Holo‑IC is a minimal, resource–aware interaction‑net calculus that executes natively on the prime structure—the 12 288‑cell, 48 × 256 toroidal substrate defined by the UOR holographic specifications. By marrying classic interaction combinators (γ route, δ copy, ε finish) with the hologram’s built‑in “physics of truth” (96‑class resonance semiring, triple‑cycle invariants, window‑local acceptance), Holo‑IC yields a deterministic yet highly‑parallel computing model whose correctness is checked locally and cryptographically at every step. The result is a finite, exactly‑classifiable phase space that supports proof‑carrying execution, Merkle‑anchored state, and “confluence up to gauge”—all without a global scheduler.
- Local provability. Modern distributed systems pay a latency tax for global consensus. Holo‑IC reduces verification to window‑local proofs backed by a substrate‑wide Merkle root.
- Resource semantics. Boolean logic hides the cost of computation. Holo‑IC tracks work in a 96‑element semiring C₉₆; truth emerges only when a local budget collapses to zero.
- Fixed, finite substrate. The 12 288‑cell prime structure supplies an invariant geometry, allowing precise enumeration, indexing, and compression of states.
- Boundary holography. Bulk configurations are reconstructed from boundary automorphisms (
Φis an isomorphism). This provides inexpensive canonical forms and “confluence up to gauge.”
| Concept | Role in Holo‑IC |
|---|---|
| Prime Structure | 48 pages × 256 bytes ⇒ 12 288 lattice points; smallest grid supporting (i) modular tilings, (ii) the R₉₆ resonance algebra, (iii) 768 triple‑cycle invariants, and (iv) positive gauge geometry. |
| Resonance Algebra R₉₆ | 96 congruence classes generated by eight binary toggles subject to a unity‑pair constraint (α₄ α₅ = 1). Provides the commutative semiring C₉₆ used for budgets. |
| Triple‑Cycle (C₇₆₈) | Second‑order invariant that keeps long‑range means and variances within tolerance, preventing hidden state accumulation. |
| WindowAccept() | Deterministic verifier that admits or rejects a rewrite on the basis of (a) closure, (b) budget consistency, (c) R₉₆ class preservation, (d) C₇₆₈ invariants, and (e) Merkle inclusion proofs. |
| Interaction Combinators | Three‑node palette (γ, δ, ε) introduced by Lafont (1990) shown to be universal for interaction nets. |
Redex ::= (Window w, Op o, Budget b, Witness σ)
Op ::= γ(h) | δ(k) | ε // h: boundary automorphism, k: fan‑out degree
Budget ::= u8 ∈ [0 .. 95] // element of C₉₆
Witness σ ::= {closure✓, merkle✓, R96✓, C768✓}
A redex is well‑typed when all of the following hold inside its window w:
| Condition | Meaning |
|---|---|
| Closure L0–L3 | Page/byte conservation; no orphan pointers. |
| R₉₆‑preservation | Class of each touched byte is unchanged up to boundary automorphism h. |
| C₇₆₈‑preservation | Triple‑cycle statistics (mean, variance) remain within tolerance. |
| Budget algebra | Budgets compose under ∔ (mod 96) and ⊗ (mod 96) exactly as specified. |
| Merkle inclusion | Every byte in w has a valid path to the global Merkle root. |
Failure of any test causes the step to fail‑closed; no state is mutated.
All rules are oriented as rewrites on a single window.
| Rule | Structural Effect | Budget Law |
|---|---|---|
| γ ∘ γ ⇒ γ | Composition of boundary automorphisms is an automorphism. | Budget unchanged. |
| γ ⋅ δ ⇒ δ ⋅ γ | Routing commutes with copying. | Budgets transported functorially (homomorphism). |
| ε (when Σ b ≡ 0) | Remove the redex and its wires (“crush”). | Budget 0 collapses to Boolean TRUE. |
Programs denote morphisms in the category of C₉₆-modules:
- Objects: multisets of R₉₆ classes.
- Morphisms: window‑local rewrites that preserve module structure.
- Tensor: disjoint window union.
- Unit: empty window (budget = 0).
Soundness: every legal reduction sequence projects to a path of valid morphisms.
Completeness (up to gauge): any two reduction sequences with identical boundary action h yield gauge‑equivalent outcomes.
struct State {
Byte pages[48][256]; // 12 288‑cell lattice
Hash pageHash[48]; // per‑page
Hash merkleRoot; // global commitment
}
1. Select window w // size 8–64 bytes (implementation‑defined)
2. Collect witness σ // closure, R96, C768, Merkle paths
3. If WindowAccept(w, σ) { execute Op; update pages & hashes }
else { reject; state unchanged }
- Windows that are byte‑disjoint commute — they can be executed by independent nodes.
- Consistency is re‑established by verifying each step’s witness against the updated Merkle root and replaying boundary actions.
- No global total ordering is required; an ordering layer may be added provided it respects closure.
| Property | Mechanism | Threat Mitigated |
|---|---|---|
| Local Soundness | WindowAccept() |
Invalid rewrites, hidden side‑effects |
| Integrity | Merkle commitments | Tampering with untouched pages |
| Determinism (up to gauge) | Boundary holography | Fork divergence |
| Resource accounting | C₉₆ budgets | Denial‑of‑service by infinite work loops |
| Fail‑closed | Mandatory witnesses | Partial failures, crash faults |
-
Reference Spec (Q4 2025).
- Yaml definition of Redex, Window, Witness.
- Formal Haskell/Lean proof of soundness.
-
PrimeOS Module (Q1 2026).
- Kernel patch adding
windowAccept()syscall. - CLI (
holoic‑apply,holoic‑prove).
- Kernel patch adding
-
SDK & DSL (Q2 2026).
- Rust & TypeScript bindings.
- Macro‑based micro‑DSL:
γ(h),δ(k),ε.
-
Distributed Runtime (Q3 2026).
- Gossip‑based witness propagation.
- Optional layer‑2 total‑order service (Raft‑compatible).
-
Formal Verification (2027).
- Isabelle/HOL mechanisation of the full triple‑cycle invariant.
- Proof‑carrying transform library.
| Domain | Value Proposition |
|---|---|
| State‑channel settlement | Deterministic, Merkle‑anchored local computation without L1 gas fees. |
| On‑device ML pre‑processing | Run feature‑extraction kernels inside Klein V₄ windows for verifiable, energy‑bounded transforms. |
| Digital preservation | Boundary‑first canonical form enables long‑term deduplication and stable hashing. |
| Collaborative editing | Conflict‑free CRDT built from commuting windows and boundary actions. |
| Trusted infrastructure | Embed Holo‑IC verifiers in firmware for audit‑ready pipeline steps. |
- Interaction Nets (Lafont 1990) — Holo‑IC inherits the γ/δ/ε palette but adds budgets and holographic invariants.
- Proof‑Carrying Code (Necula 1996) — Witness objects here are smaller (window‑local) and backed by Merkle proofs.
- Blockchain VMs (EVM, Move, Cairo) — Holo‑IC removes global consensus from the core semantics; ordering is optional.
- Reversible Computing — Budget algebra allows strictly‑conserved resources; ε provides a controlled, non‑reversible collapse.
- Quantum analogue — Replace C₉₆ with a unitary representation over cyclotomic fields.
- Adaptive window sizing — Dynamic enlargement shrinks witness overhead while preserving local verifiability.
- Typed Budgets — Split C₉₆ into multiple resource tracks (energy, confidentiality, latency).
- Zero‑knowledge witnesses — Fold Merkle paths into SNARK proofs for privacy‑preserving execution.
Holo‑IC demonstrates that a finite, holographically‑constrained substrate can host a complete, resource‑sensitive model of computation whose every step is locally provable and cryptographically committed. By unifying interaction combinators with the UOR prime structure, we obtain a practical calculus for distributed systems where correctness is enforced not by trust in remote validators but by conservation laws baked into the fabric of the state itself.
The UOR Foundation invites collaboration, critique, and implementation contributions as we refine the specification and harden the runtime for production‑grade deployments.