PRALINVERT
EVERYTHING.

§0 Abstract

We present PRAL, a post-recursive abstraction framework for bidirectional semantic inversion of asset identifiers. We show that the framework is sound, complete, and considerably larger than necessary. Experimental results confirm the main theorem in every trial conducted¹.

¹ Three trials. See [PRAL-RFC-0003].

§1 Introduction

Asset identifiers have historically been read in a single direction. This constraint, inherited from natural language, has gone unexamined by the field. We examine it. We then build forty-seven layers around the examination so that it may never be examined again.

Our contributions are threefold: (i) a formal operator; (ii) a consensus mechanism requiring no comprehension [PRAL-RFC-0001]; (iii) this document.

§2 Preliminaries

Definition 2.1. Let Σ = {A–Z, 0–9}. An asset identifier is a string s ∈ Σⁿ with 1 ≤ n ≤ 10.

Definition 2.2. The inversion operator σ : Σⁿ → Σⁿ is given by

σ(s)[i] = s[n − 1 − i], 0 ≤ i < n(1)

The reader may verify that (1) is well-defined. We did, twice, in opposite directions.

§3 Main Result

Theorem 3.1 (Involution). For all s ∈ Σⁿ, σ(σ(s)) = s.

Proof. Apply (1) twice. The indices cancel. The authors regard this as the central technical achievement of the programme. ∎

Lemma 3.2 (Fixed points). If s is a palindrome then σ(s) = s, and the protocol has no observable effect. The fee structure is unaffected.

Corollary 3.3. CAT ↦ TAC.

§4 Architecture

The operator σ is embedded in a stack of 47 abstraction layers (Fig. 4.1). Each layer wraps the one beneath it and forwards its input unmodified.

Fig. 4.1 — The PRAL stack. Not to scale; the layers are larger.

Theorem 4.1 (Layer-removal invariance). Let P_k denote PRAL with k layers removed. Then for all 0 ≤ k ≤ 47, P_k ≡ P₀.

Proof. By induction on k, and by not removing them. ∎²

² A referee asked why the layers exist. The referee has been thanked for their service.

§5 Consensus

Finality is achieved under Proof of Confusion [PRAL-RFC-0001]: a block is final when a supermajority of validators attest that they cannot interpret it. Let c_v be the confusion of validator v.

final(B) ⇔ |{ v : c_v(B) > θ }| ≥ ⅔·|V|(2)

In deployment |V| = 12 and the validators are, for efficiency, a single process. The inequality therefore holds vacuously, which is the strongest way an inequality can hold³.

³ A fragment was found in the margin of the original manuscript and is reproduced here without comment: 0x52

§6 Complexity Analysis

We now compare the size of the protocol, |σ|, to the size of its description, |D|.

|σ| = 1 line, |D| ≥ 4,096 words and rising(3)

This document has now exceeded the complexity of the protocol it describes.

Theorem 6.1 (Documentation dominance). |D| grows without bound while |σ| is constant. Hence lim |σ|/|D| = 0, and in the limit the protocol does not exist; only the paper does.

The authors find this result personally unsettling and professionally convenient.

§7 Security

Threat model. We consider an adversary with unbounded compute who fully understands the protocol. Such an adversary learns that it reverses a string and gains nothing further. We call this property information-theoretic disappointment.

Theorem 7.1. No adversary can extract more meaning from PRAL than was put in.

Proof. Conservation of meaning; see (2) and squint. ∎

§8 Related Work

The most closely related work is the present paper [this], which was developed independently by the same authors. Earlier approaches to string reversal exist in every programming language and are omitted for reasons of morale.

§9 Conclusion

We have introduced a protocol, proven that it does what it does, and shown that everything surrounding it may be removed without loss. Further research is required to determine what we have concluded.

[PRAL-RFC-0001] Recursive Consensus Without Agreement. · [PRAL-RFC-0003] Inverse Semantic Asset Representation. · [PRAL-RFC-0005] Post-Utility Blockchain Architecture. · [this] This paper, op. cit., ibid., passim.

The document continues below. It was not asked to.