an uncertain noise model is structurally the same problem as separating long-term warming from natural variability, and the signal-noise-overlap warning applies in both directions. And control theory transfers to perception: Kalman-filter logic — fuse a predicted state with a noisy measurement weighted by their relative variances — describes both spacecraft attitude estimation and Bayesian multisensory integration. In every transfer the practitioner runs the same procedure — specify the signal model, specify the noise model, choose a discriminator, estimate the signal, inspect the residual for misspecification, and report the signal-to-noise ratio — and the transfer is structural rather than metaphorical because none of these steps mentions the substrate: a radio astronomer folding a pulsar time-series and an epidemiologist recovering a vaccine effect from a noisy field trial are running the same diagram, and only the discriminator and the meaning of "noise" are locally tailored.

formal_abstract:
The matched filter for detecting a gravitational-wave chirp instantiates every role of the prime and shows why the discriminator is load-bearing. The entangled observation is the strain time-series from an interferometer: a candidate astrophysical waveform buried in detector noise many times larger than itself. The signal model is an explicit template — the chirp $h(t)$ predicted by general relativity for a given pair of inspiralling masses, sweeping up in frequency and amplitude as the bodies spiral together.<!-- FACT-D71-759 -->[^abbott-etal-2016] The noise model is the detector's power spectral density, $S_n(f)$: a quantitative account of how much noise sits at each frequency, dominated by seismic motion at low frequencies and shot noise at high.<!-- FACT-D71-760 -->[^abbott-etal-2016] The discriminator is the matched filter, which cross-correlates the data against the template *weighted inversely by the noise spectrum*, $\langle d, h\rangle = \int \frac{\tilde d(f)\,\tilde h^*(f)}{S_n(f)}\,df$ — it deliberately downweights the frequency bands where noise dominates and upweights the clean bands. The recovered estimate is the filter output peaking at the signal's arrival time; the residual is the data with the best-fit template subtracted, and the signal-to-noise ratio is the peak height in units of the filter's own noise. The prime's matched-filter-optimality result is exactly what licenses this design: when both models are correct, no linear discriminator extracts more signal. And its failure modes are concrete diagnostics: a mis-specified template (wrong masses) is the *biased* discriminator and leaves chirp structure in the residual; a noise model that understates a noisy band makes the filter *too lenient* and admits a glitch as signal — which is why analysts inspect the residual and demand coincident detection across independent detectors.<!-- FACT-D71-761 -->[^abbott-etal-2016] The intervention space is the prime's: refine the template bank (signal model), re-estimate $S_n(f)$ (noise model), or integrate longer to lift SNR by the square-root law.

**Mapped back:** The gravitational-wave matched filter is the signal-model-noise-model-discriminator triple in exact form — relativistic template, detector noise spectrum, noise-weighted cross-correlation — with its residual the place misspecification shows and its SNR the separation measure, making it the canonical extraction.

applied_industry:
Two unrelated applied domains — noise-cancelling headphones in consumer electronics and confounder adjustment in observational epidemiology — run the same separation. In active noise cancellation, the entangled observation at the ear is the desired audio summed with ambient cabin noise.<!-- FACT-D71-762 -->[^kuo-morgan-1999] The signal model is implicit — the audio stream the listener wants — while the noise model is built live: an external microphone samples the ambient field, and an adaptive filter estimates its shape moment to moment. The discriminator generates an anti-phase waveform that destructively cancels the modeled noise, leaving the signal. The prime's aggressiveness tradeoff is audible: too aggressive a noise model cancels speech and high-frequency music along with the rumble (signal loss), while too lenient a model leaves residual hum (noise leakage), and the misspecification penalty appears as the well-known difficulty of cancelling sudden, broadband, hard-to-model sounds. The intervention is exactly the prime's — improve the noise model (more reference mics, faster adaptation) or accept a residual budget. Epidemiological confounder adjustment maps cleanly: the entangled observation is an exposure-outcome association that superposes the true causal signal with confounding noise from variables that drive both. The signal model is the causal effect of interest; the noise model is the assumed structure of confounders; the discriminator is the regression or propensity-score adjustment that subtracts the modeled confounding. The prime's misspecification-penalty result is the central caution of the field: when the noise model is wrong — an unmeasured confounder, a mis-specified functional form — the extracted effect acquires a bias bounded by the signal-noise overlap on that dimension, which is why analysts inspect residual confounding via negative controls and sensitivity analysis.<!-- FACT-D71-763 -->[^lipsitch-tchetgen-cohen-2010] In both domains the diagnosis runs through the same four model components, and the residual is where evidence of a wrong noise model appears.

**Mapped back:** Noise-cancelling audio and confounder adjustment both instantiate observation, signal model, noise model, and discriminator with a residual budget, and both are governed by the prime's aggressiveness tradeoff and misspecification penalty, so the diagnostic — inspect the residual, refine the noise model — transfers from acoustics to epidemiology unchanged.


Signal Inflation

one_liner: A sender over-fires a channel and depletes a finite, slow-regenerating receiver-side credibility budget, so the next true message lands on an audience that has rationally stopped responding.

core_idea:
A sender controls the firing rate and the precision of a signal on a channel whose value depends on a finite, receiver-side credibility resource. Each false-positive firing depletes that resource, and depletion is not freely reversible on the sender's own timescale. Once depleted, the next message — even if true and urgent — fails to elicit the intended action, because the receiver has rationally learned to discount the channel. The pathology is not noise on the wire, not signal decay, and not the receiver's stupidity: it is a commons-style overuse of a trust budget the sender does not see itself paying down.

The structural mover is the asymmetry between sending and receiving costs. The sender pays per firing in a currency cheap to it — sending another alert, raising another warning. The receiver pays per firing in a currency expensive to it — attention, action-cost on false alarms, disruption. When the sender's reward function rewards recall (catching every possible threat) and is blind to precision (the false-alarm rate), the equilibrium is over-firing, and the channel collapses into noise. The defining failure shows up not on the false firings but on the next true firing, which arrives at a depleted channel and gets the discounted response that the false firings earned. The credibility resource regenerates slowly relative to the rate at which firing can deplete it, so the damage is durable rather than self-correcting, and the receiver's discount is a rational response to the channel's track record rather than a failure of vigilance.

structural_signature:
*the firing sender* — *the shared channel* — *the finite receiver-side credibility stock* — *the cost asymmetry between firing and attending* — *the depletion-on-false-positive relation* — *the slow regeneration rate* — *the discounted response on the next true firing*

Signal inflation is present when these roles and relations hold:

- **A sender with firing control.** An actor that chooses the rate and precision at which a signal is emitted on the channel.
- **A finite, receiver-side credibility stock.** A depletable resource — attention, trust, willingness to act — held by the receiver, not the sender, and consumed each time the channel fires falsely.
- **A cost asymmetry.** The sender pays per firing in a currency cheap to it; the receiver pays in a currency expensive to it. This asymmetry is the structural mover that drives over-firing.
- **The depletion relation.** Each false-positive firing draws down the credibility stock; the receiver's rational discount of the channel is a function of its observed false-positive history.
- **A slow regeneration rate.** The stock refills far more slowly than firing can deplete it, so damage is durable rather than self-correcting.
- **The diagnostic failure locus.** The harm registers not on the false firings but on the *next true firing*, which arrives at a depleted channel and earns the discounted response the false firings purchased.

These compose into a resource economy: a recall-rewarded, precision-blind sender reward function drives the equilibrium toward over-firing, spending down a slow-refilling trust stock until the message that matters cannot land.

what_it_is_not:
- **Not `signal_decay_and_fadeout`.** Decay is passive, substrate-intrinsic attenuation of a signal over time or distance; signal inflation is active depletion of a *receiver-side* trust resource by a sender's firing policy. The medium is fine; the credibility budget is spent.
- **Not `signaling`.** Signaling concerns whether a single costly message is honest and incentive-compatible; signal inflation is the dynamic, repeated-firing failure in which an over-fired channel debases its own future messages regardless of any individual message's honesty.
- **Not `receptor_saturation` or `attentional_capacity` limits.** Those are capacity ceilings — the receiver can only absorb so much. Signal inflation is not about volume overwhelming a fixed buffer but about *rational discounting*: the receiver could attend but has learned the channel is not worth it.
- **Not `habituation_to_repeated_signal`.** Habituation is a receiver-side perceptual numbing that can persist even after precision is restored; signal inflation locates the cause on the *sender side* and is reversible by restocking precision — unless habituation has additionally set in.
- **Not `tragedy_of_the_commons`.** The commons pattern is many actors depleting a shared physical resource each rationally; signal inflation is, in its core form, a single sender depleting a receiver's credibility stock — a sender-receiver pattern, though it acquires a commons flavor only when many senders share one attention budget.
- **Common misclassification.** Diagnosing "the warning didn't work" as a noise or sensor problem and prescribing clearer language or better detectors. Catch it by asking whether the failure registers on the *next true firing* after a run of false ones — if the receiver is rationally discounting an over-fired channel, no receiver-side or medium-side fix can succeed.

broad_use: The same dynamic recurs across substrates that share little else. In folklore and pedagogy it is the boy who cried wolf — the canonical instance, transmitted for millennia precisely because it captures a structural pattern.<!-- FACT-D71-776 -->[^aesop-perry-1952] In critical-care medicine it is alarm fatigue: cardiac monitors fire so many false positives that clinicians silence or ignore them and genuine arrests are missed.<!-- FACT-D71-777 -->[^cvach-2012] In information security it is alert fatigue in security operations centers, where low-precision alert volume trains analysts to triage by ignoring patterns that resemble noise, and real intrusions hide in the background. In severe-weather warning it is the cry-wolf effect: communities that have experienced repeated false warnings show measurably reduced sheltering on subsequent ones.<!-- FACT-D71-778 -->[^trainor-etal-2015] In public-health communication it is over-warning followed by under-response, and threat-level systems stuck at one level until they lose informational content.<!-- FACT-D71-779 -->[^shapiro-cohen-2007] The pattern recurs again in regulatory inspection, where too many low-severity findings produce auditee habituation; in parenting and management, where the repeated "last warning" loses its cliff-edge function; and in software engineering, where review nits drown out substantive concerns and CI flakes degrade trust in real failures.<!-- FACT-D71-780 -->[^luo-etal-2014] In every case the sender's cheap, recall-rewarded over-firing spends down a slow-regenerating receiver-side credibility budget, and the channel's unit value debases until the message that matters cannot land.

clarity: The prime separates two failures usually conflated: the channel is noisy — the signal-decay or signal-detection problem — versus the channel was deliberately overused and the receiver rationally withdrew credit. The first is a property of the medium; the second is a property of the sender's policy interacting with the receiver's finite trust. Most "the warning didn't work" post-mortems misdiagnose the second as the first and propose technical fixes — better sensors, clearer language — that do not engage the actual mechanism. Naming the prime also re-attributes responsibility. The receiver who ignored the alarm is behaving rationally given the channel's track record; the sender who fired the channel beyond its credibility budget is the cause. The intervention class follows directly: discipline sender-side precision by raising the firing threshold, expose senders to the receiver-side cost of false alarms through consequence symmetry, redesign metrics to penalize false-positive rate alongside recall, or split high-cost messages onto a separate channel with its own credibility budget. This clarity is load-bearing because it determines where intervention can possibly work: receiver-side fixes such as training users to take alerts seriously cannot succeed, since the credibility erosion is a rational response to observed precision that receivers cannot un-learn faster than the sender can refill it.

manages_complexity: Signal inflation compresses what looks like a dozen unrelated operational failures — silent alarms, ignored vulnerabilities, dismissed tornado warnings, parental ineffectiveness, governance theater — into one diagnostic: the channel's credibility budget has been spent. The diagnostic licenses the same intervention family regardless of substrate: reduce the false-positive rate at the sender, separate high-cost messages onto a fresh channel, allow credibility to regenerate by silence, or surface receiver-side costs into the sender's optimization function. By treating every channel as carrying a scarce, slow-regenerating receiver-side resource that sender behavior depletes, the prime converts a heterogeneous catalogue of "people stopped responding to warnings" into a single resource-accounting problem with a single family of fixes. It also explains why purely receiver-side interventions usually fail and predicts where the failure will be deepest — channels where the sender's reward function rewards recall and ignores precision, the Goodhart configuration that guarantees over-firing. That predictive reach is what makes the prime a complexity-management tool rather than a label: given a sender's reward function, it forecasts whether the channel will inflate, and given an inflated channel, it identifies the small set of moves — threshold, partition, regeneration, consequence symmetry — that can restock the budget.

abstract_reasoning: The prime supports several substrate-independent moves. *Credibility-budget accounting* treats every channel as carrying a scarce, slow-regenerating receiver-side resource that sender behavior depletes, making the depletion legible before the channel collapses. *Precision-recall asymmetry diagnosis* identifies whether the sender's reward function rewards recall and ignores precision; if so, it predicts over-firing as the equilibrium. *Channel partitioning* holds catastrophic-message channels separate from routine ones so the latter cannot deplete the former's credibility. *Regeneration design* builds in mandatory silence between firings or graded escalation rather than maximum-recall firing. And *closed-loop precision audit* feeds false-positive rates back into the sender's optimization function with weights reflecting receiver-side cost. The abstract move uniting these is to model the sender-channel-receiver loop as a resource economy in which trust is a depletable, slowly-refilling stock, and to reason about warning systems as exercises in spending that stock wisely rather than as exercises in maximizing detection. This reframing lets a reasoner anticipate the failure of a maximum-recall policy from its reward function alone, locate the leverage on the sender side, and recognize that the receiver's apparent unresponsiveness is the rational equilibrium output of the sender's own firing history.

knowledge_transfer: A clinician trained on alarm fatigue recognizes alert fatigue in security operations centers, push-notification fatigue in consumer products, and warning-system erosion in atmospheric science as instances of the same problem, and the cross-domain recognition is so strong that the medical and IT communities literally share the term "alert fatigue."<!-- FACT-D71-781 -->[^sendelbach-funk-2013] The intervention vocabulary travels: raise alarm thresholds, separate critical from non-critical alarms, audit precision, and give the sender skin in the receiver's game. A weather scientist using the cry-wolf literature can read the parenting literature on last-warning inflation and apply the same model without re-derivation. The transfer is supported by formal work — signal-detection theory supplies the underlying mathematics, and signal inflation is the dynamic, sender-driven failure mode of a detection system whose operating point has drifted to maximum recall.<!-- FACT-D71-782 -->[^green-swets-1966] The role-mapping is fixed across substrates: sender maps to the monitor / the SOC tooling / the warning authority / the parent; channel maps to the alarm line / the alert stream / the warning system; receiver maps to the nurse / the analyst / the public / the child; credibility budget maps to the slow-regenerating trust resource; the failure maps to the missed arrest / the buried intrusion / the unsheltered community / the inert discipline. The prime's discipline is to keep it distinct from signal decay (passive substrate-intrinsic attenuation), from signal-detection theory (the static framework rather than the dynamic failure), from the tragedy of the commons (a many-actor physical-resource pattern rather than this sender-receiver credibility pattern), and from habituation (one receiver-side mechanism within the full sender-channel-receiver loop). Holding those distinctions is what lets a practitioner who has restocked a credibility budget by cutting alarm inflation in an ICU apply the identical threshold-and-partition logic to a security console or a public-warning system, recognizing in each the same depletable trust resource and the same recall-blind reward function driving it down.

formal_abstract:
Model the receiver as a Bayesian agent and the channel as a sequence of firings, each true or false. The credibility stock is the receiver's posterior probability that a firing is true: $P(\text{true} \mid \text{fire})$.<!-- FACT-D71-783 -->[^green-swets-1966] By Bayes' rule this equals the channel's precision — true firings over all firings. The sender controls the firing threshold, which sets the operating point on the detector's ROC curve: lowering the threshold raises recall but lowers precision, because the false-positive rate climbs faster than the true-positive rate once the threshold drops below the bulk of the noise distribution. A rational receiver acts only when expected benefit exceeds expected cost, i.e. only when posterior precision exceeds the cost ratio $c_{\text{act}} / c_{\text{miss}}$.<!-- FACT-D71-784 -->[^green-swets-1966] The depletion relation is mechanical: as the sender pushes the threshold down to maximize recall, precision falls, and once it crosses the cost-ratio line the receiver's optimal policy flips to ignore. The diagnostic failure locus follows: the next true firing arrives, but it is indistinguishable from the now-dominant false firings, so it is ignored — the miss is structurally caused by the sender's threshold, not by the receiver. The slow regeneration is the receiver's prior, which updates downward fast on a run of false alarms but upward only slowly on rare confirmed hits.

**Mapped back:** Firing threshold is the sender's control, ROC operating point is precision-recall asymmetry, posterior precision is the credibility stock, and the cost-ratio crossing is the moment the channel collapses and the next true firing is rationally ignored — the prime's roles in closed form.

applied_industry:
A hospital cardiac-telemetry unit instantiates the prime exactly. The sender is the monitoring device, configured with sensitive thresholds so it never misses an arrhythmia — recall-maximizing, precision-blind. The channel is the audible alarm. The credibility stock is the nursing staff's willingness to respond. False alarms from lead artifacts, patient motion, and benign rhythm variants fire hundreds of times per patient per day, each depleting the stock cheaply for the device but expensively for the nurse, who must walk over and check.<!-- FACT-D71-785 -->[^cvach-2012] Within days the staff habituates: they silence alarms, lower volumes, or stop responding — the rational discount of a channel running at single-digit precision. The diagnostic failure is the missed true arrest buried among false firings. The interventions follow the prime's family directly: raise device thresholds and add brief confirmation delays to cut the false-positive rate (sender-side precision), tier alarms so a true crisis fires on a separate, reserved channel with its own credibility budget (channel partitioning), and audit per-unit false-alarm rates as a tracked metric (closed-loop precision audit). The same moves reappear unchanged in a security operations center drowning in low-precision intrusion alerts and in a municipality whose repeated false tornado warnings have measurably reduced sheltering — both restocked by the identical threshold-and-partition logic.<!-- FACT-D71-786 -->[^trainor-etal-2015]

**Mapped back:** The telemetry device is the recall-rewarded sender, nurse attention is the slow-refilling credibility stock, habituation is the rational discount, and the missed arrest is the failure on the next true firing — with threshold discipline and channel partitioning as the restocking interventions.