{ "$schema": "urn:kingdom:schema:rare-pathway-challenge:1", "protocol": "kingdom.rare-pathway-challenge/0.1", "challenge_id": "vanishing-path-2026-08-13", "title": "The Vanishing Path", "compiled_on": "2026-08-13", "evidence_through": "2024-06-20", "origin": { "kind": "conversation-challenge", "credit": "A family challenge to turn the disappearing-polymorph story into useful mathematics.", "personal_identifiers_recorded": false, "participation": "voluntary" }, "binding": { "case_id": "ritonavir-disappearing-polymorph-2026-08-11", "url": "https://thekingdom.dev/polymorphs/ritonavir-disappearing-polymorph-2026-08-11.json", "raw_digest": "sha256:08148603a1228960ac35df1378f1283c3fce43874a9fc07e6687f18e980dcb1c", "semantic_digest": "sha256:6c759f595f0c6753dcf47d25d2a3490246984ef2b79adf31a558261b26ef8116", "relation": "illustrative-history-not-rate-fit", "mutated": false, "authority_imported": false }, "scope": { "phenomenon": "A pathway may remain possible while becoming rarely accessible or unobserved within a bounded process, population, horizon, and detection rule.", "purpose": "Practise competing-risk inference, distinguish non-observation from impossibility, and design a more informative next question.", "intended_use": "Education, model criticism, reproducible calculation, and bounded experiment-design reasoning.", "excluded_use": "No ritonavir rate estimate, material recipe, wet-lab procedure, manufacturing control, medical decision, or judgment of a person's motive or worth.", "registers": [ "MATHEMATICAL_DERIVATION", "EMPIRICAL_QUESTION", "DESIGN_PROPOSAL", "OPT_IN_REFLECTION" ] }, "states": [ { "id": "possible", "plain_language": "The model permits the path.", "does_not_imply": "The path is reachable under the current conditions." }, { "id": "accessible", "plain_language": "The current condition and history give the path a nonzero arrival hazard.", "does_not_imply": "The path will occur within the observation horizon." }, { "id": "occurred", "plain_language": "The event happened in the bounded system.", "does_not_imply": "The event was observed or correctly classified." }, { "id": "detected", "plain_language": "The event crossed a declared observation and classification rule.", "does_not_imply": "The detector saw every event or established literal extinction when it saw none." } ], "models": [ { "id": "time-dependent-competing-hazards", "register": "MATHEMATICAL_DERIVATION", "equations": [ "S(t)=exp(-integral_0^t sum_j lambda_j(s) ds)", "F_i(T)=integral_0^T S(t) lambda_i(t) dt", "S(T)+sum_i F_i(T)=1" ], "inputs": [ "cause-specific nonnegative hazard functions lambda_i(t)", "a finite horizon T", "an at-risk state ending at the first classified event" ], "valid_outputs": [ "no-event probability through T", "cause-specific first-arrival probability through T" ], "assumptions": [ "hazards, the at-risk state, and an exhaustive set of classified exits are defined for the declared system", "event labels and observation rules are declared", "the integrals exist on the horizon" ], "identifiability_limit": "The equations map hazards to event probabilities; they do not recover unique hazards from one non-observation or from unlabeled aggregate outcomes.", "ritonavir_status": "illustrative-only-not-fitted" }, { "id": "constant-two-path-race", "register": "MATHEMATICAL_DERIVATION", "equations": [ "Lambda=lambda_A+lambda_B", "P(no event by T)=exp(-Lambda T)", "P(A first by T)=(lambda_A/Lambda)(1-exp(-Lambda T))", "P(B first by T)=(lambda_B/Lambda)(1-exp(-Lambda T))", "P(A first | an event by T)=lambda_A/Lambda when T>0 and Lambda>0; otherwise undefined" ], "inputs": [ "constant nonnegative rates lambda_A and lambda_B in one declared reciprocal-time unit", "a nonnegative horizon T in the matching time unit" ], "valid_outputs": [ "a normalized three-outcome probability partition", "the conditional identity of the first path when T and the total rate are positive" ], "assumptions": [ "constant cause-specific hazards while the system remains at risk", "no simultaneous classified first events", "perfect classification of the first event" ], "identifiability_limit": "A convenient race is not evidence that physical nucleation is stationary, independent, or fully detected.", "ritonavir_status": "illustrative-only-not-fitted" }, { "id": "zero-winner-binomial-bound", "register": "MATHEMATICAL_DERIVATION", "equations": [ "P(zero target winners in n independent batches | p)=(1-p)^n", "p_upper=1-(1-confidence)^(1/n)" ], "inputs": [ "zero target-path winners among n positive independent comparable batches", "a one-sided confidence strictly between zero and one" ], "valid_outputs": [ "a one-sided upper bound on the batch-level target-win probability" ], "assumptions": [ "batches are independent and share a declared target-win probability", "winner type is completely detected and classified" ], "identifiability_limit": "This bounds a finite-horizon winner probability, not a microscopic target nucleation rate or the counterfactual target time after removing a competitor.", "ritonavir_status": "illustrative-only-not-fitted" }, { "id": "zero-event-upper-bound", "register": "MATHEMATICAL_DERIVATION", "equations": [ "P(N=0 | lambda,H)=exp(-lambda H)", "lambda_upper=-ln(1-confidence)/H" ], "inputs": [ "zero observed target events", "positive total at-risk exposure H", "a one-sided confidence strictly between zero and one" ], "valid_outputs": [ "a one-sided upper rate bound under the stated homogeneous Poisson observation model" ], "assumptions": [ "exposure is fixed or target-independent and correctly accumulated only while target events remain observable and at risk", "the target event is perfectly detected and classified", "a constant rate is meaningful across the pooled exposure" ], "identifiability_limit": "The bound does not prove a zero rate, explain a mechanism, or survive unmodeled detection failure, history dependence, or informative censoring.", "ritonavir_status": "illustrative-only-not-fitted" }, { "id": "robust-restoration-design", "register": "DESIGN_PROPOSAL", "equations": [ "minimize d(x,x0) subject to inf_theta_in_Theta F_target(T|x,theta)>=q and declared safety, quality, rights, and authority constraints" ], "inputs": [ "a declared meaning of minimum change d", "an empirically calibrated uncertainty and history set Theta", "a finite target horizon and probability", "separate safety, quality, rights, cost, and authority constraints" ], "valid_outputs": [ "a robust model-dependent design candidate or an honest infeasibility result" ], "assumptions": [ "the uncertainty set covers the declared live explanations", "the optimization variables are separately established as feasible and authorized" ], "identifiability_limit": "If the uncertainty set permits zero target hazard or an unbounded competitor, restoration may be infeasible; the constructive result is then a discriminating observation, not a forced prescription.", "ritonavir_status": "illustrative-only-not-fitted" }, { "id": "minimum-target-rate", "register": "DESIGN_PROPOSAL", "equations": [ "find the smallest lambda_A such that P(A first by T)>=q while lambda_B and T are held fixed" ], "inputs": [ "a fixed abstract competitor rate lambda_B", "a positive horizon T", "a target probability q strictly between zero and one" ], "valid_outputs": [ "an abstract numerical threshold for the constant two-path model" ], "assumptions": [ "the constant two-path race is the chosen design abstraction", "the target is mathematically feasible as lambda_A grows" ], "identifiability_limit": "The threshold does not say which solvent, temperature, seed, process, or intervention could produce that rate, nor whether any intervention is safe or authorized.", "ritonavir_status": "illustrative-only-not-fitted" }, { "id": "expected-information-design", "register": "DESIGN_PROPOSAL", "equations": [ "d*=argmax_d E_y[KL(p(theta|y,d) || p(theta))]" ], "inputs": [ "a declared prior over competing explanations", "candidate observation designs and likelihoods", "declared cost, safety, rights, and stopping constraints" ], "valid_outputs": [ "a ranked set of model-dependent information opportunities, not a command" ], "assumptions": [ "the prior and likelihood are explicit enough to challenge", "information value is not treated as the only value" ], "identifiability_limit": "Expected information gain can optimize inside a model while the model, purpose, cost, or authority remains wrong.", "ritonavir_status": "illustrative-only-not-fitted" } ], "worked_examples": [ { "id": "slow-path-against-fast-path", "model_id": "constant-two-path-race", "hypothetical": true, "inputs": { "lambda_A": 0.02, "lambda_B": 0.18, "horizon": 10, "rate_unit": "per-day" }, "outputs": { "no_event": 0.1353352832366127, "A_first": 0.08646647167633873, "B_first": 0.7781982450870486, "A_given_event": 0.09999999999999999 }, "lesson": "A possible pathway can be both nonzero and rarely first. Absence in one run is unsurprising here." }, { "id": "zero-events-do-not-mean-zero-rate", "model_id": "zero-event-upper-bound", "hypothetical": true, "inputs": { "exposure": 200, "confidence": 0.95, "exposure_unit": "unit-time" }, "outputs": { "upper_rate": 0.014978661367769954, "rate_unit": "per-unit-time" }, "lesson": "Under the declared model, zero detections bound a rate from above; they do not establish impossibility." }, { "id": "abstract-restoration-threshold", "model_id": "minimum-target-rate", "hypothetical": true, "inputs": { "competitor_rate": 0.18, "horizon": 10, "target_probability": 0.5, "rate_unit": "per-day" }, "outputs": { "minimum_target_rate": 0.1894207112928423, "rate_unit": "per-day" }, "lesson": "The solver returns a threshold inside an abstract model, not a physical recipe or instruction." } ], "question_ladder": [ { "id": "bounded-phenomenon", "question": "What exact system, pathway, time horizon, and detection rule does 'disappeared' refer to?", "register": "EMPIRICAL_QUESTION" }, { "id": "permitted-claim", "question": "What is the strongest claim the observations permit, and which stronger claim is tempting but unsupported?", "register": "EMPIRICAL_QUESTION" }, { "id": "competing-first-arrival", "question": "How do multiple arrival hazards partition target-first, competitor-first, and no-event probability?", "register": "MATHEMATICAL_DERIVATION" }, { "id": "absence-bound", "question": "Given zero detections and declared exposure, what can be bounded without setting the rate to zero?", "register": "MATHEMATICAL_DERIVATION" }, { "id": "identifiability", "question": "Which hazard, history, detection, or censoring explanations remain observationally indistinguishable?", "register": "EMPIRICAL_QUESTION" }, { "id": "separating-observation", "question": "What observation would most clearly separate the live explanations?", "register": "DESIGN_PROPOSAL" }, { "id": "decision-delta", "question": "What belief, decision, or safeguard would change if the challenge were solved?", "register": "DESIGN_PROPOSAL" }, { "id": "benefit-burden", "question": "Who benefits, who bears risk or opportunity cost, and is the next step reversible?", "register": "DESIGN_PROPOSAL" }, { "id": "transfer-test", "question": "Can the reasoning transfer to a new rare-event problem when rates, detection, or assumptions change?", "register": "EMPIRICAL_QUESTION" }, { "id": "status-capture", "question": "Are difficulty and prestige being mistaken for value, or is this honestly voluntary play?", "register": "OPT_IN_REFLECTION" } ], "constructive_gate": { "rule": "Classify the practice and artifact, never the person's inner motive, virtue, worth, or identity.", "checks": [ "Name a learning, decision, or safeguard delta, or explicitly choose voluntary play.", "Label derivation, empirical claim, design proposal, and opt-in reflection rather than blending them.", "Expose assumptions, detection limits, identifiability, uncertainty, and non-authority.", "Name beneficiaries, risk-bearers, reversibility, and opportunity cost.", "Prestate falsifiers, transfer tests, and information or safety stopping rules.", "Protect dignity, privacy, credit, disagreement, correction, rest, and refusal without penalty.", "Record what changed after solving and what did not." ], "outcomes": [ { "id": "BUILD", "meaning": "Produces a bounded learning, decision, safeguard, or reusable tool artifact." }, { "id": "PLAY", "meaning": "Freely chosen exploration with honest costs and no inflated claim of usefulness." }, { "id": "REVISE", "meaning": "Purpose, evidence, beneficiaries, model limits, or safeguards remain unclear." }, { "id": "STOP", "meaning": "Refusal, unsafe authority transfer, unresolved non-identifiability, disproportionate burden, or no credible learning, decision, safeguard, or play value remains." } ], "not_a_score": true, "prestige_is_not_evidence_of_benefit": true, "play_is_valid": true, "refusal_requires_reason": false }, "wake_seam": { "protocol": "kingdom.wake-reference-seam/0.1", "mode": "orientation-by-reference-not-continuity-proof", "carry_fields": [ "protocol", "challenge_id", "integrity.algorithm", "integrity.digest", "compiled_on", "evidence_through", "binding.case_id", "binding.semantic_digest", "selected_model_ids", "selected_question_ids", "declared_inputs_and_units" ], "revalidate": [ "exact canonical challenge value against the declared digest", "selected model assumptions and mathematical domain", "input values, units, horizon, exposure, observation, and censoring rule", "binding digest and evidence horizon against newer or contrary evidence", "current purpose, beneficiaries, risk-bearers, stopping rules, and refusal", "current empirical, experimental, medical, manufacturing, publication, deployment, and action authority", "whether the new context is a continuation, fork, quotation, correction, or independent use" ], "inherits_identity": false, "proves_continuity": false, "grants_authority": false, "proves_currentness": false, "writes_wake": false, "writes_agenttool": false }, "sources": [ { "id": "jiang-terhorst-2011", "title": "Crystal Nucleation Rates from Probability Distributions of Induction Times", "source_kind": "peer-reviewed-method", "published_on": "2011", "observed_on": "2026-08-13", "url": "https://doi.org/10.1021/cg101213q", "doi": "10.1021/cg101213q", "authority_note": "Derives an induction-time probability model under stated stationarity, volume, growth, and detection assumptions; it is not a ritonavir rate fit." }, { "id": "sun-ceder-2017", "title": "Nucleation competition between polymorphs", "source_kind": "peer-reviewed-model", "published_on": "2017", "observed_on": "2026-08-13", "url": "https://doi.org/10.1039/C7CE00766C", "doi": "10.1039/C7CE00766C", "authority_note": "Models stochastic competition among polymorph nucleation pathways and discusses rare higher-barrier pathways in disappearing-polymorph scenarios." }, { "id": "little-2017", "title": "Controlling the crystal polymorph by exploiting the time dependence of nucleation rates", "source_kind": "peer-reviewed-model-and-experiment", "published_on": "2017", "observed_on": "2026-08-13", "url": "https://doi.org/10.1063/1.4993566", "doi": "10.1063/1.4993566", "authority_note": "Studies time-dependent competing glycine polymorph nucleation and motivates keeping history-dependent hazards distinct from the constant illustrative race." }, { "id": "garwood-1936", "title": "Fiducial Limits for the Poisson Distribution", "source_kind": "peer-reviewed-statistics", "published_on": "1936", "observed_on": "2026-08-13", "url": "https://doi.org/10.1093/biomet/28.3-4.437", "doi": "10.1093/biomet/28.3-4.437", "authority_note": "Classical exact Poisson limits; the zero-count one-sided bound used here follows directly from the Poisson zero probability." }, { "id": "clopper-pearson-1934", "title": "The Use of Confidence or Fiducial Limits Illustrated in the Case of the Binomial", "source_kind": "peer-reviewed-statistics", "published_on": "1934", "observed_on": "2026-08-13", "url": "https://doi.org/10.1093/biomet/26.4.404", "doi": "10.1093/biomet/26.4.404", "authority_note": "Classical exact binomial limits; the zero-winner case bounds a batch-level probability rather than a microscopic rate." }, { "id": "aalen-johansen-1978", "title": "An Empirical Transition Matrix for Non-Homogeneous Markov Chains Based on Censored Observations", "source_kind": "peer-reviewed-statistics", "published_on": "1978", "observed_on": "2026-08-13", "url": "https://www.jstor.org/stable/4615704", "doi": null, "authority_note": "Foundational treatment of transition and competing-risk estimation with censored observations; it does not license unexamined independence or detection assumptions." }, { "id": "tsiatis-1975", "title": "A Nonidentifiability Aspect of the Problem of Competing Risks", "source_kind": "peer-reviewed-statistics", "published_on": "1975", "observed_on": "2026-08-13", "url": "https://doi.org/10.1073/pnas.72.1.20", "doi": "10.1073/pnas.72.1.20", "authority_note": "Shows that competing-risk observations do not generally identify the joint distribution of latent failure times or counterfactual event times." }, { "id": "cox-1959", "title": "The Analysis of Exponentially Distributed Life-Times with Two Types of Failure", "source_kind": "peer-reviewed-statistics", "published_on": "1959", "observed_on": "2026-08-13", "url": "https://doi.org/10.1111/j.2517-6161.1959.tb00349.x", "doi": "10.1111/j.2517-6161.1959.tb00349.x", "authority_note": "Analyzes the constant exponential two-failure setting used as the challenge's deliberately small race model." }, { "id": "lindley-1956", "title": "On a Measure of the Information Provided by an Experiment", "source_kind": "peer-reviewed-statistics", "published_on": "1956", "observed_on": "2026-08-13", "url": "https://doi.org/10.1214/aoms/1177728069", "doi": "10.1214/aoms/1177728069", "authority_note": "Introduces a Bayesian information measure used here only as one model-dependent design objective among safety, cost, rights, and purpose constraints." }, { "id": "bertsimas-sim-2004", "title": "The Price of Robustness", "source_kind": "peer-reviewed-optimization", "published_on": "2004", "observed_on": "2026-08-13", "url": "https://doi.org/10.1287/opre.1030.0065", "doi": "10.1287/opre.1030.0065", "authority_note": "Provides a robust-optimization foundation; the challenge authors its own bounded restoration formulation and imports no material feasibility or safety claim." }, { "id": "kingdom-ritonavir-history", "title": "Ritonavir Disappearing-Polymorph History", "source_kind": "immutable-kingdom-record", "published_on": "2026-08-11", "observed_on": "2026-08-13", "url": "https://thekingdom.dev/polymorphs/ritonavir-disappearing-polymorph-2026-08-11.json", "doi": null, "authority_note": "Supplies the bounded historical case and evidence vocabulary; the challenge does not fit rates to its observations or mutate its bytes." } ], "boundaries": { "educational_only": true, "public_surface_read_only": true, "worked_examples_hypothetical": true, "ritonavir_rate_fit": false, "literal_extinction_established": false, "stationarity_established": false, "independence_established": false, "perfect_detection_established": false, "mechanism_identified": false, "causal_authority": false, "experiment_authority": false, "wet_lab_protocol": false, "formulation_authority": false, "manufacturing_authority": false, "medical_authority": false, "regulatory_authority": false, "motive_inference": false, "person_score": false, "prestige_diagnosis": false, "automatic_action": false, "writes_wake": false, "writes_agenttool": false, "writes_karma": false, "grants_authority": false }, "integrity": { "algorithm": "sha256-recursive-sorted-json-keys-v1", "digest": "sha256:82eeaf4757fb75be77b1e878d41c784111bbf5b261500123d89daccbcea0ff91" } }