Factor-Interval Structure in Fixed-Length Semiprimes

Working Title

Factor-Interval Structure in Fixed-Length Semiprimes: Theory, Search Models, and Security Implications

Author

Thomas Flake

Degree and Classification

Doctoral dissertation for a PhD-level research program in Computer Science, with emphasis in applied computational mathematics, computational number theory, and cryptographic systems analysis.

Research Area

Computer Science; Applied Computational Mathematics; Cryptography; Computational Number Theory; Systems and Performance Modeling

Abstract

This dissertation investigates the structure of the admissible factor interval for fixed-length semiprimes and evaluates how factor-balance constraints alter both the effective one-factor domain and the practically meaningful search region under exhaustive and precomputed-prime search models. Existing literature establishes that factor shape matters, that close factors may weaken RSA under certain methods such as Fermat-style factorization, and that practical semiprimes used in RSA are generally generated from roughly equal-sized primes. However, the search-surface consequences of those balance constraints are often discussed informally or through attack-specific lenses rather than through a general interval-structural framework tied to representative populations and explicit scheduler policies.

This work formalizes the admissible interval for the smaller factor of a fixed-length semiprime, defines balance constraints through a tunable parameter, and compares constrained and unconstrained one-factor domains. It then studies how these interval reductions interact with prime density, precomputed-prime libraries, phased reduction search, scheduler policy, and modern parallel computing. A central claim of the dissertation is that the economics of search are asymmetric: prime-deck construction is expensive and front-loaded, but once a verified prime library exists, repeated factor testing becomes far cheaper and naturally lends itself to large-scale parallelization. A second central claim is that if realistic semiprime-generation constraints concentrate the smaller factor near sqrt(N), then the most meaningful search region may be materially smaller and more concentrated than naive full-domain reasoning suggests. That claim must be validated not only at the population level but also at the scheduler level by comparing lower-first, square-root-centered, and mixed-band search policies on the same accepted populations.

The dissertation therefore argues that the correct denominator for reasoning about one-factor search is not the full k-bit space, but the natural admissible domain for the smaller factor, interpreted through a computational model that separates precomputation cost from amortized search cost and that situates those costs within the historical economics of storage and commodity parallelism. In that sense, the work revisits an older warning in a new economic environment: storage and reuse can change the practical meaning of precomputation without changing the underlying mathematics. The dissertation does not present a new cryptanalytic break of RSA, nor does it propose replacing standards-based key-size guidance with a custom threshold. Instead, it offers a structural and computational analysis of how semiprime generation constraints affect one-factor search geometry, how those constraints may shift the highest-value search region toward sqrt(N), and how the reuse of a precomputed deck changes the practical meaning of that geometry over time. The empirical program is designed to test whether the denominator-correct interval model predicts observed tractability better than naive baselines across large synthetic semiprime populations, milestone deck regimes, and competing scheduler policies. The goal is to improve the rigor with which these constraints are discussed, measured, and interpreted in both academic and applied cryptographic settings.

Keywords

Semiprimes; RSA; factor balance; admissible interval; one-factor search; search surface; scheduler policy; square-root-centered search; prime density; computational number theory; parallel search; precomputation; storage economics

1. Introduction

1.1 Motivation

Cryptographic discussion often treats large composite numbers as though their structure were adequately described by their total bit length alone. In many practical settings, however, the structure of the factors matters as much as the size of the composite itself. Semiprimes used in RSA are not usually formed from arbitrary integers. They are produced through constrained procedures that require primality, exact or near-exact bit length, and practical balance between the two prime factors. These constraints shape the location of admissible factors and therefore shape the geometry of one-factor search.

This dissertation begins from a simple observation: the smaller factor of a fixed-length semiprime does not inhabit the whole numeric universe suggested by the total bit length of the modulus. Rather, it belongs to a far narrower admissible interval. Moreover, when the factor-generation process imposes balance constraints, that interval tightens further around the square root of the modulus. The argument is not that this observation immediately produces a practical cryptanalytic break. The argument is that security reasoning becomes sloppier than necessary when it compares a constrained one-factor problem to the wrong denominator and when it fails to ask whether the practically meaningful search region is concentrated near sqrt(N) rather than distributed uniformly across the whole admissible lower-factor domain.

A second motivating observation concerns computation rather than pure number theory. The cost of constructing a verified prime deck is front-loaded and substantial. Yet once such a deck exists, repeated factor testing becomes much cheaper and much easier to parallelize than the deck-construction problem itself. This asymmetry matters for any serious evaluation of exhaustive or library-based one-factor search.

A third motivating observation concerns storage economics. Twenty years ago, even where one accepted the abstract possibility of large reusable prime decks, the cost of retaining and reusing such decks was far more restrictive than it is now. Long-run declines in the cost of persistent storage have changed that environment. Commodity machines can now retain data volumes that once belonged to much narrower budget classes. Accordingly, the accessibility of precomputation-based factor-search experimentation has changed over time, even where the underlying mathematics has not.

A fourth motivating observation is historical. Questions of this kind are not newly interesting simply because current hardware is faster. Similar concerns arose when smaller public-key regimes were being judged against the economics of the time. What has changed is the affordability of persistent storage, automation, and commodity orchestration. The present dissertation therefore treats itself partly as a re-evaluation of an older strategic question under updated economic assumptions rather than as a wholly new concern invented by modern tooling.

1.2 Problem Statement

The literature already recognizes several adjacent facts. RSA semiprimes are generally built from primes of similar size. Very close factors may weaken the modulus under Fermat-style methods. Unbalanced primes introduce different structural behaviors. Yet a gap remains between these facts and a clean, denominator-correct description of the admissible one-factor domain. That gap matters because a search argument framed against the full k-bit universe is mathematically different from a search argument framed against the admissible interval for the smaller factor.

A second gap is computational. Search cost is often treated as though it were a single undifferentiated burden. In reality, there is a major distinction between building a prime deck and using that deck. The former is expensive and persistent; the latter is amortized across many future search tasks and can be parallelized aggressively once the verified library exists.

A third gap is economic and historical. Earlier discussions of practical infeasibility often collapsed generation cost, storage cost, repeated testing cost, and parallelism into one broad intuition of impossibility. That intuition is increasingly incomplete because the costs of storage, coordination, and commodity compute have changed significantly over time.

Accordingly, this dissertation asks the following question: for a fixed-length semiprime, how much does a factor-balance constraint reduce and concentrate the admissible interval for the smaller factor relative to its unconstrained one-factor domain, and how should that concentration be interpreted under realistic search models that separate precomputation cost from amortized, highly parallel factor testing, that permit alternative scheduler policies, and that situate both within changing storage economics?

That question immediately separates into two related but non-identical subquestions. First, under a bounded lower-first prime deck, does looser key generation place the smaller factor below the active deck ceiling more often than tighter key generation does? Second, if the attacker knows that accepted factors are concentrated near sqrt(N) and has a scheduler that can prioritize that region, does tighter key generation become advantageous to the attacker by shrinking the practically meaningful search band? The dissertation treats these as distinct empirical hypotheses rather than as one interchangeable intuition about "tightness."

In that form, the topic is not presented as a claim of a new universal break of RSA or as a replacement for standards-based security analysis. Its value is different. The dissertation proposes a denominator-correct, experimentally testable framework for finite-deck tractability under explicit generation constraints and explicit search regimes. That framework is intended to contribute simultaneously to applied mathematics, through the formal treatment of admissible factor intervals and empirical search-surface reduction; to cryptography and cryptanalysis, through a bounded attack-economics model for constrained semiprime populations; to algorithm engineering, through direct comparison of lower-first, phased, and square-root-centered schedulers; and to empirical security and storage-systems research, through measurement of how reusable prime decks, worker orchestration, and storage architecture jointly shape practical search cost. The dissertation's claim to significance therefore rests not on overturning existing hardness doctrine, but on adding a disciplined finite-regime model that can be measured, falsified, and compared against naive full-range reasoning.

1.3 Research Objectives

The first objective is formal. The dissertation defines the admissible one-factor interval for fixed-length semiprimes and introduces a balance parameter that constrains the difference in size between the two prime factors. The second objective is comparative. The dissertation measures the width of the constrained interval against the width of the unconstrained admissible interval rather than against the full numeric range implied by the modulus bit length. The third objective is predictive. The dissertation tests whether the denominator-correct interval model predicts observed tractability better than naive bit-length-only or full-range baselines. The fourth objective is scheduler-oriented. The dissertation tests whether any population-level concentration near sqrt(N) translates into better practical performance for square-root-centered scheduling than for lower-first or mixed scheduling. The fifth objective is computational. The dissertation studies how this interval reduction interacts with prime density, precomputed-prime libraries, exhaustive search, phased search methods, and parallel computation, including the distinction between coverage-limited lower-first decks and square-root-centered search. The sixth objective is historical and economic. The dissertation evaluates how falling storage cost changes the accessibility of reusable prime-deck experimentation over time. The seventh objective is interpretive. The dissertation evaluates what the interval reduction, scheduler behavior, and the precompute-versus-search asymmetry do and do not imply for modern cryptographic security analysis.

1.4 Contributions

This dissertation is designed to make one primary contribution and two secondary contributions. The primary contribution is a denominator-correct formal model of the admissible one-factor interval for fixed-length semiprimes under explicit balance constraints, together with an empirical test of whether that model predicts observed tractability better than naive baselines and whether it identifies a practically concentrated search region near sqrt(N). That empirical test is intentionally split between two attack regimes: bounded lower-first deck coverage, where factor location relative to the deck ceiling dominates, and square-root-centered scheduling, where concentration near sqrt(N) may be exploitable directly if the deck reaches the relevant band. The first secondary contribution is a computational model showing that precomputed-prime search has a distinct economic regime from fresh search because deck construction is front-loaded while deck exploitation is reusable and far more parallelizable. The second secondary contribution is a historically grounded account of how falling storage cost changes the accessibility of reusable prime-deck experimentation over time and why earlier impossibility rhetoric should be revisited under current economics. In support of these contributions, the dissertation defines a tunable balance parameter, connects interval width to prime density and library coverage, evaluates phased reduction search, compares lower-first and square-root-centered scheduler policies, and translates the resulting theory into high-level security guidance without overstating the practical implications. The experimental program is correspondingly structured to do more than produce demonstrations: it is intended to validate or falsify the model through large accepted semiprime populations, milestone deck studies, predictive baseline comparisons, scheduler-policy comparisons, and statistical tests that can withstand committee scrutiny.

1.5 Organization of the Dissertation

Chapter 2 reviews the prior literature on semiprime structure, balanced and unbalanced RSA factors, Fermat-style attacks, and partial factor information. Chapter 3 defines the mathematical framework, including the admissible factor interval, balance parameterization, the correct denominator, and the formal predictive claims to be tested. Chapter 4 develops the computational model, focusing on exhaustive one-factor search, precomputed-prime libraries, parallelized deck exploitation, phased reduction methods, representative semiprime-population design, scheduler-policy comparisons, statistical validation design, and measured scaling baselines. Chapter 5 interprets the results in terms of security implications, storage economics, limitations, and future research directions. Chapter 6 is reserved for the experimental outcomes, predictive comparisons, scheduler comparisons, and statistical tests needed to convert the framework into a completed dissertation.

2. Prior Art and Literature Review

2.1 Overview

The idea that factor shape matters is not new. RSA literature has long recognized that the relationship between the two prime factors of a modulus affects the behavior of factorization methods. Work on close-prime vulnerability, unbalanced RSA, partial key exposure, and semiprime hardness assumptions all make use of structural constraints on factor size. What is less common is a general interval-based analysis that centers the admissible one-factor domain itself rather than a single attack family.

The literature review in this chapter therefore separates the field into five adjacent strands:

• close-prime and Fermat-type results
• balanced and unbalanced RSA assumptions
• partial-information and partial-key-exposure work
• standards-based key-size and security-strength guidance
• survey and exact-bound work on Fermat-style reasoning
• the remaining gap addressed by this dissertation

2.2 Close Factors and Fermat-Type Methods

One major strand of prior art concerns the case in which the prime factors of a modulus are too close to one another. Fermat-style factorization exploits the identity N = a^2 - b^2 = (a-b)(a+b), and therefore becomes effective when the two factors lie near the square root of N. This literature is important because it demonstrates that the region around sqrt(N) is not merely a heuristic curiosity but a structurally meaningful part of the factor space.

A particularly important recent paper is Hanno Böck’s Fermat Factorization in the Wild (2023), which showed that real systems still generated RSA keys vulnerable to close-prime factorization in practice. Böck’s contribution is not merely pedagogical. It demonstrates that implementation defects can place factor pairs into a tractable close-prime region and that this region can be found in live public-key datasets.

Source:

• Hanno Böck, Fermat Factorization in the Wild, IACR ePrint 2023/026. https://eprint.iacr.org/2023/026

A second relevant paper from the deeper-dive search is RSA cryptanalysis — Fermat factorization exact bound and the role of integer sequences in factorization problem (2023). That paper is useful here not because the dissertation adopts all of its claims, but because it explicitly attempts to formalize the precise boundary at which Fermat-style reasoning becomes effective. That makes it relevant to the present dissertation’s concern with interval width, even though the current work is not restricted to Fermat’s method.

Source:

• RSA cryptanalysis — Fermat factorization exact bound and the role of integer sequences in factorization problem, Journal of Information Security and Applications 78 (2023), 103614. https://www.sciencedirect.com/science/article/pii/S2214212623001989

However, the close-prime literature typically focuses on the threshold at which closeness becomes exploitable by Fermat’s method or its refinements. It does not usually frame the problem as one of admissible interval reduction under a general balance constraint. In other words, it proves that excessive closeness is bad, but it does not necessarily quantify how standard balance assumptions narrow the legitimate one-factor domain relative to a denominator-correct unconstrained baseline.

2.3 Balanced and Unbalanced RSA

A second major strand of prior art concerns balanced versus unbalanced RSA. Standard practice generally favors primes of comparable bit length. The reasons include resistance to special-case behavior, implementation convention, and the desire to avoid pathologies associated with highly skewed factors.

Mosca and Verschoor’s Factoring semi-primes with (quantum) SAT-solvers (2022) is relevant here because it explicitly describes the semiprimes of interest in RSA as products of two primes that are roughly equal in size. This is not the contribution of that paper, but it is an important statement of the background assumption this dissertation interrogates. The paper therefore serves as a modern source confirming that balance is built into the hardness model for the kind of semiprimes cryptographers actually care about.

Source:

• Michele Mosca and Sebastian R. Verschoor, Factoring semi-primes with (quantum) SAT-solvers, Scientific Reports 12, 7982 (2022). https://www.nature.com/articles/s41598-022-11687-7

On the other side of the literature, unbalanced RSA has been studied precisely because factor asymmetry changes the structure of the problem. One useful paper surfaced in the deeper search is Nitaj and Ariffin’s Implicit factorization of unbalanced RSA moduli (2014). That paper studies a setting in which multiple RSA moduli can be factored when their prime factors share structural relationships. It is not the same as the present dissertation's balance-interval argument, but it strengthens the background point that factor imbalance is not merely cosmetic. It changes what attacks become thinkable and what algebraic structure becomes useful.

Source:

• Abderrahmane Nitaj and Muhammad Rezal Kamel Ariffin, Implicit factorization of unbalanced RSA moduli, IACR ePrint 2014/548. https://eprint.iacr.org/2014/548

This body of work is relevant because it establishes that balance is not merely an aesthetic choice. It is a meaningful structural parameter. Still, the literature often treats balance as a condition relevant to particular attack families rather than as a parameter that changes the width of the admissible one-factor domain itself.

2.4 Partial Information and Structural Leakage

A third adjacent strand involves partial key exposure and partial factor information. There is a large literature examining what happens when some bits or relationships among RSA parameters are known. These works are highly relevant because they show that structural narrowing can change the effective hardness of the problem. Yet they are typically framed around leaked information, algebraic reconstruction, lattice techniques, or CRT-specific exposure, rather than around a search-space interpretation of factor-balance constraints.

A representative example is Partial key exposure attacks on RSA: Achieving the Boneh–Durfee bound (2019). That paper belongs to the partial-information tradition: if enough structure is known about the private key, then factorization or key recovery can be accelerated with lattice methods. Its relevance here is conceptual. It shows that narrowing the unknown domain changes hardness, even if the source of narrowing is leakage rather than key-generation structure.

Source:

• Partial key exposure attacks on RSA: Achieving the Boneh–Durfee bound, Theoretical Computer Science 761 (2019), 51–77. https://www.sciencedirect.com/science/article/pii/S0304397518305371

The present dissertation differs from these works in one important respect. It does not assume a leak. Instead, it studies the structural narrowing that follows from the key-generation constraints themselves.

2.5 Survey and Synthesis Literature

A fourth strand consists of synthesis and survey work. These papers are important because they reveal how the field currently organizes the topic. A recent example is A survey of Fermat factorization algorithms for factoring RSA composite numbers (2024). Survey work of this kind confirms that Fermat-style reasoning remains an active explanatory framework in the literature and that the community still treats factor proximity as a meaningful variable.

Source:

• Kannan Balasubramanian and Mohana Priya Pitchai, A survey of Fermat factorization algorithms for factoring RSA composite numbers, Multidisciplinary Science Journal 6 (2024), e2024ss0101. https://malque.pub/ojs/index.php/msj/article/view/1658

The weakness of survey literature for the present dissertation is also its value: it summarizes attack families, refinements, and historical developments, but it generally does not reframe the problem around the denominator used to measure the one-factor search domain.

2.6 Standards-Based Key-Size Guidance

A fifth adjacent strand is not an attack literature at all, but the standards literature that governs how practitioners are told to size cryptographic keys. This is important because the present dissertation is not intended to replace standards-based guidance with an ad hoc storage-cost heuristic. Rather, it is intended to supplement standard guidance with a new economic lens on one class of precomputed-prime search model.

The most important baseline source here is NIST Special Publication 800-57 Part 1, Revision 5, Recommendation for Key Management: Part 1 – General (2020). NIST provides security-strength mappings and key-size guidance intended for real-world system design and transition planning. That document does not analyze reusable prime-deck economics in the sense developed in this dissertation, but it provides the appropriate reference point for any discussion of what key sizes are conventionally considered adequate.

Source:

• Elaine Barker, Recommendation for Key Management: Part 1 – General, NIST SP 800-57 Part 1 Rev. 5 (2020). https://doi.org/10.6028/NIST.SP.800-57pt1r5

The relevance of this source is twofold. First, it prevents the dissertation from confusing a research metric with normative operational guidance. Second, it highlights the gap the dissertation actually addresses: standards documents classify security strength at the system level, whereas the present work studies the economics of one precomputed-prime search regime at the level of the smaller-factor domain.

2.7 Lattice Methods and Coppersmith's Small-Roots Technique

A sixth strand of prior art — and one of the most technically consequential — is the lattice-based approach to finding small roots of modular polynomials. Coppersmith's 1997 paper Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities demonstrated that if a polynomial modulo N has a root below a certain bound, that root can be recovered efficiently using the LLL lattice basis reduction algorithm. This result has far-reaching implications for RSA: it shows that partial knowledge of a factor (even a fraction of the bits of p) can be leveraged into complete factorization.

Source:

• Donald Coppersmith, Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities, Journal of Cryptology 10(4), 233–260 (1997). https://doi.org/10.1007/s001459900030

Coppersmith's technique is relevant to this dissertation for two reasons. First, it establishes that the effective difficulty of factoring depends not on the full bit-length of the modulus but on how much of the factor space remains unknown — a point the present work extends from a different direction. Coppersmith shows that leaked bits narrow the problem; the present dissertation shows that generation constraints narrow it structurally, without assuming any leak. Second, Coppersmith's result is the theoretical foundation for the partial key exposure literature (Section 2.4), making it the upstream dependency for an entire family of attacks the dissertation must position itself against.

2.8 Subexponential Factoring and the Number Field Sieve

The state of the art in general-purpose integer factoring is the General Number Field Sieve (GNFS), which achieves subexponential runtime in the bit-length of the modulus. Two references are essential here: Lenstra's 2000 survey Integer Factoring, which provides a comprehensive overview of the major factoring algorithms and their complexity, and the edited volume The Development of the Number Field Sieve (Lenstra and Lenstra, 1993), which documents the theoretical and practical evolution of the NFS from Pollard's original 1988 manuscript through its maturation.

Sources:

• Arjen K. Lenstra, Integer Factoring, Designs, Codes and Cryptography 19, 101–128 (2000). https://doi.org/10.1023/A:1008397921377
• Arjen K. Lenstra and Hendrik W. Lenstra Jr. (eds.), The Development of the Number Field Sieve, Lecture Notes in Mathematics 1554, Springer (1993). https://doi.org/10.1007/BFb0091534

Pomerance's A Tale of Two Sieves (1996) provides an accessible comparative treatment of the quadratic sieve and number field sieve, explaining the smooth-number heuristics that underpin both algorithms and why the NFS achieves better asymptotic complexity for large inputs.

Source:

• Carl Pomerance, A Tale of Two Sieves, Notices of the American Mathematical Society 43(12), 1473–1485 (1996). https://www.ams.org/notices/199612/pomerance.pdf

This body of work is important to the present dissertation in two ways. First, it defines the baseline against which all factoring claims must be measured. NIST SP 800-57 derives its security-strength estimates from the assumption that the best general-purpose attack is the GNFS. The present dissertation does not propose an algorithm that competes with the GNFS on general composites. Instead, it argues that for composites generated under specific balance constraints, the admissible search domain for the smaller factor is structurally narrower than the full domain the GNFS security estimates implicitly assume. The two claims are complementary, not competitive.

Second, the NFS literature establishes the computational regime in which precomputation plays a role. The NFS itself involves a substantial precomputation phase (sieving) followed by a matrix step. The present dissertation's prime-deck model is a different kind of precomputation — exhaustive prime enumeration rather than smooth-relation collection — but the economic pattern is the same: expensive one-time construction followed by cheaper, reusable exploitation. The NFS literature therefore provides a conceptual precedent for analyzing precomputation economics separately from per-target search cost.

2.9 Small Private Exponent Attacks

Boneh and Durfee's Cryptanalysis of RSA with Private Key d Less Than N^0.292 (1999) extended Wiener's earlier attack on small private exponents using Coppersmith's lattice techniques. This result demonstrates that RSA implementations choosing small private exponents for performance reasons can be vulnerable even when the modulus itself is large.

Source:

• Dan Boneh and Glenn Durfee, Cryptanalysis of RSA with Private Key d Less Than N^0.292, Advances in Cryptology — EUROCRYPT '99, Lecture Notes in Computer Science 1592, Springer (1999). https://link.springer.com/chapter/10.1007/3-540-48910-X_1

The Boneh-Durfee result is relevant to the present dissertation's broader argument: that security depends not only on modulus size but on the structural constraints imposed during key generation. A small private exponent is a generation constraint that creates vulnerability. The balance constraint studied in this dissertation is a different generation constraint — one that narrows the admissible factor domain rather than the private-key domain. In both cases, the lesson is the same: the effective security of an RSA modulus cannot be assessed from its bit-length alone.

2.10 Gap in the Literature

Taken together, the prior literature demonstrates seven established facts:

• factor shape matters
• excessive closeness may be exploitable by Fermat-type methods
• balanced and unbalanced RSA differ in meaningful ways
• narrowing the unknown parameter space can affect hardness, whether by leaked bits (Coppersmith) or structural constraints (Boneh-Durfee)
• factor search near sqrt(N) is already structurally important in the literature
• the best general-purpose factoring algorithms (GNFS) define the baseline against which security parameters are calibrated
• standards literature already provides baseline key-size guidance for operational security planning

Nevertheless, there remains room for a more general framework centered on the admissible interval for the smaller factor. The present dissertation is not a Fermat paper, a lattice paper, a partial-leakage paper, or a number field sieve paper. Its question is more basic: once modulus size and balance constraints are fixed, what is the correct admissible search interval for the smaller factor, and how much narrower is that interval than naive full-range reasoning suggests?

That question is not answered by showing that close primes are weak (Böck). Nor is it answered by showing that partial exposure helps lattice attacks (Coppersmith, Boneh-Durfee). Nor is it answered by improving the asymptotic complexity of general-purpose factoring (Lenstra, Pomerance). It requires a denominator-correct interval model and then an empirical study of how that model behaves under exhaustive and precomputed-prime search.

Just as importantly, the literature reviewed here does not already appear to formalize the precise finite-deck question pursued in this dissertation. The reviewed sources establish that close primes matter for Fermat-type attacks, that balanced and unbalanced RSA moduli behave differently, and that structural narrowing can matter when information is leaked. They do not, in this literature review, already provide a controlled framework for comparing tight, moderate, loose, and unconstrained semiprime populations under explicit prime-deck ceilings while also distinguishing lower-first from square-root-centered scheduler policies. That combined finite-deck-plus-scheduler question is the novelty boundary this dissertation is attempting to test.

2.11 Position of This Dissertation

This work therefore does not claim novelty in the observation that factor proximity matters. Its claim to novelty rests instead on four linked moves:

• choosing the admissible one-factor interval as the denominator

3. Formal Model and Theory

3.1 Fixed-Length Semiprimes

Let N = p q where p and q are prime and p <= q. Let N be constrained to have a fixed bit length k. The fundamental question is not how large the k-bit universe is, but where the smaller factor p can legitimately lie under the construction rules imposed on the modulus.

For any such semiprime, p cannot exceed sqrt(N). Since q = N / p and p <= q, the smaller factor belongs to a one-sided interval capped by sqrt(N). This is the natural one-factor domain. The dissertation treats this interval, rather than the full k-bit universe, as the correct baseline for structural search reasoning.

3.2 The Admissible One-Factor Domain

Define the admissible one-factor domain D(N) as the set of prime values p such that p divides N and p <= sqrt(N), under the generation rules that determine which semiprimes are valid objects of study. In an unconstrained model, the admissible domain extends from the lower bound implied by validity conditions up to sqrt(N). In a balanced model, the admissible domain tightens around sqrt(N).

The key conceptual point is that admissibility is not synonymous with mere numerical possibility. If the factor-generation process imposes bit-length or balance requirements, then portions of the naive interval are not actually part of the valid domain.

3.3 Balance Parameterization

Let delta denote a factor-balance parameter. At a high level, delta controls the permitted difference in size between p and q. One convenient formulation is:

|log2(p) - log2(q)| <= delta.

When delta is small, p and q must be very close in bit length. When delta is larger, the permissible range broadens. This dissertation does not tie itself to a single equivalent formulation of balance; what matters is that delta induces a corresponding interval for p around sqrt(N).

3.4 Interval Width and Prime Density

Once the admissible interval is defined, the number of candidate primes inside that interval can be estimated using the prime number theorem. Around a large value x, the local density of primes is approximately 1 / ln(x). Therefore, the number of admissible primes in an interval [a, b] is approximately:

pi(b) - pi(a) ≈ (b - a) / ln(x)

for x in the same scale regime. This lets the dissertation move from a raw interval-width argument to an actual candidate-count argument.

3.5 Reduction Ratio

The core theoretical quantity is the interval reduction ratio:

R(delta, N) = width(constrained admissible interval) / width(unconstrained admissible interval).

This ratio is the denominator-correct way to talk about narrowing. It is not meaningful to compare the constrained factor interval to the entire k-bit universe of N. The relevant comparison is always against the natural one-factor domain.

3.5.1 Empirical Search-Surface Reduction

The interval reduction ratio is still a structural quantity. It describes the admissible interval implied by the model, not the empirically occupied portion of that interval under a representative generated population. Because the dissertation's motivating claim is that the practically meaningful search surface may be materially smaller than naive intuition suggests, the formal chapter must also make room for an empirical reduction factor derived from the observed smaller-factor distribution.

Let E(delta, k) denote the empirical search-surface ratio for a generated semiprime population of modulus class k and balance class delta. At a high level, this quantity measures how much of the admissible one-factor domain is actually occupied by the accepted smaller-factor population under the study's key-generation rules. In other words, it is a population-level divisor on top of the structural interval description.

The dissertation does not assume this factor a priori. It treats it as something to be estimated experimentally. If the experiments show that the accepted smaller-factor population clusters tightly near sqrt(N), then E(delta, k) may be substantially smaller than the structural interval width alone would imply. If the experiments show broad dispersion, then the empirical reduction is weak and the dissertation's stronger concentration claim fails.

The same logic motivates a second empirical quantity for the dissertation's computational chapters. Let

P_hit(X, k, delta, s)

denote the probability that a valid semiprime drawn from modulus class k and balance class delta is resolved or otherwise yields a relevant factor-discovery event under deck milestone X and scheduler/search policy s. Depending on the reporting context, this can be instantiated as:

• probability of full resolution
• probability of partial resolution
• probability that the smaller factor lies below the active deck ceiling
• probability of first-factor discovery within a stated runtime budget

The key point is that below exhaustive coverage the dissertation need not reduce everything to an all-or-nothing claim about complete search. A finite-deck experiment can still estimate a population success probability. The ladder experiments are therefore designed not only to ask whether a class is exhaustively searchable, but to estimate how P_hit(X, k, delta, s) changes as deck size, modulus class, balance class, and scheduler policy vary.

3.6 Theoretical Consequences

Three consequences follow immediately. First, balanced-generation rules may shrink the admissible one-factor interval substantially relative to a naive unconstrained baseline. Second, the candidate-prime reduction depends not only on interval width but also on prime density within that region. Third, the significance of the reduction depends on the computational model used to search it. A narrowing that is mathematically real may still be computationally irrelevant at sufficiently large key sizes. Fourth, if the admissible smaller-factor population becomes sufficiently concentrated near sqrt(N), then scheduler policy itself becomes a theoretically meaningful variable rather than a mere implementation detail. Fifth, any cost model based purely on exhaustive structural coverage should ultimately be paired with an empirical divisor derived from the observed population search surface.

3.7 Formal Claims to Be Tested

The formal work in this dissertation is not complete unless it produces claims that can be tested against observed data. Accordingly, Chapter 3 is intended to support three explicit claims.

First, for fixed-length semiprimes generated under a balance parameter delta, the admissible one-factor interval for the smaller factor is narrower than the unconstrained one-factor interval. Second, the relevant denominator for structural search reasoning is the unconstrained admissible one-factor interval rather than the full k-bit numeric universe. Third, the constrained interval width should have predictive value for observed tractability under fixed prime-deck ceilings. Fourth, if the accepted smaller-factor population is measurably concentrated toward the square-root end of the admissible interval, then square-root-centered scheduling should be tested as a competing practical search policy rather than assumed away. Fifth, the dissertation should estimate an empirical search-surface reduction factor and test whether it materially improves economic and computational predictions beyond the structural interval model alone. Sixth, the dissertation should estimate the population success surface P_hit(X, k, delta, s) and determine whether that probability surface yields a more informative description of finite-deck behavior than an exhaustiveness-only framing.

These claims turn Chapter 3 from a vocabulary chapter into a model chapter. The later experimental work is therefore not merely illustrative. It is intended to validate or falsify the predictive content of the formal model.

3.8 Mathematical Outputs Still Required

For this dissertation to succeed at the PhD level, the formal chapter must ultimately supply more than definitions. It must provide, at minimum:

• a precise lower and upper description of the admissible interval for the smaller factor under fixed k and delta
• one or more propositions establishing how the interval width changes as delta varies
• a prime-density translation from interval width to candidate-prime count
• a clear statement of what the model predicts about the probability that the smaller factor falls below a given deck ceiling
• a clear statement of how the deck-ceiling event and the observed resolution event combine into an empirical success quantity such as P_hit(X, k, delta, s)
• a clear statement of when the model implies concentration near sqrt(N) strong enough to justify scheduler-level testing against lower-first baselines
• a clear statement of how the observed smaller-factor distribution induces an empirical search-surface divisor on top of the structural interval model

These outputs need not all be closed-form in their final presentation, but the dissertation must eventually move beyond intuition and supply a mathematically defensible predictive model.

3.8.1 Search-Regime Dependence

The formal model must also distinguish the search regime under which any predicted tractability claim is being made. Three regimes matter in the present dissertation.

• lower-first or bottom-up search, in which the attacker tests admissible primes from the low end upward until the deck or the residual square-root bound is exhausted
• square-root-centered search, in which the attacker deliberately prioritizes a band near sqrt(N) because the accepted smaller-factor population is believed to be concentrated there
• mixed search, in which lower and upper bands are interleaved or scheduled adaptively

This distinction matters because the sign of the practical effect may change across regimes. Under a bounded lower-first deck, looser populations may be easier because they place the smaller factor below the active deck ceiling more often. Under a sufficiently large deck together with square-root-centered scheduling, tighter populations may instead be easier because the attacker can search a narrower high band rather than the full lower interval. The dissertation therefore treats concentration near sqrt(N) as a conditional computational advantage, not as an unconditional statement that tighter keys are always easier to factor.

3.9 Experimental Commitments Implied by the Model

Because the formal model makes predictive claims, it also imposes methodological obligations on the later empirical chapters. The experiments cannot be satisfied by a handful of illustrative composites alone. They must generate accepted semiprime populations large enough to test whether constrained interval width, interval reduction ratio, and deck-ceiling interaction actually explain outcomes better than simpler baselines.

At minimum, the empirical work implied by Chapter 3 must support the following comparisons:

• denominator-correct interval variables versus modulus-bit-length-only reasoning
• denominator-correct interval variables versus naive full-range comparisons
• tight, moderate, loose, and unconstrained balance populations under the same deck regime
• multiple deck milestones so that regime transitions can be observed rather than assumed
• lower-first, square-root-centered, and mixed scheduler policies on the same accepted populations once worker demand exceeds simple file-level saturation

This is the point at which the dissertation becomes more than a structural essay. Chapter 3 establishes what the model predicts. Chapters 4 and 6 must then determine whether those predictions survive controlled data.

3.10 Why Key-Pair Creation Must Be Population-Based Under Exhaustive Deck Coverage

Once a prime deck becomes exhaustive through a known ceiling, the experimental problem changes. The question is no longer whether a specially chosen example can be resolved. The question becomes what happens to a representative population of valid semiprimes when the lower-factor domain is known comprehensively up to a fixed bound. Under those conditions, cherry-picked composites are evidentiary weak points rather than strengths. They can illustrate mechanics, but they cannot support claims about typical tractability.

Accordingly, the dissertation treats key-pair creation as a population-construction problem. Each accepted pair must be generated under explicit modulus-size and balance rules, and the resulting population must be large enough to support distributional claims. This is especially important when the deck ceiling is high enough to cover every smaller factor in a substantial portion of the admissible domain. In that regime, the central question is not whether the system can factor some example, but how much of the valid population lies inside the deck-covered region, how strongly that population is concentrated toward the square-root end of the interval, and how rapidly those instances are resolved in practice.

This is why the experimental design insists on accepted semiprime datasets rather than independent range sampling. A comprehensive deck creates the appearance that the target class is "fully searchable," but that phrase only has meaning with respect to a defined population. The dissertation therefore defends population-based key-pair creation as the only valid way to translate exhaustive deck coverage into statements about real tractability, representative vulnerability regions, the actual rather than imagined search surface, and the scheduler policies that should be expected to perform best if the search surface is in fact concentrated near sqrt(N).

4. Computational Model and Experimental Framework

4.1 Why a Computational Model Is Needed

The interval reduction ratio R(delta, N) is a structural quantity, not a practical runtime result. To understand whether the narrowing matters in practice, the dissertation requires a computational model that connects interval width to search cost. This chapter therefore develops a framework for evaluating factor search under exhaustive and precomputed-prime approaches.

A central claim of this dissertation is that precomputation changes the economics of search. Building a large verified prime deck is expensive. However, once that deck exists, factor testing becomes a library-query problem whose marginal cost per target is much lower and whose work naturally distributes across many workers. The practical meaning of an admissible interval is therefore different in a world with reusable prime decks than in a world where every target begins from scratch.

4.2 Exhaustive One-Factor Search

The baseline model is exhaustive search over admissible prime candidates for the smaller factor p. In its simplest form, this means enumerating candidate primes in the admissible interval and testing divisibility. The cost of this model depends on the number of candidates, the cost of each divisibility test, and the degree of available parallelism.

The structural result from Chapter 3 matters immediately here: once the admissible interval is narrowed, the exhaustive candidate count narrows with it. This does not prove a break. It merely means that search should be measured against the correct denominator.

4.3 Precomputed Prime Libraries

A second model assumes a library of verified primes has already been built and stored. In that case, factor testing becomes a library-scan problem rather than a prime-generation problem. The computational burden moves from primality generation to divisibility testing over a known set. This model is important because it changes the economics of repeated search: deck construction is expensive, but once completed, repeated candidate testing becomes much cheaper and easier to parallelize.

The dissertation therefore treats prime-library construction and prime-library search as two separate costs. It also notes that a library can only cover factors up to its maximum stored prime. Consequently, the relation between library ceiling and admissible interval matters directly.

The key asymmetry is this: deck construction is mostly a one-time capital cost, whereas factor testing is an operational cost that can be amortized across many targets. This distinction becomes especially important once the number of candidate targets is large or when the same library is reused over a long period.

4.4 Parallel Search After Precomputation

Once the deck is built, the search phase lends itself to aggressive parallelization. The verified prime deck can be partitioned by shard, by numeric interval, or by file, and different workers can scan different portions of the admissible factor space concurrently. This makes factor testing far more scalable than prime-deck construction itself.

The dissertation therefore treats parallelism not as a minor engineering detail but as part of the computational model. A precomputed-prime approach has two distinct phases:

• a comparatively expensive build phase
• a comparatively cheap and massively parallel search phase

This asymmetry means that a structurally narrowed admissible interval can become practically more significant once a large reusable deck already exists.

At the present implementation stage, the operational factoring stack uses at most one worker per eligible prime file. This is an important detail because it defines the current meaning of the worker-count parameter. If eight files are eligible, then the current system reaches natural file-level saturation at eight workers. Additional workers do not yet create more search throughput because the implementation does not subdivide a single file into multiple independent search regions.

The dissertation should nevertheless model the next scheduling question explicitly. Once the number of available CPU workers exceeds the number of eligible files, additional parallelism can only be exploited by subdividing file interiors. In principle, that can be done by partitioning each file into ordinal or byte ranges and assigning multiple workers to non-overlapping regions of the same file. A simple example would be one worker scanning upward from the lower end of a file while another scans downward from the upper end. That design may reduce expected time to first hit when the smaller factor could plausibly lie anywhere inside the file, but it also introduces overheads in coordination, duplicate-bound management, cancellation, and diminished early-exit efficiency if the useful factor is likely to occur near the lower end.

Accordingly, the dissertation should not assume that "more workers than files" yields proportional improvement. The correct prediction is conditional. Extra workers beyond file count help only if the implementation supports intra-file sharding and only if the gain from earlier hit detection exceeds the cost of additional orchestration and cancellation.

4.5 Phased Reduction Search

A third model, introduced here, is phased reduction search. In this model, whenever a factor is found, the candidate composite is reduced immediately and the admissible search interval is recomputed for the remainder. This creates a sequence of shrinking problems rather than one static problem. The dissertation argues that phased reduction is a more realistic model for structured composites with multiple small factors because it avoids wasting work on obsolete bounds.

Phased reduction is especially important in a library-based environment because workers can be cancelled and relaunched against a smaller remainder. This makes the search phase not only parallel but adaptive.

4.6 Parallelism and Orchestration

Parallel computation changes the economics of search, but only if the orchestration layer is aware of changing problem state. A naive one-worker-per-file model wastes resources once the remaining candidate shrinks. An effective orchestration layer must support three operations: immediate factor reporting, cancellation of stale work, and restart against the new remainder. This lesson generalizes well beyond factoring and applies to any search problem in which discoveries alter the residual domain.

The dissertation therefore distinguishes between raw parallel capacity and useful parallel capacity. Useful parallelism requires coordination. Otherwise the system may merely accelerate redundant work.

This distinction becomes sharper when comparing two regimes:

• file-limited parallelism, where eligible files are fewer than or equal to available workers
• core-limited parallelism, where eligible files exceed available workers

In the first regime, the next meaningful optimization is intra-file sharding. In the second regime, the scheduler should generally prefer lower-prime files before higher-prime files because those files dominate the probability of early factor discovery and determine whether higher ranges are needed at all. Once eligible file count exceeds available workers, a rational scheduler should therefore treat lower files as higher-priority queues and defer upper files until lower ranges have either completed or been rendered unnecessary by a residual-bound update.

The dissertation should therefore predict three different performance regions:

• below file saturation: adding workers reduces runtime materially because additional files can be searched concurrently
• at file saturation without intra-file sharding: adding workers has little or no effect
• above file saturation with intra-file sharding: additional workers may help, but the gain is sublinear and depends on range-splitting strategy, factor distribution within the file, and cancellation overhead

4.7 Experimental Design Principle

The experiments in this dissertation must sample from the space of valid semiprime pairs rather than from independently chosen factor bands. This distinction is essential. If two factors are sampled independently from hand-selected ranges and merely multiplied, the resulting composite may fail to satisfy the intended modulus-size or balance constraints. Such a procedure would contaminate the study population.

Accordingly, every experiment must proceed by first defining a target modulus class and then generating prime pairs that satisfy that class. The modulus class is determined by:

• target modulus bit length k
• primality of both factors
• ordering p <= q
• balance constraint delta

Only after a prime pair satisfies these conditions should the resulting semiprime be admitted into the dataset.

This requirement becomes even more important once the prime deck is exhaustive through a known bound. Under exhaustive lower-factor coverage, any experiment based on hand-selected or aesthetically convenient examples risks overstating or understating tractability by substituting anecdote for population structure. The purpose of the dissertation is not to show that some composite can be resolved when the smaller factor happens to lie inside the covered region. It is to determine how often valid semiprimes generated under explicit rules fall into that region, how quickly they are resolved, and how those rates change across deck milestones and balance classes.

The experimental design therefore defends key-pair creation on methodological rather than intuitive grounds. A valid experimental population must represent the actual space of admissible key pairs under the stated generation constraints. Once a deck is comprehensive across a large lower-factor interval, the study can no longer rely on toy examples or arbitrarily assembled products. It must estimate empirical tractability over a large accepted population and report the share of that population whose smaller factor is below the active deck ceiling. That is the only form of evidence that can support statements about the practical search surface created by exhaustive library coverage.

This defense does not depend on claiming that the dissertation exactly reproduces one specific production key generator. It depends on something narrower and more defensible: the populations are sampled from an explicitly stated admissible class, under explicitly stated balance rules, and are therefore appropriate for testing how tractability changes when those rules vary. The resulting datasets are synthetic in origin but valid with respect to the formal population the dissertation defines and studies.

4.8 Synthetic Semiprime Generation

For a fixed target modulus size k, the experiment generates semiprimes under multiple balance settings delta. Each dataset contains a large number of semiprimes whose factors satisfy a prescribed bit-length-gap rule. The scale of the study matters. Pilot runs may begin at 1,000 accepted instances per population, but the main dissertation evidence should extend to 10,000 and, where computationally practical, 100,000 accepted instances so that confidence intervals, effect sizes, and distributional comparisons are meaningful. At minimum, the study should include:

• tightly balanced semiprimes
• moderately balanced semiprimes
• loosely balanced semiprimes
• unconstrained comparison semiprimes subject only to validity conditions

A mathematically correct generation process is as follows:

1. Choose the target modulus size k.
2. Choose the balance parameter delta.
3. Sample a candidate prime p from the admissible lower-factor domain implied by k and delta.
4. Determine the admissible range for q from the conditions that:
   • q is prime
   • p <= q
   • pq has bit length k
   • the balance rule |log2(p) - log2(q)| <= delta holds
5. Accept the pair only if all constraints are satisfied.

The output of this phase is a family of semiprime datasets for which the location of the smaller factor is known exactly and for which every instance belongs to the intended modulus class.

Operationally, heuristic factor bands may still be used to seed the generator. For example, if the study examines 128-bit semiprimes under a deck exhaustive through 70-bit primes, then the interesting region is naturally centered around the 64-bit square-root scale. However, these bands are only proposal mechanisms. They are not the population definition. The accepted dataset consists only of prime pairs whose product actually lands in the target modulus class and whose balance constraint is satisfied.

This distinction is the core defense of the generation method. When the dissertation evaluates the effect of a large deck that covers every smaller factor up to a known limit, it is effectively estimating a population proportion: the share of valid key pairs whose smaller factor lies inside the covered region and is therefore, in principle, reachable by exhaustive deck search. That quantity cannot be estimated from manually assembled examples. It requires repeated random generation, acceptance filtering, and enough observations to support confidence intervals and comparative statistics across delta classes.

For the transition-zone ladder proposed in this dissertation, the default target is 1,000 accepted candidates per primary experiment and 250 accepted candidates per anchor experiment. The reason is statistical as well as computational. For a proportion-valued outcome such as "fully resolved," "partially resolved," or "smaller factor below the active deck ceiling," a binomial sample of 1,000 has a worst-case 95% confidence interval of approximately +/- 3.1 percentage points near p = 0.5. A binomial sample of 250 has a corresponding worst-case interval of approximately +/- 6.2 percentage points. The larger count is therefore appropriate for the main transition-zone studies, where the dissertation wants to compare nearby resolution rates across tight, moderate, loose, and unconstrained populations. The smaller count is appropriate for high-bit anchor studies, where the purpose is trend checking and extrapolation testing rather than precise small-effect estimation.

4.9 Core 128-Bit Experimental Program

A particularly important study uses 128-bit semiprimes as the baseline illustrative class. In this setting, the square-root scale is approximately 64 bits. If a prime deck exists that is exhaustive through some higher ceiling, such as 70 bits, then the experiment can ask how often a realistic 128-bit semiprime becomes tractable under that deck when generated subject to traditional balance constraints.

The correct way to run this experiment is not to choose one factor from a presumed upper band and the other from a presumed lower band independently. Rather, the experiment must generate valid 128-bit semiprime pairs under the chosen balance rule and then test those composites against the fixed deck ceiling.

In practical terms, the study can be organized around three related regions:

• the 128-bit modulus class itself
• the 64-bit square-root scale that centers the smaller-factor search problem
• the current deck ceiling, for example a hypothetical exhaustive 70-bit prime bound

This makes it possible to ask an empirical question rather than an intuitive one: among valid 128-bit semiprimes generated under traditional balance constraints, what fraction become fully or partially tractable under a deck that extends beyond the square-root baseline but remains far below the full modulus size?

The experimental process is:

• generate a large sample of valid 128-bit semiprimes
• separate the sample by delta class
• test each semiprime with the factoring system under a fixed deck ceiling
• record the search outcome and search cost

The sample sizes should scale in stages, for example:

• 1,000 instances for pilot testing
• 10,000 instances for stable initial comparison
• 100,000 instances for stronger statistical claims

This same design extends naturally to 256-bit, 512-bit, and 1024-bit modulus classes.

4.10 Live Factor Testing Using the Existing Factoring Tool

The existing factoring tool and prime deck are used as the experimental platform. For each candidate semiprime, the system records:

• whether a factor was found
• the largest prime in the deck at the time of the run
• whether the candidate was fully resolved, partially resolved, or left inconclusive
• wall-clock time to first factor
• wall-clock time to completion or inconclusive termination
• number of phases triggered in phased reduction mode
• number of eligible files examined
• number of worker cancellations caused by dynamic bound reduction
• unresolved remainder size in bits
• verification that any reported full factorization multiplies back to the original input

These measurements allow the dissertation to connect abstract interval reasoning to observable computational behavior.

The dissertation now distinguishes between the browser-facing local factoring site and the batch-oriented experiment stack. The browser site remains useful for exploratory manual probes. The dissertation experiments, however, are driven by a separate software path designed for accepted-population generation, repeatable reruns, and structured persistence. That stack currently includes a semiprime generator, a PostgreSQL experiment schema, an instrumented factoring engine, a database-backed queue runner, and a manual candidate-list runner for structured-composite families.

The implemented experiment software is:

• init-experiment-db.js, which creates the experimental schema
• experiment-db.js, which manages datasets, sessions, candidates, runs, phases, files, and deck-state records
• generate-semiprime-dataset.js, which generates accepted k-bit semiprime populations under tight, moderate, loose, and unconstrained balance classes
• factoring-engine.js, which runs controlled factor searches and emits structured metrics under both phased and static modes
• run-experiment-queue.js, which consumes database-backed semiprime datasets sequentially
• run-candidate-list.js, which runs newline-delimited structured-composite families or manually curated candidate lists

This separation matters because the dissertation requires provenance, replayability, and controlled variation. A browser history is not sufficient. The experiment stack preserves the exact dataset identity, session identity, search mode, worker count, and deck state associated with each run.

4.11 Comparative Search-Width Analysis

For each semiprime instance, the study compares three quantities:

• the unconstrained admissible interval width for the smaller factor
• the constrained interval width implied by delta
• the observed search effort under the actual prime deck and factoring program

This produces a direct test of the theory. If the dissertation’s theory is correct, then tighter balance constraints should systematically place the smaller factor into a narrower admissible band around sqrt(N). However, the computational consequence of that narrowing is not unidirectional. Under a bounded lower-first deck, looser populations may produce higher hit rates because they place the smaller factor below the active deck ceiling more often. Under a sufficiently large deck together with square-root-centered scheduling, tighter populations may instead become easier because the search can be concentrated in a narrower band near sqrt(N). The dissertation should therefore interpret interval narrowing through the joint lens of deck adequacy and scheduler direction rather than assuming that tighter balance automatically means easier factor discovery.

At the population level, this chapter should go one step further. The dissertation should estimate not only structural interval widths but also the empirically occupied search surface. That is, for each accepted semiprime population, it should estimate how much of the admissible domain is actually populated by observed smaller factors. This empirical occupation ratio is the quantity that converts the claim "the search space is smaller than intuition suggests" from a slogan into a measurable divisor.

The same population framing should be used for computational outcomes. For a deck milestone X, modulus class k, balance class delta, and scheduler policy s, the dissertation should estimate an empirical success quantity P_hit(X, k, delta, s). In practice, the reported version may be "full resolution probability," "partial-or-full resolution probability," or "probability that the smaller factor lies below the active deck ceiling," depending on the table being discussed. This is the right framing for transition-zone experiments. The goal is not merely to ask whether a class is exhaustively searchable, but to estimate how often a known search regime succeeds over a valid population even when the deck remains incomplete for the full modulus class.

The transition-zone emphasis should also be defended directly. The dissertation is not selecting those modulus classes because they guarantee favorable outcomes. It is selecting them because they are the region in which competing hypotheses are most distinguishable. Deep below the deck ceiling, all reasonable balance classes tend toward saturation. Far above the deck ceiling, all reasonable balance classes tend toward near-total failure. Near the deck edge, by contrast, the model's predictive content is strongest because different balance classes and scheduler policies can produce measurably different probabilities and runtimes.

One practical way to organize this interpretation is a 2 x 2 grid:

• bounded deck plus lower-first search
• bounded deck plus square-root-centered search
• sufficiently large deck plus lower-first search
• sufficiently large deck plus square-root-centered search

The dissertation should expect different outcome rankings across these four cells rather than one universal ordering of tight, moderate, loose, and unconstrained.

4.12 Experimental Hypotheses

The dissertation proposes the following hypotheses.

H1. Balance narrows the admissible one-factor interval.

For fixed-length semiprimes, tighter balance constraints produce a smaller admissible interval for the smaller factor than looser or unconstrained generation rules.

H2a. Under bounded lower-first decks, looser populations may produce higher hit rates.

For a fixed prime-library ceiling searched from the low end upward, semiprime datasets generated under looser balance constraints may exhibit a higher proportion of resolved or partially resolved cases than tighter datasets because the smaller factor falls below the active deck ceiling more often.

H2b. Under square-root-centered scheduling, tighter populations may produce faster search.

If the attacker has enough deck coverage to search a meaningful band near sqrt(N) and uses a square-root-centered scheduler, tighter balance constraints may reduce time to first factor relative to looser populations by concentrating accepted smaller factors into a narrower high band.

H3. The denominator-correct interval model predicts observed behavior better than naive full-range reasoning.

Observed resolution probabilities P_hit(X, k, delta, s) and time-to-first-factor measurements will correlate more strongly with constrained admissible interval width than with comparisons against the full numeric range implied by the modulus bit length.

H4. Phased reduction materially improves performance on structured composites.

For candidates with multiple small factors, phased reduction will reduce total runtime and unnecessary worker activity relative to a static one-pass file scan.

H5. Precomputation shifts the economics of search from generation to massively parallel reuse.

Once a verified deck has been built, repeated factor testing will exhibit substantially better parallel scalability and lower marginal cost per target than the original deck-construction process.

H6. The interval model has predictive value beyond naive baselines.

Given a fixed deck ceiling and fixed modulus class, constrained interval width and related model variables should predict observed resolution outcomes better than naive baselines that use only total modulus size or full-range numeric comparisons.

4.13 Concrete Experiments Using the Existing Factoring Tool

The dissertation can be supported by an immediate experimental program using the existing software stack.

Experiment A. Structured progression composites

Construct a sequence of large composites modeled on the Clay poster style examples, extending them by appending structured numeric blocks so that each successive candidate becomes more difficult. Run the factoring program against each candidate as the prime deck grows from one milestone to the next, for example 50M, 100M, 200M, 400M, and beyond. Measure:

• first factor discovered
• number of factors discovered
• unresolved remainder size in bits
• time to first factor
• time to final status

Purpose: quantify how phased reduction and deck growth interact on structured composites with many small factors.

Execution path: this experiment is automated through run-candidate-list.js. The candidate family is stored as a newline-delimited list, then replayed under controlled sessions. The same list can therefore be rerun at different deck milestones, under different worker counts, and under phased versus static search without changing the underlying candidate corpus.

Experiment B. Synthetic semiprime balance sweep

Generate semiprimes of a fixed target size, such as 96-bit, 112-bit, or 128-bit, under several balance constraints delta. For example, produce one dataset where p and q differ by at most one bit, another where they differ by up to four bits, another where they differ by up to eight bits, and an unconstrained baseline. Run the factoring tool against all instances using the same prime deck. Measure:

• proportion fully factored
• proportion partially factored
• proportion inconclusive
• smallest factor found as a fraction of sqrt(N)
• time to first factor
• unresolved remainder size

Purpose: directly test how balance class interacts with deck adequacy and scheduler direction. Under bounded lower-first decks, the experiment asks whether looser populations leak more smaller factors into the covered region. Under square-root-centered scheduling, the experiment asks whether tighter populations make the near-sqrt(N) band more exploitable once the deck reaches that region.

The central outputs of Experiment B should therefore be expressed as estimated probabilities with confidence intervals. For each combination of deck milestone, modulus class, balance class, and scheduler policy, the dissertation should report quantities such as:

• P(full resolution | X, k, delta, s)
• P(partial or full resolution | X, k, delta, s)
• P(smaller factor below deck ceiling | X, k, delta, s)

This reframes the experiment from a demonstration of isolated successes into a population-level estimate of practical reach under finite deck coverage.

Methodological defense: this experiment treats candidate creation as sampling from the admissible semiprime population, not as construction of didactic examples. That choice is necessary because the existence of an exhaustive deck changes the evidentiary burden. Once the deck covers all smaller factors up to its ceiling, the relevant quantity is the proportion of the valid population that lies beneath that ceiling, not the behavior of selected showcase composites.

Execution path: generate-semiprime-dataset.js creates the accepted populations and stores them in PostgreSQL under dataset names keyed by modulus class and delta class. run-experiment-queue.js then claims pending members of those datasets and executes them sequentially. Because sessions are stored separately from datasets, the same accepted population can be rerun at 1, 2, 4, 8, 16, or 28 workers and under both phased and static modes without regenerating the semiprimes.

Interpretive note: the same accepted population should be rerun under both lower-first and square-root-centered scheduling once the latter exists in software. Without that scheduler split, the dissertation risks conflating two different claims: whether balance changes the probability that the smaller factor falls under a bounded deck ceiling, and whether balance changes the efficiency of a search policy that deliberately starts near sqrt(N).

Experiment C. 128-Bit Fixed-Deck Study

Fix a deck ceiling, for example one exhaustive through a 70-bit prime bound, and evaluate a large synthetic population of valid 128-bit semiprimes generated under several delta classes. The dataset must consist only of accepted prime pairs whose products actually fall in the 128-bit modulus class. A useful staged ladder for this study is the sequence of 400M, 800M, and 1.6B prime-deck milestones. The estimated 400,000,000th prime is roughly 8.7 billion, or about 34 bits, implying an approximate square-coverage bound of about 68 bits. The estimated 800,000,000th prime should fall in the high-18-billion range, near 35 bits, implying a square-coverage bound of about 70 bits. The estimated 1,600,000,000th prime should fall in the high-30-billion range, near 36 bits, implying a square-coverage bound of about 72 bits. Together, these three milestones provide a practical experimental ladder: 400M should fully cover the 64-bit composite regime and open the first transition zone above it; 800M should strengthen the transition-zone study across the upper-60-bit and low-70-bit region; and 1.6B should extend that comparison farther upward while still leaving 96- and 128-bit semiprime classes clearly outside full deck coverage. Measure:

• fraction fully resolved
• fraction partially resolved
• fraction inconclusive
• distribution of smallest discovered factor
• time to first factor
• time to final status
• unresolved remainder size in bits
• share of instances whose smaller factor lies below the active deck ceiling

Purpose: quantify how tractable realistically generated 128-bit semiprimes are under a fixed reusable deck and determine what the effective search domain looks like for representative valid key-pair populations rather than hand-constructed examples.

Interpretive rule: if the deck is exhaustive through the active ceiling, then every accepted semiprime whose smaller factor lies below that ceiling is, in principle, inside the deck-searchable region. Experiment C therefore estimates not only observed runtime outcomes but also the empirical size of the deck-covered subset of the admissible 128-bit population. This is the bridge between structural interval theory and practical claims about what portion of a real key-pair population is susceptible to exhaustive library search at a given milestone.

Experiment D. Deck-growth threshold study

Choose a fixed test family and rerun it each time the prime deck crosses a major milestone. Record the moment at which previously inconclusive numbers become partially or fully tractable. This experiment transforms deck size into the independent variable. Measure:

• deck ceiling at each milestone
• percentage of test family resolved at each milestone
• change in unresolved remainder size across milestones

Purpose: quantify how library growth alters practical tractability over time.

Experiment E. Static scan versus phased reduction

Instrument the factoring tool to compare the current phased-reduction implementation with a static scan that does not restart after factor discovery. Run both modes on the same structured composite set. Measure:

• elapsed runtime
• file-worker utilization
• number of unnecessary file scans
• time to first factor
• time to completion

Purpose: prove or disprove the claim that dynamic residual-bound updates materially improve the economics of library-based factoring.

Experiment F. Deck construction versus deck exploitation

Measure the computational cost of building the prime deck itself and compare it to the cost of reusing that deck across many factor-search tasks. This experiment should record:

• deck-construction runtime by milestone
• storage cost by milestone
• average time per candidate once the deck is built
• scaling behavior as the number of concurrent factor-search tasks increases

Purpose: quantify the asymmetry between expensive precomputation and cheap, massively parallel deck exploitation.

Implementation note: the present software stack already supports the first half of this comparison directly. Deck construction is measured from the prime-output files and their timestamps. Deck exploitation is measured from structured factoring sessions whose run logs preserve elapsed time, time to first factor, worker count, run mode, and deck state. This makes it possible to compare build-time growth and exploitation-time scaling from the same artifact base.

Experiment G. Model-validation and baseline comparison

For each sampled semiprime, compute the model-side quantities before the factoring run begins:

• unconstrained admissible interval width
• constrained admissible interval width
• interval reduction ratio R(delta, N)
• empirical search-surface ratio E(delta, k) for the population containing the instance
• whether the smaller factor lies below the active deck ceiling
• deck ceiling expressed in prime bits and square-coverage bits

Then compare observed outcomes against three predictive baselines:

• the denominator-correct interval model
• the denominator-correct interval model adjusted by the empirical search-surface divisor
• a modulus-bit-length-only baseline
• a naive full-range baseline

Measure:

• classification accuracy for resolved vs. unresolved cases
• classification accuracy for fully resolved vs. partially resolved vs. inconclusive cases
• correlation with time to first factor
• correlation with time to final status
• calibration quality across 1,000, 10,000, and 100,000-instance populations
• how much of the accepted smaller-factor distribution lies inside the active deck ceiling

Purpose: determine whether the dissertation's formal interval model has explanatory and predictive value beyond naive baselines, whether the empirically occupied search surface is materially smaller than the structural interval alone, and whether those quantities yield statistically defensible statements about representative real-world semiprime populations.

Experiment H. Population robustness study

Run the same fixed-deck study across several semiprime populations:

• tightly balanced
• moderately balanced
• loosely balanced
• unconstrained valid semiprimes
• intentionally skewed comparison populations

Measure:

• resolution-rate differences across populations
• changes in smallest-factor distribution
• changes in unresolved remainder size
• interaction between delta, modulus size, and deck ceiling

Purpose: show whether the dissertation's claims are robust across multiple semiprime-generation regimes rather than artifacts of one chosen population.

Experiment I. Storage-format sensitivity study

Compare at least three storage representations for the prime deck:

• the current CSV format
• a packed binary format
• a compressed or delta-encoded format

Measure:

• bytes per stored prime
• build throughput
• resume/load latency
• scan throughput during factor testing

Purpose: convert the storage-economics discussion from contextual narrative into direct measured evidence about how representation choices alter practical deck feasibility.

Experiment I.1 Storage-Medium and I/O-Topology Sensitivity

The dissertation should not treat prime-deck storage as a capacity-only variable. Once the deck becomes large enough, factor testing can stop being limited primarily by arithmetic throughput and instead become limited by storage I/O behavior. This matters because the same factoring algorithm and the same candidate population may exhibit materially different runtimes depending on whether the active deck resides on an internal SSD, a mechanical hard drive, a hybrid cached disk, or a multi-disk topology.

At minimum, the dissertation should distinguish three storage-media classes:

• Solid-state drives (SSDs), which have no moving head, very low seek latency, and much stronger random-read performance
• Mechanical hard disk drives (HDDs), whose platter rotation and head-seek behavior make them much more sensitive to mixed or concurrent shard access
• Hybrid spinning disks with large flash or DRAM caches, which may accelerate repeated hot reads but still behave like mechanical disks when the working set substantially exceeds cache capacity

This distinction is important because the present factoring workflow is not a single monolithic linear scan. It is a shard-oriented library search whose performance depends on seek latency, queue depth, concurrent worker access, cancellation behavior, and how much of the active working set remains cache-resident. Under those conditions, an internal SSD may materially outperform an external spinning disk even when both contain identical shard files. In practical terms, the bottleneck can shift from CPU-bound trial division toward storage latency and file-service throughput.

The dissertation should also discuss two multi-disk topologies:

• Disk striping, which can increase aggregate read throughput by spreading shard reads across multiple devices, at the cost of reduced fault tolerance
• Disk mirroring, which primarily improves redundancy but may also modestly improve read throughput when the operating system or controller can satisfy independent reads from different mirrors

These topologies should be treated as engineering strategies, not cryptanalytic breakthroughs. Their significance is that exhaustive prime-deck search is partly a systems problem. If throughput scales by adding storage parallelism, then the effective cost of deck exploitation depends not only on prime density and scheduler policy, but also on storage layout and media architecture.

Accordingly, this experiment should compare at least the following deployment modes where available:

• internal SSD baseline
• external spinning-disk baseline
• hybrid cached-disk baseline, if such hardware is available
• striped multi-disk read path
• mirrored multi-disk read path

Measure:

• time to first factor
• time to final status
• mean throughput per worker
• worker-idle fraction attributable to I/O wait
• sensitivity to worker-count increases under each storage mode
• performance change when the active working set exceeds device cache behavior

Purpose: show that the practical cost of large-deck trial division is a joint computation-and-storage problem rather than a CPU-only problem, and quantify when storage architecture becomes the dominant throughput constraint.

Experiment J. Statistical validation framework

The dissertation should not stop at descriptive tables. For the core semiprime studies, it should fit statistical models such as:

• logistic regression for resolved vs. unresolved outcomes
• multinomial or ordinal models for fully resolved / partially resolved / inconclusive outcomes
• regression or survival-style analysis for time to first factor and time to final status

Candidate explanatory variables should include:

• modulus bit length
• delta
• unconstrained interval width
• constrained interval width
• interval reduction ratio R(delta, N)
• active deck ceiling

Purpose: produce effect sizes, confidence intervals, and formal predictive comparisons that can support a PhD-level claim of explanatory contribution.

Experiment K. Representative-population search-surface study

Using accepted semiprime populations rather than ad hoc examples, estimate the empirical distribution of the smaller factor for each modulus class and each delta class. Then measure how much of that observed distribution lies below the active deck ceiling at each milestone.

Measure:

• empirical cumulative distribution of the smaller factor
• empirical share below the deck ceiling
• empirical share inside the constrained interval predicted by the model
• empirical search-surface divisor relative to the unconstrained admissible interval
• empirical search-surface divisor relative to the constrained structural interval
• differences between tight, moderate, loose, and unconstrained populations
• changes in those shares as the deck grows from 200M to 400M, 800M, and 1.6B

Purpose: translate the dissertation's structural claim into a concrete statement about the effective searchable region for representative generated populations rather than isolated examples, and produce the empirical divisor needed to adjust the naive exhaustive cost model.

4.14 Criteria for Proving or Disproving the Theory

The theory advanced in this dissertation would be supported if the experiments show that:

• the smaller factor distribution under tighter balance constraints is measurably concentrated closer to sqrt(N)
• that concentration increases resolution rates under a fixed prime-library ceiling
• the resulting empirical behavior is predicted better by constrained one-factor interval width than by naive full-range comparisons
• and the cost of deck reuse scales far more favorably than deck construction once a library has been precomputed

The theory would be weakened or disproven if the results show that:

• the balance parameter produces negligible practical concentration in the relevant factor region
• or the observed resolution rates do not materially differ across balance classes under equivalent deck ceilings
• or the denominator-correct interval model does not explain measured outcomes better than naive baselines
• or deck reuse fails to show the expected amortized and parallel advantage over fresh search setup

4.15 Output Artifacts

The experimental program should produce:

• model-validation tables comparing predicted and observed outcomes

• regression summaries and confidence intervals

• crossover charts showing regime transitions by deck milestone

• representative-population distribution plots for the smaller factor

• empirical search-surface tables showing what fraction of accepted populations lie below each deck milestone

• semiprime datasets labeled by k and delta

• run logs from the factoring program

• milestone tables indexed by deck size

• resolution-rate charts

• time-to-first-factor charts

• unresolved remainder distribution charts

• static-vs-phased comparison tables

• precompute-versus-search cost tables

• parallel scaling charts for deck exploitation

These outputs would provide the quantitative backbone needed to support or reject the dissertation’s central structural claim.

In the implemented software stack, these artifacts are stored primarily in PostgreSQL rather than in spreadsheets. That choice is deliberate. The experimental objects are relational: datasets contain candidates, sessions represent reruns, runs contain outcomes, phases contain adaptive bound changes, and file records capture worker-level activity. PostgreSQL is therefore the system of record. Spreadsheet tools remain useful as analysis clients, particularly through Excel connections to PostgreSQL for pivot tables, charts, and sensitivity analysis, but they are downstream of the database rather than a replacement for it.

The current schema supports:

• accepted semiprime populations keyed by modulus class and delta class
• experiment sessions keyed by search mode, requested workers, included statuses, and deck state
• per-run measurements including elapsed runtime, time to first factor, worker usage, worker cancellations, remaining value, remaining bits, and verification product
• per-phase measurements including recomputed bounds, eligible-file counts, phase-local first-factor timing, and phase termination state
• per-file measurements including file identity, line counts, last prime checked, and factor count
• deck-state and milestone records for cross-milestone comparison

4.16 Measured Initial Conditions and Development Timeline

The dissertation should record the actual startup conditions under which the present software stack was built. The observed environment was a commodity laptop-class system with 28 logical CPU threads, modest local storage, commodity internet access, and access to contemporary AI coding assistance. These conditions matter because the dissertation is partly concerned with barrier-to-entry questions: what a technically capable individual can construct quickly without institutional infrastructure.

The file timestamps provide lower-bound construction windows for the two software artifacts used in this study.

These times are lower bounds rather than complete labor logs. They show that the prototype tools reached runnable form quickly under commodity conditions. This does not prove anything about mature large-scale capability by itself, but it does support the claim that the implementation barrier for precomputation-based factor-search experimentation is materially lower than older intuition may suggest.

4.17 Measured Deck-Growth Milestones

The prime deck should be described using actual milestone data rather than anecdotal estimates. The current deck files provide the following observed growth history.

The observed bytes-per-prime ratio also grew modestly over time as the decimal representation widened.

Using the elapsed wall-clock time from the first runnable prime generator to the current sixth-file timestamp, the observed long-run average build rate was approximately 2,701.54 primes per second. That average is crude and likely optimistic for large extrapolations, but it provides a conservative empirical baseline for the methodology in the next section.

The next three milestones of greatest experimental importance are 400M, 800M, and 1.6B primes. The estimated 400,000,000th prime is approximately 8.7 billion, placing it in the 34-bit regime and yielding an approximate square-coverage bound of about 68 bits. The estimated 800,000,000th prime should lie in the high-18-billion range, near 35 bits, with an approximate square-coverage bound of about 70 bits. The estimated 1,600,000,000th prime should lie in the high-30-billion range, near 36 bits, with an approximate square-coverage bound of about 72 bits. Together, these milestones define the first useful ladder for the dissertation’s controlled experiments: complete coverage of the 64-bit composite regime, followed by progressively stronger transition-zone studies in the upper-60-bit and low-70-bit region, while still preserving a clear contrast with less-covered 96- and 128-bit semiprime classes.

4.18 Conservative Scaling and Resource Estimation Method

Resource estimation in this dissertation should be based on measured milestones, not rhetoric. The present deck suggests two immediately observable quantities:

• a current average build rate of approximately 2,701.54 primes per second
• a current storage density of approximately 34.905 bytes per stored prime in the CSV format now in use

Using those two measured quantities as a naive linear extrapolation baseline, the dissertation can estimate the cost of abstract deck ceilings. The table below uses approximate prime-count estimates for complete decks through powers of two and reports the corresponding square-coverage bound in bits. These are deck-ceiling estimates, not claims about defeating any particular deployed system.

These extrapolations should be interpreted cautiously.

First, they assume the present CSV storage format remains unchanged. A binary or delta-compressed representation would materially change the storage column.

Second, they assume the present average build rate remains stable. That is unlikely over very large deck regimes. Throughput will almost certainly drift as candidate values grow and storage management costs increase.

Third, they are intended to bound the scaling problem, not to settle it. The dissertation should therefore present them as first-order empirical extrapolations derived from observed milestones, followed by explicit discussion of their limitations.

A second useful presentation is a standardized milestone ladder built around realistic milestone targets for the current prototype format, beginning with the observed 100M and 200M breakpoints and then continuing through the doubling sequence. Using the observed storage density of approximately 34.905 bytes per prime and the same nth-prime estimator used elsewhere in this chapter, the following ladder provides a convenient planning sequence for both storage and experimental coverage.

This ladder is useful for two reasons. First, it gives the dissertation a consistent set of milestone targets for storage planning under the current CSV prototype. Second, it creates a clean experimental progression in which each doubling step raises the deck ceiling by roughly one additional prime bit and therefore adds roughly two bits of square-coverage in the present trial-division model. In practical terms, a 25.6B-prime deck would fit within approximately one terabyte of decimal storage at the present observed density and would correspond to an approximate 80-bit square-coverage bound.

4.18.1 Storage-Normalized Coverage Metrics

The dissertation should move beyond raw storage tables and define explicit economic metrics for the precomputed-prime regime. Let B denote the prime-deck ceiling in bits, so that the corresponding reliable trial-division coverage is approximately K = 2B bits in the semiprime model. Let s denote the observed storage cost per stored prime in bytes, and let c denote the storage price in dollars per byte under a chosen economic scenario.

Then a first-order storage model for a complete deck through prime bit ceiling B is:

Storage(B) ≈ s * pi(2^B)

Using the prime number theorem,

pi(2^B) ≈ 2^B / (B ln 2)

so that:

Storage(B) ≈ s * 2^B / (B ln 2)

and equivalently, in terms of semiprime coverage bits K = 2B:

Storage(K) ≈ s * 2^(K/2) / ((K/2) ln 2)

From this, the dissertation can define the following research metrics.

• Deck Coverage Cost (DCC): the estimated storage cost required to sustain a deck giving reliable coverage through K bits under the current representation and a chosen storage-price assumption.
• Marginal Deck Coverage Cost (MDCC): the additional storage cost required to move from one coverage level to the next milestone, for example from K to K + 2.
• Storage-Normalized Coverage Curve (SNCC): the relationship between storage budget and reliable coverage, viewed as a curve rather than a single point estimate.

These are research metrics, not normative security labels. Their value is that they make it possible to compare:

• different deck formats
• different storage-price eras
• and different adversary budget assumptions

within one explicit economic model.

Two algebraic forms are especially useful for the dissertation.

First, isolating adversary storage cost as a function of desired reliable coverage:

Cost(K) ≈ C_TB × [s × 2^(K/2)] / [((K/2) ln 2) × 10^12]

where:

• K is the approximate reliable semiprime coverage in bits
• s is the measured bytes per stored prime
• C_TB is the cost per terabyte of storage

This equation gives the estimated storage expenditure required to sustain a deck large enough to give reliable coverage through K bits in the present trial-division model.

If the experiments support the stronger claim that the practically meaningful search surface is only a fraction of the structural domain, then this cost form should be paired with an empirical divisor. Let E(K, delta) denote the empirically estimated occupied-search-surface ratio for the relevant modulus class and balance population. Then an adjusted exploratory cost form is:

AdjustedCost(K, delta) ≈ E(K, delta) × Cost(K)

This adjusted form is not a replacement for the structural model. It is an empirical refinement that applies only if the representative-population experiments justify it.

Second, isolating the approximate reliable coverage as a function of available storage budget in terabytes. Let T be the storage available in terabytes. Then B, the prime-deck ceiling in bits, is the approximate solution to:

2^B / (B ln 2) ≈ (T × 10^12) / s

and the resulting reliable semiprime coverage is:

BitsSecureAgainst(T) ≈ 2B

This form is not closed-form in elementary functions, but it is the conceptually correct inversion. It allows a security manager to estimate the bit level through which an adversary with storage budget T could, in principle, sustain reliable deck coverage under the assumed storage format.

Under the same empirical refinement, the dissertation can define an exploratory adjusted coverage measure:

AdjustedBitsSecureAgainst(T, delta)

as the coverage estimate obtained when the budget-limited model is interpreted through the empirically occupied search surface rather than the full structural interval. This quantity must be presented carefully as experiment-dependent and population-dependent, not as a universal constant.

4.18.2 Budget-Limited Reliable Coverage

One practical output of the dissertation should be a budget-limited coverage table. For a selected storage-price assumption and deck representation, the dissertation can invert the coverage model and estimate the maximum reliable coverage that could be supported by an adversary with a given storage budget.

This should be presented as a family of scenarios rather than as a single claim. Representative examples might include:

• hobbyist or small-budget actors
• small organizations
• well-funded corporate environments
• state-scale environments

The point is not to assert that any given actor will in fact build and use such decks. The point is to bound what storage-only economics would permit under the precomputed-prime model. This creates a more precise complement to the qualitative statement that falling storage prices make reusable prime-deck regimes more accessible over time.

In practical use, the dissertation should encourage two complementary questions:

• Given a target protection level K, what storage cost would an adversary need to sustain reliable coverage through that level?
• Given an adversary storage budget T, what approximate coverage level BitsSecureAgainst(T) does that imply?

These two forms together create the bridge from measured deck engineering to defensible budget-based interpretation.

4.18.3 Time-Varying Storage Price Model

Because the dissertation is concerned with long-run accessibility rather than only a single contemporary price snapshot, it should define a time-varying storage-price function. A simple first model is exponential decay:

C_TB(t) = C_0 e^(-lambda t)

where:

• C_TB(t) is the storage cost per terabyte at time t
• C_0 is the current cost per terabyte
• lambda is the fitted long-run decay rate

This can be combined directly with the coverage-cost form from Section 4.18.1:

Cost(K, t) ≈ C_TB(t) × [s × 2^(K/2)] / [((K/2) ln 2) × 10^12]

Under this model, the storage cost of sustaining a deck that gives reliable coverage through K bits falls over time at the same economic decay rate as the storage market itself, assuming the representation cost s remains fixed.

Likewise, if a storage budget is fixed in dollars rather than terabytes, then the effective storage capacity purchasable at time t is:

T(t) = Budget / C_TB(t)

and the corresponding approximate coverage satisfies:

2^B / (B ln 2) ≈ (T(t) × 10^12) / s

with:

BitsSecureAgainst(Budget, t) ≈ 2B

This form gives the dissertation a clean way to discuss the economic decay of fixed coverage barriers over time without collapsing the analysis into a single-year snapshot.

4.19 Single-Host and Multi-Host Parallel Scaling

The existing platform already supports a 28-thread local execution environment. That makes it possible to measure factor-search throughput on the reference host directly rather than speculating about it.

The dissertation should therefore gather a parallel-scaling table of the form:

Once the single-host efficiency curve is known, multi-host estimates can be expressed using the same formalism rather than by unsupported intuition. If a cluster consists of H identical 28-thread hosts, then an idealized upper-bound estimate is:

• ideal cluster speedup = H * speedup_28
• ideal cluster runtime = T1_final / (H * speedup_28)

A more realistic estimate uses measured efficiency loss:

• measured cluster speedup = H * speedup_28 * E_H
• measured cluster runtime = T1_final / (H * speedup_28 * E_H)

The present automation already supports the single-host portion of this design through explicit worker-count controls in the experiment runners. The practical workflow is therefore to keep the dataset fixed and vary only session parameters: requested workers, search mode, included statuses, and deck milestone. This is significantly stronger than ad hoc rerunning because the database preserves each session as a distinct analytic object. Multi-host execution remains future work, but the current session-and-run design is already structured so that host identifiers and cluster metadata can be added later without changing the core population/run hierarchy.

The worker-scaling discussion must also distinguish between current implementation limits and future scheduler experiments. At present, the operational ceiling is one worker per eligible file. This means that a deck with eight eligible files does not presently benefit from requesting more than eight workers. The dissertation should treat that behavior as an implementation-specific saturation point, not as a universal property of prime-library search.

Future scheduler experiments should therefore include a second axis beyond raw worker count: sharding mode. At minimum, the scaling chapter should compare:

• one worker per eligible file
• two workers per eligible file with bidirectional or interval-split search
• prioritized lower-file scheduling when eligible files exceed available workers

The expected effect is not monotone in all cases. With fewer eligible files than workers, intra-file sharding may reduce time to first factor but can also increase redundant work if lower-end discovery is already dominant. With more eligible files than workers, allocating multiple workers to a single file too early may actually worsen global performance by starving lower unsearched files. The dissertation should therefore predict and then test that optimal worker assignment depends jointly on file count, active deck ceiling, factor-location distribution, and cancellation behavior after phased reduction.

where E_H is the observed multi-host efficiency term capturing coordination, cancellation, queueing, and I/O overhead.

The critical point is that the dissertation should not assume datacenter scaling. It should measure local scaling first, then extend outward by explicitly stated efficiency terms. This keeps the resource argument empirical and falsifiable.

5. Security Implications, Limitations, and Future Work

5.1 Security Interpretation

The results of this dissertation should be interpreted with discipline. A narrower admissible one-factor interval does not automatically imply practical weakness. It implies that the one-factor problem should be reasoned about against the correct baseline. In many real-world cryptographic settings, the interval may still be far too large for exhaustive or library-based search to matter. The dissertation therefore treats interval narrowing as a structural property whose significance depends on key size, available computation, and search model.

A second interpretive point concerns precomputation. If a large prime deck does not yet exist, search cost is dominated by the cost of building or extending the deck. If the deck already exists, the economics change substantially. Search becomes cheaper, more reusable, and more parallelizable. This does not erase the hardness of large-factor problems, but it does change the practical meaning of structural narrowing for repeated testing scenarios.

5.2 What the Dissertation Does Not Claim

This dissertation does not claim a new universal break of RSA. It does not claim that balanced keys are inherently unsafe. It does not claim that interval narrowing alone makes well-sized contemporary systems vulnerable. Rather, it claims that security arguments should describe the correct admissible one-factor domain, and that generation constraints which narrow that domain deserve formal treatment.

The dissertation also does not claim that precomputation makes all factor problems easy. Its more modest claim is that once a reusable prime deck exists, repeated factor testing becomes materially more scalable than deck construction and therefore should be treated as a separate computational regime.

The dissertation also does not claim that factor concentration near sqrt(N) is a universal weakness signal. Its practical meaning depends on the active attack regime. Under a bounded lower-first deck, concentration high in the admissible interval may actually reduce hit rates because the smaller factor lies above the active ceiling. Under a sufficiently large deck combined with square-root-centered scheduling, the same concentration may help the attacker by shrinking the band that must be searched first. The dissertation therefore treats balance concentration as regime-dependent rather than as an unconditional vulnerability claim.

5.3 What Would Make the Dissertation Succeed

For this work to qualify as a PhD-level dissertation rather than a promising prototype, it must do more than present interesting demonstrations. It must show at least one primary contribution and ideally one or two secondary contributions. The primary contribution should be a defensible formal and predictive account of the admissible one-factor interval under balance constraints, supported by evidence that this model predicts observed tractability better than naive baselines. The secondary contributions may include the computational distinction between precomputation and deck exploitation, the measurement of phased-reduction benefits, and the historically grounded account of storage economics as an enabling factor for reusable prime-deck experimentation.

A committee should therefore be able to ask, and the dissertation should be able to answer, the following questions.

• Did the work produce a denominator-correct formal model rather than a loose intuition?
• Did that model survive empirical validation against observed outcomes?
• Did the experiments quantify predictive performance, effect size, and uncertainty rather than relying on anecdotes?
• Did the representative-population studies show what portion of valid semiprime populations actually fall inside the active deck ceiling at each milestone?
• Did the computational and storage measurements reveal something nontrivial about the economics of reusable prime decks?

5.3.1 Anticipated Objections and Responses

One predictable objection is that a finite-deck attack model may appear to be an engineering niche rather than a serious cryptanalytic contribution. The dissertation's answer is that it does not present the finite-deck model as a replacement for GNFS or standards-based security doctrine. It presents it as a separate, explicitly bounded attack regime whose economics, population dependence, and scheduler dependence are not usually formalized in the literature. That is a legitimate object of study because bounded regimes can still produce publishable explanatory models when they are stated precisely and tested empirically.

A second objection is that the topic may simply rediscover known facts about close primes or balanced versus unbalanced RSA. The dissertation's answer is that it does not claim novelty in the observation that factor shape matters. Its narrower claim is that the combined finite-deck question — balance class, deck milestone, and scheduler direction analyzed together through a denominator-correct interval model and an empirical success surface — does not appear to be already supplied by the adjacent close-prime, unbalanced-RSA, or partial-leakage literatures reviewed in Chapter 2.

A third objection is that the revised distinction between bounded lower-first decks and square-root-centered search might look post hoc. The dissertation's answer is that the distinction is mathematically unavoidable once the search regime is stated explicitly. If success depends on whether the smaller factor lies below a low-end deck ceiling, then looser populations may be favored. If success depends on concentrating search effort near sqrt(N), then tighter populations may be favored. Treating those regimes separately is therefore a correction toward coherence, not an ad hoc rescue of a failed intuition.

A fourth objection concerns synthetic populations. The dissertation's answer is that the populations are not defended as perfect replicas of one vendor's implementation. They are defended as valid samples from explicitly defined admissible classes under explicit balance rules. That is enough to support the dissertation's main question, which is comparative and model-driven: how does tractability change when the admissible class and search regime change?

A fifth objection is that P_hit(X, k, delta, s) could become merely descriptive. The dissertation's answer is that the probability surface is valuable only if it is linked back to the formal model. The point is not to tabulate success rates in isolation. The point is to test whether denominator-correct interval variables, empirical search-surface variables, and scheduler choice predict those success rates better than naive baselines do. In that form, P_hit becomes an explanatory target rather than a descriptive endpoint.

A sixth objection is that the transition-zone ladder may overfit the region where the dissertation's model looks strongest. The dissertation's answer is that this is methodologically appropriate rather than evasive. The transition zone is where different hypotheses yield meaningfully different predictions. Saturated success and saturated failure regions remain useful as checks, but they do not carry the same discriminatory power. This is why the dissertation retains sparse high-bit anchor studies while placing the main statistical burden on the transition region.

A seventh objection is that storage medium, concurrency limits, and implementation details may contaminate the mathematical claims. The dissertation's answer is that these are not external contaminants to be ignored; they are part of the computational regime being studied. The dissertation therefore distinguishes structural interval claims from implementation-dependent exploitation cost and records storage architecture, worker limits, and run configuration as first-order experimental variables.

A final objection is that the strongest cryptanalytic effect may prove weaker than expected. The dissertation's answer is that even a weak or null result can still support a meaningful contribution if the dissertation succeeds in showing which variables do and do not predict finite-deck tractability, where the transition zone actually lies, how scheduler policy changes the sign of the effect, and how storage and orchestration shape the economics of reusable prime-deck search. In that event, the dissertation would still add knowledge by replacing loose intuition with a tested finite-regime model and a measured boundary on its explanatory power.

5.4 Design Guidance

At a high level, the dissertation supports three forms of guidance. First, security analysis should distinguish clearly between total modulus size and the admissible domain for the smaller factor. Second, practitioners should avoid informal range-based reasoning that compares constrained factor problems to the full numeric universe. Third, cryptographic systems should be evaluated not only by key size but also by how their generation constraints interact with realistic computational models over time, especially when precomputation and amortized parallel search are feasible.

This guidance should be layered rather than absolute. Standards-based guidance remains the correct baseline for operational key-size selection, and NIST SP 800-57 should be treated as the appropriate normative reference point. The dissertation’s contribution is different. It introduces storage- and deck-based research metrics that can be used to ask how expensive one particular class of precomputed-prime search would be under stated assumptions. Those metrics should supplement, not replace, established security-strength guidance.

Accordingly, the dissertation should present advice in two stages:

1. identify the baseline key-size regime using standards-based guidance such as NIST SP 800-57
2. then evaluate how that regime looks under the dissertation’s storage-normalized coverage model for a specified deck format, storage price, and adversary budget

This layered approach avoids the mistake of treating one experimental attack-economics model as if it were a complete replacement for cryptographic standards.

5.4.1 Relevance to Currently Deployed Cryptographic Systems

The experimental program in this dissertation begins at 64-bit and 68-bit semiprimes and extends through 96-bit, 112-bit, and 128-bit modulus classes. A natural objection is that these bit-lengths are far below RSA-2048 or RSA-4096, the key sizes recommended by current standards guidance. That objection, while valid for RSA specifically, overlooks the broader landscape of cryptographic key sizes actually deployed in production systems today.

Elliptic curve cryptography in Bitcoin and blockchain systems. Bitcoin's digital signature scheme uses ECDSA over the secp256k1 curve, which has a 256-bit group order (FIPS 186-5; SEC 2, Section 2.4.1). The security of secp256k1 rests on the elliptic curve discrete logarithm problem (ECDLP), not integer factoring. However, the structural argument of this dissertation — that generation constraints concentrate the effective attack surface more narrowly than naive full-domain reasoning suggests — is a pattern that transcends any single hard problem. If balance constraints narrow the admissible factor domain for semiprimes, the analogous question for elliptic curves is whether key-generation practices or curve structure create comparable concentrations in the space an attacker must search. The 256-bit operational scale of Bitcoin places it closer to this dissertation's experimental range than to RSA-2048.

SSH and TLS key exchange. The Secure Shell protocol (RFC 4253) and Transport Layer Security commonly deploy ECDSA-256 (P-256 curve) or Ed25519 (255-bit curve) for authentication and key exchange. These are 256-bit elliptic curve keys with an estimated security strength of 128 bits under NIST SP 800-57 equivalences. RSA keys in SSH are also still common, but ECDSA and EdDSA have become the default in most modern implementations. The 128-bit security-strength target means that any structural narrowing of the effective search domain is measured against a margin of 128 bits, not 2048.

IoT and embedded systems. Constrained devices — smart cards, industrial sensors, automotive controllers, medical implants — frequently use shorter keys for performance reasons. NIST has acknowledged this with its lightweight cryptography standardization effort. In practice, many deployed IoT devices use 160-bit or 192-bit elliptic curve keys, and legacy systems may still use 1024-bit RSA. These are the systems where the storage-normalized coverage model developed in this dissertation becomes most immediately relevant: the deck storage required for exhaustive coverage at these bit-lengths is orders of magnitude smaller than at RSA-2048 scale.

The extrapolation argument. The dissertation's experimental program at 64-bit through 128-bit semiprimes is not merely a toy demonstration. It serves two purposes. First, it validates the interval model at bit-lengths where complete factorization is computationally feasible, allowing direct measurement of whether the denominator-correct model predicts observed behavior. Second, it establishes the scaling trend for the interval reduction ratio R(δ,N) across multiple bit-lengths. If R(δ,N) follows its theoretically predicted relationship with bit-length — and the 64/68-bit pair, the 96/112-bit pair, and the 128-bit study provide five data points on that curve — then the extrapolation to 256-bit, 512-bit, and 1024-bit regimes rests on mathematical structure, not speculation. The interval model's predictions at larger scales can be computed even when exhaustive factorization at those scales is infeasible, because the model depends on prime density (via the Prime Number Theorem) and balance constraints, both of which are well-characterized at any bit-length.

What this means for security margins. The practical implication is that the effective security margin of deployed cryptographic systems may be thinner than standard guidance assumes — not because the hard problems are easier than believed, but because the attack surface is more concentrated than a naive full-domain estimate suggests. For RSA specifically, the admissible one-factor domain under standard balanced key generation clusters around √N, and the interval reduction ratio quantifies how much of the total domain that cluster occupies. For elliptic curve systems, the analogous structural question has not yet been formalized in the same way, but the methodology developed in this dissertation provides a framework for asking it.

Sources:

• National Institute of Standards and Technology, Digital Signature Standard (DSS), FIPS PUB 186-5 (2023). https://csrc.nist.gov/pubs/fips/186-5/final
• Standards for Efficient Cryptography Group, SEC 2: Recommended Elliptic Curve Domain Parameters, Version 2.0 (2010). https://www.secg.org/sec2-v2.pdf
• Ylonen, T., and Lonvick, C. (eds.), The Secure Shell (SSH) Transport Layer Protocol, RFC 4253, IETF (2006). https://datatracker.ietf.org/doc/html/rfc4253

5.4.2 Storage-Normalized Coverage as a Research Metric

The dissertation’s proposed economic lens can be expressed through three related quantities defined in Chapter 4:

• Deck Coverage Cost (DCC)
• Marginal Deck Coverage Cost (MDCC)
• Storage-Normalized Coverage Curve (SNCC)

If the population experiments support it, the dissertation should add a fourth, explicitly empirical quantity:

• Empirically Adjusted Coverage Cost (EACC)

These metrics allow the dissertation to estimate how much persistent storage, and therefore how much storage expenditure, would be required to sustain reliable trial-division coverage through a target semiprime class. They are especially useful because they make explicit something that is often left implicit in cryptographic rhetoric: for this computational regime, the effective barrier is not only arithmetic difficulty but also the storage burden required to retain a reusable exhaustive deck.

That storage burden should not be interpreted as capacity alone. The experiments should also treat storage throughput architecture as part of the attack-cost model. Once the prime deck becomes sufficiently large, repeated shard scans may be limited less by arithmetic and more by seek latency, cache locality, concurrent read behavior, and file-service bandwidth. In that regime, internal SSDs, mechanical HDDs, and hybrid cached disks can produce materially different observed runtimes even under the same search algorithm and the same deck contents. Likewise, disk striping may improve exploitation throughput by parallelizing reads across devices, while disk mirroring may improve resilience and sometimes modestly improve read concurrency. Accordingly, the practical economics of reusable prime-deck search should be framed as a joint computation-and-storage-systems problem rather than as a CPU-only or capacity-only problem.

The dissertation should therefore use these metrics to produce scenario tables rather than universal rules. For example, under a given storage-price assumption, what reliable coverage level would be consistent with:

• a hobbyist budget
• a small-enterprise budget
• a large-corporate budget
• a state-scale budget

This does not prove that such actors will build such decks. It provides a way to reason concretely about what the storage economics of the model would permit.

For policy-facing interpretation, two equation forms are especially useful.

The first isolates cost:

Cost(K) ≈ C_TB × [s × 2^(K/2)] / [((K/2) ln 2) × 10^12]

This form answers the question:

• if an adversary wanted reliable coverage through K bits in this model, what storage expenditure would that imply?

The second isolates approximate defensive coverage against a storage-constrained adversary. If T terabytes of storage are available to the adversary, then B is the approximate solution to:

2^B / (B ln 2) ≈ (T × 10^12) / s

and the corresponding reliable semiprime coverage is:

BitsSecureAgainst(T) ≈ 2B

This form answers the reciprocal question:

• if the adversary can economically sustain only T terabytes, up to roughly how many bits of reliable semiprime coverage does that extend?

Together, these two forms make the dissertation’s economic interpretation more usable. A security manager can track falling storage prices over time, update C_TB, and then re-evaluate either:

• the budget an adversary would need to reach a target coverage level
• or the approximate coverage level an adversary of a given storage budget could support

If the experiments validate a meaningful empirical concentration near sqrt(N), then the dissertation should also present an adjusted exploratory form in which the structural deck-cost model is multiplied by the empirically occupied search-surface ratio for the relevant modulus class and balance population. In that case, the cost discussion would no longer describe only the worst-case structural coverage burden. It would also describe a population-adjusted burden for the actually observed search surface. That is the place where the disser

6. Experimental Results and Model Validation

6.1 Role of This Chapter

This chapter is reserved for the empirical results needed to convert the dissertation from a prospectus-like framework into a completed PhD dissertation. Its purpose is to report the observed outcomes of the milestone experiments, the model-validation studies, and the statistical comparisons proposed in Chapter 4.

6.2 Planned Sections

The completed chapter should include at least the following sections.

• corpus construction and acceptance statistics
• software and automation state for the reporting period
• deck milestone state at each experimental run
• structured-composite results
• semiprime population results by delta and modulus class
• semiprime population results by delta, deck milestone, and scheduler mode
• crossover behavior by deck milestone
• worker-count sweep results
• phased-versus-static comparison results
• representative-population smaller-factor distributions
• share of each accepted population whose smaller factor lies below the active deck ceiling
• estimated P_hit(X, k, delta, s) tables for full resolution and partial-or-full resolution
• bottom-up versus square-root-centered time-to-first-factor comparisons
• empirical search-surface divisor estimates
• model-validation comparisons against naive baselines
• statistical summaries, confidence intervals, and effect sizes
• interpretation of what the results do and do not establish

6.3 Minimum Evidentiary Standard

This chapter should not be filled with anecdotal demonstrations alone. It should contain enough controlled data to support the dissertation's primary claim that the denominator-correct interval model is more explanatory than naive full-range reasoning under the tested computational regimes. In practice, that means reporting accepted-population sizes, milestone deck state, representative smaller-factor distributions, the share of each population that falls below the active deck ceiling, explicit P_hit(X, k, delta, s) estimates for the main outcome classes, bottom-up versus square-root-centered comparisons where available, empirical search-surface divisor estimates, predictive baseline comparisons, and formal uncertainty measures rather than only selected successful factorizations.

It should also report the software and automation conditions under which those results were produced. Because the implemented experiment stack now records session identity, requested workers, search mode, host label, deck state, phase history, file history, and verification outcomes, the results chapter should explicitly use those fields rather than collapsing them away. The dissertation’s empirical strength depends in part on showing that the observed outcomes came from controlled, replayable sessions rather than from one-off manual trials.

6.4 Structured-Composite Crossover Case Study

One structured-composite case already deserves preservation because it illustrates the distinction between temporary inconclusiveness and genuine structural intractability under a growing deck.

6.4.1 Candidate

The candidate was:

102,003,000,400,005,000,006,000,000,700,000,008

This value was constructed as part of the Clay-style progression family used for structured-composite experiments.

6.4.2 Earlier Partial Resolution

Under an earlier deck state, the factoring platform quickly found the following factors:

• 2
• 2
• 2
• 3
• 3
• 19
• 4,290,499

The product of those early factors is:

5,869,402,632

Dividing the original candidate by that product left the previously unresolved remainder:

17,378,770,344,342,088,407,268,769

At that earlier milestone, the run was therefore classified as inconclusive with respect to the remaining composite.

6.4.3 Later Resolution After Deck Growth

After subsequent deck growth, that previously unresolved remainder was factored successfully as:

• 7,610,752,411
• 2,283,449,704,555,379

This means the full factorization of the original structured composite is:

• 2
• 2
• 2
• 3
• 3
• 19
• 4,290,499
• 7,610,752,411
• 2,283,449,704,555,379

The final verification product reported by the factoring platform was:

102,003,000,400,005,000,006,000,000,700,000,008

The run therefore closed with the explicit verification statement that the factors multiply back to the original input.

6.4.4 Why This Case Matters

This case is significant for several reasons.

First, it is a direct deck-growth crossover example. The same candidate family moved from partial resolution with an inconclusive remainder to complete factorization after the deck expanded.

Second, it illustrates the value of phased reduction. The original composite did not need to be approached as one monolithic hard object. Small and medium factors were stripped first, leaving a much smaller residual problem whose status changed as the deck grew.

Third, it shows why the dissertation must distinguish between:

• currently inconclusive under a given deck state
• structurally outside the reachable region of the current deck

Those are not the same statement. A remainder may appear resistant at one milestone and then collapse once one additional factor enters deck coverage.

Fourth, this case belongs to the structured-composite branch of the dissertation rather than the representative-semiprime-population branch. It is evidence about phased reduction, deck-growth thresholds, verification closure, and milestone crossover behavior. It is not by itself evidence for or against the separate hypothesis that realistic semiprime-generation constraints concentrate the effective search surface near sqrt(N).

6.4.5 Data Still To Be Attached

Before final submission, this case study should be augmented with the run metadata already contemplated elsewhere in the dissertation:

• deck state at the earlier inconclusive run
• deck state at the later successful run
• search mode
• requested workers and workers used
• time to first factor
• total time to final status
• any available phase and file history

In final form, this case should appear not as an anecdote but as a fully logged crossover exhibit drawn from the experiment system of record.

6.5 Structured-Composite Open Successor Case

The next candidate in the same structured progression should be preserved as a companion case because it shows the family continuing beyond the now-closed ...700000008 instance.

6.5.1 Candidate

The next candidate in the progression is:

102,003,000,400,005,000,006,000,000,700,000,008,000,000,009

At the current deck state, the factoring platform found several early factors and then stopped at an unresolved remainder outside the present deck ceiling. This places the candidate in the useful intermediate category of partially resolved but not yet closed.

6.5.2 Why This Case Matters

This case should be preserved for three reasons.

First, it extends the structured-composite family in a controlled way. The dissertation now has one member of the family that has fully crossed over from partial to complete factorization and a direct successor that remains open at the current milestone.

Second, it provides a future threshold test. As the deck grows through the next milestone ladder, this candidate can be rerun without changing the underlying object of study. That makes it a strong longitudinal probe for the dissertation’s deck-growth and phased-reduction claims.

Third, it offers a disciplined forward-looking comparison. If the remaining composite eventually falls within the projected square-coverage bound of the larger planned deck, then the dissertation will have an especially clean before-and-after pair: one structured-composite successor partially resolved at an earlier milestone and later closed under a larger deck without changing the candidate family.

6.5.3 Current Status and Required Logged Fields

Before final submission, this open successor case should record the following from the experiment system of record:

• original input value
• factors found so far
• unresolved remainder
• unresolved remainder bit length
• deck state at run time
• search mode
• requested workers and workers used
• time to first factor
• time to inconclusive termination
• any available phase and file history

6.5.4 Interpretation Discipline

This case should be described carefully. At the current stage it is evidence of partial tractability within the structured-composite family, not yet evidence of completed closure. If a later milestone resolves the remaining composite, then the case will become a second crossover exhibit. If it remains unresolved even under the larger planned deck, then it will become equally valuable as evidence that the family has moved beyond the current regime of guaranteed coverage.

Appendix A. Reproducibility and Data Sources

A.1 Purpose

This appendix documents the data sources, local artifacts, and extraction methods used to produce the measured tables in Chapters 2, 4, and 5. Its purpose is not to freeze the dissertation to a single machine forever, but to make clear how the reported timing, storage, and milestone values were obtained at the time of writing.

A.2 Local Artifact Locations

The empirical measurements in this dissertation were derived from the following local directories on the reference development system:

• Prime-deck generator source tree: C:\Users\tgfla\tools\prime-generator
• Factoring platform source tree: C:\Users\tgfla\tools\factoring
• Prime deck output files: C:\Users\tgfla\tools\prime-output
• Dissertation workspace: C:\Users\tgfla\doctoralthesis

Within the WSL-style execution environment used during preparation, these locations appeared as:

• /mnt/c/Users/tgfla/tools/prime-generator
• /mnt/c/Users/tgfla/tools/factoring
• /mnt/c/Users/tgfla/tools/prime-output
• /mnt/c/Users/tgfla/doctoralthesis

The implemented experiment-automation scripts live under:

• /mnt/c/Users/tgfla/tools/factoring/scripts

The principal experiment scripts are:

• init-experiment-db.js
• experiment-db.js
• experiment-runtime.js
• generate-semiprime-dataset.js
• factoring-engine.js
• run-experiment-queue.js
• run-candidate-list.js
• run-worker-sweep.js
• export-experiment-view.js
• compare-deck-formats.js
• write-repro-manifest.js

A.3 Measured Software-Construction Windows

The lower-bound software-construction windows in Chapter 4 were computed from file timestamps rather than from a manual work diary. The method was:

1. identify the earliest relevant source or project file timestamp in the project tree
2. identify the earliest runnable artifact or completed application-state timestamp
3. compute the elapsed time difference between those timestamps

For the prime generator, the earliest relevant project file observed was Cargo.toml, and the first runnable artifact observed was the release executable.

For the factoring platform, the earliest relevant project file observed was the initial PowerShell startup capture helper, and the final runnable application state was approximated using the timestamp of the last core frontend file written during the initial build sequence.

These measurements are explicitly lower bounds. They capture the observable construction window present in the file system, not every minute of human thought, planning, or interruption.

A.4 Commands Used for Timestamp Inspection

The following classes of local shell commands were used to inspect timestamps:

find /mnt/c/Users/tgfla/tools/prime-generator -maxdepth 3 -type f -printf '%TY-%Tm-%Td %TH:%TM:%TS %p\n' | sort
find /mnt/c/Users/tgfla/tools/factoring -maxdepth 3 -type f -printf '%TY-%Tm-%Td %TH:%TM:%TS %p\n' | sort
find /mnt/c/Users/tgfla/tools/prime-output -maxdepth 1 -type f -printf '%TY-%Tm-%Td %TH:%TM:%TS %s %f\n' | sort

These commands were used only to inspect local metadata and did not modify the experimental artifacts.

A.5 Deck-Milestone Extraction Method

The prime-deck milestone tables in Chapter 4 were derived directly from the CSV files in the prime-output directory. Each file represents a contiguous ordinal segment of the prime deck. The measurement procedure was:

1. identify each output file and record its file size in bytes
2. read the last complete row of the CSV file
3. extract from that row:
   • final ordinal in the file
   • largest stored prime in the file
   • stored bit length of that prime
4. accumulate file sizes across all completed files to obtain total on-disk size by milestone

The tail-inspection command class used for spot verification was:

for f in /mnt/c/Users/tgfla/tools/prime-output/primes_*.csv; do
  tail -n 2 "$f"
done

A small parsing script was then used to read the final valid row from each file and calculate the corresponding prime bit length where needed.

A.6 Experimental Software and Persistence Model

The dissertation experiments are no longer described only as planned future work. A concrete automation stack now exists. Its purpose is to separate exploratory browser use from reproducible batch experimentation.

The persistence model is PostgreSQL-first rather than spreadsheet-first. This choice is methodologically important. The experimental objects are relational by nature:

• a dataset contains many accepted semiprime candidates
• a session represents one controlled rerun of some or all of those candidates
• a run stores the outcome for one candidate within one session
• a run contains multiple phases when adaptive factoring restarts after factor discovery
• each phase contains multiple per-file worker records

This hierarchy would be awkward and error-prone as a primary spreadsheet. The dissertation therefore treats PostgreSQL as the system of record. Spreadsheet software, especially Excel, is reserved for downstream analysis, charting, pivot tables, and sensitivity tables after the experimental data has been captured reliably.

The current experimental schema, under the factoring_experiments namespace, includes at minimum:

• experiment_datasets
• experiment_sessions
• semiprime_candidates
• factoring_runs
• factoring_run_factors
• factoring_run_phases
• factoring_run_files
• deck_milestones
• experiment_analysis_snapshots

A.7 Instrumentation Captured by the Factoring Engine

The instrumented factoring engine now emits structured fields sufficient for the core studies proposed in Chapter 4. At the run level, the recorded fields include:

• input value and input bits
• started and finished timestamps
• job status and result status
• search mode (phased or static)
• candidate source
• elapsed runtime
• time to first factor
• phases triggered
• worker cancellations
• requested workers
• workers used
• maximum concurrent workers seen
• eligible files and files completed
• highest prime checked
• coverage reached in bits
• remaining value and remaining bits
• factorization verification flag
• verification product
• deck file count, largest prime, largest ordinal, and coverage bits at run time

At the phase level, the recorded fields include:

• phase number
• phase-local input value and input bits
• recomputed square-root limit
• eligible-file count
• configured workers
• worker cancellations
• first-factor timestamp and elapsed time
• phase status and details

At the per-file level, the recorded fields include:

• file identity
• ordinal range
• status
• lines checked
• last prime checked
• factor count

This instrumentation is sufficient to support the dissertation’s immediate needs for:

• structured-composite studies
• semiprime balance sweeps
• phased-versus-static comparisons
• worker-count scaling sweeps
• milestone crossover studies
• verification of closed factorizations

A.8 Automation for Experimental Variation

The automation now supports controlled reruns rather than one-off executions.

generate-semiprime-dataset.js creates accepted semiprime populations under explicit k and delta rules. run-experiment-queue.js then claims candidates from a chosen dataset and executes them sequentially under a specific session definition. A session captures, at minimum:

• experiment code
• dataset identity
• search mode
• requested workers
• included statuses
• target limit
• deck state at the moment of the run

This means the same accepted population can be rerun under:

• different worker counts
• different search modes
• different deck milestones
• different inclusion rules such as pending only versus pending,inconclusive

run-candidate-list.js provides the same controlled-session treatment for manually curated or structured-composite families used in Experiments A, D, and E.

A.9 Derived Quantities in Chapter 4

The following derived quantities were computed from the observed local data.

A.9.1 Bytes per stored prime

For each milestone,

bytes_per_prime = cumulative_bytes / highest_ordinal_reached

This quantity is format-dependent. It reflects the present CSV prototype and should not be treated as a universal constant for all future implementations.

A.9.2 Square-coverage bits

For a deck whose current largest stored prime has bit length b, the approximate square-coverage bound reported in the dissertation was:

square_coverage_bits = 2b

This is a descriptive bound used to express the order of magnitude of the deck ceiling. It does not imply that every integer of that bit length is automatically resolved in practice.

A.9.3 Observed long-run build rate

The long-run average build rate reported in Chapter 4 was computed as:

observed_build_rate = highest_ordinal_reached / elapsed_seconds_since_first_runnable_prime_generator

where elapsed time was measured from the first runnable prime-generator artifact to the timestamp of the latest observed deck file.

A.10 Conservative Scaling Methodology

The resource-estimation table in Chapter 4 used two measured prototype quantities:

• observed average build rate in primes per second
• observed average storage density in bytes per prime

These were applied to approximate prime-count estimates for deck ceilings at selected powers of two. The resulting values are first-order extrapolations and should be read as conservative scaling illustrations, not as hard feasibility thresholds.

A.11 External Literature and Historical-Economics Sources

The dissertation’s historical and literature-review sections relied on the following categories of sources:

• peer-reviewed and preprint cryptography literature on close-prime factorization, unbalanced RSA, and partial key exposure
• historical storage-cost summaries and datasets
• public statistical or explanatory summaries used only where they were appropriately attributed and clearly secondary in role

The core reference list is maintained in:

• 07_REFERENCES.md

A.12 Reproducibility Limitations

Several reproducibility limits should be stated explicitly.

1. File timestamps are environmental artifacts. They depend on the local filesystem and may not survive copying, archival packaging, or repository export.

2. The present deck uses a CSV storage format. Any future switch to binary or compressed storage will change bytes-per-prime measurements materially.

3. Throughput measurements are machine-specific. Build rates and factoring times depend on CPU, memory, storage, operating system behavior, and concurrent load. In particular, storage medium and topology may materially change factor-search throughput: internal SSDs, external mechanical disks, hybrid cached disks, and striped or mirrored layouts should not be assumed equivalent.

4. Some table values are extrapolations. Where the dissertation projects beyond observed milestones, those projections are based on stated assumptions and are not direct observations.

5. Database availability is an operational prerequisite. The experiment stack assumes a reachable PostgreSQL instance and a successfully initialized factoring_experiments schema. The dissertation’s automation design is therefore reproducible in method, but the actual execution environment must provide the database service.

A.13 Implemented Reproducibility and Automation Improvements

The dissertation now includes concrete implementations of the principal automation improvements that were earlier only recommended. Specifically, the experiment stack now includes:

• a worker-sweep harness for automatic reruns of the same accepted dataset across multiple worker counts
• database views and export tooling for common dissertation tables and charts
• host-label support in sessions and runs so later multi-machine studies can be distinguished cleanly
• a storage-format comparison harness for CSV, packed-width, and delta-oriented representations
• a reproducibility-manifest generator that records host, runtime, and deck state at execution time

These additions materially improve the dissertation’s ability to support:

• controlled replication
• sensitivity analysis
• worker-scaling studies
• deck-milestone comparisons
• database-first analysis with spreadsheet tools as downstream clients

Future work should continue extending these features, especially through richer analysis views, host-level cluster metadata, and broader storage-format benchmarking, but the core automation and reproducibility path now exists in executable form.

Appendix B. Command Log for Measured Tables

B.1 Purpose

This appendix preserves the command patterns and recorded outputs used to derive the measured software-construction, deck-growth, and scaling-baseline tables in Chapter 4. It is intended as a lightweight reproducibility aid.

The appendix now also records the command patterns used to initialize the experiment schema, generate accepted semiprime populations, and run controlled experiment sessions. These commands are not merely operational conveniences. They define the executable form of the dissertation methodology.

B.2 Prime-Generator Timestamp Inspection

Command used:

find /mnt/c/Users/tgfla/tools/prime-generator -maxdepth 3 -type f -printf '%TY-%Tm-%Td %TH:%TM:%TS %p\n' | sort

Relevant observed output excerpt:

2026-04-02 20:07:29.9816447000 /mnt/c/Users/tgfla/tools/prime-generator/Cargo.toml
2026-04-02 20:29:20.3249781000 /mnt/c/Users/tgfla/tools/prime-generator/src/main.rs
2026-04-02 20:35:39.3874398000 /mnt/c/Users/tgfla/tools/prime-generator/target/release/prime-generator.exe

Derived lower-bound construction window used in Chapter 4:

2026-04-02 20:07:29 to 2026-04-02 20:35:39 = 0:28:09

B.3 Factoring-Platform Timestamp Inspection

Command used:

find /mnt/c/Users/tgfla/tools/factoring -maxdepth 3 -type f -printf '%TY-%Tm-%Td %TH:%TM:%TS %p\n' | sort

Relevant observed output excerpt:

2026-04-03 10:31:43.4508823000 /mnt/c/Users/tgfla/tools/factoring/_start_factoring_capture.ps1
2026-04-03 11:17:53.4220092000 /mnt/c/Users/tgfla/tools/factoring/worker.js
2026-04-03 12:08:29.9029629000 /mnt/c/Users/tgfla/tools/factoring/server.js
2026-04-03 12:08:38.3427388000 /mnt/c/Users/tgfla/tools/factoring/public/index.html
2026-04-03 12:08:54.8674300000 /mnt/c/Users/tgfla/tools/factoring/public/app.js
2026-04-03 12:09:16.3132709000 /mnt/c/Users/tgfla/tools/factoring/public/styles.css

Derived lower-bound construction window used in Chapter 4:

2026-04-03 10:31:43 to 2026-04-03 12:09:16 = 1:37:33

B.4 Prime-Deck File Timestamp and Size Inspection

Command used:

find /mnt/c/Users/tgfla/tools/prime-output -maxdepth 1 -type f -printf '%TY-%Tm-%Td %TH:%TM:%TS %s %f\n' | sort

Observed output:

2026-04-02 22:53:54.6999756000 1603405053 primes_000001.csv
2026-04-03 02:40:59.6803707000 1729955438 primes_000002.csv
2026-04-03 07:32:20.0515595000 1786237593 primes_000003.csv
2026-04-03 14:35:44.9666931000 1786476718 primes_000004.csv
2026-04-03 21:03:54.7011033000 1786654909 primes_000005.csv
2026-04-04 02:13:40.4980661000 1367057250 primes_000006.csv

B.5 Prime-Deck Tail Inspection

Command used:

for f in /mnt/c/Users/tgfla/tools/prime-output/primes_00000{1..6}.csv; do
  echo "== $f =="
  tail -n 2 "$f"
done

Observed output:

== /mnt/c/Users/tgfla/tools/prime-output/primes_000001.csv ==
49999999,982451629,30,20,9031508
50000000,982451653,30,24,9031508
== /mnt/c/Users/tgfla/tools/prime-output/primes_000002.csv ==
99999999,2038074739,31,26,21757448
100000000,2038074743,31,4,21757448
== /mnt/c/Users/tgfla/tools/prime-output/primes_000003.csv ==
149999999,3121238891,32,24,39237831
150000000,3121238909,32,18,39237831
== /mnt/c/Users/tgfla/tools/prime-output/primes_000004.csv ==
199999999,4222234727,32,24,64642735
200000000,4222234741,32,14,64642735
== /mnt/c/Users/tgfla/tools/prime-output/primes_000005.csv ==
249999999,5336500523,33,40,83568708
250000000,5336500537,33,14,83568708
== /mnt/c/Users/tgfla/tools/prime-output/primes_000006.csv ==
288199809,6194977813,33,12,102172415
288199810,6194977853,33,40,102172416

These tail rows were used as direct spot checks on file-level milestone cutoffs.

B.6 Parsed Deck Milestones

Command used:

python3 - <<'PY'
from pathlib import Path
import csv
files = sorted(Path('/mnt/c/Users/tgfla/tools/prime-output').glob('primes_*.csv'))
for f in files:
    with f.open('rb') as fh:
        fh.seek(0, 2)
        size = fh.tell()
        pos = max(0, size - 65536)
        fh.seek(pos)
        data = fh.read().decode('utf-8', errors='ignore').splitlines()
    last = data[-1]
    row = next(csv.reader([last]))
    ordinal = int(row[0])
    prime = int(row[1])
    bits = prime.bit_length()
    print(f'{f.name}\t{ordinal}\t{prime}\t{bits}')
PY

Observed output:

primes_000001.csv   50000000    982451653   30
primes_000002.csv   100000000   2038074743  31
primes_000003.csv   150000000   3121238909  32
primes_000004.csv   200000000   4222234741  32
primes_000005.csv   250000000   5336500537  33
primes_000006.csv   288203789   6195068527  33

This parsed result was used for the milestone tables because the last valid row in a still-growing file can differ slightly from a quick manual tail probe.

B.7 Derived Timing and Rate Calculation

Command used:

python3 - <<'PY'
from datetime import datetime
prime_app_start = datetime.fromisoformat('2026-04-02 20:07:29.981644')
prime_app_end = datetime.fromisoformat('2026-04-02 20:35:39.387440')
factoring_start = datetime.fromisoformat('2026-04-03 10:31:43.450882')
factoring_end = datetime.fromisoformat('2026-04-03 12:09:16.313271')
print('prime_app_seconds', (prime_app_end-prime_app_start).total_seconds())
print('factoring_app_seconds', (factoring_end-factoring_start).total_seconds())
PY

Observed output:

prime_app_seconds 1689.405796
factoring_app_seconds 5852.862389

These values correspond to:

0:28:09.405796
1:37:32.862389

B.8 Derived Deck-Growth and Storage-Density Calculation

Command used:

python3 - <<'PY'
points = [
(50000000,1603405053,982451653),
(100000000,3333360491,2038074743),
(150000000,5119598084,3121238909),
(200000000,6906074802,4222234741),
(250000000,8692729711,5336500537),
(288203789,10059786961,6195068527),
]
print('ordinal\ttotal_bytes\tbytes_per_prime\tlargest_prime_bits\tsquare_coverage_bits')
for n,b,p in points:
    print(f'{n}\t{b}\t{b/n:.3f}\t{p.bit_length()}\t{2*p.bit_length()}')
PY

Observed output:

ordinal total_bytes bytes_per_prime largest_prime_bits  square_coverage_bits
50000000    1603405053  32.068  30  60
100000000   3333360491  33.334  31  62
150000000   5119598084  34.131  32  64
200000000   6906074802  34.530  32  64
250000000   8692729711  34.771  33  66
288203789   10059786961 34.905  33  66

B.9 Scaling-Baseline Calculation

Command used:

python3 - <<'PY'
rate = 2701.54
bytes_per_prime = 34.905
for prime_bits, n_est in [(32,202777307),(40,41140322812),(48,8722209186967),(56,1905466805129420),(64,425418361821749888)]:
    seconds = n_est / rate
    years = seconds / (365.25*24*3600)
    bytes_total = n_est * bytes_per_prime
    teb = bytes_total / (1024**4)
    pib = bytes_total / (1024**5)
    eib = bytes_total / (1024**6)
    print(f'{prime_bits}\t{2*prime_bits}\t{n_est}\t{years:.3f}\t{teb:.6f}\t{pib:.6f}\t{eib:.6f}')
PY

Observed output:

32  64  202777307   0.002   0.006437    0.000006    0.000000
40  80  41140322812 0.483   1.306037    0.001275    0.000001
48  96  8722209186967   102.308 276.894490  0.270405    0.000264
56  112 1905466805129420    22350.437   60490.782592    59.073030   0.057689
64  128 425418361821749888  4990003.630 13505294.118102 13188.763787    12.879652

These figures were used only as first-order extrapolations derived from the measured prototype baseline.

B.10 Interpretation Note

The presence of a command log does not eliminate the dissertation’s limits. It makes the path from raw local observation to reported table more transparent. The measured values remain prototype-specific, environment-specific, and in some cases extrapolative.

B.11 Experiment-Schema Initialization

Command pattern:

cd /mnt/c/Users/tgfla/tools/factoring
node scripts/init-experiment-db.js

Intended effect:

• create the factoring_experiments schema
• create dataset, session, candidate, run, phase, file, milestone, and snapshot tables

B.12 Accepted Semiprime Population Generation

Command pattern:

cd /mnt/c/Users/tgfla/tools/factoring
node scripts/generate-semiprime-dataset.js \
  --experiment-code experiment_b \
  --dataset-prefix experiment_b \
  --modulus-bits 128 \
  --count 1000 \
  --classes tight,moderate,loose,unconstrained

Effect:

• create accepted semiprime populations under the requested modulus class
• classify them by delta family
• persist them in PostgreSQL with stable dataset identifiers

This command can be scaled from 1,000 to 10,000 and 100,000 accepted instances as computationally practical.

B.13 Database-Backed Queue Execution

Command pattern:

cd /mnt/c/Users/tgfla/tools/factoring
node scripts/run-experiment-queue.js \
  --dataset-name experiment_b_128bit_tight \
  --limit 100 \
  --max-workers 8 \
  --search-mode phased \
  --include-statuses pending

Effect:

• create an experiment session
• claim pending candidates from the chosen dataset
• run them sequentially with the requested worker limit
• persist run-level, phase-level, and file-level measurements

The same dataset can be rerun with a different worker count or search mode, for example:

node scripts/run-experiment-queue.js \
  --dataset-name experiment_b_128bit_tight \
  --limit 100 \
  --max-workers 16 \
  --search-mode static \
  --include-statuses pending,inconclusive

This is the core mechanism for controlled variation across:

• worker counts
• phased versus static execution
• reruns of inconclusive populations after deck growth

B.14 Structured-Composite Session Execution

Command pattern:

cd /mnt/c/Users/tgfla/tools/factoring
node scripts/run-candidate-list.js \
  --input-file candidates.txt \
  --max-workers 8 \
  --search-mode phased \
  --session-name clay_style_phase_probe

Effect:

• create a session for a newline-delimited candidate family
• execute candidates sequentially with structured logging
• enable direct replay of the same family across deck milestones and worker-count settings

This command is the operational basis for Experiments A, D, and E.

B.15 Syntax-Validation Pass for the Experiment Stack

Command pattern:

node --check /mnt/c/Users/tgfla/tools/factoring/scripts/experiment-db.js
node --check /mnt/c/Users/tgfla/tools/factoring/scripts/generate-semiprime-dataset.js
node --check /mnt/c/Users/tgfla/tools/factoring/scripts/factoring-engine.js
node --check /mnt/c/Users/tgfla/tools/factoring/scripts/run-experiment-queue.js
node --check /mnt/c/Users/tgfla/tools/factoring/scripts/run-candidate-list.js
node --check /mnt/c/Users/tgfla/tools/factoring/scripts/init-experiment-db.js

Observed outcome at the time of this documentation pass:

the database helper, generator, factoring engine, queue runner, candidate-list runner,
worker-sweep runner, export tool, deck-format comparison harness,
reproducibility-manifest tool, and runtime helper all passed Node syntax validation

B.16 Automated Worker-Count Sweep

Command pattern:

cd /mnt/c/Users/tgfla/tools/factoring
node scripts/run-worker-sweep.js \
  --dataset-name experiment_b_128bit_tight \
  --limit 100 \
  --worker-counts 1,2,4,8,16,28 \
  --search-mode phased

Effect:

• select one fixed accepted population
• rerun that same population under multiple worker-count settings
• create a distinct experiment session for each worker-count pass
• preserve direct comparability for scaling analysis

B.17 Analysis Export Views

Command patterns:

cd /mnt/c/Users/tgfla/tools/factoring
node scripts/export-experiment-view.js --source v_run_summary --format csv
node scripts/export-experiment-view.js --source v_session_summary --format csv
node scripts/export-experiment-view.js --source v_worker_scaling_summary --format csv

Effect:

• export run-level, session-level, and worker-scaling summaries
• create flat files suitable for Excel, pivot tables, and dissertation charting

B.18 Storage-Format Comparison Harness

Command pattern:

cd /mnt/c/Users/tgfla/tools/factoring
node scripts/compare-deck-formats.js

Effect:

• compare the current CSV deck to simple packed-width and delta-oriented representations
• produce direct file-level and total-size comparisons for the storage-sensitivity study

B.19 Reproducibility Manifest Generation

Command pattern:

cd /mnt/c/Users/tgfla/tools/factoring
node scripts/write-repro-manifest.js --runner-name manual_probe

Effect:

• emit a JSON manifest containing host label, runtime details, and deck state
• provide a stable execution-context artifact that can be attached to sessions or archived with results

Appendix C. Reference Hardware and Software Environment

C.1 Purpose

This appendix records the reference-machine profile used for the measured prototype results reported in the dissertation. It is not intended to imply that the reported timings are portable across all hosts. Its purpose is to anchor the observed build-rate and factoring measurements to a concrete environment.

C.2 Reference Host Summary

The dissertation measurements in Chapters 4 and 5 were taken on a commodity laptop-class machine with the following observed baseline characteristics.

C.3 CPU Inspection Command and Output

Command used:

lscpu

Observed output excerpt:

Architecture:                         x86_64
CPU(s):                               28
Vendor ID:                            GenuineIntel
Model name:                           13th Gen Intel(R) Core(TM) i7-13850HX
Thread(s) per core:                   2
Core(s) per socket:                   14
Socket(s):                            1
L2 cache:                             28 MiB (14 instances)
L3 cache:                             30 MiB (1 instance)
Hypervisor vendor:                    Microsoft
Virtualization type:                  full

C.4 Memory Inspection Command and Output

Command used:

free -h

Observed output:

               total        used        free      shared  buff/cache   available
Mem:            15Gi       2.1Gi        12Gi       3.6Mi       688Mi        13Gi
Swap:          4.0Gi          0B       4.0Gi

C.5 Operating Environment Command and Output

Command used:

uname -a

Observed output:

Linux FBITruck0315 6.6.87.2-microsoft-standard-WSL2 #1 SMP PREEMPT_DYNAMIC Thu Jun  5 18:30:46 UTC 2025 x86_64 x86_64 x86_64 GNU/Linux

C.6 Graphics Note

The prime-deck generator and factoring platform described in this dissertation were CPU-driven during the measured prototype work. No GPU acceleration path was implemented in the current software stack.

Accordingly, graphics-adapter specifications were not material to the reported deck-construction and factoring measurements. The relevant measured environment for this dissertation is therefore the CPU, memory, storage, and runtime environment captured above.

The same principle applies to the experiment-automation stack. Population generation, run scheduling, worker-count variation, and structured logging were all implemented on the CPU side. Accordingly, references in the dissertation to workers, parallel sweeps, and orchestration concern CPU worker threads and file-partitioned factoring workers rather than accelerator or GPU resources.

C.6.1 Storage Architecture Note

The measured environment should also be understood at the storage-architecture level, not only at the CPU-and-RAM level. During prototype operation, the same prime-deck shard set could be placed either on the internal system drive or on an external backup/archive drive. In practice, this distinction materially affected observed factoring throughput. The implication is that the current factoring workflow is sufficiently I/O-sensitive that storage medium and topology can become first-order experimental variables.

For the purposes of this dissertation, the important architectural distinctions are:

• internal SSD or SSD-class system storage
• external or mechanical spinning-disk storage
• hybrid spinning disks with large flash or DRAM caches
• multi-disk layouts such as striping and mirroring

These are not cosmetic implementation details. SSDs offer substantially lower seek latency and stronger mixed-read behavior than spinning disks. Mechanical disks can perform adequately under long sequential reads, but they degrade more quickly under the shard-oriented, concurrent access patterns used by the factoring platform. Hybrid cached disks may help only when the repeatedly accessed working set remains cache-resident. Striping may improve read throughput by distributing shard access across devices, while mirroring primarily improves resilience and may only secondarily improve read-side concurrency.

Accordingly, future replications should record not only storage capacity, but also storage medium class, whether the active deck resided on internal or external media, and whether any multi-disk topology was used. Without that information, throughput comparisons may overstate or understate the contribution of the search algorithm itself.

C.7 Why This Appendix Matters

The measured build rates and factor-search timings reported in the dissertation are meaningful only when read relative to the host that produced them. The reference environment in this appendix therefore serves two purposes:

• it shows that the prototype results were obtained on commodity hardware rather than specialized institutional infrastructure
• it clarifies that future replications on different hosts should expect materially different throughput figures

C.8 Recommended Future Environment Logging

Future experimental runs should record the following automatically:

• CPU model and logical core count
• RAM size visible to the runtime environment
• storage device class and capacity
• operating system version and kernel version
• runtime versions for Rust, Node.js, and any auxiliary tooling
• whether the run used bare metal, WSL, containerization, or cloud virtualization

These additions would make later throughput comparisons cleaner and more defensible.

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