QH256 and the K501 Information Space - Evolutionary Reference Definition v2.0

QH256 and the K501 Information Space Introduction, Development Context, Research Motivation, and Publication Guide Version: 1.0 Date: 15 August 2026 Author and developer: Patrick R. Miller (alias: Iinkognit0) Project: K501 Information Space Publication context: Companion introduction for Zenodo and related scholarly dissemination 1. Introduction The present document introduces QH256 and its relationship to the K501 Information Space. It is intended to accompany the formal QH256 specification and its reference bibliography by providing the broader context in which the formal construction was developed. The central objective is to make the distinction between established scientific foundations and K501-specific development explicit. QH256 is not proposed as a replacement for existing many-valued logics, bilattices, Belnap-Dunn semantics, or established theories of incomplete and inconsistent information. Rather, the current specification places QH256 within that scientific landscape and defines a K501-specific finite information-state structure built around a 128-cell, 256-bit representation. The resulting system is intended to provide a deterministic representation of four informational conditions—UNKNOWN, FALSE, TRUE, and GUARD—while preserving a strict distinction between accumulated state and the historical evidence from which that state is derived. The broader motivation is the construction of an information space in which information can accumulate without destructive rewriting, contradictions can remain explicitly represented, and system state can remain reconstructible from canonical historical data. This publication therefore presents two related but distinct layers: QH256 as a formal information-state algebra. K501 as the larger architectural and reconstructive information-space framework in which such state structures may be used. 2. The K501 Information Space The K501 Information Space is the larger project context from which QH256 emerged. Its architectural direction is based on several principles that are reflected in the formal specification: append-only information accumulation, preservation of historical states, explicit distinction between canonical data and derived state, separation of evidence from interpretation, deterministic state transformation, reconstructibility, and preservation of uncertainty where sufficient evidence is not available. Within this architecture, a current state is not treated as a replacement for its history. A derived aggregate may describe the information currently accumulated, but it does not automatically contain the complete sequence of events, sources, timestamps, or evidence that produced it. The formal specification therefore states an explicit boundary: [ QH256 \neq \text{Canonical History}. ] Instead, QH256 is a deterministic information-state representation or aggregate, while the complete historical evidence remains in the append-only information space. This distinction is fundamental to the K501 design. 3. Why QH256 Emerged The original motivation for QH256 is the need for a compact formal state structure that can represent more than a simple binary distinction while remaining directly implementable on ordinary binary computing hardware. The resulting construction uses 128 cells, with two evidence bits per cell: [ QH256 = Q^{128}, \qquad Q={00,01,10,11}. ] The four states are canonically mapped as follows: Encoding K501 state Evidence interpretation 00 UNKNOWN neither positive nor negative evidence 01 FALSE negative evidence 10 TRUE positive evidence 11 GUARD positive and negative evidence The complete space therefore contains: [ 4^{128}=2^{256} ] distinct states. At the representation level, QH256 is binary: [ QH256 \cong \mathbb{F}_2^{256}. ] The many-valued character is therefore semantic rather than dependent on non-binary hardware. This distinction is important. QH256 is not presented as a claim that binary computing is inadequate, nor as a claim that ternary or quaternary hardware is required. It is a structured semantic interpretation of a binary state space. 4. The Scientific Context The formal QH256 specification explicitly situates its four-valued structure in the established family of many-valued and paraconsistent information logics. The closest reference framework is the Belnap-Dunn tradition, including the four-valued structure commonly associated with Neither, False, True, and Both, together with its distinction between a truth-oriented ordering and an information or knowledge ordering. The QH256 correspondence is: Belnap/FDE-oriented interpretation QH256 Neither UNKNOWN False FALSE True TRUE Both GUARD This connection is intentional and explicit. The QH256 specification therefore distinguishes between what belongs to established mathematics and what is introduced by K501. The four-valued logical basis, information ordering, truth ordering, paraconsistent treatment of conflicting information, and related algebraic structures belong to the established research context. The specific name QH256, the 128-cell construction, the canonical K501 encoding, the operational meaning of GUARD, and the append-only state dynamics are K501-specific definitions. This separation is part of the scientific methodology of the project. 5. GUARD as the K501 Extension The most characteristic K501 contribution in the present formulation is the treatment of the 11 state. At the logical level: [ 11=(1,1) ] means that both positive and negative evidence are present. In the established four-valued literature this corresponds to the Both state. K501 names this state GUARD and assigns it an additional operational meaning within append-only state accumulation. A conflict is not erased by replacing the state with one side of the conflict. For ordinary evidence accumulation: [ TRUE \sqcup_K FALSE = GUARD. ] Once both evidence components are present, the state is not reduced to TRUE or FALSE through destructive overwrite. This gives the K501 state accumulator a monotone information dynamic: [ q_{new}=q_{old}\sqcup_K e ] with: [ q_{old}\leq_K q_{new}. ] The objective is not to declare every GUARD state true. On the contrary, the formal specification explicitly distinguishes positive evidence from absence of contradiction. GUARD represents a conflict that remains visible. This is one of the points at which the existing logical foundation is connected to K501's append-only archival philosophy. 6. Development of the Concept The QH256 work should be understood as an evolving research construction rather than as an isolated mathematical object appearing independently of the K501 project. The present specification describes a progression from an initial 256-bit state concept toward a formally defined finite information-state algebra. That development includes several successive clarifications: the state space was fixed as 128 two-bit cells; the cardinality was formally derived as (2^{256}); the binary representation was related to (\mathbb{F}_2^{256}); the two bits were reinterpreted as positive and negative evidence components; the GUARD state was connected to the established Both state of four-valued information logic; information fusion was defined as component-wise logical OR; knowledge and truth orderings were separated; append-only monotonicity was formalized; canonical history was explicitly separated from derived aggregate state; serialization, implementation assumptions, and machine-testable invariants were defined. The current scientific formulation is consequently more precise than a purely conceptual 256-bit state proposal. It provides a formal object that can be analyzed, implemented, and tested independently of broader philosophical claims. The specification itself records this status carefully: QH256 is formally specified and mathematically defined, is directly implementable in a small C-level representation, and still requires exhaustive validation of the reference implementation and its algebraic properties. 7. The Role of the Author and Developer The QH256 work presented here is authored and developed by Patrick R. Miller, also known under the alias Iinkognit0, within the broader K501 Information Space project. The work is developed as a continuous research and engineering effort in which formal definitions, implementation questions, archival principles, and reproducibility requirements are considered together. The authorial role is therefore not limited to proposing a mathematical notation. It includes the development of the surrounding information-space architecture, the definition of K501-specific rules, the separation of canonical history from derived state, and the preparation of formal definitions and reference implementations for verification. The present publication does not claim institutional affiliation or external scientific validation beyond the sources and status distinctions explicitly documented in the accompanying specification and bibliography. 8. Research Goal The immediate goal of QH256 is not to replace classical binary computation, nor to claim a universal superiority over existing four-valued logical systems. The more precise research goal is to investigate whether a compact, deterministic, evidence-oriented information state can provide useful properties when connected to an append-only, reconstructible information architecture. The central questions are therefore structural: Can incomplete, positive, negative, and conflicting evidence be represented compactly? Can such information be accumulated monotonically without destructive rewriting? Can the resulting state be reconstructed from canonical evidence? Can logical operations and information-fusion operations remain formally distinct? Can the representation be implemented and tested with a small deterministic software core? What additional properties emerge when a Belnap-style information structure is embedded in an append-only archival architecture? The current specification deliberately does not answer the final questions in advance. Claims of mathematical novelty, algorithmic superiority, hardware superiority, or practical superiority over established alternatives remain open research questions. The purpose of the formal specification is to make those questions testable. 9. QH256 and Reconstructible Information A central architectural principle of K501 is that a compact current state should not be confused with the history that produced it. Suppose two historical frames contain: Frame A → TRUE Frame B → FALSE The corresponding QH256 cell becomes: TRUE ⊔K FALSE = GUARD The resulting GUARD state records that both evidence directions are present. It does not, by itself, reconstruct the identities, timestamps, sources, or ordering of Frame A and Frame B. Those remain properties of the canonical historical archive. This yields a deliberate architectural separation: [ \text{Canonical Archive} = \text{evidence and history} ] [ QH256 = \text{deterministic aggregate state}. ] This separation permits the state representation to remain small while preserving the possibility of complete reconstruction from the underlying information space. It also prevents a common category error in information systems: treating a derived state, cache, embedding, summary, or aggregate as if it were the primary source of truth. 10. Determinism and Reproducibility QH256 is designed so that its core transformations are finite and deterministic. The principal merge operation is: [ \operatorname{MERGE}_K ((t_a,f_a),(t_b,f_b)) (t_a\lor t_b,\;f_a\lor f_b). ] Consequently, the merge is associative, commutative, and idempotent. These properties are important not only mathematically but operationally. They allow repeated ingestion of the same evidence to remain idempotent. They permit aggregation in different technical groupings without changing the final aggregate state. They also provide direct targets for automated testing. The reference representation uses two 128-bit evidence planes, requiring 256 bits or 32 bytes of state. The formal specification further defines machine-testable invariants including state dimension, cell encoding, merge monotonicity, merge idempotence, merge commutativity, merge associativity, and involutive negation. The current status of these properties is therefore suitable for a transition from formal specification to implementation and exhaustive validation. 11. What This Publication Does and Does Not Claim This introduction and the accompanying specification distinguish three broad categories. Established scientific foundations include many-valued logic, Belnap-style four-valued information, Dunn/FDE-related semantics, bilattice structures, truth and knowledge orderings, and paraconsistent treatments of incomplete and inconsistent information. K501-specific definitions include QH256 itself, the 128-cell organization, the canonical 00/01/10/11 encoding, the name GUARD, the append-only merge semantics, the separation of QH256 state from canonical history, and the associated reconstruction model. Open or unvalidated questions include whether QH256 is mathematically novel, whether its particular organization offers measurable algorithmic advantages over established alternatives, whether it has practical advantages in real information systems, and whether the K501 GUARD dynamics provide general benefits beyond existing four-valued approaches. Maintaining these distinctions is essential to the publication's credibility. 12. Relationship to the Companion Documents This introduction should be read together with two companion artifacts. QH256 — K501 Formal Scientific Specification v2.0 The formal specification provides the mathematical definition, semantic interpretation, order structures, merge operation, logical operators, invariants, serialization model, implementation-level reference structures, and explicit epistemic status of the current QH256 construction. Its current publication status is: EDITORIAL VERSION / RELEASE CANDIDATE with scientific status: SPECIFIED / MATHEMATICALLY DEFINED implementation status: C-IMPLEMENTABLE and validation status: CORE ALGEBRA TO BE EXHAUSTIVELY TESTED. fileciteturn0file0L5-L15 QH256 — K501 References v2.0 The accompanying bibliography provides the scholarly reference layer supporting the established mathematical and logical context, including Belnap, Dunn, Fitting, contemporary work on Belnap-Dunn extensions, many-valued logic, bilattices, and implementation-related standards. The bibliography is maintained as a separate artifact so that citation data can evolve without modifying the scientific definition itself. The separation of specification and bibliography also supports the K501 principle that canonical definitions and derived or supporting metadata should remain distinguishable. 13. Toward Implementation and Validation The present QH256 definition is intended to support a disciplined progression: [ \text{SPECIFIED} \rightarrow \text{IMPLEMENTED} \rightarrow \text{TESTED} \rightarrow \text{VALIDATED}. ] The immediate technical milestone identified by the specification is a QH256 reference implementation together with an exhaustive cell-algebra test matrix and subsequent property tests over the 128-cell structure. This progression is important because formal clarity and implementation correctness are separate achievements. A mathematical definition can be correct while an implementation is incorrect. An implementation can work on examples while still failing an invariant. A validated implementation can still leave broader scientific questions open. K501 therefore treats these as distinct evidence layers rather than collapsing them into a single status. 14. Long-Term Perspective QH256 is currently one formal component within a broader research direction. The broader K501 objective is the development of an information space in which information can remain persistent, traceable, and reconstructible while derived state can be generated deterministically from canonical data. Within that architecture, QH256 provides a compact algebraic mechanism for representing a particular class of informational states. The long-term research problem is consequently larger than the 256-bit representation itself: How can formal information states, historical evidence, deterministic aggregation, provenance, and reconstructible archival structures be composed into a coherent information space without losing the distinction between what was recorded, what was derived, and what remains unknown? QH256 provides one concrete formal object through which that question can be investigated. 15. Conclusion The QH256 project stands at the boundary between established logical theory and a K501-specific information architecture. Its mathematical foundations are connected to well-established work on many-valued logic, Belnap-Dunn four-valued semantics, bilattices, and reasoning with incomplete or conflicting information. Its specific construction—128 two-bit cells, the UNKNOWN / FALSE / TRUE / GUARD encoding, monotone evidence accumulation, and explicit separation from canonical history—is defined as part of the K501 Information Space. The aim is not to replace existing theory with a new label. The aim is to construct a precise, testable, and reproducible formal layer that can be evaluated within a larger append-only information architecture. At the current stage, the most important achievement is therefore not a claim of finality, but a clear research boundary: Established mathematics provides the foundation. K501 defines the architecture and the specific extension. Implementation provides the next evidence layer. Validation determines what survives scientific testing. This introduction, the formal scientific specification, and the accompanying bibliography together form the current publication package for the QH256–K501 work. Author and Project Patrick R. Miller Alias: Iinkognit0 K501 Information Space GitHub: https://github.com/k501-Information-Space DEV Community: https://dev.to/k501is Mastodon: https://mastodon.social/@K501 Publication date: 15 August 2026 Introduction version: 1.0 QH256 — K501 Formal Scientific Specification Evolutionary Reference Definition v2.0 — Editorial Version — Date: 15 August 2026 Document status: EDITORIAL VERSION / RELEASE CANDIDATE Scientific specification version: 2.0 Editorial revision: Editorial Revision 1 Editorial time anchor: Unix 1786819032 — 15 August 2026, 18:37:12 UTC / 20:37:12 CEST Author and developer: Patrick R. Miller (alias: Iinkognit0) Project: K501 Information Space Project links: GitHub: https://github.com/k501-Information-Space DEV Community: https://dev.to/k501is Mastodon: https://mastodon.social/@K501 Publication statement This document is the Editorial Version / Release Candidate of the English-language QH256 — K501 Formal Scientific Specification v2.0. It is derived from the supplied K501 reference specification and prepared for scholarly dissemination on Zenodo and related research platforms. The purpose of this editorial revision is to improve scientific precision, bibliographic stability, terminology, status labeling, and reproducibility. The revision does not intentionally change the QH256 state space, canonical cell encoding, formal operations, order relations, K501 append-only merge rule, or mathematical equations defined in the reference specification. Where wording has been tightened, the change is editorial or evidential rather than a change of formal definition. The document explicitly separates four epistemic/status classes: ESTABLISHED: results, structures, or historical facts supported by the cited literature or standards. K501-SPECIFIED: definitions and architectural rules introduced by the K501 project. IMPLEMENTATION-DEFINED: concrete representation or engineering choices associated with the reference implementation. OPEN / UNVALIDATED: claims that require exhaustive testing, comparative experiments, benchmarking, or further mathematical study. This separation is intentional and is part of the editorial preparation for publication. 1. Scientific classification Current research distinguishes among several many-valued systems. Many-valued logics are not characterized merely by the number of values they contain, but by their value space, operations, designated values, and consequence relation. This distinction is essential for QH256. [Marcos, Přenosil & Egré, 2026; see References.] Historically relevant systems include: [ \text{Łukasiewicz 3VL} ] with TRUE, INDETERMINATE, and FALSE, as well as [ \text{Kleene/SQL-style 3VL} ] for incomplete information. For QH256, however, the four-valued family associated with Belnap and Dunn is the substantially closer reference. There the four values are commonly interpreted as Neither, False, True, and Both. The structure carries both a truth order and an information/knowledge order. This family is explicitly used for incomplete and inconsistent information and for information systems and databases. Recent work continues to extend Belnap-Dunn/FDE and related four-valued systems, including first-order information logics and truth-functional/modal expansions. Accordingly, QH256 is not scientifically isolated. It is positioned within an established research family. 2. QH256 — formal foundational definition We first define a cell-state space [ Q={00,01,10,11}. ] The canonical correspondence is: [ 00 = U = UNKNOWN ] [ 01 = F = FALSE ] [ 10 = T = TRUE ] [ 11 = G = GUARD. ] A QH256 value consists of 128 such cells: [ \boxed{ QH256 = Q^{128} } ] Thus: [ |Q|=4 ] and therefore: [ |QH256| = 4^{128} = (2^2)^{128} = 2^{256}. ] Proposition QH-01 (Cardinality): The QH256 state space has exactly (2^{256}) distinct states. Consequently, any unambiguous binary representation of the complete state space requires 256 bits. This is a direct consequence of the finite cardinality of the defined state space and is not an empirical postulate. 3. Representation as (\mathbb F_2^{256}) Because every cell consists of two bits: [ Q \cong \mathbb F_2^2. ] Therefore: [ QH256 \cong (\mathbb F_2^2)^{128} \cong \mathbb F_2^{256}. ] This is an important clarification. QH256 is completely binary-realizable at the representation level. It requires neither ternary nor quaternary hardware. The many-valued character initially lies in the interpretation of the bit pairs, not in a claim about physically fundamental states. This is methodologically consistent with research on many-valued logic: a many-valued logical semantics can be implemented on a binary technical substrate. The historical existence of ternary computers such as Setun demonstrates only that other physical representations are possible; it does not establish their general technical superiority. 4. The decisive semantic reformulation The two bits of a cell are no longer regarded merely as an arbitrary 2-bit encoding. We interpret them as two evidence components: [ q=(t,f) ] with [ t\in{0,1} ] for positive evidence and [ f\in{0,1} ] for negative evidence. Thus: Bit pair ((t,f)) State 00 0, 0 — UNKNOWN 01 0, 1 — FALSE 10 1, 0 — TRUE 11 1, 1 — GUARD This is the central formal interpretation. Structurally, it is essentially identical to the representation used in Belnap-Dunn research for the four epistemic states as subsets of ({0,1}): no evidence, positive evidence, negative evidence, or evidence in both directions. 5. Meaning of GUARD This is where the K501 evolution occurs. We define: [ G=(1,1). ] This means: Positive and negative evidence are both present. This corresponds to the established Belnap state Both. K501 deliberately uses the name GUARD for this state. It is therefore neither a fifth state nor a mystical additional truth value. It is: [ \boxed{ GUARD = BOTH\text{-evidence} } ] plus a K501 operational rule: A conflict state in the append-only K501 state accumulator is not reset to TRUE or FALSE by overwriting. This is the K501 extension. Belnap provides the mathematical foundation. K501 defines a more restrictive state dynamic on top of it. 6. Two levels must be distinguished QH256 therefore has two explicitly separated levels. Logical level [ G \equiv BOTH ] that is, [ t=1,\quad f=1. ] K501 operational level [ G = GUARD ] means additionally: The accumulated state contains conflicting evidence and may not be reduced to a simpler state by ordinary state replacement. Thus: [ G \rightarrow G ] under normal append-only fusion. A historical frame sequence can still be analyzed later. However, an old state is not mutated. This is important because information-theoretic/logical representation and K501 archival semantics perform different tasks. 7. QH256 is not a replacement for AIONARC history This is a mandatory specification boundary. A QH256 state stores: the current aggregated information state. It does not automatically store: which frames, sources, or events generated that state. Example: Frame A → TRUE Frame B → FALSE produces: QH256 cell = GUARD However, the cell value GUARD cannot reconstruct by itself that: A was TRUE B was FALSE Therefore: [ \boxed{ QH256 \neq Canonical\ History } ] but rather: [ \boxed{ QH256 = derived/aggregated\ state } ] or, if explicitly defined as a canonical snapshot: [ QH256 = canonical\ state\ snapshot ] while the complete evidence remains in the append-only archive. This is fully consistent with K501 CANONICAL_DATA_FIRST. 8. The information space We now define an information order: [ \leq_K ] by component-wise evidence inclusion. For [ a=(t_a,f_a) ] and [ b=(t_b,f_b) ] define: [ a\leq_K b ] if and only if: [ t_a\leq t_b ] and [ f_a\leq f_b. ] This yields: GUARD (11) / \ TRUE (10) FALSE (01) \ / UNKNOWN (00) This is the information order. It does not describe “more truth”. It describes: more evidence being present. This is the same functional role as the Knowledge Order in the Belnap bilattice. 9. Information content The four states therefore have the following precise meanings: [ UNKNOWN=(0,0) ] no evidence. [ FALSE=(0,1) ] negative evidence. [ TRUE=(1,0) ] positive evidence. [ GUARD=(1,1) ] positive and negative evidence. Thus: [ UNKNOWN \leq_K FALSE ] [ UNKNOWN \leq_K TRUE ] [ FALSE \leq_K GUARD ] [ TRUE \leq_K GUARD. ] FALSE and TRUE are incomparable with respect to (\leq_K). This is logically appropriate: positive evidence is not “more” than negative evidence, and vice versa. 10. Information fusion The operation most important for K501 is defined as: [ \boxed{ MERGE_K(a,b) } ] with: [ MERGE_K((t_a,f_a),(t_b,f_b))=(t_a\lor t_b,\;f_a\lor f_b). ] This is bitwise OR on both evidence components. The complete table is: (\sqcup_K) U F T G U U F T G F F F G G T T G T G G G G G G This operation is: [ commutative ] [ associative ] and [ idempotent. ] Because bitwise OR has all three properties, [ (Q,\leq_K,\sqcup_K) ] is a join-semilattice. This is especially valuable for AIONARC. 11. Why this property is decisive for append-only operation Let: [ q_0=UNKNOWN. ] Then: [ q_1=MERGE_K(q_0,TRUE)=TRUE. ] Later: [ q_2=MERGE_K(q_1,FALSE)=GUARD. ] Still later: [ q_3=MERGE_K(q_2,TRUE)=GUARD. ] There is therefore no information loss. Formally: [ q_t\leq_K q_{t+1}. ] For every regular merge step: [ \boxed{ q_t\leq_K q_{t+1} } ] This is the desired K501 monotonicity. 12. State transitions For an individual QH256 cell this gives: UNKNOWN ──TRUE──────→ TRUE │ │ │ FALSE │ FALSE ↓ ↓ FALSE ──TRUE──────→ GUARD │ ↑ └──── FALSE ─────────┘ TRUE ──TRUE──────────→ TRUE FALSE ──FALSE────────→ FALSE GUARD ───────────────→ GUARD In simplified form: [ U\rightarrow F ] [ U\rightarrow T ] [ F\rightarrow G ] [ T\rightarrow G ] and [ G\rightarrow G. ] No ordinary merge operation produces: [ G\rightarrow F ] or: [ G\rightarrow T. ] That would be information loss. 13. Truth Order In addition, QH256 adopts the established truth order of the Belnap system. Define: [ a\leq_T b ] if and only if: [ t_a\leq t_b ] and [ f_a\geq f_b. ] Thus: [ F\leq_T U\leq_T T ] and simultaneously: [ F\leq_T G\leq_T T. ] UNKNOWN and GUARD are incomparable with respect to the Truth Order. This means: TRUE / \ UNKNOWN GUARD \ / FALSE Whereas the Knowledge Order is: GUARD / \ TRUE FALSE \ / UNKNOWN These two distinct orders are a central feature of bilattice theory. 14. QH256 negation Negation is defined at the evidence level by swapping the two components: [ \boxed{ NOT(t,f)=(f,t) } ] Thus: [ NOT(U)=U ] [ NOT(F)=T ] [ NOT(T)=F ] [ NOT(G)=G. ] x NOT(x) UNKNOWN UNKNOWN FALSE TRUE TRUE FALSE GUARD GUARD This corresponds to the established Belnap/FDE negation. Moreover: [ NOT(NOT(x))=x. ] Proof: Swapping the two bits twice returns the original state. Therefore the negation is involutive. 15. QH256 conjunction Conjunction is not defined arbitrarily. We adopt the FDE/Belnap structure: [ \boxed{ AND((t_a,f_a),(t_b,f_b))=(t_a\land t_b,\;f_a\lor f_b) } ] that is, [ t_{out}=t_a\land t_b ] [ f_{out}=f_a\lor f_b. ] Examples: [ TRUE\land TRUE=TRUE ] [ TRUE\land FALSE=FALSE ] [ TRUE\land UNKNOWN=UNKNOWN ] [ TRUE\land GUARD=GUARD ] [ FALSE\land GUARD=FALSE. ] The resulting truth table corresponds to the established Belnap matrix. 16. QH256 disjunction Analogously: [ \boxed{ OR((t_a,f_a),(t_b,f_b))=(t_a\lor t_b,\;f_a\land f_b) } ] that is, [ t_{out}=t_a\lor t_b ] [ f_{out}=f_a\land f_b. ] Hence: [ FALSE\lor FALSE=FALSE ] [ TRUE\lor FALSE=TRUE ] [ UNKNOWN\lor FALSE=UNKNOWN ] [ TRUE\lor GUARD=TRUE ] [ FALSE\lor GUARD=GUARD. ] This structure likewise corresponds to the Belnap/FDE model. 17. Why we do NOT define a QH256 implication for the time being This is an important specification decision. Many-valued logics contain different possible implication operators. Łukasiewicz uses semantics different from FDE. Other systems employ still other implication notions. Research therefore shows precisely that the selection of connectives is part of the definition of the logic itself. Accordingly: [ \boxed{ QH256\text{-Core initially contains no primitive }\rightarrow } ] We define only: [ NOT ] [ AND ] [ OR ] and [ MERGE_K. ] A later implication must receive its own proof and comparison framework. This prevents semantic drift. 18. Designated states A formal four-valued logic also requires a definition of designated values. These belong to the definition of a logical matrix in the relevant literature. For the QH256/FDE-compatible core we define: [ D={TRUE,GUARD}. ] Why? Because both states contain positive evidence: [ TRUE=(1,0) ] [ GUARD=(1,1). ] By contrast, [ UNKNOWN=(0,0) ] contains no positive evidence, while [ FALSE=(0,1) ] contains negative evidence only. Thus QH256 proposition semantics is compatible with the established Belnap designation of T and B. 19. But “designated” does not mean “proved” For K501 this distinction must be stated explicitly. [ GUARD\in D ] does not mean: The statement is indisputably correct. It means: The state value contains positive evidence. Because negative evidence is simultaneously present, GUARD is a conflict state. Thus TRUE is not flattened into GUARD. The K501 model may therefore continue to distinguish: TRUE = positive and not contradicted GUARD = positive and contradicted This is a substantially more precise semantics than a simple three-valued notion of truth. 20. Paraconsistency Here QH256 acquires an important scientific property. In classical logic, a contradiction may, under the principle of explosion, entail everything. FDE/Belnap is specifically constructed so that contradictory information does not automatically license every arbitrary conclusion. The literature correspondingly characterizes FDE as paraconsistent and explicitly applies it to inconsistent information states. QH256 adopts this property at the logical level. Accordingly: [ TRUE\land FALSE ] produces: [ GUARD ] but not: [ EVERYTHING=TRUE. ] The contradiction is localized in the affected information state. This fits the AIONARC conception of a locally reconstructible information space. 21. QH256 differs from FDE, however Up to this point the cell logic is largely FDE-compatible. The actual K501 extension is the state dynamics. FDE primarily answers: How should four-valued information be handled logically? K501 additionally asks: How may an append-only system change these states over time? For this purpose we define: [ MERGE_K ] as monotone evidence accumulation. This yields: [ UNKNOWN\leq_K TRUE/FALSE\leq_K GUARD. ] This append-only transition concept is not identical to the logical Truth Order. That separation should be retained as K501-specific. 22. QH256 is therefore a two-layer system Formally: [ \boxed{ QH256 = (Q^{128},\leq_K,\sqcup_K,\neg,\land,\lor) } ] with: [ Q={U,F,T,G}. ] Here: (\sqcup_K) describes knowledge/evidence fusion. (\land,\lor,\neg) describe logical operations. (\leq_K) describes information increase. (\leq_T) describes the Truth Order. This separation is mathematically clean and directly connected to bilattice theory. 23. A fundamental K501 theorem For every regular append-only fusion: [ \boxed{ q_{new}=q_{old}\sqcup_K e } ] and therefore: [ \boxed{ q_{old}\leq_K q_{new}. } ] It follows that: A QH256 information state can, under regular evidence accumulation, contain an equal amount or more information, but never less. This is a mathematical formulation of: APPEND_ONLY and NO_REWRITE. 24. Idempotence [ q\sqcup_K q=q. ] This means that processing the same evidence twice does not produce a new state. Example: [ TRUE\sqcup_K TRUE=TRUE. ] This is essential for reproducible ingestion. A replay of an already processed event must not change the QH256 state. 25. Commutativity [ a\sqcup_K b=b\sqcup_K a. ] Thus the order of technical fusion is irrelevant. Example: [ TRUE\sqcup_K FALSE = FALSE\sqcup_K TRUE = GUARD. ] This does not mean that historical events lack an order. The history remains time-ordered. Only the aggregated information state is commutative with respect to the merge operation. This is an important K501 distinction: [ \text{History order}\neq\text{Merge order}. ] 26. Associativity [ (a\sqcup_K b)\sqcup_K c = a\sqcup_K(b\sqcup_K c). ] Thus, for example, 1,000 frames may first be processed in ten blocks and then merged into a single aggregate without changing the final state. This is important for a scalable runtime. 27. Determinism Because the QH256 operations are defined as finite, total state transformations: [ QH256\times QH256\rightarrow QH256 ] with fixed operation definitions, they are deterministic. For any fixed operation (f) and identical inputs (x), repeated evaluation produces the same output (f(x)). No probabilistic component is required by the algebra itself. QH256 is therefore fundamentally compatible with the K501 requirement PURE_LOGIC at the algebraic level. 28. Bit-plane representation For practical implementation, the cell representation is not even the most efficient form. Instead, we define two 128-bit evidence masks: [ T[127:0] ] and [ F[127:0]. ] For cell (i): [ t_i=T_i ] [ f_i=F_i. ] The cell encoding is: [ code_i=(T_i