{ "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T10:02:25Z", "providers": [ "Codex" ], "breadth": "recalled by Codex without web access - no source was read", "missingProviders": [], "pass": 2, "cost": { "grok": { "seconds": 22.9, "error": "Reading additional input from stdin...\nOpenAI Codex v0.153.4\n--------\nworkdir: R:\\02_PROJECTS\\02_Meta_Models_Platforms\\Ver.cy\\current\\thing-registry-backlog\nmodel: gpt-6-astra\nprovider: openai\napproval: never\nsandbox: read-only\nreasoning effort: none\nreasoning summaries: none\nsession id: 01a09510-9382-7972-94da-fd2d431e6e11\n--------\nuser\nDescribe what is already known about one registered thing. Answer as JSON only, no prose around it.\n\nThing: Gompertz function\nSense to describe: (none recorded)\nDomain code: XCT.REL\nAlso known as: (none)\n\n\nContext for this batch of 1001 things:\n# Batch 043: 10", "usd": 0.0, "recall": true }, "codex": { "seconds": 59.3, "error": "Reading additional input from stdin...\nOpenAI Codex v0.153.4\n--------\nworkdir: R:\\02_PROJECTS\\02_Meta_Models_Platforms\\Ver.cy\\current\\thing-registry-backlog\nmodel: gpt-6-astra\nprovider: openai\napproval: never\nsandbox: read-only\nreasoning effort: none\nreasoning summaries: none\nsession id: 01a09510-938c-7f91-bc72-66bcc63103f9\n--------\nuser\nYou are drafting a Vercy meta-model for one registered thing. Answer as JSON only, no prose around it.\n\nThing: Gompertz function\nRegistry id: vr.tr.gompertz-function\nPlane / domain: XCT / XCT.REL\nRegistry definition: (none recorded)\nNames folded into this entr" } }, "enrichment": { "pass": 3, "engine": "claude", "producedAt": "2026-10-06T12:00:00Z", "ingestedAt": "2026-10-06T08:42:05+00:00", "fields": [ "agent_conduct", "ethics", "failure_modes_and_hazards", "owners", "visual_identification" ] }, "gate": { "passed": true, "version": "1.0", "at": "2026-10-07T10:10:25+00:00", "evidence": [ "completeness 1.00", "sense: Wikidata description", "2 live sources", "review by codex: pass" ] } }, "metaModel": { "id": "THING-Q1011785", "registryId": "vr.tr.gompertz-function", "name": "Gompertz function", "version": "0.1.0-research.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.REL" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q1011785/", "model": { "registry_id": "vr.tr.gompertz-function", "name": "Gompertz function", "purpose": "Enable an agent to recognise a Gompertz function, interpret its parameters and growth behaviour, and judge whether its use for a particular dataset or decision is justified.", "definition": "The Gompertz function is an asymmetric sigmoidal function, commonly written y(t) = A exp(−b exp(−kt)) with A, b, k > 0, that describes growth from a limiting value of zero toward an upper asymptote A.", "scope_statement": "This model owns the identity, parameterisations, mathematical behaviour, estimation and application boundaries of the Gompertz function registered as vr.tr.gompertz-function in XCT / XCT.REL, using y(t) = A exp(-B exp(-kt)) with A, B, k > 0 as the reference increasing form.", "in_scope": [ "Canonical increasing Gompertz functions and explicitly mapped equivalent parameterisations", "Asymptotes, derivatives, inflection points and asymmetric growth behaviour", "Parameter meanings, units, coordinate dependence and admissible values", "Initial conditions and the associated Gompertz growth differential equation", "Fitting, parameter identifiability, uncertainty and application suitability", "Explicitly identified offsets, reversals and modified Gompertz formulations" ], "out_of_scope": [ "The Gompertz probability distribution as a complete statistical model", "Gompertz and Gompertz-Makeham mortality laws as demographic models", "Logistic, Richards and other growth-function families except for boundary comparisons", "Biological or economic mechanisms inferred solely from a fitted curve", "Data collection systems and general-purpose optimisation software" ], "distinguishing_features": [ "For the reference form, log(-log(y/A)) = log(B) - kt wherever 0 < y < A; this double-log relationship is linear in the independent variable.", "The increasing reference curve has an inflection at y = A/e, whereas the standard zero-baseline logistic curve has its inflection at A/2.", "Its relative growth rate y'/y = Bk exp(-kt) decreases exponentially, while absolute growth first rises and then falls.", "Its differential equation is y' = ky log(A/y), distinguishing it from the standard logistic equation y' = ry(1-y/A).", "Its zero and A asymptotes are approached at opposite ends of the real line, with a curve that lacks logistic point symmetry about its inflection." ], "characteristics": [ { "name": "Declared functional form", "kind": "category", "unit_or_values": "Exact equation and parameter convention; reference form, equivalent form or explicitly modified form", "why_it_matters": "The name Gompertz does not by itself determine parameter meanings or whether a baseline or lag convention has been added." }, { "name": "Asymptotic level A", "kind": "measurement", "unit_or_values": "A > 0; same units as y in the reference form", "why_it_matters": "Defines the upper asymptote and scales predictions, but can be poorly constrained before saturation is observed." }, { "name": "Position parameter B", "kind": "measurement", "unit_or_values": "B > 0; dimensionless in the reference form", "why_it_matters": "Determines y(0) = A exp(-B) and changes when the origin of t changes." }, { "name": "Rate parameter k", "kind": "measurement", "unit_or_values": "k > 0; inverse units of t", "why_it_matters": "Controls the time scale and the exponential decline in relative growth rate." }, { "name": "Inflection coordinate", "kind": "measurement", "unit_or_values": "t_i = log(B)/k; same units as t", "why_it_matters": "Locates maximum absolute growth and helps establish which growth phases the observations cover." }, { "name": "Maximum absolute slope", "kind": "measurement", "unit_or_values": "Ak/e; units of y per unit of t", "why_it_matters": "Provides an interpretable growth measure that is distinct from k." }, { "name": "Coordinate and response interpretation", "kind": "relation", "unit_or_values": "Independent variable, origin, units, response quantity and any transformation", "why_it_matters": "A curve fitted to a logarithmic response describes different behaviour from one fitted to the original response." }, { "name": "Evidence-supported use", "kind": "state", "unit_or_values": "Unassessed, descriptively supported, predictively validated or unsuitable for the declared use", "why_it_matters": "Separates mathematical membership in the family from empirical support for an application." } ], "affordances": [ "Recognise equivalent Gompertz equations and translate their parameters into a declared convention.", "Compute levels, slopes, inflection coordinates and admissible threshold-crossing times.", "Fit a Gompertz curve with an explicit observation model and estimate uncertainty.", "Check whether observations constrain the asymptote, rate and inflection sufficiently for the intended use.", "Compare Gompertz behaviour with logistic or other candidate growth curves.", "Flag extrapolations or mechanistic interpretations that exceed the evidence." ], "agent_conduct": { "may": [ "Fit and explain the Gompertz function for growth or mortality data, stating parameters and fit quality.", "Compare it with other sigmoid models when choosing a model for a dataset." ], "must_not": [ "Present a fitted curve as a reliable forecast outside the range of the data.", "Use mortality fits to judge an individual person's lifespan or insurability without a lawful and fair basis.", "Hide poor fit or alternative models that would change a conclusion." ], "requires_human": [ "Using a fitted model in medical, epidemic or insurance decisions about people." ] }, "ethics": { "considerations": [ "Growth and mortality models drive decisions about tumours, epidemics and insurance that affect people's lives." ], "affected_parties": [ "Patients", "Insured people", "Researchers" ] }, "owners": { "steward": "Nobody owns it; it is a mathematical function.", "master_systems": [] }, "visual_identification": [ "An S-shaped curve that rises slowly at first, steeply in the middle and approaches its upper limit more gradually than it left zero." ], "failure_modes_and_hazards": [ "Extrapolating a fitted curve can badly misjudge future growth or decline.", "Parameter estimates can be unstable with short or noisy data." ], "what_it_is_for": [ "The Gompertz function is used to represent asymmetric growth curves in biological populations, organism growth and other processes approaching a limiting size.", "Analysts fit it to observations to estimate growth timing and an upper asymptote, with predictions depending on the parameter convention and suitability of the model." ], "facets_not_applicable": { "interfaces": "No generally applicable standard, law or official classification governs the Gompertz function itself." } }, "sources": [ { "id": "wikidata", "title": "Wikidata item Q1011785: Gompertz function", "url": "https://www.wikidata.org/wiki/Q1011785", "url_status": "live", "what_it_supports": "identity and sense of the item", "checked_by": "wikidata-api" }, { "id": "wikipedia-en", "title": "Wikipedia: Gompertz function", "url": "https://en.wikipedia.org/wiki/Gompertz_function", "url_status": "live", "what_it_supports": "general description of the item", "checked_by": "wikidata-api sitelink" } ], "structure": { "bundles": [ { "id": "functional-identity", "name": "Functional identity", "description": "Establish the exact function represented by the Gompertz name.", "rationale": "Equivalent conventions and modified equations can otherwise cause incorrect parameter interpretation.", "layers": [ { "id": "equation-and-convention", "name": "Equation and convention", "description": "Record the equation, restrictions and coordinate conventions.", "findings": [ { "id": "reference-form-membership", "name": "Reference form membership", "description": "The standard increasing form is y(t) = A exp(-B exp(-kt)) with positive A, B and k; departures require explicit interpretation.", "questions": [ { "text": "What exact equation, parameter restrictions and independent-variable domain define this instance?", "kind": "definition", "id": "reference-form-membership-q01" }, { "text": "Which source establishes the chosen parameter convention, including any offset or response transformation?", "kind": "provenance", "id": "reference-form-membership-q02" } ] } ] }, { "id": "equivalence-and-neighbours", "name": "Equivalence and neighbours", "description": "Distinguish reparameterisation from a change of function family or modelling role.", "findings": [ { "id": "parameter-mapping", "name": "Parameter mapping", "description": "The form y(t) = A exp(-exp(-k(t-t_i))) is equivalent to the reference form when B = exp(kt_i). Distribution and mortality uses require separate semantic identification.", "questions": [ { "text": "Can the proposed equation be mapped algebraically to the reference form, and what is the mapping?", "kind": "definition", "id": "parameter-mapping-q01" }, { "text": "Does this use concern a growth response, a distribution quantity or a mortality hazard, and which neighbouring model owns the additional semantics?", "kind": "boundary", "id": "parameter-mapping-q02" } ] } ] } ] }, { "id": "growth-geometry", "name": "Growth geometry", "description": "Describe the constraints that the Gompertz shape imposes on a trajectory.", "rationale": "Recognition and suitability depend on asymmetric growth and limiting behaviour, not merely a sigmoid appearance.", "layers": [ { "id": "limits-and-inflection", "name": "Limits and inflection", "description": "Locate the limiting levels and the transition between acceleration and deceleration.", "findings": [ { "id": "asymmetric-sigmoid", "name": "Asymmetric sigmoid", "description": "On the full real line, the reference curve increases from zero toward A and inflects at (log(B)/k, A/e); a restricted observation interval may contain only one growth phase.", "questions": [ { "text": "Where are the inflection and asymptotes relative to the observed interval?", "kind": "measurement", "id": "asymmetric-sigmoid-q01" }, { "text": "Does the response support a zero lower asymptote and Gompertz asymmetry, or is another formulation needed?", "kind": "boundary", "id": "asymmetric-sigmoid-q02" } ] } ] }, { "id": "rates-and-dynamics", "name": "Rates and dynamics", "description": "Connect the curve with its absolute and relative growth rates.", "findings": [ { "id": "gompertz-rate-law", "name": "Gompertz rate law", "description": "The reference curve satisfies y' = ky log(A/y), with exponentially declining relative growth and maximum absolute slope Ak/e.", "questions": [ { "text": "What absolute and relative growth rates does the fitted function imply across the relevant interval?", "kind": "measurement", "id": "gompertz-rate-law-q01" }, { "text": "Is this differential equation being used as a descriptive identity or as an independently justified process law?", "kind": "boundary", "id": "gompertz-rate-law-q02" } ] } ] } ] }, { "id": "estimation-and-evidence", "name": "Estimation and evidence", "description": "Establish how observations support the function and its parameters.", "rationale": "A plausible fitted trajectory can coexist with weakly determined asymptotes and unreliable extrapolation.", "layers": [ { "id": "observation-and-fitting", "name": "Observation and fitting", "description": "Specify the response scale, observation errors and fitting procedure.", "findings": [ { "id": "fit-semantics", "name": "Fit semantics", "description": "Fitting raw responses, log responses or double-log transformed responses generally implies different error assumptions and objectives.", "questions": [ { "text": "What response scale, error model, weighting and parameter constraints define the fit?", "kind": "definition", "id": "fit-semantics-q01" }, { "text": "How are zero values, censoring, repeated observations and measurement limits handled when present?", "kind": "action", "id": "fit-semantics-q02" } ] } ] }, { "id": "identifiability-and-validation", "name": "Identifiability and validation", "description": "Assess whether available data constrain the parameters and support the intended predictions.", "findings": [ { "id": "parameter-support", "name": "Parameter support", "description": "Limited coverage of growth phases can leave the asymptote and associated parameters weakly constrained despite small fitting residuals.", "questions": [ { "text": "How do uncertainty estimates and parameter dependence change when the asymptote is fixed, freed or informed by additional observations?", "kind": "measurement", "id": "parameter-support-q01" }, { "text": "What residual checks and held-out predictions distinguish this fit from plausible alternative growth curves?", "kind": "action", "id": "parameter-support-q02" } ] } ] } ] }, { "id": "prediction-and-use-boundaries", "name": "Prediction and use boundaries", "description": "Translate the function into justified calculations and decisions.", "rationale": "Threshold calculations and extrapolations require domain checks and evidence beyond algebraic correctness.", "layers": [ { "id": "evaluation-and-inversion", "name": "Evaluation and inversion", "description": "Support forward predictions and inverse threshold calculations.", "findings": [ { "id": "threshold-reachability", "name": "Threshold reachability", "description": "For 0 < y_star < A, the reference curve reaches y_star at t = (log(B) - log(-log(y_star/A)))/k; thresholds at the asymptotes have no finite crossing time.", "questions": [ { "text": "Is the requested threshold reachable within the declared domain and supported prediction interval?", "kind": "boundary", "id": "threshold-reachability-q01" }, { "text": "How will parameter uncertainty be propagated into the predicted crossing time, especially near A?", "kind": "action", "id": "threshold-reachability-q02" } ] } ] }, { "id": "application-validity", "name": "Application validity", "description": "Set the empirical and interpretive limits of using the curve.", "findings": [ { "id": "conditional-extrapolation", "name": "Conditional extrapolation", "description": "Long-range predictions assume the selected Gompertz formulation remains applicable; a fit alone does not establish a causal growth mechanism or guarantee a real-world ceiling.", "questions": [ { "text": "What evidence supports continuing the same growth regime beyond the observed interval?", "kind": "provenance", "id": "conditional-extrapolation-q01" }, { "text": "Which observed departures, such as decline, multiple growth waves or a shifting ceiling, should trigger replacement or extension of the model?", "kind": "action", "id": "conditional-extrapolation-q02" } ] } ] } ] }, { "id": "evidence-and-external-alignment", "name": "Evidence and external alignment", "description": "What the world already says about this thing, gathered so the model can be checked against it.", "rationale": "A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.", "layers": [ { "id": "reported-evidence", "name": "Reported evidence", "description": "Findings from the breadth pass, kept separate from the structural claims.", "findings": [ { "id": "evidence-confidence-notes", "name": "Check these first", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "No sense is recorded; this description assumes the mathematical growth function rather than the mortality law or probability distribution.", "All stated parameter identities apply to y(t) = A exp(−b exp(−kt)) with positive parameters and no additive baseline.", "Application-specific parameterizations and empirical suitability require checking; this account is from recall, without source consultation." ], "questions": [ { "id": "evidence-confidence-notes-q01", "text": "Which of these check these first hold for the sense of Gompertz function this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-kinds", "name": "Kinds and varieties", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "Increasing Gompertz growth curve", "Decreasing Gompertz curve", "Modified Gompertz parameterizations expressed using lag time and maximum growth rate" ], "questions": [ { "id": "evidence-kinds-q01", "text": "Which of these kinds and varieties hold for the sense of Gompertz function this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-real-world-use", "name": "Real-world use", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "Fitting tumour growth trajectories", "Modelling microbial population growth", "Describing animal and plant growth", "Modelling cumulative technology adoption or market penetration" ], "questions": [ { "id": "evidence-real-world-use-q01", "text": "Which of these real-world use hold for the sense of Gompertz function this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-measurements", "name": "Typical measurements", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ { "quantity": "Upper asymptote A", "typical_range": "Positive; application-dependent", "unit": "Same as the response variable" }, { "quantity": "Rate parameter k", "typical_range": "Positive for the stated increasing form; no universal typical range", "unit": "Inverse unit of t" }, { "quantity": "Position parameter b", "typical_range": "Positive in the stated form; depends on the origin of t", "unit": "Dimensionless" }, { "quantity": "Response at the inflection point", "typical_range": "Exactly A/e, approximately 0.368A, for the stated form", "unit": "Same as the response variable" }, { "quantity": "Maximum absolute growth rate", "typical_range": "Exactly Ak/e for the stated form", "unit": "Response unit per unit of t" } ], "questions": [ { "id": "evidence-measurements-q01", "text": "Which of these typical measurements hold for the sense of Gompertz function this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-failure-modes-and-hazards", "name": "Failure modes and hazards", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "Data covering only early growth may poorly constrain the upper asymptote and produce unreliable extrapolations.", "A good curve fit does not establish the biological or social mechanism generating the observations.", "The fixed asymmetry and single inflection point can misrepresent trajectories with different shapes, multiple growth phases, or changing conditions.", "Parameter meanings differ across parameterizations, making direct comparisons unsafe without checking the equations.", "Confusing Gompertz growth with the Gompertz mortality law mixes a sigmoidal response with an exponentially increasing hazard." ], "questions": [ { "id": "evidence-failure-modes-and-hazards-q01", "text": "Which of these failure modes and hazards hold for the sense of Gompertz function this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-neighbours", "name": "Neighbouring kinds and how to tell them apart", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ { "name": "Logistic function", "difference": "The standard logistic growth curve is symmetric about its inflection point at half its upper asymptote; the stated Gompertz curve is asymmetric and inflects at A/e." }, { "name": "Richards growth curve", "difference": "Richards models introduce an additional shape parameter; the Gompertz curve occurs as a limiting case in common parameterizations." }, { "name": "Exponential growth", "difference": "Exponential growth has a constant relative growth rate and no finite upper asymptote; Gompertz growth has an exponentially decreasing relative growth rate." }, { "name": "Gompertz mortality law", "difference": "The mortality law specifies a hazard that increases exponentially with age, rather than a sigmoidal growth trajectory." }, { "name": "Gompertz distribution", "difference": "The distribution specifies probabilities through an exponentially varying hazard on a defined support; the growth function alone is not a probability-distribution specification." } ], "questions": [ { "id": "evidence-neighbours-q01", "text": "Which of these neighbouring kinds and how to tell them apart hold for the sense of Gompertz function this model covers, and on what evidence?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Which authoritative sources should anchor the registry definition and distinguish the Gompertz function from the related distribution and mortality law?", "Should baseline-shifted, decreasing and application-specific modified Gompertz equations be owned here as variants or linked to separate registered concepts?", "Which parameter conventions used across application fields require explicit mappings, particularly conventions labelled growth rate or lag time?", "What evidence and uncertainty criteria should govern acceptance of asymptote estimates and threshold predictions in each intended application?" ], "statistics": { "bundles": 5, "layers": 9, "findings": 14, "questions": 22 } }