Interference, Renormalization & Community

Tools

`coarsen`

Apply a coarsening (many-to-one quotient map) to a system. Computes the coarsened state space, information lost as conditional entropy H(fine|coarse), Landauer heat of coarsening (kT ln 2 per lost bit...

`renormalization_flow`

Compute renormalization group flow across scales. Identifies fixed points, classifies operators as relevant (grow under RG), irrelevant (shrink), or marginal (unchanged), and determines flow direction...

`community_dominance`

Analyze community structure for dominance properties (LEDGER section 25). Checks nondegradation (attenuation non-increasing under composition), strict domination pairs, diversity amplification, and co...

`skyrms_nadir`

Compute the Skyrms nadir for a mediation scenario (LEDGER section 25). The nadir is the worst outcome each party can be forced to accept -- the threat point of bargaining. Returns the nadir point, che...

`moa_fabric`

Analyze a heterogeneous mixture-of-agents (MoA) fabric (LEDGER section 26). Computes the optimal weave (assignment of agents to subtasks), coverage score, redundancy factor, and expected quality of th...

`thm_sliver_fractal`

Scale-invariant interference boundary: a support-preserving coarse quotient cannot erase the paid cost of contagious fine-scale interference; if a fine nontrivial fork is contagious and the coarse ima...

`thm_renormalization_coarsening`

Constructive many-to-one renormalization closure: a finite bounded graph quotient can collapse a genuinely many-to-one fine support into a coarse node or coarse graph while preserving aggregate `λ`, `...

`thm_renormalization_reuse`

Recursive one-node reuse closure: once a verified quotient has been collapsed onto its coarse support, the next-stage coarse graph interface can be synthesized automatically from aggregate rates, and ...

`thm_recursive_coarsening_synthesis`

Recursive coarsening synthesis: for any finite bounded verified subgraph `G`, Betti can synthesize the quotient operator `Φ_G`, compute each collapsed coarse node `N'` by aggregating the fine-node `λ`...

`thm_skyrms_nadir`

Two metacognitive walkers converging via failure on a shared void surface: each failure strictly increases void density. Growing void shifts complement toward center. Inter-walker distance is non-incr...

`thm_skyrms_three_walker`

Mediator as player on the convergence site: Skyrms walker's payoff = negative distance delta. Each failure enriches at least one void boundary. Site void density is non-decreasing. All three walkers e...

`thm_moa_fabric_lowering`

Heterogeneous paired-kernel MoA lowering preserves mirrored per-lane kernel counts, active backend-layer diversity, ready-backend cardinality bounds, zero-active exactness at the all-zero witness, and...

`thm_moa_fabric_cannon`

The `gnode` cannon/helix scheduler arms each wave before launch, advances the rotating cursor by wave width modulo lane count, includes honest zero-lane/zero-layer boundary behavior, and preserves Aeo...

`thm_moa_fabric_pair`

Mirrored primary/shadow kernel pairs accept immediately on agreement or on an early sufficient primary before the delayed shadow fires, otherwise escalate to explicit disagreement handling, never repo...

`thm_moa_fabric_waste`

Winner/loser/vent/skipped accounting is conserved across layered race/fold/vent execution, monotone sequence framing preserves Aeon's binary stream order, zero skipped hedges always stay within budget...

`thm_moa_fabric_coupled`

Backend-diverse mirrored kernels inherit stability from aligned twins, and paired downstream fabrics remain geometrically stable under bounded imported arrival pressure while the handoff stays below d...

`thm_offset_penalty_real`

Offset penalty is real [LEDGER: THM-OFFSET-PENALTY-REAL]

`thm_offset_penalty_bounded`

Offset penalty is bounded [LEDGER: THM-OFFSET-PENALTY-BOUNDED]

`thm_diversity_gain_reduces_deficit`

Diversity gain reduces deficit [LEDGER: THM-DIVERSITY-GAIN-REDUCES-DEFICIT]

`thm_launch_offset_honest`

Launch offset is honest [LEDGER: THM-LAUNCH-OFFSET-HONEST]

`thm_rg_fixed_points`

Iterated coarsening as renormalization group flow: cumulative heat is monotone along trajectories, information loss is additive (chain rule), RG fixed points have injective pushforward (singleton fibe...

`thm_community_attenuates_failure`

Community CRDT context reduces the scheduling deficit of any failure topology. Each CRDT-synced observation (win counts, latency means, failure rates, disagreement records) acts as one unit of shared ...

`thm_community_monotone_nondegradation`

Community memory cannot increase the expected cost of an already-optimal backend. If the scheduling deficit is already non-positive (good hand), community context keeps it non-positive. Good hands sta...

`thm_community_strict_domination`

When community context is positive and the failure topology has positive deficit (bad hand), the community-reduced deficit is strictly less than the bare deficit. Combined with nondegradation: Pareto ...

`thm_community_bridges_tares`

Community context reduces unbridged tares (failure modes not covered). More community knowledge means fewer uncovered failure modes. Sufficient context bridges all tares. [LEDGER: THM-COMMUNITY-BRIDGE...

`thm_community_bule_convergence`

The Bule deficit (scheduling gap in Bule units) decreases monotonically with community context. Each CRDT sync round reduces the deficit by exactly one Bule when there is remaining deficit. Sufficient...

`thm_community_dominance_theory`

Master theorem: for any failure topology with decisionStreams < failurePaths and positive community context: (1) failure deficit is real, (2) community attenuates failure strictly, (3) Bule deficit mo...

`thm_community_dominance_model`

Model-checked community fabric: safety (Bule bounded, non-negative, monotone, adaptive dominates static, nondegradation stable), liveness (eventually converges, eventually zero deficit), domination (p...

`thm_community_is_skyrms_walker`

The community CRDT plays the role of Skyrms Walker S. The scheduling deficit IS the negotiation deficit. Community attenuation IS Skyrms mediation. One CRDT sync round = one mediation round = one Bule...

`thm_bule_zero_is_nadir`

The Skyrms nadir (inter-walker distance = 0) is exactly the state where buleDeficit = 0. Biconditional: buleDeficit = 0 iff interWalkerDistance = 0. At the nadirContext, Bule = 0. Below nadirContext, ...

`thm_nadir_algebraic`

The nadir can be computed algebraically: nadirContext = walkerA_dims + walkerB_dims - 1. At that context Bule = 0. Below that context Bule > 0. The solution is closed-form. No void walking needed. Set...

`thm_skyrms_nadir_is_bule_zero`

Master theorem: three equivalent characterizations of the Skyrms nadir. (I) Community IS the mediator: deficit identification + attenuation = mediation. (II) Bule=0 IS the nadir: convergence at nadirC...

`thm_skyrms_nadir_bule_model`

Model-checked equivalence of three-walker convergence and Bule convergence: Bule nonneg, bounded, nadir constant, community tracks rounds, attenuation = convergence, algebraic nadir correct, solve-don...

`thm_diversity_optimality`

Composition theorem: diversity is the monotonically optimal, thermodynamically necessary condition for information-preserving computation in fork/race/fold systems. Five pillars: (1) Monotonicity — ad...

`thm_diversity_beats_mono`

K languages alive > 1 language alive [LEDGER: THM-DIVERSITY-BEATS-MONO]

`thm_dead_stays_dead`

Extinct language has zero option value [LEDGER: THM-DEAD-STAYS-DEAD]

`thm_alive_has_value`

Surviving language has positive option value [LEDGER: THM-ALIVE-HAS-VALUE]

`thm_sliver_costs_now`

Sliver node pays more than winner [LEDGER: THM-SLIVER-COSTS-NOW]

`thm_nash_kills`

Nash equilibrium eliminates K-1 languages [LEDGER: THM-NASH-KILLS]

`thm_exploration_amortizes`

Option value exceeds sliver cost after T topologies [LEDGER: THM-EXPLORATION-AMORTIZES]

`thm_skyrms_optimal_myopic`

Skyrms optimal for T=0 [LEDGER: THM-SKYRMS-OPTIMAL-MYOPIC]

`thm_forest_optimal_farsighted`

Forest optimal for T large [LEDGER: THM-FOREST-OPTIMAL-FARSIGHTED]

`thm_fixed_point_is_oscillation`

Neither purity nor diversity is the fixed point; the oscillation is [LEDGER: THM-FIXED-POINT-IS-OSCILLATION]

`thm_disagreement_is_breathing`

Forest-Skyrms disagreement = sliver budget = oscillation amplitude [LEDGER: THM-DISAGREEMENT-IS-BREATHING]

`thm_buleyean_oscillation`

Both forces positive = system alive [LEDGER: THM-BULEYEAN-OSCILLATION]

`thm_two_phases_one_orbit`

Purity and diversity are two phases of one period-2 orbit [LEDGER: THM-TWO-PHASES-ONE-ORBIT]

`thm_the_identity`

Optimal - Skyrms = Exploration [LEDGER: THM-THE-IDENTITY]

`thm_zero_exploration_optimal`

Exploration = 0 implies Skyrms = Optimal [LEDGER: THM-ZERO-EXPLORATION-OPTIMAL]

`thm_positive_exploration_gap`

Exploration > 0 implies Optimal < Skyrms [LEDGER: THM-POSITIVE-EXPLORATION-GAP]

`thm_exploration_is_sliver`

Exploration budget = K - 1 [LEDGER: THM-EXPLORATION-IS-SLIVER]

`thm_monoculture_zero`

K = 1 implies zero exploration [LEDGER: THM-MONOCULTURE-ZERO]

`thm_total_gap_decomposition`

Total gap = God Gap + Exploration [LEDGER: THM-TOTAL-GAP-DECOMPOSITION]

Interference, Renormalization & Community

Tools

coarsen

renormalization_flow

community_dominance

skyrms_nadir

moa_fabric

thm_sliver_fractal

thm_renormalization_coarsening

thm_renormalization_reuse

thm_recursive_coarsening_synthesis

thm_skyrms_nadir

thm_skyrms_three_walker

thm_moa_fabric_lowering

thm_moa_fabric_cannon

thm_moa_fabric_pair

thm_moa_fabric_waste

thm_moa_fabric_coupled

thm_offset_penalty_real

thm_offset_penalty_bounded

thm_diversity_gain_reduces_deficit

thm_launch_offset_honest

thm_rg_fixed_points

thm_community_attenuates_failure

thm_community_monotone_nondegradation

thm_community_strict_domination

thm_community_bridges_tares

thm_community_bule_convergence

thm_community_dominance_theory

thm_community_dominance_model

thm_community_is_skyrms_walker

thm_bule_zero_is_nadir

thm_nadir_algebraic

thm_skyrms_nadir_is_bule_zero

thm_skyrms_nadir_bule_model

thm_diversity_optimality

thm_diversity_beats_mono

thm_dead_stays_dead

thm_alive_has_value

thm_sliver_costs_now

thm_nash_kills

thm_exploration_amortizes

thm_skyrms_optimal_myopic

thm_forest_optimal_farsighted

thm_fixed_point_is_oscillation

thm_disagreement_is_breathing

thm_buleyean_oscillation

thm_two_phases_one_orbit

thm_the_identity

thm_zero_exploration_optimal

thm_positive_exploration_gap

thm_exploration_is_sliver

thm_monoculture_zero

thm_total_gap_decomposition

Discovery Endpoints

Paper Reference