---
title: 'Conquest: Dynamics, Strategy, and Computation'
url: https://www.emergentmind.com/topics/conquest
type: topic
---

# Conquest: Dynamics, Strategy, and Computation

Conquest denotes the acquisition of control under conditions of opposition, scarcity, or asymmetry. In the literatures considered here, it appears as the fall of a besieged city under coupled resource–population–army dynamics, as the probabilistic capture of territory in formal games, as the long afterlife of military takeover in language and genealogy, as the conquest of distance and space through microwave and satellite communication, and as the proper name **CONQUEST**, a large-scale density functional theory code whose aim is to make first-principles calculations feasible for systems from thousands to millions of atoms [2504.01649][1512.04333][2410.09283][1803.11497][2205.08941].

## 1. Conquest as a dynamical military regime

In the mathematical treatment of the Siege of Syracuse (214–212 BC), conquest is modeled as the terminal regime of a prolonged siege in which internal resources, defender population, and attacker persistence are jointly coupled [2504.01649]. The state variables are \(R(t)\) for resources, \(P(t)\) for the population of Syracuse, and \(A(t)\) for the size of the Roman army, with calibrated initial values \(A(0) \approx 35{,}000\), \(P(0) \approx 75{,}000\), and \(R(0)=100.0\). The governing ODE system is
\[
\frac{dR}{dt} = -\alpha R + \beta,
\]
\[
\frac{dP}{dt} = -\gamma P + \delta R - \kappa P A,
\]
\[
\frac{dA}{dt} = \mu A - \lambda A P.
\]
Here \(\lambda\) aggregates defensive effectiveness, including fortifications and Archimedes’ war machines; \(\mu\) is Roman reinforcement; \(\kappa\) is attack-induced population loss; and \(\alpha,\beta,\gamma,\delta\) govern depletion, replenishment, mortality, and resource-supported survival.

The critical quantity is the defense threshold
\[
\lambda_c = \frac{\alpha\gamma\mu}{\delta\beta}.
\]
The consistent reading given in the paper is that \(\lambda > \lambda_c\) corresponds to survival or repulsion of the siege, whereas \(\lambda < \lambda_c\) corresponds to eventual collapse and conquest. The article explicitly notes an internal inconsistency in one paragraph that reverses these inequalities; the threshold formula and the abstract support the \(\lambda < \lambda_c\) conquest regime. Under baseline parameters \(\alpha = 0.05\), \(\beta = 1.0\), \(\gamma = 0.001\), \(\delta = 0.01\), \(\kappa = 0.000043\), \(\mu = 0.0001\), and \(\lambda = 0.00002\), the trajectories over roughly 730 days show \(R(t)\) declining toward \(\beta/\alpha = 20\), \(P(t)\) declining significantly, and \(A(t)\) remaining roughly persistent. In that framework, conquest is not identical to tactical success in a single assault; it is the long-run attractor of a feedback loop in which blockade, mortality, and persistence outlast defensive capacity.

This formulation shifts the analytical center of conquest from battlefield brilliance to system-level imbalance. The paper’s own interpretation is that Roman success became “inevitable” once the relevant parameters placed Syracuse below the threshold. A plausible implication is that siege conquest is best understood as a control problem over attrition, logistics, and resilience rather than as a sequence of isolated engagements.

## 2. Conquest as a probabilistic and strategic objective in games

In probabilistic game analysis, conquest is a precisely defined terminal event. In the board game *RISK*, “conquer odds” are the probability that an attacker eliminates all defenders and takes a territory, assuming repeated engagements under the game’s dice rules [1512.04333]. For standard play, the paper derives a practical threshold: the attacker needs **two more than 86% of the defending armies** to have an over 50% chance to conquer an enemy territory. In rough form,
\[
\hat{A} \gtrsim 0.86\,\hat{D} + 2.
\]
Using normal approximation, the authors show that the transition from low to high conquer probability is rapid around this threshold. In this usage, conquest is a stochastic event with a sharply defined phase-like crossover.

A more explicitly strategic formulation appears in “Cooperate to Compete,” a multi-agent conquest game inspired by *Risk* [2604.25088]. The environment contains 12 territories organized into 4 regions, including 2 special chokepoint territories. Each of 4 players receives a secret objective: conquer two assigned non-adjacent regions, and the first player to complete that objective wins. The game combines reinforcement, attack, support, fortification, fog-of-war, and private cheap-talk negotiation with non-binding agreements. Conquest therefore depends not only on tactical capture but also on alliance formation, betrayal, and information asymmetry.

The empirical results show that conquest in such settings is structurally mixed-motive. Humans were significantly less reliable partners than LM-based agents, and were more aggressive negotiators, accepting deals without a counteroffer only 56.3% of the time compared to 67.6% for LM-based agents. Through targeted prompting, the authors raised agent win rates from 22.2% to 32.7%. Across over 1,100 games, more than 16,000 private conversations, 15.2 million tokens, and over 150,000 player actions, conquest emerged as a function of coordination flexibility rather than isolated local optimization.

These two game-theoretic usages are complementary. In *RISK*, conquest is a terminal probability determined by attrition statistics. In C2C, it is a territorial race mediated by negotiation, support transfers, and strategic deception. Together they indicate that conquest in formal games can be either a closed-form probabilistic event or an emergent property of multi-agent coordination under secret, asymmetric objectives.

## 3. Conquest and post-conquest transformation of society and language

Military conquest also appears as a long-duration process that reorganizes categories of identity, property, and institutional meaning. In the computational study of Medieval Latin charters, the Norman Conquest of 1066 is treated as a “lexical shock” whose effects can be measured in semantic change across the DEEDS corpus of over 70,000 Latin charters, of which 40,000 relate to England and 17,000 are dated [2410.09283]. The corpus is partitioned into ANG (589–1065), NOR (1066–1153), and PLA (1154–1272). The study identifies changes such as **comes**, which shifts from a broad administrative “official” to the specialized title “earl” or “count,” and **proprius**, which moves from “with one’s own hand” toward a legal-proprietary sense of what is “one’s own.” The AN transition from ANG to NOR shows stronger semantic change than the later NP transition from NOR to PLA, and contextual embeddings outperform static embeddings in detecting those shifts.

The genealogical and mtDNA study of early colonial Costa Rica shows a different post-conquest mechanism: incorporation, erasure, and reclassification [1605.06208]. The conquest campaigns are dated to 1561–1563, after which colonial rule centered on Cartago. The paper reconstructs a 17-generation matrilineal chain from a present-day Costa Rican woman back to doña Isabel Jiménez, daughter of the conquistador Alonso Jiménez. Sequencing of mitochondrial DNA from living matrilineal descendants yielded the motif 16111T, 16223T, 16290T, 16319A, 16362C, identified as haplogroup A, one of the classic Amerindian founding lineages, and reported by the authors as never found in individuals of European or African origin. The paper concludes that Isabel Jiménez was mestiza and that her maternal ancestry was indigenous. In related autosomal work cited there, the reported admixture proportions for Costa Rica are 61.04% European, 29.91% Amerindian, and 9.05% African.

Taken together, these studies present conquest as a generator of durable transformations rather than a completed military act. In one case, conquest reorganizes legal vocabulary and documentary semantics; in the other, it produces a colonial elite whose documentary classification as “Spanish” obscures Amerindian maternal ancestry. The combined evidence suggests that conquest frequently survives its military phase through altered institutions of naming, ownership, and social memory.

## 4. Conquest as the conquest of distance and space

In the history of telecommunication, conquest is used literally as the conquest of distance and, later, of space itself [1803.11497]. The key enabling device is the traveling-wave tube (TWT), a vacuum electron amplifier in which an electron beam co-propagates with an RF wave along a slow-wave structure, usually a metallic helix. For a helix, waves are slowed down by a factor \(2\pi a_\varnothing/d\), and in space tubes this typically yields an axial phase velocity on the order of one-tenth of the speed of light. The appendix formulates the underlying wave–particle interaction with the Hamiltonians
\[
H(p,q)=\frac{p^2}{2}-U\cos q
\]
for a single electron in the wave frame, and
\[
H(p,z,I,\varphi)=\sum_r \frac{p_r^2}{2} + \sum_j \omega_j I_j - \sum_j \sum_r U_j \cos(k_j z_r-\varphi_j)
\]
for the self-consistent many-electron, many-wave system. Conservation of total momentum,
\[
P=\sum_r p_r + \sum_j k_j I_j,
\]
makes explicit that when electrons lose momentum, the wave gains it and is amplified.

Historically, the TWT became decisive in the microwave radio-relay systems that extended television and telephony across continents, and then in the active communications satellites that created continuous global links. Telstar 1, launched in 1962, carried a 4 GHz helix TWT amplifier delivering about 2 W and 40 dB gain; Relay 1 used an 11 W tube; Syncom demonstrated the geosynchronous model foreseen by Clarke. In later decades, commercial GEO satellites, broadband systems, navigation payloads, Earth-observation platforms, and deep-space probes all depended on TWTAs for high-gain, wide-bandwidth, high-frequency transmission. Voyager 1 and 2, for example, carried TWTAs that enabled communication over distances now exceeding 20 billion km, while the James Webb Space Telescope uses a Ka-band TWT around 26 GHz, 50 W, more than 50 dB gain, and 55% efficiency.

In this literature, conquest no longer refers to seizure of territory but to extension of communicative reach. The device’s significance lies in making long-distance, high-bandwidth radio communication practical in vacuum and at microwave frequencies. This suggests a broader technical sense of conquest: the expansion of operational control over previously inaccessible spatial scales.

## 5. **CONQUEST** as a large-scale electronic-structure code

As a proper noun, **CONQUEST** is a density functional theory code designed from the outset for extremely large-scale electronic-structure calculations on massively parallel platforms [2205.08941][2002.07704]. It implements both exact diagonalisation and linear-scaling solvers for the ground state, uses local basis sets—pseudo-atomic orbitals (PAOs) and systematically convergent B-splines or “blips”—and relies on sparse matrix storage and operations to preserve locality throughout the calculation. With full PAO bases and diagonalisation, systems up to about 1,000 atoms can be modeled with relatively modest resources; multi-site support functions extend exact calculations to about 10,000 atoms with similar resources; and linear scaling has been demonstrated on over 1,000,000 atoms, with essentially perfect weak scaling and runs on nearly 200,000 cores.

Its central representation is density-matrix based. In a localized support-function basis, the density matrix is written as
\[
\rho(\mathbf{r},\mathbf{r}')=\sum_{i\alpha,j\beta}\phi_{i\alpha}(\mathbf{r})\,K_{i\alpha,j\beta}\,\phi_{j\beta}(\mathbf{r}')^*.
\]
In linear-scaling mode, CONQUEST avoids global diagonalisation and instead uses density matrix minimization with an auxiliary matrix \(L\), enforcing weak idempotency through
\[
K = 3LSL - 2LSLSL.
\]
Because support functions are strictly localized and \(L\) is truncated beyond a finite cutoff, the Hamiltonian, overlap, and density matrices are sparse, and the cost of SCF and related operations scales as \(\mathcal{O}(N)\) for suitably localized systems. The code calculates total energy, forces, and stresses, supports structural optimization and molecular dynamics in the NVE, NVT, and NPT ensembles, interfaces with LibXC for LDA and GGA functionals, and includes dispersion treatments such as DFT-D2/3, Tkatchenko–Scheffler, and vdW-DF.

This is a distinct technical sense of conquest. Here the term names an infrastructure for making fully quantum-mechanical calculations feasible at scales usually inaccessible to conventional \(\mathcal{O}(N^3)\) Kohn–Sham DFT. The proper noun preserves the vocabulary of scale and reach that also appears in the historical and telecommunication senses of conquest.

## 6. Algorithms, accuracy, and applications of **CONQUEST**

The defining algorithmic innovations of CONQUEST are its localized support functions, its multi-site support function (MSSF) formalism, and its coupling to specialized sparse eigensolvers [1704.03203][1806.07610][2003.05327][2607.07472]. Multi-site support functions are linear combinations of PAOs on a target atom and neighbors within a cutoff,
\[
\phi_{i\alpha}(\mathbf{r})=\sum_{k}^{\text{neighbors}}\sum_{\mu\in k} C_{i\alpha,k\mu}\,\chi_{k\mu}(\mathbf{r}),
\]
with coefficients initially obtained by local filter diagonalization and, in later work, variationally optimized. This allows a near-minimal number of support functions per atom while retaining the accuracy of much larger primitive PAO sets.

The performance consequences are large. In graphene/Rh(111), a 3,088-atom calculation compared primitive PAOs with MSSFs: matrix construction changed from 155.7 s to 1455.4 s, diagonalization from 37,647.7 s to 700.8 s, and total SCF step time from 37,803.5 s to 2,156.3 s [1806.07610]. In hydrated DNA, CONQUEST combined with the Sakurai–Sugiura method (SSM) extracted 6,093 states in the range \([-0.4,0.4]\) Hartree for a 3,439-atom system, and also handled Ge hut clusters on Si(001) as large as 194,573 atoms with a matrix dimension of 1,751,157 [1704.03203]. For perovskite oxides, modest PAO basis sets reproduced the plane-wave charge density to a total integrated error of better than 0.5%, and the multi-mode antiferroelectric distortion of PbZrO\(_3\) and its energetics were reproduced by better than 99% [2003.05327].

More recent work extends the code’s linear-scaling formalism to stress and dynamics. Stress is derived analytically for both exact diagonalisation and \(\mathcal{O}(N)\) density-matrix methods, including the terms associated with the electron-number constraint in linear scaling. Across a range of materials with different band gaps, stress errors fall below 0.1 GPa for modest density-matrix cutoffs, and large-scale NPT molecular dynamics is reported as stable and accurate [2607.07472]. In that sense, CONQUEST is not merely a static solver: it is a scalable platform for geometry, stress, and finite-temperature dynamics in complex materials.

Across these applications, a common structural theme reappears. Whether conquest denotes the fall of Syracuse, the capture of a territory, the reorganization of post-1066 legal vocabulary, the conquest of space by communications engineering, or the scaling frontier of first-principles simulation, the term consistently marks a threshold crossing from local contest to sustained control. The cited literature does not reduce these cases to a single theory, but it repeatedly associates conquest with persistence, asymmetry, and the ability to convert local interactions into durable system-level outcomes.

Source: https://www.emergentmind.com/topics/conquest