Buleyean Probability Theory

`complement_distribution`

Compute the Buleyean complement distribution: probabilities derived entirely from what hypotheses are NOT. Given a set of hypotheses with rejection counts, P(H_i) is proportional to (total_rejections ...

`bule_value`

Compute the Bule value B for a hypothesis space. B measures how much of the void has been walked: B = (total_possible_observations - observations_made) / total_possible_observations. B=0 means fully w...

`bayesian_bridge`

Demonstrate how Buleyean probability reduces to Bayesian inference at B=0. When the void has been fully walked, the complement distribution converges to the Bayesian posterior. Given a prior and likel...

`solomonoff_bridge`

Demonstrate how Buleyean probability reduces to Solomonoff induction at B=B_max. Before any empirical observation, the void boundary is initialized from Kolmogorov complexity. Simpler hypotheses (shor...

`update_evidence`

Bayesian-style evidence update through the Buleyean lens. When an observation eliminates or weakens hypotheses, increase their rejection counts. Returns the updated complement distribution, the inform...

`sufficient_statistic`

Compute the sufficient statistic from the void boundary. The rejection history is a sufficient statistic for decision-making under irreversibility: given the rejection counts, the full temporal histor...

`three_traditions`

Compute the answer to a probabilistic question using all three traditions -- Bayesian, frequentist, and Solomonoff -- and show they are cross-sections of the same Buleyean framework at different Bule ...

`buleyean_logic`

Compute Buleyean logic connectives. Buleyean logic generalizes Boolean logic: truth values are natural numbers (Bule counts) rather than {true, false}. 0 = proved (ground state). n > 0 = n rejections ...

`buleyean_proof`

Run a Buleyean proof by rejection. Given a set of options and a sequence of rejections, compute which options survive. Truth emerges from the complement -- the child never asserts "toy = true"; the ch...

`buleyean_pulse`

Compute the Buleyean Pulse -- the +1/-1 oscillation between hope and truth. Buleyean Probability is the +1 (the sliver that keeps every option alive). Buleyean Logic is the -1 (the rejection that driv...

`personality_weighted_complement`

Compute the Buleyean complement distribution modulated by the Five-Bule personality profile (Try/Choose/Commit/LetGo/Learn). Each personality parameter controls a training hyperparameter: Try=eta (ape...

`buleyean_unification`

Demonstrate the grand unification: Buleyean RL, Buleyean Logic, and Buleyean Probability are three views of a single operation -- rejection. v_i++ in RL = bules-- in Logic = recordRejection in Probabi...

`convergence_unification`

Demonstrate that all three Buleyean systems converge to the same ground state: void_walkers_converge (RL), n_rejections_reach_ground (Logic), buleyean_coherence (Probability). Boolean is the K=2 conve...

`thm_quadrant_roundtrip`

Quadrant ↔ primitive bijection [LEDGER: THM-QUADRANT-ROUNDTRIP]

`thm_fork_increases_beta1`

Fork increases beta-1 [LEDGER: THM-FORK-INCREASES-BETA1]

`thm_fold_decreases_beta1`

Fold decreases beta-1 [LEDGER: THM-FOLD-DECREASES-BETA1]

`thm_fork_vent_conjugate`

Fork and vent are conjugate [LEDGER: THM-FORK-VENT-CONJUGATE]

`thm_race_fold_conjugate`

Race and fold are conjugate [LEDGER: THM-RACE-FOLD-CONJUGATE]

`thm_canonical_cycle`

Fork → race → fold → vent is complete and injective [LEDGER: THM-CANONICAL-CYCLE]

`thm_buleyean_pulse`

The master pulse quadrant theorem [LEDGER: THM-BULEYEAN-PULSE]

`thm_buleyean_uniform_boundary`

Uniform boundary implies equal weights [LEDGER: THM-BULEYEAN-UNIFORM-BOUNDARY]

`thm_fisher_curvature_monotone`

Fisher curvature is monotone in alternatives [LEDGER: THM-FISHER-CURVATURE-MONOTONE]

`thm_solomonoff_gap`

Solomonoff gap is complexity difference [LEDGER: THM-SOLOMONOFF-GAP]

`thm_retrocausal_at_zero`

Retrocausal bound at zero distance [LEDGER: THM-RETROCAUSAL-AT-ZERO]

`thm_fisher_master`

Fisher manifold master theorem [LEDGER: THM-FISHER-MASTER]

`thm_complement_reverses`

Complement reverses ordering [LEDGER: THM-COMPLEMENT-REVERSES]

`thm_double_complement_preserves`

Double complement preserves rejection ordering [LEDGER: THM-DOUBLE-COMPLEMENT-PRESERVES]

`thm_sign_alternation`

Sign alternation at extremes [LEDGER: THM-SIGN-ALTERNATION]

`thm_weight_spread_bounded`

Weight spread is bounded [LEDGER: THM-WEIGHT-SPREAD-BOUNDED]

`thm_complement_oscillation_master`

Master complement oscillation [LEDGER: THM-COMPLEMENT-OSCILLATION-MASTER]

`thm_sliver_is_plus_one`

Sliver is +1 [LEDGER: THM-SLIVER-IS-PLUS-ONE]

`thm_proof_step_is_minus_one`

Proof step is -1 [LEDGER: THM-PROOF-STEP-IS-MINUS-ONE]

`thm_sliver_proof_inverse`

Sliver and proof are inverses [LEDGER: THM-SLIVER-PROOF-INVERSE]

`thm_hope_enables_proof`

Hope enables proof [LEDGER: THM-HOPE-ENABLES-PROOF]

`thm_sliver_bridges`

Sliver bridges probability and logic [LEDGER: THM-SLIVER-BRIDGES]

`thm_sliver_of_hope`

Master: sliver of hope [LEDGER: THM-SLIVER-OF-HOPE]

`thm_three_are_one`

Buleyean RL rejection (v_i++), Buleyean Logic rejection (bules--), and Buleyean Probability update (recordRejection) are the same operation because bules = T - v_i (complement) [LEDGER: THM-THREE-ARE-...

`thm_void_is_proof_trace`

The void boundary (RL rejection history per token) is structurally identical to the proof trace (Logic rejection reason sequence). Same data, same sufficient statistic [LEDGER: THM-VOID-IS-PROOF-TRACE...

`thm_complement_is_target`

The Buleyean complement distribution (Probability) is identically the KL training target (RL) and the ground state indicator (Logic). All three select by NOT-rejected [LEDGER: THM-COMPLEMENT-IS-TARGET...

`thm_five_map_to_five`

Try=Fork=eta, Choose=Race=temperature, Commit=Fold=commitGain, LetGo=Vent=decayRate, Learn=Reject=feedbackGain. The five personality words, five proof primitives, and five RL hyperparameters are the s...

`thm_convergence_unification`

All three systems converge to the same ground state: void_walkers_converge (RL), n_rejections_reach_ground (Logic), buleyean_coherence (Probability). Boolean is the K=2 convergence endpoint. Phi is th...

`thm_grand_unification_qed`

The grand unification proof topology has 54 nodes, 23 edges, 15 Bules, 15 rejections, zero diagnostics. All terminals at bules = 0. QED [LEDGER: THM-GRAND-UNIFICATION-QED]

`thm_monoculture_parallel`

++ state is parallel [LEDGER: THM-MONOCULTURE-PARALLEL]

`thm_heat_death_parallel`

-- state is parallel [LEDGER: THM-HEAT-DEATH-PARALLEL]

`thm_forest_antiparallel`

+- state is antiparallel [LEDGER: THM-FOREST-ANTIPARALLEL]

`thm_skyrms_antiparallel`

-+ state is antiparallel [LEDGER: THM-SKYRMS-ANTIPARALLEL]

`thm_antiparallel_preserves`

Antiparallel preserves both purity and diversity [LEDGER: THM-ANTIPARALLEL-PRESERVES]

`thm_forest_flips_skyrms`

Forest phase flips to Skyrms phase [LEDGER: THM-FOREST-FLIPS-SKYRMS]

`thm_skyrms_flips_forest`

Skyrms phase flips to Forest phase [LEDGER: THM-SKYRMS-FLIPS-FOREST]

`thm_period_two`

flip . flip = id (exact period 2) [LEDGER: THM-PERIOD-TWO]

`thm_mono_flips_heat`

Monoculture flips to heat death [LEDGER: THM-MONO-FLIPS-HEAT]

`thm_heat_flips_mono`

Heat death flips to monoculture [LEDGER: THM-HEAT-FLIPS-MONO]

`thm_antiparallel_lower`

Antiparallel has lower energy [LEDGER: THM-ANTIPARALLEL-LOWER]

`thm_antiparallel_ground`

Antiparallel is ground state (energy 0) [LEDGER: THM-ANTIPARALLEL-GROUND]

`thm_parallel_excited`

Parallel is excited state (energy 1) [LEDGER: THM-PARALLEL-EXCITED]

`thm_ground_iff_antiparallel`

Ground state iff antiparallel [LEDGER: THM-GROUND-IFF-ANTIPARALLEL]

`thm_compiler_family_ground`

Forest and Skyrms are ground; monoculture and heat death are excited [LEDGER: THM-COMPILER-FAMILY-GROUND]

`thm_barbelo_is_sliver`

Barbelo = 1 (the +1) [LEDGER: THM-BARBELO-IS-SLIVER]

`thm_primitives_is_fibonacci`

F(5) = 5 (operation count is Fibonacci) [LEDGER: THM-PRIMITIVES-IS-FIBONACCI]

`thm_kenoma_from_primitives`

5 choose 2 = 10 (Kenoma) [LEDGER: THM-KENOMA-FROM-PRIMITIVES]

`thm_kenoma_decomposition`

Kenoma = Sophia + Barbelo (10 = 9 + 1) [LEDGER: THM-KENOMA-DECOMPOSITION]

`thm_void_is_fibonacci_gap`

F(10) - F(9) = F(8) = 21 (Void) [LEDGER: THM-VOID-IS-FIBONACCI-GAP]

`thm_pleroma_is_fib_kenoma`

F(10) = 55 (Pleroma) [LEDGER: THM-PLEROMA-IS-FIB-KENOMA]

`thm_triple_coincidence`

F(10) = T(10) = 55 (Luo Ming uniqueness) [LEDGER: THM-TRIPLE-COINCIDENCE]

`thm_fibonacci_gap`

F(n) - F(n-1) = F(n-2) (void between numbers is the sequence) [LEDGER: THM-FIBONACCI-GAP]

`thm_gap_reaches_barbelo`

Every gap between named numbers reaches 1 [LEDGER: THM-GAP-REACHES-BARBELO]

`thm_gnostic_number_system`

Complete: nine names + structure + coincidence [LEDGER: THM-GNOSTIC-NUMBER-SYSTEM]

Buleyean Probability Theory

Tools

complement_distribution

bule_value

bayesian_bridge

solomonoff_bridge

update_evidence

sufficient_statistic

three_traditions

buleyean_logic

buleyean_proof

buleyean_pulse

personality_weighted_complement

buleyean_unification

convergence_unification

thm_quadrant_roundtrip

thm_fork_increases_beta1

thm_fold_decreases_beta1

thm_fork_vent_conjugate

thm_race_fold_conjugate

thm_canonical_cycle

thm_buleyean_pulse

thm_buleyean_uniform_boundary

thm_fisher_curvature_monotone

thm_solomonoff_gap

thm_retrocausal_at_zero

thm_fisher_master

thm_complement_reverses

thm_double_complement_preserves

thm_sign_alternation

thm_weight_spread_bounded

thm_complement_oscillation_master

thm_sliver_is_plus_one

thm_proof_step_is_minus_one

thm_sliver_proof_inverse

thm_hope_enables_proof

thm_sliver_bridges

thm_sliver_of_hope

thm_three_are_one

thm_void_is_proof_trace

thm_complement_is_target

thm_five_map_to_five

thm_convergence_unification

thm_grand_unification_qed

thm_monoculture_parallel

thm_heat_death_parallel

thm_forest_antiparallel

thm_skyrms_antiparallel

thm_antiparallel_preserves

thm_forest_flips_skyrms

thm_skyrms_flips_forest

thm_period_two

thm_mono_flips_heat

thm_heat_flips_mono

thm_antiparallel_lower

thm_antiparallel_ground

thm_parallel_excited

thm_ground_iff_antiparallel

thm_compiler_family_ground

thm_barbelo_is_sliver

thm_primitives_is_fibonacci

thm_kenoma_from_primitives

thm_kenoma_decomposition

thm_void_is_fibonacci_gap

thm_pleroma_is_fib_kenoma

thm_triple_coincidence

thm_fibonacci_gap

thm_gap_reaches_barbelo

thm_gnostic_number_system

Discovery Endpoints

Paper Reference