BBG: polynomial authenticated state

abstract

BBG (Big Badass Graph) is the authenticated state layer for cyber. the entire cybergraph — particles, axons, neurons, tokens, temporal state, private records — commits to a single polynomial:

$$\text{BBG\_root} = \text{Lens.commit}(\text{BBG\_poly})$$

32 bytes. one polynomial. all state. every query is a polynomial opening. cross-index consistency is structural — different evaluation dimensions of the same polynomial cannot disagree. LogUp is unnecessary. a 240-byte checkpoint (BBG_root + universal accumulator + height) proves all history from genesis in 10–50 μs.

the architecture follows from zheng's sumcheck foundation: the proof system operates on multilinear polynomials natively. making the STATE a polynomial means state reads ARE the proof system's native operation. this is not optimisation — it is alignment between state and proof architecture.

1. three laws

law 1: bounded locality. operation cost $\propto$ what it touches, not total state size. at $10^{15}$ particles, a single cyberlink costs $O(\log n)$ polynomial path updates. global recomputation is physically impossible and architecturally forbidden.

law 2: constant-cost verification. verifying any claim about the graph costs $O(1)$: one Lens opening (~200 bytes, 10–50 μs). independent of graph size, history length, or computation complexity. a light client with 240 bytes has the same verification power as a full node.

law 3: structural security. guarantees come from mathematical structure, not protocol correctness. polynomial binding prevents lying — a committed polynomial evaluates to a unique value at each point. the Brakedown lens is post-quantum (code-based, no pairings). privacy comes from the mutator set (SWBF bitmap prevents double-spend by construction).

2. five primitives

derived: axon = $H(\text{from}, \text{to})$. aggregate of all cyberlinks between two particles. the axon is public; individual cyberlinks are private.

3. the polynomial state

BBG_poly

all state encodes as evaluations of a single multivariate polynomial:

$$\text{BBG\_poly}(\text{index}, \text{key}, t) = \text{value}$$

three dimensions:

• index $\in \{0..9\}$ — which data domain (particles, axons_out, axons_in, neurons, locations, coins, cards, files, time, signals)
• key $\in \mathbb{F}_p$ — namespace key (particle CID, neuron ID, denomination hash, etc.)
• $t$ $\in \mathbb{N}$ — block height (temporal dimension)

committed via Brakedown lens:

$$\text{BBG\_root} = \text{Brakedown.commit}(\text{BBG\_poly}) \quad \text{(32 bytes)}$$

what each dimension encodes

why one polynomial

nine independent data structures (the old NMT approach) force redundant computation:

old (NMTs):    cyberlink touches 4-5 trees → 4.5 × O(log n) hemera hashes
               cross-index: LogUp proves trees agree (~1,500 constraints)
               total: ~107,500 constraints per cyberlink

new (one poly): cyberlink updates polynomial at 4-5 evaluation points
               cross-index: STRUCTURAL (same polynomial, different dimensions)
               total: ~3,200 constraints per cyberlink

the polynomial makes cross-index consistency FREE. axons_out and axons_in are different evaluation dimensions of BBG_poly. they CANNOT disagree because they are the same committed object. LogUp — which cost ~6M constraints per block — is eliminated entirely.

state reads

a state read IS a polynomial evaluation:

"what is the energy of particle P?"
= Brakedown.open(BBG_root, (particles, P, t_now))
= one Lens opening: ~200 bytes proof, O(√N) field operations, 10-50 μs

"all outgoing axons from particle P?"
= Brakedown.open(BBG_root, (axons_out, P, t_now))
= one Lens opening: ~200 bytes, completeness guaranteed by Lens binding

compare with the hash-tree approach: $O(\log n) \times 32$ bytes Merkle path, $O(\log n)$ Hemera hashes to verify. the polynomial approach is O(1) proof size and O(√N) field operations — no hashing.

state updates

a cyberlink updates the polynomial at multiple evaluation points:

cyberlink (ν, p, q, τ, a, v, t):
  BBG_poly(particles, p, t)    ← energy update for source
  BBG_poly(particles, q, t)    ← energy update for target
  BBG_poly(axons_out, p, t)    ← outgoing axon update
  BBG_poly(axons_in, q, t)     ← incoming axon update
  BBG_poly(neurons, ν, t)      ← focus deduction

each update: O(log n) polynomial path operations × ~100 field ops
total: ~3,200 constraints per cyberlink

with Brakedown (Merkle-free lens), the update cost is O(N) for batch recommit at block boundary. no hemera hashing for state verification — 0 calls per block (was 144,000 in the NMT approach).

4. private state

individual cyberlinks are private. the polynomial state handles this:

commitment polynomial $A(x)$: all committed private records. $A(c_i) = v_i$ for commitment $c_i$ with value $v_i$. membership proof: one Lens opening — O(1).

nullifier polynomial $N(x) = \prod(x - n_i)$: all spent nullifiers. $N(n) = 0$ iff nullifier $n$ is spent. non-membership proof: one Lens opening showing $N(c) \neq 0$ — O(1).

old (SWBF + MMR):
  membership:      O(log N) hemera hashes (AOCL MMR)
  non-membership:  128 KB witness (SWBF bitmap) + O(log N) MMR walk
  update:          bitmap flip + periodic archive
  total:           ~40,000 constraints per spend

new (polynomial):
  membership:      one Lens opening — O(1)
  non-membership:  one Lens opening — O(1)
  update:          N'(x) = N(x) × (x - n_new) — O(1) polynomial extend
  witness:         32 bytes (Lens commitment, was 128 KB)
  total:           ~5,000 constraints per spend

privacy is preserved: Lens opening proofs are zero-knowledge. opening $A(c_i)$ reveals nothing about other commitments. opening $N(n)$ reveals nothing about other nullifiers.

5. temporal state

the temporal dimension $t$ in BBG_poly enables continuous-time queries:

"what was π of particle P at block 1000?"
= Brakedown.open(BBG_root, (particles, P, 1000))
= one Lens opening — no separate time index needed

the old approach used a time.root NMT with 7 namespaces (steps, seconds, hours, days, weeks, moons, years). the polynomial absorbs time as a native dimension — any historical query is one evaluation.

with gravity commitment: recent + high-$\pi$ queries are cheapest (low-degree polynomial terms). old + low-$\pi$ queries cost more (high-degree terms). verification cost follows the exponential — important facts are cheaper to verify.

6. algebraic DAS

Data Availability Sampling uses the same polynomial infrastructure. the erasure-coded block is a bivariate polynomial $P(\text{row}, \text{col})$. each DAS sample is one Lens opening:

sample: Brakedown.open(block_commitment, (row_i, col_i)) → value + proof

old (NMT-based DAS):
  per sample:  O(log n) × 32 bytes NMT path, O(log n) hemera hashes
  20 samples:  ~25 KiB bandwidth, ~471K constraints

algebraic DAS:
  per sample:  ~200 bytes Lens opening, O(√N) field ops
  20 samples:  ~4 KiB bandwidth, ~3K constraints

improvement: 157× fewer constraints, 6× less bandwidth

the same lens serves state queries AND availability sampling. one commitment scheme for everything.

7. signal-first architecture

BBG_poly is DERIVED DATA. the source of truth is the signal log:

$$\text{BBG\_poly}(t) = \text{fold}(\text{genesis\_poly}, \sigma[0..t])$$

each signal updates the polynomial at specific evaluation points. the fold is deterministic. any node can reconstruct BBG_poly at any height by replaying signals.

consequences:

• crash recovery: download checkpoint (240 bytes) + replay signals since checkpoint
• storage proofs: prove signal availability (DAS), derive everything else
• the irreducible minimum per node: signal log + latest checkpoint
• BBG_poly is a materialised view, not primary data

see signal-first for the full design.

8. sync

one mechanism at three scales. five verification layers (structural-sync):

a light client joins:

1. download checkpoint                    ~240 bytes
2. verify (one zheng decider)             10-50 μs
3. sync namespaces (Lens openings)         ~200 bytes each
4. DAS sample (algebraic)                 ~4 KiB
5. maintain (fold each block)             ~30 field ops / block

total: < 10 KiB, 10-50 μs, ZERO trust

this is Verified Eventual Consistency (VEC): convergence guaranteed (CRDT), completeness verifiable (lens), availability verifiable (DAS). no consensus protocol needed.

9. π-weighted everything

$\pi^*$ (cyberank from tri-kernel) is the master distribution. the entire stack follows it:

one distribution governs proof cost, storage, availability, decay, and query performance. the universal law predicts this: given finite resources, exponential allocation minimises total cost.

10. the numbers

cost of one cyberlink in the permanent, verified, globally-available knowledge graph:

proof:           ~30 field ops per nox step (proof-carrying)
identity:        ~164 constraints (folded hemera sponge)
public state:    ~3,200 constraints (polynomial update)
private state:   ~5,000 constraints (polynomial mutator set)
total overhead:  ~8,400 constraints

11. state transitions

six transaction types modify BBG_poly:

every transaction produces a zheng proof via proof-carrying. every proof folds into the block accumulator via HyperNova (~30 field ops per fold).

12. privacy model

PRIVATE (polynomial commitments):          PUBLIC (BBG_poly dimensions):
  who linked what (individual cyberlinks)    axon weights (aggregate conviction)
  individual conviction amounts               particle energy, π*
  neuron linking history                      neuron summaries (focus, karma, stake)
  market positions                            token supplies
  UTXO values and owners                     axon market state

anonymous cyberlinks: a neuron proves identity ($H(\text{secret}) \in$ neuron set), stake sufficiency, nullifier freshness — without revealing which neuron. ~13,000 constraint zheng proof. the graph sees edges and weights. not authors.

13. honest assessment

the dependency chain: nebu → Hemera → nox → zheng → BBG. nothing runs until the stack beneath it runs.

see structural-sync for the sync theory, zheng for the proof system, nox for the VM, Hemera for the hash, tri-kernel architecture for focus, knowledge capacity for limits, link production for the intelligence problem, algebraic state commitments for why polynomial state is natural