This first post in our open research space describes where Subspace consensus stands in a global taxonomy, abstractly and formally. Thanks to @rg3l3dr for putting this together.
This post is written primarily for potential collaborators to develop a common understanding of our definitions and interests.
ProofofArchivalStorage (PoAS) Consensus
Simply put, proofofarchivalstorage (PoAS) consensus is an adaption of Nakamoto’s longestchain protocol in which the proofofwork (PoW) puzzle is replaced with a proofofreplication (PoRep).
In PoAS, the input data to the PoR is the history of the blockchain itself (i.e., the past blocks).
PoAS consensus is designed to solve one part of a very specific problem for proofofcapacity (PoC) blockchains, the farmer’s dilemma, as described in the Subspace Technical Whitepaper.
“farmers must decide whether to allocate scarce storage resources to maintain the chain state and history or maximize the amount of space they pledge towards consensus. Rational farmers will always choose the latter, at best becoming light clients, while at worst encouraging pooled farming under a few trusted operators.”
Terminology
We are solely interested in Nakamoto consensus (longestchain) protocols. We treat these abstractly as black box protocols, which include some sybilresistant (resourceconstrained) cryptographic puzzle that can be sampled, audited, or challenged in a manner that may be modeled as a Poisson arrival process. Example puzzles include PoW (computation), PoS (capital) and PoC (storage).
We use proofofcapacity as an abstract umbrella term, capturing proofs of storage, space, replication, reterievability, and spacetime. We view them as primitives that can be used to construct or implement a specific PoC consensus protocol. We broadly divide PoC into two categories, somewhat prejoritavely as:

proofsof(useless)space,
where the data being stored is randomly generated (Burst, Spacemint, Chia, Spacemesh);  proofsof(useful )storage,
where the data being stored on disk is some transformation of meaningful, realworld data (Filecoin, Subspace).
With respect to proofsofstorage, we broadly follow the framework and terminology provided in the PoReps paper by Ben Fisch. Specifically, a proofofreplication is a useful proofofstorage, which composes a proofofspace (PoSpace) with a proofofretreivability (PoRet) — more formally defined as a proofofretrievablecommitment (PoRC).
For any PoRep to be secure, the transformed data must be indistinguishable from random, achieved through a proofofspace. A similar idea is captured by Publically Incompressible Encodings (PIEs) since truly random data is, by definition, informationtheoretically incompressible.
Using this framework, we define generic PoAS as:
 A Nakamoto (longestchain) consensus protocol
 Employing a PoC resource puzzle for spacebound sybil resistance
 Where the space reflects some useful storage (PoRep)
 And the specific data being replicated is the archival history of the blockchain