Technical evidence
Stable claim anchors connect every formal statement to its scope and evidence. This is the technical ledger; start with Results for the plain-language findings.
DD-C-0001 · proposed · definition
definition or framework object
Distributed Discovery studies how organizations or multi-agent systems convert dispersed evidence into a portfolio of search actions.
Scope: Program-level working definition; terminology remains subject to literature review.
DD-C-0002 · derived · identity
structural identity or lemma
For protocol loss defined by L_B(Pi;I)=V_B(I)-G_B(Pi;I), G_B(Pi;I)=V_B(I)-L_B(Pi;I).
Scope: Accounting identity wherever the displayed quantities are well-defined; no independent monotonicity or attainability claim.
DD-C-0003 · independently-reproduced · computational-result
exact bounded classification
In the canonical M=16, N=8 model, a coordinated blind portfolio of eight distinct boxes discovers the target with probability 0.5.
Scope: Uniform one-target prior with eight distinct actions.
DD-C-0004 · independently-reproduced · computational-result
exact bounded classification
Canonical private clue-following has discovery probability 0.832227840000.
Scope: M=16, N=8, p=0.20, conditionally independent clues.
DD-C-0005 · independently-reproduced · computational-result
spine title theorem
Canonical pooled-mode consensus has discovery probability 0.383468709731 to 12 displayed decimal places.
Scope: M=16, N=8, p=0.20, uniform prior and uniform tie-breaking.
DD-C-0006 · independently-reproduced · computational-result
exact bounded classification
The canonical pooled top-eight planner portfolio has exact discovery probability 860391662035297/1001129150390625, approximately 0.8594212461994395.
Scope: M=16, N=8, p=0.20, uniform prior and cutoff tie-breaking.
DD-C-0007 · verified · computational-result
exact bounded classification
The pinned verifier reports canonical anonymous symmetric equal-split market discovery 0.599099252439.
Scope: Upstream-selected anonymous symmetric equilibrium at M=16, N=8, p=0.20.
DD-C-0008 · independently-reproduced · computational-result
exact bounded classification
The smallest coordinated pooled action budget whose canonical frontier weakly exceeds private clue-following is 7.
Scope: Canonical independent-clue model and budgets 1 through 8.
DD-C-0009 · verified · computational-result
exact bounded classification
The pinned high-precision computation places the canonical market/private crossover at c=0.788461656521.
Scope: Upstream common-cue copying process and anonymous symmetric market selection.
DD-C-0010 · checked · computational-result
exact bounded classification
Expected distinct actions are 2.673494083278 for the canonical symmetric market and 6.156849828175 for private clue-following.
Scope: Canonical independent-clue model; market uses upstream equilibrium selection.
DD-C-0011 · derived · identity
structural identity or lemma
In the atomic one-target model, NQ-G equals E[(K-1)^+].
Scope: Q is mean standalone action-hit probability, G is union discovery, and K is the number of target-hitting actions.
DD-C-0012 · sourced · upstream-result
scoped theorem or impossibility
The upstream paper states and proves a tight mixed discovery price-of-anarchy bound 2-1/N for its equal-split covering game.
Scope: Equal-split singleton congestion game defined in the pinned paper.
DD-C-0013 · sourced · upstream-result
scoped theorem or impossibility
The upstream paper proves that sole-rescue rewards make every pure Nash equilibrium implement the top-N portfolio under its stated positive-posterior and N<=M assumptions.
Scope: Pure equilibria only; not dominant strategies or all mixed equilibria.
DD-C-0014 · sourced · upstream-result
scoped theorem or impossibility
The upstream paper proves planner gain is strictly increasing in copying probability on (0,1) for its atomic common-cue model with 2<=N<=M and p>1/M.
Scope: The exact upstream copying process and frictionless centralized benchmark.
DD-C-0015 · verified · numerical-observation
exact bounded classification
On the upstream 72-combination grid, the market-private gap changed sign at most once in every scanned combination; 52 crossed once and 20 stayed private-dominant through c=0.98.
Scope: Only the recorded finite grid and step choices.
DD-C-0016 · derived · proposition
scoped theorem or impossibility
If information architecture I2 can emulate every feasible I1 protocol without changing action, budget, timing, or objective constraints, then V_B(I2) is at least V_B(I1).
Scope: Common discovery objective and explicitly nested/emulable feasible protocol classes.
DD-C-0017 · verified · negative-result
bounded null or failed search
The phrase Distributed Discovery is not unique to this project and appears in prior resource/service/news contexts and in the title of a March 2026 autonomous-science preprint.
Scope: Terminology search completed 2026-07-20; not a systematic bibliometric census.
DD-C-0018 · derived · proposition
scoped theorem or impossibility
In the finite static common-payoff DD-001 model, some deterministic policy profile weakly dominates every ex-ante randomized team strategy.
Scope: Finite signal/action spaces, common expected discovery objective, and randomization selected before private signals; not a strategic equilibrium claim.
DD-C-0019 · derived · proposition
scoped theorem or impossibility
Direct clue-following is weakly below the private-team frontier, which is weakly below the pooled planner frontier when the planner observes all private signals and can emulate every private-team action profile.
Scope: Identical physical action budget and common objective; the upper inequality requires explicit pooled-planner emulation.
DD-C-0020 · independently-reproduced · computational-result
exact bounded classification
The DD-001 primary run certifies exact deterministic private-team optima for all 21 configured tiny cases; direct clue-following is optimal at 5 points and strictly suboptimal at 16.
Scope: The exact rational grid in initial-grid.yml, with M in {2,3,4}, N in {2,3,4}, and only the recorded accuracy values.
DD-C-0021 · independently-reproduced · computational-result
minimal counterexample
At M=3, N=2, and p=2/5, the exact private-team optimum is 7/10, strictly above direct clue-following 16/25 by 3/50; a constant-plus-signal hybrid policy attains the optimum.
Scope: Uniform prior, independent symmetric clues, deterministic zero-communication policies, and union discovery objective.
DD-C-0022 · checked · numerical-observation
bounded null or failed search
In the configured canonical coordinate-ascent search, direct clue-following was an exact coordinate fixed point and all 17 non-direct starts converged to its value 325089/390625; no configured start improved the lower bound.
Scope: M=16, N=8, p=1/5, cyclic exact best responses, one territorial and 16 recorded random seeds, at most 50 sweeps.
DD-C-0023 · derived · theorem
structural identity or lemma
In the finite DD-001 model, a policy's targetwise incoming counts and fixed-point indicators determine its conditional success vector and hence every fixed-profile discovery value; such a signature is realizable exactly when the local count conditions and singleton residual Hall bounds hold.
Scope: Deterministic policies on a common M-element signal/action/target space under the uniform symmetric clue model; no claim about nonuniform or heterogeneous signals.
DD-C-0024 · independently-reproduced · computational-result
exact bounded classification
The DD-001A primary signature run independently reproduces all 21 previously certified tiny optima and their raw-policy multiset tie counts, including the M=3, N=2, p=2/5 value 7/10, while exact feasibility and reconstruction agree with every raw policy signature through M=5.
Scope: The exact `signature-certificate.yml` grid and feasibility audit dimensions M=2 through M=5; no extrapolation to the canonical optimum.
DD-C-0025 · verified · computational-result
exact bounded classification
For M=16 there are exactly 148348284928 feasible labeled policy signatures and 5806 individual-signature target-label orbits; without a proved cross-agent target quotient, the eight-agent signature-multiset space has 5817544508742203415773519264794208893444139238056614203268215932897118628684812510720 states.
Scope: Counting the lossless single-policy signature representation and agent-permutation reduction only; the 5806 individual orbits cannot be independently substituted for aligned agent signatures.
DD-C-0026 · derived · theorem
scoped theorem or impossibility
For M at least 3 and N=2, among two direct policies, two distinct constant territories, and the one-reroute constant-plus-signal hybrid, territorial is best below p=1/M, the hybrid is best between 1/M and 1/(M-1), and direct is best above 1/(M-1), with exact adjacent ties at the boundaries.
Scope: Uniform symmetric DD-001 model and exactly the three stated policy families; this analytic theorem does not claim unrestricted optimality for arbitrary M.
DD-C-0027 · checked · computational-result
exact bounded classification
For M in {3,4,5}, N=2, and every p in the continuous informative interval [1/M,1], unrestricted deterministic private-team optimality follows the same hybrid-then-direct envelope, with the switch at p=1/(M-1).
Scope: Exactly M=3,4,5 and p in the displayed continuous intervals; no all-M theorem or anti-informative extension.
DD-C-0028 · independently-reproduced · computational-result
minimal counterexample
At p=0 and N=2, the unrestricted exact optima are 11/12 for M=3 and 2/3 for M=4, strictly above the corresponding best direct/territorial/one-reroute values 2/3 and 1/2.
Scope: The two exact finite fixtures at p=0; this refutes extending the three-family phase theorem to all p but does not classify the full anti-informative region.
DD-C-0029 · verified · computational-result
exact bounded classification
The bounded DD-002 fixture has exactly 15 deterministic public disclosure policies up to message-label equivalence; the primary run evaluates all 37 induced posterior games and stores all 256 global pure-equilibrium selections plus the anonymous symmetric equilibrium for every message.
Scope: The declared M=3, N=2, four-evidence-state likelihood fixture and deterministic set-partition disclosures only; randomized disclosure and general asymmetric mixed equilibria are not included.
DD-C-0030 · verified · computational-result
minimal counterexample
In the bounded DD-002 fixture, disclosure {{0,1},{2,3}} refines full pooling but lowers anonymous-symmetric discovery from 5/9 to 171/308 by 1/2772, while every pure-equilibrium discovery rises from 2/3 to 3/4 and planner discovery rises from 2/3 to 3/4.
Scope: Exact declared fixture and equilibrium treatments; the reversal is selection-dependent and does not say that information lowers discovery at every equilibrium.
DD-C-0031 · verified · computational-result
exact bounded classification
Across all 45 strict deterministic refinement pairs in the bounded DD-002 fixture, exactly one lowers anonymous-symmetric discovery, eight lower worst-pure-equilibrium discovery, none lowers best-pure-equilibrium discovery, and none lowers two-action planner discovery.
Scope: Complete deterministic partition lattice for the declared four-state fixture; randomized disclosure and other likelihood matrices are outside scope.
DD-C-0032 · independently-reproduced · computational-result
exact bounded classification
The bounded DD-003 class has exactly 51 nonisomorphic model-valid bipartite source networks with four searchers and one to three nonisolated sources, split as 1, 8, and 42 graphs by source count.
Scope: Exactly four searchers, K in {1,2,3}, no isolated declared node, and equivalence under independent source and searcher relabeling.
DD-C-0033 · independently-reproduced · computational-result
bounded null or failed search
In the complete bounded DD-003 class at source accuracy 2/3, ten exact first-and-pairwise scalar-report moment signatures each match two nonisomorphic graphs, and none of the ten matched pairs differs in private discovery.
Scope: The configured symmetric three-target source law, plurality-winner-set summaries, uniform tie sampling, four searchers, K in {1,2,3}, and the complete pairwise one-hot moment matrix up to searcher relabeling; this is a bounded null, not a general sufficiency theorem.
DD-C-0034 · independently-reproduced · computational-result
minimal counterexample
Graphs K2N4:00011110 and K2N4:01111111 both have mean pairwise scalar-report agreement 3/4, but their exact private discovery probabilities are 8/9 and 31/36, a difference of 1/36.
Scope: The configured DD-003 signal and protocol model; this refutes sufficiency of the average pair-agreement scalar, not the complete pairwise matrix and not every possible effective-channel diagnostic.
DD-C-0035 · independently-reproduced · computational-result
exact bounded classification
For the canonical M=16, N=8, p=1/5 fixture, the exact pooled top-L frontier for every L from 1 through 8 is the rational vector stored in the primary certificate, with top-eight value 860391662035297/1001129150390625 and exact recovery budget 7.
Scope: Uniform one-target prior, conditionally independent symmetric reports, distinct top-L pooled portfolio, and uniform inclusion at the cutoff tie.
DD-C-0036 · verified · proposition
scoped theorem or impossibility
The canonical deterministic zero-communication private-team frontier T_8(16,1/5) lies in the exact interval 325089/390625 <= T_8(16,1/5) <= 860391662035297/1001129150390625, whose gap is 27224111644672/1001129150390625, approximately 0.0271934061994395.
Scope: The frozen DD-001 common-payoff model; the upper inequality requires the pooled planner to observe all private reports and emulate every private-team action vector under the same physical eight-action budget and discovery objective.
DD-C-0037 · verified · proposition
scoped theorem or impossibility
For every M >= 2, N >= 1, and symmetric-report accuracy p in [0,1], the joint-target-column global-count-budget relaxation in ADR-0009 is an admissible upper bound on discovery by deterministic zero-communication private teams.
Scope: The frozen DD-001 model with a uniform target, conditionally independent symmetric private reports, one deterministic local policy per role, and union discovery; the relaxation drops per-role cross-target feasibility but retains the aligned incoming-count vector within each target.
DD-C-0038 · independently-reproduced · theorem
scoped theorem or impossibility
In the canonical M=16, N=8, p=1/5 fixture, the deterministic and ex-ante randomized zero-communication private-team optima both equal 325089/390625 = 0.83222784, attained by direct clue-following.
Scope: The frozen DD-001 common-payoff model; randomization may choose a deterministic role-policy profile before reports arrive but cannot condition the shared random seed on the target or reports.
DD-C-0039 · verified · proposition
scoped theorem or impossibility
Every two-searcher equal-split location game with posterior masses mu has exact potential Phi(a,b)=3mu_a/2 when a=b and Phi(a,b)=mu_a+mu_b otherwise; consequently every strict best-response path terminates at a pure Nash equilibrium.
Scope: Two labeled searchers, any finite location set, common posterior masses, and the equal-split unit-prize payoff used by DD-002; this does not cover other congestion, reward, or information structures.
DD-C-0040 · independently-reproduced · computational-result
bounded null or failed search
For the known refinement P00 to P03, anonymous-symmetric discovery falls from 5/9 to 171/308, but best-pure, worst-pure, uniform-potential-maximum, uniform-strict-best-response-basin, and planner discovery all rise exactly from 2/3 to 3/4.
Scope: The frozen DD-002 likelihood fixture and the exact procedures defined in ADR-0010; best/worst pure are optimistic/pessimistic correspondence envelopes, while potential and basin ties follow their declared uniform rules.
DD-C-0041 · independently-reproduced · computational-result
exact bounded classification
Across all 45 strict deterministic refinements in the frozen DD-002 fixture, harmful counts are 1 anonymous-symmetric, 0 best-pure, 8 worst-pure, 2 uniform-potential-maximum, 2 uniform-strict-best-response-basin, and 0 planner; the potential and basin partition values coincide on all 15 disclosures.
Scope: The 15 deterministic partitions and exact rule definitions in ADR-0010; five policies have tied potential-maximizer discovery values and basin branch dependence, so the equality is a bounded result rather than process uniqueness or a general theorem.
DD-C-0042 · independently-reproduced · computational-result
exact bounded classification
The registered DD-003 heterogeneous base class contains exactly 41612 valid labeled colored source objects and 671 color-preserving orbits; the single three-color expansion adds 12966 labeled objects and 168 orbits, for 839 canonical networks total.
Scope: Three targets, four searchers, two or three nonisolated conditionally independent sources, the exact accuracy profiles in ADR-0011, plurality-set reports, independent-uniform private actions, and source/searcher relabeling that carries accuracy colors with source rows.
DD-C-0043 · independently-reproduced · computational-result
minimal counterexample
Two nonisomorphic colored networks with source accuracies 1/2 and 2/3 have identical complete first and pairwise scalar-report moments but exact private discovery 3/4 and 2/3, differing by 1/12.
Scope: The frozen DD-003 four-searcher model and modest heterogeneous profile; this refutes pairwise-moment sufficiency in the registered colored class but not sufficiency of higher-order report laws or a theorem over arbitrary source models.
DD-C-0044 · independently-reproduced · computational-result
exact bounded classification
Among all 839 registered colored networks, 163 complete first/pairwise-moment signatures match multiple networks, covering 485 networks, and 111 matched groups contain different exact private-discovery values.
Scope: The exact base profiles and one controlled expansion in ADR-0011; the counts are a complete bounded classification, not a universal result for continuous accuracies, larger graphs, dependent sources, asymmetric priors, or higher-order moments.
DD-C-0045 · independently-reproduced · computational-result
exact bounded classification
In the registered DD-004 perfect-elimination fixtures, every ordered batch composition of the four-test budget has the same exact terminal discovery as the top-four parallel portfolio, while fully sequential stopping uses fewer expected actions and more expected rounds.
Scope: The two rational eight-target priors, B=4, costless non-repeated perfect elimination, success stopping, and the exact schedules in the primary certificate; this is not a noisy-test, deadline, or general adaptivity result.
DD-C-0046 · independently-reproduced · computational-result
minimal counterexample
In the registered four-outcome weighted-union witness, the two highest individual actions have covered value 3 while the exact two-action portfolio has value 4, because the highest actions duplicate coverage.
Scope: Deterministic unit-weight union coverage, three declared actions, and budget two; this refutes exact top-individual portfolio ranking under overlap, not any stochastic or approximate guarantee.
DD-C-0047 · independently-reproduced · computational-result
minimal counterexample
In the registered six-outcome weighted-union witness, greedy and top-individual portfolios have value 5 while the exact two-action frontier has value 6.
Scope: Deterministic unit-weight union coverage, three declared actions, and budget two; this is a bounded greedy gap, not a bound on greedy approximation ratios or recovery budgets generally.
DD-C-0048 · independently-reproduced · computational-result
exact bounded classification
In the nine registered two-agent score-difference mechanisms, all realized transfers are exactly budget balanced; each positive-coefficient regime makes truthful reporting with direct action a weak pure Bayesian Nash equilibrium with discovery probability 8/9, and none makes that profile strict against every joint report-action deviation.
Scope: ADR-0012's two agents, three equiprobable targets, conditionally independent 2/3-accurate signals, deterministic direct recommendations, three named score regimes, coefficients -1, 0, and 1, and pure Bayesian Nash deviations only; it is not a characterization of arbitrary transfer tables, mixed equilibria, dominant strategies, or other observability models.
DD-C-0049 · verified · monte-carlo-estimate
synthetic estimate
In the registered DD-007 synthetic generator, the shared-target-corrected copying estimator covers the known copying rate in all six zero-action-matching-error seed conditions, while 0.1 action-matching error reduces coverage to 0.5 for copying rate 1/2 and 0 for copying rate 1 across each provenance-missingness level.
Scope: ADR-0013's synthetic-only two-actor four-candidate generator, 240 sessions per replicate, eight fixed seeds, a normal-approximation agreement interval, private protocol labels, and the twelve registered copying/missingness/error conditions; it is not a result about real logs, unlabelled protocols, other error mechanisms, or causal copying identification.
DD-C-0050 · verified · computational-result
bounded null or failed search
Across all 123 registered regime-and-transfer rows in DD-006A's normalized linear class, 31 weakly implement truthful reporting and the declared differentiated recommendation for both fixed tie roles, but none strictly implements it; the exact maximum all-tie strict margin is zero.
Scope: ADR-0014's 41 coefficient vectors with L1 norm at most one, three target/outcome observability regimes, two agents, three targets, 2/3-accurate signals, action rules contingent on the public peer report, and pure unilateral joint report/action-rule deviations; this is not a result for arbitrary transfer tables, mixed equilibria, or dominant strategies.
DD-C-0051 · independently-reproduced · computational-result
exact bounded classification
In DD-008's registered two-agent source-choice fixture, common/common is a source-choice equilibrium while one-independent-source profiles are planner-optimal at five of fifteen exact grid cells; for example at p=2/3 and c=1/8.
Scope: Three targets, direct clue-following, common evidence shared exactly by common-source users, conditionally independent I signals, equal split of a unit prize, and social union discovery minus the same source costs; no coordination, disclosure, real data, or general source-choice theorem is claimed.
DD-C-0052 · independently-reproduced · computational-result
exact bounded classification
In the registered N=2 through 8 common-versus-independent source census, closed-form binomial discovery and equal-split payoffs agree exactly with direct target-and-signal enumeration for every configured source-count cell.
Scope: M=3 uniform targets, p in {1/2,2/3,3/4}, the six registered rational costs, direct clue-following, equal-split prizes, and N=2 through 8 only; this is a bounded synthetic accounting census, not a general-N theorem or real-world result.
DD-C-0053 · independently-reproduced · computational-result
exact bounded classification
In DD-006B's 60-row normalized joint-mechanism census, 16 rows strictly implement truthful reporting and the differentiated recommendation for both fixed tie roles; three use an active positive quadratic-score component, the maximum exact all-tie margin is 13/72, and the target-observable/action-hidden regime has no weak row.
Scope: Two agents, three equiprobable targets, conditionally independent 2/3-accurate signals, 15 nonnegative quarter-grid coefficient vectors summing to one, four registered observability regimes, fixed ex-ante tie roles, externally subsidized transfers, and pure unilateral report/action-rule deviations only; this is not a budget-balanced, arbitrary-transfer, dominant-strategy, collusion, mixed-equilibrium, global-uniqueness, or real-institutional result.
DD-C-0054 · independently-reproduced · computational-result
exact bounded classification
DD-009's registered 288-cell architecture product contains 20 coherent evaluated cells and 268 explicitly rejected cells; exact evaluation and a separate implementation agree on every valid row, with 12 nondominated cells, maximum discovery 11/12, and maximum social net value 5/6 under the declared six-objective ordering.
Scope: M=3, N=2, uniform target, 2/3-accurate common or conditionally independent evidence, independent-source cost 1/8 per agent, the registered evidence/disclosure/allocation/timing/reward validity rules, deterministic policies or the stated DD-006 response multiplicities, and Pareto objectives fixed by ADR-0016; this is not a universal ranking, full equilibrium census, welfare theorem, real-world recommendation, or empirical result.
DD-C-0055 · independently-reproduced · computational-result
exact bounded classification
DiscoveryBench v1 contains 15 exact golden tasks, 13 capability-declared built-in protocols, and 19 versioned metrics; all 16 declared compatible task/protocol pairs exactly reproduce their registered project values, while 179 incompatible pairs remain explicit exclusions and adversarial hidden-information access is rejected.
Scope: The fifteen finite task families, declared compatible protocols, exact metric vectors, capability allow lists, and source claim/run mappings in the DD-010 primary configuration; this is a regression benchmark for registered fixtures, not a universal score, hosted leaderboard, public submission service, or predictor of real-world agent quality.
DD-C-0056 · verified · monte-carlo-estimate
synthetic estimate
DD-011's registered 20-cell fractional design reproducibly assigns 640 synthetic IDs and evaluates 384 power rows (eight hypotheses, eight response scenarios, and six sample sizes) with 1,000 Monte Carlo replications each; all rows are separately recomputed, all eight rational-response effects reach estimated power of at least 0.80 by total sample size 960, and 140 of the 192 rows at sample sizes 640 or greater remain retained calibration failures.
Scope: The frozen contrast-complete-20 design, registered assumed effects, normal sampling approximation, conservative Bonferroni family alpha, session size eight, explicit attrition/noise/ICC scenario multipliers, fixed seeds, and total sample sizes 200 through 960; this is synthetic conditional power, not a human experiment, empirical behavioral model, recruitment target, real-world feasibility result, or evidence that any treatment works on people.
DD-C-0057 · verified · theorem
spine title theorem
For every finite N at least 2 and accuracy p strictly between zero and one in the frozen common-source game, the common-to-independent gross payoff thresholds A_k=p(1-p)E[1/(1+X_k)-1/(N-k+X_k)] are strictly decreasing from A_0=p(1-p)(N-1)/N to A_(N-1)=0 and exactly characterize all weak pure source-count equilibria; planner thresholds are B_k=p(1-p)^(k+1) through k=N-2 and B_(N-1)=0, so the all-common equilibrium/planner-independence trap holds exactly on [p(1-p)(N-1)/N,p(1-p)), an interval of width p(1-p)/N.
Scope: Arbitrary finite N at least 2, M=3 uniform targets, homogeneous conditional accuracy 0<p<1, one free common source, costly conditionally independent sources, direct clue-following, equal splitting of a unit prize, nonnegative common independent-source cost, pure source-count equilibria, and union discovery net of acquisition costs only; no heterogeneous sources, learning, disclosure, allocation choice, mixed equilibrium, empirical behavior, or real-world welfare claim.
DD-C-0058 · independently-reproduced · negative-result
minimal counterexample
Universal private under-acquisition is false even inside the frozen common-source model; at N=3, p=4/5, and c=13/375, the private threshold at k=1 is 14/375 while the planner threshold is 4/125=12/375, yielding the unique source-count equilibrium k=2 and unique planner optimum k=1.
Scope: The exact DD-008B model at the single stated rational parameter fixture; this is a counterexample to a universal source-count ordering, not a claim that over-acquisition is typical, empirically observed, or robust to other prize and source structures.
DD-C-0059 · verified · theorem
spine title theorem
For every finite N at least 2 and accuracies p and q strictly between zero and one in the frozen access-gated follow/private model, discovery is G_N(0)=1-(1-p)^N and G_N(k)=1-(1-q)(1-p)^(N-k) for k at least 1; its optimum is uniquely one shared-signal reader when q>p, uniquely zero readers when q<p, and exactly zero or one reader when q=p.
Scope: Arbitrary finite N at least 2, three uniform targets, one shared clue with accuracy q, conditionally independent private clues with homogeneous accuracy p, uniform wrong labels, ex-ante binding access, shared-clue following by readers, private-clue following by non-readers, and union discovery; this is not an unrestricted disclosure, policy, correlated-signal, robustness, dynamic, behavioral, or empirical theorem.
DD-C-0060 · verified · theorem
scoped theorem or impossibility
In the frozen equal-split attention game, the private gain Delta_k from becoming reader k+1 is strictly decreasing in k and exactly characterizes every weak and strict pure attention-count equilibrium; all-ignore is a weak equilibrium exactly when q<=p, while all-attend is weak exactly when q>=Np/(1+(N-1)p), so all-ignore cannot occur when the unique social optimum is one reader.
Scope: The DD-C-0059 model plus equal splitting of a unit success prize, risk-neutral ex-ante role choice, anonymous pure attention counts, and gross expected prizes without attention cost; no mixed equilibrium, heterogeneous preferences, arbitrary sharing, behavioral attention, or unrestricted policy response is covered.
DD-C-0061 · independently-reproduced · computational-result
exact bounded classification
In DD-012's 175-cell rational census, equal-split attention has 63 excessive-attention cells, three equilibrium-multiplicity cells with a welfare difference, and 24 cells where all-attend is an equilibrium although one reader is uniquely optimal; sole-rescue, marginal-coverage, universal-pooling, and a binding public-reader license exactly match the social-optimum correspondence in all 175 cells, while off-shared-success yields unique all-ignore in all cells and the assigned-reader subsidy has no socially optimal weak equilibrium in any cell.
Scope: N=2 through 8 and p,q in {1/3,1/2,2/3,3/4,5/6} under DD-C-0059's model and the seven precisely registered reward/access rules, observability assumptions, enforcement, and subsidy conventions; this is not a mechanism-optimality, budget-balance, collusion, robustness, human-behavior, or empirical result.
DD-C-0062 · verified · theorem
scoped theorem or impossibility
For every finite N at least 2 and interior accuracies p and q in the frozen access-gated follow/private model, full-precision shared evidence is binding-optimal for exactly one recipient when q>p, no recipients when q<p, and either zero or one recipient when q=p; every audience of size at least two is strictly worse.
Scope: Arbitrary finite N at least 2, three uniform targets, homogeneous conditionally independent private clues, one symmetric shared clue, binding ex-ante delivery, the registered follow-shared/follow-private actions, and union discovery; no voluntary use, arbitrary action rule, robustness value, correlated signal, dynamic, behavioral, or empirical claim is included.
DD-C-0063 · verified · theorem
scoped theorem or impossibility
Within DD-013's symmetric garbling family with delivered accuracy g in the registered grid and 1/3<=g<=q, the binding audience optimum weakly dominates every garbled positive-audience design; equality occurs only at a full-precision one-recipient design when q>=p, and all other 2,520 registered garbling rows are strictly worse.
Scope: The DD-C-0062 model and only the 2,625 feasible `(N,p,q,g,m)` rows with g drawn from {1/3,1/2,2/3,3/4,5/6}, g<=q, and positive audience m; this is not a Blackwell-order, arbitrary experiment, endogenous-precision, communication-cost, persuasion, or empirical theorem.
DD-C-0064 · verified · theorem
scoped theorem or impossibility
With full public access and ex-ante choice between the registered follow-shared and follow-private modes, universal pooling—each role receives 1/N exactly when the team discovers—implements the binding-optimal reader-count correspondence for every finite N at least 2 and interior p,q, is strict when p differs from q, is ex-post budget balanced against the unit discovery prize, and needs no external subsidy.
Scope: The DD-C-0062 signal and discovery model, risk-neutral agents, full access, pure ex-ante two-mode unilateral deviations, team discovery observability, and the stated equal pooling rule; arbitrary signal-contingent policies, mixed equilibrium, collusion, identity selection, hidden discovery, dynamics, and real institutions are excluded.
DD-C-0065 · independently-reproduced · computational-result
exact bounded classification
In DD-013's 1,050 registered audience settings and 4,025 voluntary profiles, the voluntary weak-equilibrium correspondence differs from binding use in 656 settings, permits excessive use relative to the global binding optimum in 273, and has welfare-relevant multiplicity in eight; all 70 insufficient-use settings have zero audience, so no positive audience has voluntary equilibrium use below the global optimum on this grid.
Scope: N=2 through 8, p,q in {1/3,1/2,2/3,3/4,5/6}, equal splitting of a unit discovery prize, an ex-ante recipient choice between only follow-shared and follow-private modes, and nonrecipients without shared access; no conditional policy, mixed equilibrium, heterogeneous payoff, attention cost, behavioral, or empirical conclusion is claimed.
DD-C-0066 · verified · theorem
scoped theorem or impossibility
In the complete deterministic agreement-respecting label-equivariant disagreement class for three labels, if p>=1/3 then replacing any third-option contrarian with a private-dominant role weakly improves discovery, strictly when p>1/3; consequently conditional policies do not improve the DD-012 unconditional planner optimum, and its zero-or-one public-dominant role classification survives.
Scope: Arbitrary finite N at least 2, three uniform targets, homogeneous conditionally independent private clues, one shared clue, uniform wrong labels, universal access, deterministic agreement-respecting label-equivariant policies, union discovery, and p>=1/3; this is not an unrestricted-policy, raw-table, correlated-signal, dynamic, behavioral, or empirical theorem.
DD-C-0067 · independently-reproduced · computational-result
exact bounded classification
In DD-014's 75-cell rational census, the 775 anonymous conditional-policy profiles contain 115 weak and 53 strict equilibria; 23 cells have a positive gap between planner discovery and the best weak-equilibrium discovery, 49 cells have only planner-optimal weak equilibria, and none has a positive conditional planner gain over the unconditional class.
Scope: N in {2,3,4}, p,q in {1/3,1/2,2/3,3/4,5/6}, the exact three-policy class in DD-C-0066, equal splitting of a unit discovery prize, pure ex-ante role choice, and all unilateral deviations among the three types; no mixed equilibrium, arbitrary policy, heterogeneous payoff, attention cost, dynamics, behavior, or empirical conclusion is included.
DD-C-0068 · independently-reproduced · negative-result
minimal counterexample
In each of DD-014's four two-agent, two-label raw-policy audit cells, all sixteen deterministic policy tables permit discovery one through complementary constant policies, strictly above the best embedded unconditional private/public profile by between 1/16 and 1/9; hence the restricted three-label conditional result cannot be promoted to an unrestricted deterministic-policy claim.
Scope: M=2, N=2, p,q in {2/3,3/4}, all 256 ordered profiles of the sixteen raw deterministic observation-to-action tables, equal-split prize, and union discovery; this is an adversarial scope audit, not the M=3 registered class or a general characterization of unrestricted equilibria.
DD-C-0069 · independently-reproduced · computational-result
exact bounded classification
DiscoveryBench v2 preserves all v1 defaults and exact vectors while extending the registry to 20 tasks, 21 capability-declared protocols, and 27 metrics; all 28 declared compatible pairs exactly reproduce their registered values and all 392 incompatible pairs remain explicit exclusions.
Scope: DiscoveryBench v1 plus the five registered DD-012 through DD-014 selective-attention fixtures at N=4, p=1/2, and q=3/4, the eight appended protocols, eight appended metrics, exact rational evaluator, capability gate, and source claim/run mappings; this is not a universal score, hosted leaderboard, external-agent evaluation, behavioral result, or empirical claim.
DD-C-0070 · verified · monte-carlo-estimate
synthetic estimate
DD-011 v2 reproducibly extends the synthetic package to 29 treatment cells, 14 hypotheses, 19 outcomes, 11 response scenarios, 928 balanced synthetic assignments, and 924 power rows with 1,000 replications each; 335 of 462 rows at sample sizes 640 or greater remain below the registered 0.80 calibration threshold.
Scope: The frozen selective-attention-29 design, declared assumed effects, normal sampling approximation, conservative Bonferroni family alpha, session size eight, explicit response multipliers, fixed seeds, and sample sizes 200 through 960; this is conditional synthetic power, not a human experiment, empirical attention model, recruitment target, behavioral finding, or evidence that any treatment works on people.
DD-C-0071 · verified · theorem
spine title theorem
In the frozen atomic threshold-opening model, a deterministic planner's optimal discovery value is the sum of the largest min(floor(N/tau),M) posterior masses, and an optimum exists that assigns exactly tau agents to each selected candidate before placing leftovers.
Scope: Any finite M, positive N, threshold tau in {1,...,N}, one hidden atomic target, one action per agent, deterministic execution, zero value above threshold, no replication value, and a normalized posterior; this is an existence and value theorem, not uniqueness, strategic implementation, or an equilibrium statement.
DD-C-0072 · verified · identity
structural identity or lemma
If every other agent independently selects candidate j with probability x_j, equal splitting gives payoff V*pi_j*E[1{K_j>=tau-1}/(K_j+1)]; at tau=1 its share factor is [1-(1-x_j)^N]/(N*x_j) with zero limit one, and at tau=2 it subtracts (1-x_j)^(N-1) and has zero limit zero.
Scope: The frozen one-candidate deviation calculation with N finite, K_j binomial(N-1,x_j), threshold opening, equal splitting among agents at the successful target, and fixed posterior pi_j; this does not characterize an equilibrium correspondence or select an equilibrium.
DD-C-0073 · independently-reproduced · computational-result
exact bounded classification
In DD-016's canonical M=16, N=8, p=1/5, tau=2 fixture, common deterministic mode, selected independent uniform mixing over tied posterior modes, private clue-following, and the paired planner have exact discovery 37916217637/98876953125, 275661897594857/576650390625000, 194017/390625, and 223779310319051/333709716796875 respectively; the selected mixture has exact expected distinct count 222783285994256/129746337890625, viable count 4605284003019928/3243658447265625, largest crowd 21276151146203888/3243658447265625, and pair-collision probability 3368373871489/4449462890625.
Scope: Uniform target, eight conditionally independent clues with correct probability 1/5 and each wrong label probability 4/75, posterior modes from pooled clue counts, the named deterministic or independent selection rules, threshold two, and exact occupancy definitions in the frozen model; this is not the full strategic equilibrium correspondence or a behavioral claim.
DD-C-0074 · independently-reproduced · computational-result
exact bounded classification
Across tau=1 through 8 in the canonical DD-016 fixture, both common deterministic mode and registered tied-mode mixing lower discovery relative to private clue-following at tau=1 and 2 and raise it at tau=3 through 8; the deterministic planner strictly exceeds the selected tied-mode result at every threshold, only one viable coalition can form at tau=5 through 8, and planner diversification strictly beats common mode at tau=1 through 4 before tying it at tau=5 through 8.
Scope: The DD-C-0073 canonical signal model and named selection rules over the complete registered integer threshold grid; all comparisons are exact within this fixture and do not assert equilibrium uniqueness, comparative statics outside the grid, human coordination, or empirical behavior.
DD-C-0075 · verified · theorem
scoped theorem or impossibility
An occupancy n is a weak pure Nash equilibrium of the frozen equal-split threshold game if and only if every agent on every occupied candidate j weakly prefers pi_j/n_j when n_j>=tau, or zero otherwise, to pi_k/(n_k+1) when n_k+1>=tau, or zero otherwise, for every alternative candidate k.
Scope: Any finite strictly positive posterior, N at least 2, threshold tau in {1,...,N}, one action per agent, deterministic threshold opening, unit prize split equally among agents at the true opened target, and pure unilateral action changes; this is not a mixed, correlated, strong, coalition-proof, or behavioral equilibrium characterization.
DD-C-0076 · verified · theorem
scoped theorem or impossibility
Independent uniform mixing over the highest-posterior candidates is a symmetric mixed Nash equilibrium for every finite strictly positive posterior when tau>=2; at tau=1 it is an equilibrium exactly when the common support payoff pi_max*[1-(1-1/m)^N]/(N/m) weakly exceeds every unsupported posterior mass, and this condition can fail.
Scope: The frozen equal-split threshold game, m posterior-maximizing candidates, symmetric independent mixing on exactly those modes, and pure one-agent deviations; this verifies one mixture and neither characterizes all mixed equilibria nor supplies a selection process.
DD-C-0077 · independently-reproduced · computational-result
exact bounded classification
In DD-017's 160 registered rational games, 52 have a weak pure Nash equilibrium with zero discovery and therefore infinite pure price of anarchy under the declared positive-planner-over-zero convention.
Scope: Eight frozen positive posterior fixtures with M in {2,3,4}, N=2 through 6, every tau=1 through N, equal-split threshold payoffs, weak pure Nash occupancy, deterministic planner benchmark, and the registered ratio convention; this is not a universal existence, asymptotic, mixed-equilibrium, or empirical result.
DD-C-0078 · independently-reproduced · computational-result
exact bounded classification
Among DD-017's 160 registered games, eight have no pairwise-strict-stable weak pure Nash equilibrium and 35 have no exact-size-tau-strict-stable weak pure Nash equilibrium.
Scope: The DD-C-0077 registry; a coalition blocks only when a joint action change makes every named member strictly better, pairwise means exactly two agents, and tau-player means exactly tau agents; no transfers, weak-Pareto blocks, arbitrary coalition sizes, strong equilibrium, coalition-proofness, or empirical stability is included.
DD-C-0079 · verified · proposition
scoped theorem or impossibility
In the frozen DD-015 autonomous game, choosing a posterior-mode action at every reached information set is a pure sequential Bayesian equilibrium under full duplicate credit.
Scope: Three targets, two or three sequential agents, the declared shared/private clue channel, visible public history, fixed-budget or stopping feedback, full unit credit whenever an agent's own action is correct, and lowest-label tie-breaking; this does not apply to equal-split, sole-rescue, marginal-credit, threshold-team, signaling, or mixed-strategy games.
DD-C-0080 · verified · computational-result
exact bounded classification
Across DD-015's 64 exact objective rows, the common-payoff planner strictly exceeds the full-credit autonomous equilibrium in 38 rows, autonomous play follows a previous action with positive probability in 32 rows and leans against a repeated prior action with positive probability in 50 rows, and stopping reduces expected actions relative to fixed budget in all 32 parameter cells.
Scope: The frozen four-by-four rational accuracy grid, N in {2,3}, three targets, visible actions, the named fixed/stopping feedback rules, the full-credit autonomous equilibrium, and the deterministic common-information prescription planner; this is a bounded exact census, not a general comparative-static, equal-split equilibrium, or behavioral result.
DD-C-0081 · verified · negative-result
bounded null or failed search
In DD-015's 32 fixed-budget parameter cells, visible-action inference improves neither discovery nor expected action dispersion relative to the history-hidden Bayes baseline; it strictly reduces both in 18 cells and ties in the remaining 14.
Scope: Three targets, N in {2,3}, the frozen rational accuracy grid, full duplicate credit, fixed budget, outcome hidden until terminal, visible-action posterior inference versus agents using only the shared and own private clue, and expected distinct actions as dispersion; this is a bounded negative result, not a theorem that visibility is generally harmful.
DD-C-0082 · verified · computational-result
exact bounded classification
In DD-015's separately labeled threshold-two planner extension, stopping preserves the fixed-budget optimal discovery value and strictly reduces expected actions in all 16 parameter cells; across all 32 objective rows the optimal policy uses, with positive probability, each of starting a singleton, joining a singleton, joining a viable team, following the shared clue, and opposing it.
Scope: Three targets, three sequential agents, threshold two, the frozen rational shared/private accuracy grid, visible actions, deterministic common-information prescriptions, and the declared fixed/stopping feedback rules; this is a planner-only bounded extension and not a decentralized, equilibrium, mechanism, coalition-proof, or general theorem claim.
DD-C-0083 · verified · computational-result
exact bounded classification
In DD-018's 50 exact mechanism-fixture rows, 40 recommended outcomes attain the two-minimum-team planner discovery value; central assignment, authoritative random matching, team tokens, exclusive coalition rights, correlated mediation, marginal coalition contribution, threshold-adjusted equal split, and the pairwise matching market attain it in all five fixtures, while universal pooling and sole-team rescue attain it in none.
Scope: Three candidates, four agents, threshold two, five frozen weakly descending common posteriors, the ten precisely registered recommendation and reward rules, and exact recommendation-support discovery; this is a bounded common-posterior allocation census, not a report-truthfulness, equilibrium-selection, general implementation, or behavioral result.
DD-C-0084 · verified · computational-result
exact bounded classification
Among DD-018's 35 rows with feasible unilateral action changes, 31 recommendations are strictly obedient and 29 withstand every strict-member pair deviation and every exact-size-two tau-player deviation; all 50 rows satisfy nonnegative participation, weak prize-budget balance, and zero external subsidy.
Scope: The DD-C-0083 registry, posterior-expected utilities under the frozen prize-eligibility rules, unilateral deviations only where individually feasible, strict improvement for every named coalition member, pairs and exact-size-two coalitions recorded separately, and zero outside option; report truthfulness, weak-Pareto blocks, arbitrary coalition sizes, strong equilibrium, and coalition-proofness are excluded.
DD-C-0085 · verified · negative-result
bounded null or failed search
In all five DD-018 fixtures, universal pooling and sole-team rescue recommend four agents on the top posterior candidate and attain strictly less discovery than the two-team planner; universal-pooling collapse is strictly unilaterally obedient in all five but pairwise strict-member stable in only one, while sole-team-rescue collapse is strictly unilaterally obedient and pairwise strict-member stable in all five.
Scope: The DD-C-0083 frozen common-posterior fixtures, lexicographic top-candidate collapse, threshold-adjusted equal split for universal pooling, the registered exactly-one-viable-team rescue prize, and strict-member unilateral/pair deviations; this is a bounded negative result, not a theorem that pooling or rescue generally fails.
DD-C-0086 · verified · computational-result
exact bounded classification
Team tokens, exclusive coalition rights, and marginal coalition contribution support the two-team planner recommendation with strict unilateral obedience and no strict-member pair or exact-size-two tau-player block in all five DD-018 fixtures, with weak prize-budget balance and zero external subsidy; nevertheless their labeled pure-Nash multiplicities are respectively 25, 25, and 21 in every fixture.
Scope: The DD-C-0083 registry and fixed assignments, binding token/right eligibility, exactly-minimal-team contribution reward, unilateral pure Nash over all 81 labeled profiles, and the declared pair/tau strict-member checks; the claim neither selects an equilibrium nor establishes report truthfulness, anonymous identity selection, arbitrary coalition stability, or a general mechanism theorem.
DD-C-0087 · independently-reproduced · computational-result
exact bounded classification
DiscoveryBench v3 preserves every v1 and v2 exact metric vector while extending the opt-in registry to 24 tasks, 29 capability-declared protocols, and 39 metrics; all 36 compatible pairs exactly reproduce their registered values and all 660 incompatible pairs remain explicit exclusions among 696 candidate pairs.
Scope: DiscoveryBench v2 plus four exact DD-016, DD-017, DD-015, and DD-018 Program V4 tasks, eight appended protocols, twelve appended metrics, exact task-level vectors, the declared capability gate, and immutable source claim/run mappings; this is not a universal score, hosted leaderboard, external-agent evaluation, synthetic-performance claim, behavioral result, or empirical claim.
DD-C-0088 · verified · monte-carlo-estimate
synthetic estimate
DD-011 v3 reproducibly extends the synthetic package to 37 treatment cells, 20 hypotheses, 23 outcomes, 14 response scenarios, 1,184 balanced synthetic assignments, and 1,680 power rows with 1,000 replications each; every v2 power row is preserved under its shared seed, and 644 of 840 rows at sample sizes 640 or greater remain below the registered 0.80 calibration threshold.
Scope: The frozen threshold-dynamic-37 design, declared assumed effects, normal sampling approximation, conservative Bonferroni family alpha, session size eight, explicit response multipliers, fixed seeds, and sample sizes 200 through 960; this is conditional synthetic power and version preservation, not a human experiment, empirical threshold or dynamic-attention model, recruitment target, behavioral finding, or evidence that any treatment works on people.
DD-C-0089 · independently-reproduced · computational-result
exact bounded classification
In the registered M=4, N=3 fixture, the exact pooled top-L profiles for the noisy point, noisy two-shortlist, guaranteed two-shortlist, explicit exclusion, and confidence-point channels are respectively (7/12,43/54,25/27), (35/64,79/96,539/576), (17/18,1,1), (16/27,26/27,1), and (295/384,175/192,31/32).
Scope: Five declared finite conditionally independent channels, uniform four-target prior, three observations, binding distinct pooled top-L actions, and L=1,2,3; not a general channel theorem, anonymous-private optimum, strategic result, threshold technology, or M=16,N=8 result.
DD-C-0090 · independently-reproduced · computational-result
minimal counterexample
The registered half-accurate noisy-point channel and guaranteed two-shortlist channel both have one-person Bayes accuracy 1/2 and direct private-portfolio discovery 7/8, yet their pooled profiles differ, (7/12,43/54,25/27) versus (17/18,1,1); therefore one-person accuracy does not determine the complete bounded action-budget discovery profile.
Scope: Exact M=4, N=3 counterexample for one-person Bayes accuracy (equivalently expected posterior maximum in this comparison) and the declared direct private baseline; it does not address mutual information, entropy reduction, a single likelihood ratio, elimination count, or every scalar summary.
DD-C-0091 · independently-reproduced · computational-result
exact bounded classification
Relative to each channel's declared direct private-portfolio baseline, the exact recovery budgets across the five registered channels are 3, 2, 1, 2, and 3 in registry order; in particular, the same-accuracy noisy point requires three pooled actions while the guaranteed two-shortlist requires one.
Scope: The DD-019 M=4, N=3 direct private-portfolio baseline with capacity L<=3 and binding pooled top-L planner authority; recovery budgets change with the baseline and do not identify an unrestricted implementation result.
DD-C-0092 · verified · identity
structural identity or lemma
In the registered target-symmetric block-sharing protocol with pooled MAP accuracy C_s and constant conditional private-action accuracy q, group discovery is G_s=1-(1-C_s)(1-q)^(N-s), and its adjacent difference is (1-q)^(N-s-1)[C_(s+1)-C_s-q(1-C_s)].
Scope: Finite targets, uniform target prior, conditionally independent block and remaining signals, target-symmetric pooled accuracy, constant conditional private-action accuracy, independent tie randomization, one pooled block action, and N-s direct private actions; no arbitrary dependence, strategic behavior, or private-team optimum.
DD-C-0093 · verified · theorem
scoped theorem or impossibility
For every finite M>=2 and symmetric noisy-point accuracy p in [1/M,1], two-signal uniformly tie-broken pooled MAP accuracy is C_2=p; hence G_1=1-(1-p)^2, G_2=p, and G_2-G_1=-p(1-p), strictly negative for p<1.
Scope: The DD-020 uniform-prior symmetric noisy-point channel, N=2, one-hit common discovery, direct-private baseline, one-action full pooling, and independent uniform posterior-MAP tie-breaking; no anti-informative p, arbitrary signal channel, or optimal-team claim.
DD-C-0094 · verified · theorem
scoped theorem or impossibility
For every finite M>=2, N>=2, symmetric noisy-point accuracy p in [1/M,1], and adjacent block sizes s and s+1, the registered discovery profile satisfies G_(s+1)<=G_s; the inequality is strict for p<1 and equality for p=1.
Scope: Uniform prior, finite symmetric noisy-point channel, conditionally independent signals, one uniformly tie-broken pooled MAP action duplicated by the sharing block, remaining direct private Bayes actions, and one-hit group discovery; no arbitrary channel, assignment, pooled action list, strategic equilibrium, or global private-team optimum.
DD-C-0095 · independently-reproduced · computational-result
exact bounded classification
Across DD-020's 73 registered (M,p) cells, N=2 through 8, and all block sizes, the 2,555-row exact census has 1,848 strictly negative adjacent discovery increments, 196 zero increments, and no positive increment; every zero is a p=1 boundary row.
Scope: M=2 through 8, N=2 through 8, the exact rational accuracy grid in census.yml, all s=1 through N, and the DD-C-0094 protocol; this is a bounded exact census and theorem regression, not an empirical frequency, arbitrary-channel result, or novelty claim.
DD-C-0096 · independently-reproduced · computational-result
minimal counterexample
In the M=4,N=3 DD-019 extension, the noisy point, noisy two-shortlist, guaranteed two-shortlist, explicit exclusion, and confidence-point incremental-sharing profiles are respectively (7/8,3/4,7/12), (387/512,11/16,35/64), (7/8,11/12,17/18), (19/27,17/27,16/27), and (485/512,113/128,295/384); the noisy point and guaranteed shortlist both have one-person accuracy 1/2 but have strictly decreasing versus strictly increasing profiles.
Scope: The five fixed DD-019 channels, uniform four-target prior, three conditionally independent observations, constant target-symmetric direct-action accuracy, one pooled MAP action for block sizes one through three, remaining direct Bayes actions, and independent uniform ties; not a general arbitrary-channel theorem or claim that one-person accuracy never predicts any other functional.
DD-C-0097 · verified · theorem
spine title theorem
In the registered finite target-symmetric block-sharing protocol, with pooled error e_s=1-C_s and constant conditional direct-action accuracy q, G_(s+1)-G_s=(1-q)^(N-s-1)[(1-q)e_s-e_(s+1)]; therefore the next sharing step helps, is neutral, or hurts exactly when e_(s+1) is respectively below, equal to, or above (1-q)e_s.
Scope: Finite targets, uniform prior, conditionally independent signals and private tie draws given the target, constant target-conditional direct accuracy, one uniformly tie-broken pooled MAP action, remaining declared direct actions, and one-hit discovery; no source dependence, strategy, private-team optimum, or novelty claim for the descriptive theorem name.
DD-C-0098 · verified · theorem
scoped theorem or impossibility
Under the registered protocol, the centralized posterior top-L pooled value at L=min(N,M) weakly exceeds the declared N-action direct-private discovery value, so the recovery budget L* always exists and satisfies L*<=min(N,M); moreover L*=1 exactly when full consensus C_N weakly exceeds the private baseline P_N.
Scope: Uniform finite target prior, conditionally independent registered signals, declared target-equivariant direct actions with independent ties, centralized authority to choose a posterior top-L target set after seeing all signals, binding distinct target actions up to min(N,M), and one-hit discovery; no decentralized, incentive-compatible, anonymous-private, or equilibrium implementation claim.
DD-C-0099 · independently-reproduced · computational-result
exact bounded classification
Across DD-021's complete 177-scenario exact registry, 126 sharing curves are strict compression-dominated, 16 are strict aggregation-dominated, 35 are all-neutral, and none are mixed; full sharing yields 78 Shared Discovery Paradox cases, 16 strict aggregation-dominated consensus cases, 83 boundary cases, and no strict no-one-action aggregation-gain case.
Scope: The frozen 59 target-symmetric channel laws over M in {3,4,5}, N in {2,3,4}, and the exact parameter grid in registry.yml; these are bounded registry counts, not frequencies over organizations or a universal channel classification theorem.
DD-C-0100 · independently-reproduced · computational-result
minimal counterexample
Under DD-021's frozen minimization order, the first same-baseline strict opposite-sign witness occurs at M=4,N=2,q=1/2 and private discovery 3/4: the noisy-point sharing step is -1/4, while the guaranteed two-shortlist step is +1/12.
Scope: Minimality within the registered channel families ordered by targets, agents, signal-alphabet size, denominator complexity, declared description rank, channel ID, and step; not minimal over all finite channels or a general characterization by one-person accuracy.
DD-C-0101 · independently-reproduced · computational-result
exact bounded classification
DD-021's minimized registered witnesses include an M=3,N=2 noisy two-shortlist Shared Discovery Paradox with q=3/8<C_N=27/64<P_N=39/64, an M=4,N=2 guaranteed two-shortlist strict consensus-dominance case with P_N=3/4<C_N=5/6, and an M=3,N=2 uninformative point case with C_N=q=1/3<P_N=5/9 and therefore no useful one-action aggregation gain.
Scope: Minimality within the frozen registry and declared order; the no-useful witness permits equality C_N=q but excludes perfect q=1, and the three witnesses do not imply that every arbitrary channel falls into a strict class.
DD-C-0102 · independently-reproduced · computational-result
exact bounded classification
DD-021's exact recovery budgets are L*=1 in 51 scenarios, L*=2 in 55, L*=3 in 64, and L*=4 in seven; the first same-accuracy recovery separation is at M=3,N=2,q=1/2, where the noisy point requires two pooled actions and the guaranteed two-shortlist requires one.
Scope: The complete registered 177-scenario grid, each channel's declared direct-private baseline, centralized posterior top-L actions for L<=min(N,M), and the frozen witness order; budgets would change with a different baseline or implementation constraint.
DD-C-0103 · verified · negative-result
bounded null or failed search
No mixed sharing curve—one having both a positive and a negative adjacent increment—appears in DD-021's complete 177-scenario registered grid.
Scope: The exact frozen 59-law registry over M in {3,4,5} and N in {2,3,4}; this is a bounded negative result and failed witness search, not proof that mixed curves cannot occur in arbitrary target-symmetric finite channels.
DD-C-0104 · verified · theorem
scoped theorem or impossibility
In DD-022's two-agent hidden-dependence game, let t=2p-1 and A=t^2+rho(1-t^2). Within anonymous label-equivariant strategies that follow a private signal with probability r, the symmetric Bayes-Nash equilibrium is r=1 when A<=3t and r=1/2+3t/(2A) when A>3t, with the separately stated uninformative boundary.
Scope: Two binary targets, uniform prior, two agents, the registered world-level common-versus-independent signal mixture, equal-split unit prize, and the declared anonymous label-equivariant symmetric behavioral class; not all equilibria, coalition-proofness, arbitrary targets, or arbitrary source laws.
DD-C-0105 · verified · theorem
scoped theorem or impossibility
In DD-022, public agreement has posterior u=1/2+t/(1+A) on the agreed label and positive-probability disagreement has posterior 1/2. In the selected provenance-blind posterior-only identical-mixing game, the favored target is chosen with probability one when u>=2/3 and with probability 3u-1 otherwise; disagreement selects probability 1/2.
Scope: The registered two-target equal-split game, positive-probability public profiles, independent private mixing, and strategies depending identically only on the common posterior; excludes signal-ownership conditioning, correlated randomization, roles, recommendations, and every-equilibrium claims.
DD-C-0106 · verified · theorem
spine title theorem
At p=3/5, discovery under DD-022's selected shared posterior-only symmetric equilibrium exceeds discovery under its private anonymous symmetric Bayes-Nash equilibrium exactly when rho lies in ((5sqrt(73)-17)/48,1), with equality at the algebraic root and at rho=1.
Scope: The exact DD-022 binary two-agent source law, equal-split prize, and the selections in DD-C-0104 and DD-C-0105; not a comparison over every equilibrium, direct clue-following alone, arbitrary p, arbitrary source dependence, or centralized assignment.
DD-C-0107 · verified · corollary
scoped theorem or impossibility
On the exact interval in DD-C-0106, sharing also strictly raises each agent's ex-ante expected payoff; total realized and expected payoff equals the discovery indicator and probability, and player symmetry gives each agent exactly one half.
Scope: Same model and equilibrium selections as DD-C-0106 with the ex-post budget-balanced equal-split unit prize; no welfare weights, transfers, coalitions, or asymmetric equilibrium payoffs.
DD-C-0108 · independently-reproduced · computational-result
exact bounded classification
DD-022's complete 42-cell rational grid contains six positive, 18 negative, and 18 neutral selected shared-minus-private discovery comparisons.
Scope: The six registered p values, seven registered rho values, binary two-agent model, and selected equilibria in DD-C-0104 and DD-C-0105; bounded exact counts, not empirical frequencies or a general-p classification.
DD-C-0109 · verified · negative-result
scoped theorem or impossibility
DD-022's positive-sharing comparison does not survive every equilibrium. At p=3/5, opposite constant-target private pure equilibria exist for every rho and attain discovery one; signal-ownership-aware symmetric shared rules can also split targets on disagreement, so the posterior-only mixed result is selection-dependent.
Scope: Complete labeled private pure Bayesian correspondence in the registered binary game and the ownership-aware disagreement subgame; this is a scope-defining negative result, not a claim about refinements, learning dynamics, coalition-proofness, or arbitrary games.
DD-C-0110 · verified · theorem
scoped theorem or impossibility
At p=3/5, the gap from DD-021's centralized binary top-two value V_2=1 to DD-022's selected shared discovery is (7+3rho)/25 for rho<=1/6 and (4+3rho)/(13+12rho) for rho>1/6, and is strictly positive throughout rho in [0,1].
Scope: DD-022 source law and posterior-only shared selection compared with the centralized ability to assign both binary targets; the benchmark is an upper value, not a decentralized or incentive-compatible implementation.