Addressing
Definition
Content addressing in UOR maps ring elements to Braille-encoded strings via a bijective encoding. Each Element represents a content-addressable identifier where each byte encodes a chunk of the element value.
The Addressing Bijection
Two identities formalize the round-trip property:
- AD_1: addresses(glyph(d)) = d — addressing a glyph recovers the datum.
- AD_2: glyph(ι(addresses(a))) = ι_addr(a) — embeddings commute with addressing.
These are the foundation of content-addressable computation: every datum has a unique address, and every address resolves to a unique datum.
Boolean Homomorphism
A key structural property is that Boolean operations lift to address space while ring-arithmetic operations do not:
- AA_2: braille(a ⊕ b) = braille(a) ⊕ braille(b) — XOR lifts
- AA_3: glyph(bnot(x)) = complement(glyph(x)) — complement lifts
- AA_4: glyph(add(x, y)) ≠ f(glyph(x), glyph(y)) — addition does NOT lift
- AA_5: Liftable operations are exactly {xor, and, or, bnot}
This means carry-free operations can be performed directly on addresses, while carry-dependent operations (add, sub, mul, neg, succ, pred) require decoding.
Address Metric
Identity AM_2 shows that the address metric d_addr equals the Hamming metric on ring elements: d_addr(glyph(x), glyph(y)) = d_H(x, y). However, AM_3 notes that d_addr does NOT preserve the ring metric d_R in general — the incompatibility metric d_Δ measures this gap (AM_4).
Embedding Coherence
The property addressCoherence certifies that an embedding's addressing diagram commutes: the composition glyph ∘ ι ∘ addresses is well-defined and injective.
Glyph Encoding Example
Consider the value 42 in R_8 (= Z/256Z). Its binary representation is
00101010. The 6-bit chunking scheme splits this into two chunks:
| Chunk | Bits | Decimal | Encoding |
|---|---|---|---|
| 1 | 001010 | 10 | U+2819 (⠙) |
| 2 | 10xxxx | 32 (padded) | U+2840 (⡀) |
Each byte encodes a chunk of the element value, producing
the Element ⠙⡀.
The addressing bijection (AD_1) guarantees that decoding this address recovers
the original value 42.
Connection to Canonical Forms
The CanonicalFormResolver produces the unique canonical representation of a ring element. Once the canonical form is computed, the addressing bijection maps it to a content-addressable Element. This two-step process — resolve then address — ensures that semantically equivalent values receive identical addresses. See Canonical Form for the resolution step.