5a. FNV-1a 32-bit hash (input → basis index / h(item))

Read the equation

This relation defines hash ← offset_basis for byte in input: hash ← (hash XOR byte) hash ← (hash × prime in the source section “5a. FNV-1a 32-bit hash (input → basis index / h(item))”. Combines binary values with exclusive OR in the stated bit-level expression. Uses a remainder or modular operation, retaining the numeric representation used in the source. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive meaning to them.

Terms & definitions

The recovered passage does not supply a separate symbol table. Read the source context for the notation used in this version; familiar letters may carry different meanings in other builds.

How it works

1. Read the displayed expression together with the surrounding source statement; it is retained as an expression rather than silently completed into an equality.

2. Combines binary values with exclusive OR in the stated bit-level expression.

3. Uses a remainder or modular operation, retaining the numeric representation used in the source.

4. Use this version’s variable definitions and update order. The same symbol may be assigned a different meaning in another build.

Assumptions & limits

• This source expression was recovered from the pinned code revision. Its presence does not establish that the associated program was built, run, or scientifically validated.

Original context

Constants: offset basis = 2166136261 (0x811c9dc5); prime = 16777619 (0x01000193).
Algorithm (1a order — XOR then multiply):
```
hash ← offset_basis
for byte in input:
    hash ← (hash XOR byte)
    hash ← (hash × prime) mod 2³²
```

**Citation.** Fowler–Noll–Vo hash, official spec (http://www.isthe.com/chongo/tech/comp/fnv/); Wikipedia, *Fowler–Noll–Vo hash function*.

### 5b. Hebbian learning (weight adaptation)

**What it is.** Unsupervised correlational plasticity — co-active units strengthen their connection ("fire together, wire together"); requires no error signal.
ΔW_ij = η · x_i · x_j (η learning rate; x_i, x_j post/pre-synaptic activations)

**Citation.** Hebb, *The Organization of Behavior* (1949); Wikipedia, *Hebbian theory*.

### 5c. Swarm consensus / flocking (multi-agent alignment)

**What it is.** Agents reach collective alignment by averaging neighbors' states within radius R — emergent order, no central controller.
Vicsek heading update: θ_i(t+1) = arg( ⟨ e^{iθ_j(t)} : j ∈ N_i ⟩ ) + η (η noise in [−π,π])
Cucker–Smale (continuous, distance-weighted): dv_i/dt ∝ Σ_j (v_j − v_i)/(1 + |x_j − x_i|)

**Citations.** Vicsek et al. (Vicsek model); Cucker & Smale flocking model; Chazelle, *The Convergence of Bird Flocking*, arXiv:0905.4241 (2009).

### 5d. Activation functions

hash ← offset_basis
for byte in input:
    hash ← (hash XOR byte)
    hash ← (hash × prime) mod 2³²

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