focusing

focusing is tru's continuous computation — the act whose output is focus ($\phi^*$). every epoch: read signals from cybergraph, build effective adjacency, run tri-kernel, emit $\phi^*$ and derivatives.

inputs

every signal contributes a header (ν, t) and one or more cyberlink tuples (p, q, τ, a, v):

tru reads two additional quantities from bbg per epoch:

• karma(ν): accumulated BTS score history for each neuron — the long-run record of honest signaling
• price(ℓ): ICBS market price per link — the market's collective epistemic assessment of link validity

stake is economic commitment. karma is epistemic track record. price is collective verdict. φ* is always computed from these three weighted inputs, never stored independently.

effective adjacency

raw stake is not the edge weight. tru constructs effective adjacency from the full truth-weighted signal:

A_eff(p, q) = Σ_{ℓ: from=p, to=q}  stake(ℓ) × karma(ν(ℓ)) × f(price(ℓ))

where stake(ℓ) = a(ℓ) × token_weight(τ(ℓ)) normalizes across denominations using the weights declared in config.tokens.

f(price) maps ICBS price to an edge multiplier in [0, 1]. a link the market believes (price → λ) carries full weight. a link the market doubts (price → 0) carries diminished weight. this is market inhibition — collective epistemic assessment prunes false connections structurally.

valence v ∈ {-1, 0, +1} does not directly enter A_eff. its effect is mediated through price(ℓ): valence is the BTS meta-prediction, and BTS scoring accumulates into karma and drives ICBS market convergence.

tri-kernel

the composite operator:

R = λ_d · D + λ_s · S + λ_h · H_τ      (λ_d + λ_s + λ_h = 1)

diffusion

column-stochastic transition matrix P = A_eff · diag(1 / col_sum(A_eff)):

φ^(t+1) = α · P^T φ^(t) + (1 - α) · u

α ∈ (0, 1): teleport parameter. u: prior (uniform or stake-weighted). teleport ensures ergodicity — probability mass occasionally restarts from the prior, preventing trapping in dense subgraphs.

locality: geometric decay with rate α. a local edit's effect reaches ε precision within O(log(1/ε)) hops.

answers: where does probability flow?

springs

screened laplacian solve. let L = diag(col_sum(A_eff)) − A_eff:

(L + μI) x* = μ x_0

μ > 0: screening parameter. x_0: reference state (often uniform). the screened green's function (L + μI)^−1 decays exponentially with graph distance — locality with exponential tail. larger μ pulls harder toward x_0; smaller μ lets structure dominate.

locality: exponential decay with rate O(exp(−μ^{1/2} · d)).

answers: what satisfies structural constraints?

heat

heat kernel approximated by chebyshev polynomial truncation at degree K. let L̃ = 2L / λ_max − I:

H_τ ≈ Σ_{k=0}^{K} c_k(τ) T_k(L̃)

c_k(τ) are the chebyshev coefficients of exp(−τ·). τ ≥ 0: temperature. high τ smooths broadly across the graph (annealing). low τ focuses locally (crystallization). the ability to adjust τ lets tru operate simultaneously across multiple scales.

locality: gaussian tail decay, O(log(1/ε)) hops.

answers: what does the graph look like at scale τ?

fixed point

the collective focus theorem guarantees: under ergodicity of P, μ > 0, and bounded τ, the composite operator R is a contraction:

‖Rφ − Rψ‖ ≤ κ ‖φ − ψ‖,   κ = λ_d α + λ_s ‖L‖/(‖L‖+μ) + λ_h e^{−τλ_2} < 1

by the banach fixed-point theorem, φ^(t) → φ* at linear rate. the fixed point is unique and satisfies:

φ* = norm[R(φ*)]        Σ_i φ*(i) = 1        φ*(i) > 0 ∀ i

computation is one coupled iteration: each step applies D, S, and H_τ to the same current φ, blends with weights λ, and normalizes — repeat to the fixed point. tru does not solve the three operators independently to their own fixed points and average the results; that is a different, weaker object that minimizes no single free energy and has no single κ (see tri-kernel §2.4). the contraction κ < 1 governs this coupled iteration, and the ct0 architecture parameters read κ from it.

the iteration is fixed-point arithmetic over the Goldilocks field, never float (see arithmetic). φ, the operators, the blend, and the normalization are all field elements; H_τ is a Chebyshev polynomial in L so it stays field-native. the loop does not run to a float threshold — it runs a fixed step count T(ε) = ⌈log(1/ε)/log(1/κ)⌉ derived from κ, so every neuron and validator computes the byte-identical φ* and the trace zheng proves has a constant length.

φ* is the boltzmann equilibrium minimizing the free energy functional:

F(φ) = λ_s E_spring(φ) + λ_h E_heat(φ) + λ_d D_KL(φ ‖ D(φ))

every cyberlink shifts φ*. learning and knowledge state are the same operation.

eigensolver

tru runs LOBPCG (locally optimal block preconditioned conjugate gradient) on the screened laplacian (L + μI) to extract the k leading eigenvectors V_k.

each particle receives a position in k-dimensional spectral space: row i of V_k is particle i's coordinate. particles that are structurally similar (densely interconnected) cluster in spectral space. these positions are emitted to mir every epoch as the geometric substrate of the R-1.0 world.

outputs

karma is not written by tru's focusing pass. karma is accumulated by plumb from BTS scores and read from bbg as input. tru reads karma; plumb writes it.

see collective focus theorem for the convergence proofs. see ct0.md for how φ* feeds into model compilation. see tri-kernel for why these three operators are the minimal sufficient basis.