Multidimensional Stacked Shards and Parallel Shards:

A Unified Hierarchical Architecture for Scalable, Verifiable Extreme Computation

DOI:

John Swygert

January 07, 2026

INDEX

1. Multidimensional Stacked Shards and Parallel Shards:
   A Unified Hierarchical Architecture for Scalable, Verifiable Extreme Computation
   
2. Multidimensional Stacked Shard Architectures:
   Deterministic Computation, Memory, and Stability Through Factorial Shard Composition

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Paper 1 

Multidimensional Stacked Shards and Parallel Shards:

A Unified Hierarchical Architecture for Scalable, Verifiable Extreme Computation

DOI:

John Swygert 

January 06, 2026

Abstract

This paper introduces a unified computational architecture that combines parallel sharding with multidimensional stacked sharding into a single coherent framework. The system is designed to address the dominant failure modes of extreme-scale computation: global state explosion, I/O bottlenecks, restart fragility, and verification cost. Instead of treating sharding as a flat partitioning strategy, we formalize shards as composable computational objects that can themselves be stacked into higher-order shards, forming a deterministic hierarchy. Parallelism provides throughput; stacking provides stability, replay efficiency, and verification compression. The architecture is orchestrated by a minimal coordination layer (the Secretary Suite) that enforces deterministic assembly rules, quorum-based verification, and variance containment. The result is a system capable of scaling to trillions of work units while remaining restart-tolerant, storage-light, and falsifiable through staged benchmarks.

1. Introduction

Modern record-scale computations—such as trillion-digit numerical calculations, large FFT-driven transforms, and long-horizon simulations—are increasingly constrained not by raw compute, but by coordination overhead. Existing approaches rely on monolithic memory footprints, massive intermediate storage, and brittle execution paths where local failures can invalidate weeks of progress.

Sharding is widely used to mitigate these issues, but almost always in a flat sense: work is divided into independent pieces, processed in parallel, and recombined at the end. This paper argues that flat sharding alone is insufficient at extreme scale.

We propose a combined architecture:

• Parallel shards for throughput.
• Stacked shards (shards-of-shards) for hierarchical stability and verification.

This architecture is multidimensional in structure but deterministic in execution, enabling both scale and control.

2. Definitions and Core Concepts

2.1 Shard

A shard is a deterministic unit of computation defined by:

• A bounded input domain (e.g., term range, index range, subtree)
• A deterministic generation rule
• A result object
• A cryptographic commitment (hash or Merkle root)
• A replay policy

A shard is regenerable: it can be recomputed from its descriptor without reliance on global state.

2.2 Parallel Sharding (Horizontal Dimension)

Parallel sharding divides work across independent compute resources:

• Shards are processed concurrently
• No shard depends on another at the same level
• Failures only affect local progress

Parallelism increases E (effective work throughput).

2.3 Stacked Sharding (Vertical Dimension)

Stacked sharding treats shards themselves as atomic units that can be combined into higher-order shards.

• Micro-shards → Shards → Super-shards → Meta-shards
• Each level aggregates verified results from the level below
• Each aggregation produces a new shard with its own commitment

This introduces a hierarchy, not just a partition.

Stacking increases Y (system stability).

2.4 Multidimensional Sharding

The architecture is multidimensional in the sense that:

• Parallelism operates within each level
• Stacking operates between levels

These dimensions are orthogonal and complementary.

3. Architectural Overview

3.1 Hierarchical Structure

Let:

• : micro-shards (leaf computations)
• : shards composed of micro-shards
• : super-shards composed of shards
• …
• : meta-shards

Each level applies a deterministic composition operator :

S_{i+1} = \mathcal{F}(S_i)

Where enforces:

• Canonical ordering
• Fixed aggregation rules
• Commitment generation
• Verification policy

3.2 Determinism Over Combinatorics

While the space of possible shard combinations is large (factorial-like), only one canonical path is chosen.

This preserves:

• Reproducibility
• Auditability
• Minimal coordination cost

The system explores a large space of combinations conceptually, but executes a single deterministic sequence.

4. Secretary Suite Coordination Layer

The Secretary Suite is a minimal orchestration layer, not a heavy scheduler.

Its responsibilities are strictly limited to:

1. Shard Assignment
   • Deterministic mapping of work to nodes
2. Commit Tracking
   • Recording shard commitments
3. Verification Quorums
   • Selective replay of shards at chosen levels
4. Failure Detection
   • Heartbeats and timeouts
5. Replay and Reassignment
   • Recompute only affected shards
6. Hierarchical Aggregation
   • Triggering stack transitions between levels

Notably absent:

• No global state accumulation
• No large intermediate storage
• No centralized memory pool

5. Verification and Replay Economics

5.1 Localized Failure Containment

Failures are handled at the lowest affected level:

• Micro-shard failure → replay micro-shard
• Shard failure → replay shard
• Super-shard failure → replay only its subtree

Higher-level shards remain valid if their commitments are intact.

5.2 Hierarchical Verification

Verification is hierarchical:

• Hashes at low levels
• Merkle roots at mid levels
• Aggregate commitments at top levels

Quorum replay is applied selectively, not globally.

This reduces verification cost from to approximately .

6. Application to Extreme Numerical Computation

6.1 Binary Splitting as a Natural Stack

Binary splitting algorithms naturally fit stacked sharding:

• Leaves = small index ranges
• Internal nodes = merged rational tuples
• Root = final result

Each subtree is a shard. Each merge is a stack transition.

Parallelism occurs at each depth; stacking occurs across depths.

6.2 Memory and Communication Benefits

Aspect

Flat Sharding

Stacked Sharding

Memory

O(D) per node

O(D/N + log D)

Checkpoints

O(D)

O(1)–O(log D)

Replay Cost

High

Localized

Verification

Global

Hierarchical

7. Formal Performance Model

Let:

• : total work size
• : nodes
• : micro-shards
• : stack levels
• : algorithm constant

Compute time:

T \approx \frac{a \cdot D \cdot (\log D)^k}{N} + O(\text{stack overhead})

Stack overhead:

O(L \cdot M \cdot \text{sync}) \ll \text{compute}

Replay overhead:

O(P_{\text{fail}} \cdot M \cdot t_{\text{shard}})

All terms are measurable and falsifiable via staged benchmarks.

8. Benchmark and Validation Plan

1. Single-node baseline
   • Measure algorithm constant
2. Parallel-only run
   • Validate scaling efficiency
3. Stacked run
   • Measure replay frequency and verification cost
4. Injected failure tests
   • Validate localized recovery
5. Extrapolation
   • Bound runtime with uncertainty intervals

9. Broader Implications

While motivated by extreme numerical computation, the architecture generalizes to:

• Large simulations
• Distributed proof systems
• Long-context AI workloads
• AGI-scale task decomposition
• Sovereign, restart-safe compute

The same shard hierarchy can represent:

• Computation
• Memory
• Identity
• Provenance

Without conflating them.

10. Conclusion

Flat sharding provides speed, but not stability.
Monolithic systems provide determinism, but not scalability.

By combining parallel shards with stacked shards, we obtain a system that is:

• Fast
• Restart-tolerant
• Verifiable
• Storage-light
• Deterministic
• Falsifiable

This architecture does not merely optimize existing systems; it changes the geometry of computation itself, replacing fragile monoliths with hierarchical, regenerable structure.

The result is not just better performance—but a fundamentally more stable way to compute at extreme scale.

References 

None

Paper 2

Multidimensional Stacked Shard Architectures

Deterministic Computation, Memory, and Stability Through Factorial Shard Composition

DOI:

John Swygert

January 067 2026

Abstract

This paper introduces a multidimensional shard architecture for the Secretary Suite that unifies two previously distinct scaling strategies: parallel shard execution and stacked shard composition. Parallel shards distribute work across independent nodes or agents, while stacked shards introduce higher-order structure by composing shards of shards in factorial layers. Together, these dimensions form a deterministic, replayable, and equilibrium-preserving computational fabric capable of scaling without centralized state, global memory pressure, or authority amplification.

Unlike conventional distributed systems that scale primarily by replication or throughput aggregation, the Secretary Suite shard model scales by structural depth as well as breadth. Each additional shard dimension increases expressive power, fault containment, and auditability without increasing agent authority or system opacity. This paper formalizes stacked shard layers, defines their interaction with parallel execution planes, and demonstrates how multidimensional shard systems preserve equilibrium under load, failure, and regeneration.

1. Purpose and Scope

The Secretary Suite already defines shards as the atomic units of knowledge, logic, and capability. This paper extends that foundation by addressing a critical scaling question:

How does a system grow in power without growing in fragility, authority, or hidden state?

The answer proposed here is not larger shards, faster shards, or smarter agents—but structured composition of shards across dimensions.

This paper is concerned exclusively with shard mechanics. It does not redefine agents, governance, digital fingerprints, or hardware acceleration. It provides a formal model for how shards themselves may be arranged, stacked, replayed, and recomposed to support large-scale computation and memory while preserving AO (Equilibrium as Law).

2. Shards Revisited: Atomic but Not Flat

A shard is atomic in authority and scope, but it is not inherently flat.

In earlier formulations, shards are treated primarily as first-order units:

• A reference shard
• A logic shard
• A tool shard
• A constraint shard

These shards are attached per task and revoked at completion. This model is sufficient for correctness, but it leaves expressive capacity unused.

A shard may itself be:

• Generated from other shards
• Verified against other shards
• Reconstructed deterministically from shard inputs

This observation motivates a higher-order structure: shards composed of shards, without violating atomic authority boundaries.

3. Parallel Shards (Horizontal Dimension)

The first and most familiar shard dimension is parallelism.

Parallel shards:

• Execute independently
• Share no mutable state
• May run concurrently across nodes
• Are aggregated only through explicit, auditable operations

Parallelism provides:

• Throughput scaling
• Failure isolation
• Localized variance containment

However, parallelism alone does not increase structural depth. It multiplies capacity, not expressiveness.

Parallel shards answer the question:

“How many things can we do at once?”

They do not answer:

“How do results themselves become structured, layered, and self-verifying?”

4. Stacked Shards (Vertical Dimension)

Stacked shards introduce a second dimension: factorial composition.

A stacked shard is not a larger shard. It is a higher-order shard whose content is defined by the deterministic composition of lower-order shards.

4.1 Definition

A stacked shard:

• References a bounded set of child shards
• Declares a composition rule
• Produces a deterministic result shard
• Contains no new authority beyond its children

It is functionally analogous to:

• A proof built from lemmas
• A function built from functions
• A computation built from partial sums

4.2 Factorial Nature

Stacking is factorial, not linear.

If first-order shards are atomic units, second-order shards encode relationships between shards, and third-order shards encode relationships between relationships.

Each level increases expressive power without increasing:

• Agent privilege
• Hidden state
• Runtime mutation

The stack grows upward, not outward.

5. Multidimensional Shard Space

When parallel and stacked dimensions coexist, the system operates in a multidimensional shard space.

• Horizontal axis: parallel shard planes
• Vertical axis: stacked shard layers

Each coordinate in this space represents:

• A specific shard
• At a specific composition level
• Executed under explicit constraints
• With full replayability

This structure allows:

• Massive distributed computation
• Hierarchical verification
• Deterministic regeneration after failure
• Local correction without global recomputation

6. Deterministic Regeneration and Replay

A critical property of stacked shards is regenerability.

Because a stacked shard is defined entirely by:

• Its child shard identifiers
• Its composition rule
• Its execution constraints

…it can be destroyed, lost, or invalidated and then reconstructed exactly.

This eliminates:

• Checkpoint dependency
• Global rollback
• Persistent intermediate storage

Failure becomes a local event. Equilibrium is preserved.

7. Equilibrium Preservation Across Dimensions

AO (Equilibrium as Law) applies identically at all shard dimensions.

• A first-order shard must maintain equilibrium within its scope
• A stacked shard must maintain equilibrium across its composition
• A parallel plane must maintain equilibrium across aggregation

If any layer violates equilibrium:

• Execution halts
• The shard is invalid
• No compensatory fabrication occurs

Stacking does not weaken constraints. It multiplies them.

8. Authority Containment and Non-Amplification

A key design invariant is preserved:

No shard, at any dimension, acquires authority.

• Stacked shards do not decide
• Parallel shards do not vote
• Aggregation does not confer priority

Authority remains:

• Task-scoped
• Agent-bounded
• Explicitly granted and revoked

This prevents the most common distributed failure mode: emergent authority through aggregation.

9. Practical Implications

Multidimensional shard systems enable:

• Large-scale mathematical computation without monolithic memory
• Hierarchical verification of results
• Incremental confidence accumulation
• Efficient recomputation after failure
• Deep audit trails with shallow storage

They are equally applicable to:

• Numerical computation
• Knowledge synthesis
• Constraint satisfaction
• Tool generation
• Evidence aggregation

10. Relationship to Existing Secretary Suite Components

This architecture:

• Extends the Shard Library without altering its primitives
• Integrates naturally with the Shard Library Funnel
• Produces rich input for Digital Fingerprint formation
• Benefits from optional hardware acceleration but does not require it

No existing paper is contradicted. This work is purely additive.

11. Failure Modes and Boundaries

Recognized failure modes include:

• Improper shard composition rules
• Excessive stacking depth without justification
• Attempted authority inference from aggregation

These are constrained structurally:

• Invalid stacks do not execute
• Over-composition is rejected
• Authority leakage is impossible by construction

12. Summary

Parallel shards scale breadth.
Stacked shards scale depth.

Together, they enable systems that grow in capability without growing in fragility, opacity, or authority.

This multidimensional shard architecture completes the shard model of the Secretary Suite by introducing a lawful, deterministic method for hierarchical computation and memory construction. It demonstrates that scale does not require centralization, persistence does not require storage, and power does not require authority.

In the Secretary Suite, intelligence is not amplified.
It is structured.

References 

None