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REVIEW 5 major objections 6 minor 6 references

Discovering a Regularity: the Case of An 800-year Law of Advances in Small-Arms Technologies

T0 review · 5 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A weighted product of four weapon attributes follows one accelerating curve from 1200 CE to today.

desk verdict A cleanly presented 800-year regularity that is likely a fitted artifact rather than a discovered law; the model-selection contradiction in Table 2 is the soft spot that needs fixing before the headline claim can be taken seriously. read the letter →

arxiv 1908.03435 v1 pith:ZGOMQSKF submitted 2019-08-09 stat.AP

Alexander Kott , Philip Perconti , Nandi Leslie This is my paper · ORCID
classification stat.AP
keywords technologytrendsFigureofRegularitysmallarmsmilitarysuperexponentialgrowthMoore'slawtechnologicalforecastingperformancemeasures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a way to find a composite measure, called a Figure of Regularity (FoR), for technologies that lack a single accepted performance measure, and demonstrates it on infantry small arms from medieval bows to modern assault rifles. It reports that a weighted product of muzzle velocity, effective range, projectile mass, and rate of fire follows a smooth accelerating curve for roughly 800 years. The fitted law is $\log(FoR) = 1.0 + 8.27\times 10^{-6}(t-1200)^2$, with $FoR = (1.1\times 10^{-6}) V^{2.0}D^{2.35}M^{0.61}R^{0.39}$, selected over four alternative temporal-growth models. If the result stands, it gives technology analysts a quantitative basis for long-range forecasting and for checking whether a research portfolio can plausibly meet the forecast.

What carries the argument

The central object is the Figure of Regularity (FoR), a multiplicative composite measure $FoR = k\prod_i x_i^{\alpha_i}$ built from a technology's attributes. The paper's mechanism is to fit the attribute exponents $\alpha_i$ simultaneously with the parameters of a set of candidate temporal-growth models—exponential, quadratic-exponential, cubic-exponential, double-exponential, and piecewise-exponential—choosing among them by the Bayesian Information Criterion and related fit statistics. This joint fitting, which generalizes the earlier weighted-attribute regression approach, is what lets the composite measure and its growth law be discovered together rather than assumed in advance.

What would settle it

Re-fit the FoR and temporal models while perturbing the pre-1600 values of maximum effective range within plausible historical uncertainty, say ±25 to 50 meters; if the exponent on effective range moves by more than a few tenths, or if the quadratic-exponential model no longer wins on the Bayesian Information Criterion, the 800-year law is an artifact of those estimates.

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Extended reading notes

Core claim

The central claim is that infantry small-arms technology, although it lacks an agreed measure of performance, has a regular 800-year trend when described by the composite Figure of Regularity $FoR = (1.1\times 10^{-6}) V^{2.0}D^{2.35}M^{0.61}R^{0.39}$, whose logarithm grows as $1.0 + 8.27\times 10^{-6}(t-1200)^2$. The authors select this quadratic-exponential (superexponential) temporal model over exponential, cubic-exponential, double-exponential, and piecewise-exponential alternatives, preferring it by parsimony, the Bayesian Information Criterion, $R^2$, and mean absolute percentage error, and they reject the plain exponential law because it assigns an implausibly small weight to rate of fire. They present the formula as a description of the data rather than a design guide, and use it to forecast an increase of about 0.80 in $\log(FoR)$ by 2050, consistent with their estimate of the combined effect of currently reported rifle-development directions.

Load-bearing premise

The historical attribute values for medieval and early modern weapons—especially maximum effective range, which for longbows and early guns is a rough historical judgment rather than a measured quantity—are accurate and comparable enough that the fitted exponents and the shape of the 800-year trend are trustworthy.

Editorial extensions

If this is right

  • Absent a discontinuity, the model implies small-arms $\log(FoR)$ will grow by about 0.80 above today's best value by 2050, reaching roughly 6.97.
  • Assuming a Moore-law exponential trend a priori is misleading for this domain, because that model assigns almost no weight to rate of fire, contradicting the historical importance of that attribute.
  • Currently discussed rifle improvements—increased velocity, effective range, and projectile mass—would together raise $\log(FoR)$ by about 0.78, close to the 0.80 forecast from the long-term trend.
  • The same joint-fitting procedure can be applied to other technology families lacking a defined performance measure, with the FoR serving as a Moore-law-like quantity for each.
  • The FoR should not be read as a design guide; the paper states explicitly that improving the FoR is not the same as designing a better weapon.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension of the paper's logic would be to fit the same FoR form to weapons data outside the paper's Western European and U.S. scope, or before 1200 CE; large changes in the fitted exponents would indicate the trend is tied to one regional trajectory rather than to small arms generally.
  • The method can be read as constructing a hidden combined measure from attributes, so similarly smooth combined measures may exist in other domains with no accepted performance metric, such as naval gunnery or hand tools.
  • The 2050 forecast assumes the same FoR formula extrapolates beyond the fitted range; a natural check is to compare the actual attributes of any new infantry rifle introduced in the 2020s and 2030s against the trend band implied by the model.
  • The large fitted exponent on effective range (2.35) suggests range has been the dominant selection pressure over the eight centuries studied, an inference about historical drivers that the paper does not itself draw.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

Summary. The paper proposes a composite measure called a Figure of Regularity (FoR) for technologies lacking a standard measure of performance, defined as a multiplicative power-law of four small-arms attributes: muzzle velocity, effective range, projectile mass, and rate of fire. Using a dataset of over 120 weapons from 1180 to 2015 CE, the authors jointly fit the FoR exponents and one of five temporal dynamics models, and select a quadratic-exponential model (Model B) as best. This yields Eq. 4, FoR = (1.1e-6) V^2.0 D^2.35 M^0.61 R^0.39, and Eq. 5, log(FoR) = 1.0 + 8.27e-6 (t-1200)^2, claimed to describe an 800-year superexponential regularity. The paper then uses Eq. 5 to forecast log(FoR) growth to 2050 and compares it to a sum of increments from plausible R&D improvements in muzzle velocity, range, and projectile mass.

Significance. If the claimed regularity held, it would be a remarkable empirical finding: a single weighted product of four weapon attributes following a smooth accelerating curve for over eight centuries, and a methodological template for studying technologies without an accepted measure of performance. The paper is clearly written and transparent about its data and fitting procedure, and the compilation of historical small-arms data in the companion technical report is a useful contribution. However, the statistical support for the specific 800-year law is substantially weaker than the presentation suggests, because the model-selection evidence in the paper's own Table 2 points to a piecewise model, the model comparison is made on in-sample fits, and the forecast feasibility check is internal to the fitted equations.

major comments (5)
  1. [Section 5 and Table 2] The claim that Model B is best 'in terms of parsimony, BIC, R2, and MAPE' is contradicted by Table 2: Model E has a lower BIC (-334 vs -326) and a higher R2 (0.961 vs 0.955) than Model B, which wins only on MAPE (0.112 vs 0.153). Since Section 2 introduces BIC as the model-selection statistic, the choice of Model B requires an explicit justification for prioritizing MAPE, which the paper does not provide. If Model E is correctly specified, the data favor a piecewise exponential with a break near 1832, not a single quadratic superexponential law from 1200 CE to the present.
  2. [Section 4.3, Model E] Model E is mis-specified as printed: both branches of the piecewise formula are identical, exp(theta1 + theta2 t_i), and the stated constraint theta4 > theta1 references a parameter theta4 that does not appear in the formula. As written, this is not a two-regime model and cannot reproduce the fitted values in Table 2. The authors must provide the correct piecewise specification and refit the model before the comparison between Models B and E can be evaluated.
  3. [Sections 4.2-4.4 and Table 2] The FoR exponents alpha_i are re-estimated for each temporal model on the same dataset, and the temporal model is selected on that same dataset. The reported R2, BIC, and MAPE therefore measure in-sample fit of a model class that was selected to fit, not evidence of an independent regularity. To support the claim of discovering an 800-year law, the paper needs an out-of-sample or holdout analysis, for example fitting on pre-1800 data and testing on post-1800 data, or a comparison against a null model that randomizes the attribute values while preserving the time series.
  4. [Section 6] The forecast and feasibility check are internal to the fitted model. The forecast log(FoR)=6.97 in 2050 uses Eq. 5 directly, and the feasibility check sums increments computed from the same fitted FoR exponents (mislabeled as Eq. 5 in the text). The close agreement between the required increment (0.80) and the sum of feasible increments (0.78) is therefore an arithmetic consequence of using the same fitted equations, not independent evidence of forecast feasibility.
  5. [Section 4.1 and Table 1] The central regularity rests on historical attribute values, especially maximum effective range and rate of fire for medieval and early modern weapons, which are rough historical judgments rather than precisely measured quantities. No sensitivity analysis is provided to show how the fitted exponents in Eq. 4 or the temporal curve in Eq. 5 would change under plausible perturbations of these values. Without such an analysis, the stability of the claimed 800-year regularity is not demonstrated.
minor comments (6)
  1. [Section 2, Eq. 2] The symbol k is used both as the number of free parameters and as the product index in Eq. 1; please use different notation to avoid confusion.
  2. [Section 2, Step 1] The text says 'Table 2 shows examples of the data points' but the relevant table is Table 1; please correct the cross-reference.
  3. [Section 4.3, Model C] The verbal description says 'the fraction is a quadratic function of time' but the model is cubic in time; please align the wording with the equation.
  4. [Section 4.4, MAPE description] The MAPE is described as an average over three conditions with different fitting and testing periods; this is a form of out-of-sample evaluation and deserves a more prominent and precise description, including the number of data points in each period.
  5. [Section 6] The text refers to 'Eq. 5' when discussing the FoR model with velocity, range, and mass terms, but those terms appear in Eq. 4; please correct the equation references throughout the section.
  6. [Section 7] The claim that this is 'the longest exponential or superexponential trend reported for any technology' would benefit from a more careful comparison with prior literature, which includes multi-century trends in other domains; as stated, the claim is too broad for the evidence presented.

Circularity Check

2 steps flagged · score 5.0 of 10

The 800-year law is a joint in-sample fit: the Section 6 forecast and feasibility check validate Eq. 5 with the same fit (Eq. 4), and Table 2 contradicts the paper's claim that Model B is best on BIC and R2 — Model E wins both.

  1. fitted input called prediction [Section 6, 'Using the Models for Analysis and Forecasts' (2050 forecast and R&D feasibility check; Eqs. 4–6)]
    "we use the temporal dynamics model, Eq. 5, which gives us the value of logFoR=6.97 in the year 2050... This means that logFoR is likely to grow by about 0.80. ... the currently known developments in small-arms technology might yield a total potential increment of logFoR about 0.08 + 0.54 + 0.16 = 0.78. This is very close to the total increment of 0.80 we forecast based on the long-term trend. ... strengthens the forecast feasibility."

    The predicted growth of 0.80 is not an independent benchmark: Eq. 5 is the temporal law fitted to the same 1200–2015 dataset, with parameters 'optimized jointly with those of the FoR model' (Section 5), so logFoR=6.97 in 2050 is a deterministic extrapolation of the in-sample fit, not an out-of-sample test. The feasibility check reuses the same jointly fitted exponents (2.0, 2.35, 0.61 from Eq. 4) multiplied by assumed improvements (10%; 70%, midpoint of '50–100%'; 80%) to obtain 0.78, and presents the closeness to 0.80 as strengthening the forecast. Both sides derive from one joint fit, so the agreement is internal consistency, not independent evidence.

  2. other [Section 5, 'Discussion of Results' (model selection claim; Table 2 in Section 4.4)]
    "Of the remaining four temporal evolution models, we assess the results associated with model B as the best overall, in terms of parsimony of the model, BIC, R2, and MAPE."

    This is not a definitional circle but a missing-support flag at the selection point of the derivation chain. The paper's own Table 2 reports for Model E: R2 0.961, BIC -334; for Model B: R2 0.955, BIC -326, MAPE 0.112 vs 0.153. Model E is better on R2 and BIC; Model B wins only on MAPE, and Section 2 states 'the lower BIC value is assumed to be one of the criteria for selecting the best-fit model.' The asserted selection of Model B 'in terms of ... BIC, R2' is therefore contradicted by the table: the choice of Eq. 5 silently privileges MAPE over BIC/R2. The same table favors a piecewise model with an 1832 break, so Eq. 5 is a discretionary selection, not the best-supported law by the paper's own criteria, weakening the 'discovered regularity' claim at its selection point.

full rationale

The core mathematical result (Eqs. 4–5) is the output of a joint least-squares fit of the FoR exponents and the quadratic temporal law over the same 1200–2015 dataset (Section 4.4), so presenting the fitted curve as a 'previously unreported regular trend' is partly reporting the fit's own success. The finding nonetheless has independent content: a weighted product of four attributes tracking a smooth superexponential curve over 800 years is not trivially guaranteed (muzzle energy alone is non-monotonic, Fig. 1), and the paper provides a genuine pseudo-out-of-sample MAPE evaluation (Table 2 notes) in which Model B performs best. No load-bearing self-citation chain is present: the data are the first author's Kott [2019] historical compilation (attribute values, not model outputs), and Kott and Perconti [2018] is background context. The score of 5 rests on two compounding problems: Section 6 reuses the same jointly fitted model both to generate the 0.80 forecast and to 'strengthen' it via a feasibility sum (0.78) computed from the same fitted exponents with flexibly chosen improvement assumptions, and Section 5's claim that Model B is best on BIC and R2 contradicts the paper's own Table 2, where Model E has better BIC and R2 (also, Model E's printed formula has identical branches, making the comparison non-transparent). These do not make the derivation equivalent to its inputs by definition, but they mean the headline 'law' is a selected in-sample fit whose forecast/feasibility section is partially circular, with the out-of-sample MAPE as the main independent support.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central result rests on four fitted parameters, several strong domain assumptions about attribute meaningfulness and data reliability, and one newly introduced composite index (FoR). No independent external benchmark validates the FoR as a true performance measure.

free parameters (4)
  • alpha2 (effective range exponent) = 2.35
    Fitted by least squares to the small-arms dataset under Model B; determines how strongly range enters the FoR.
  • alpha3 (projectile mass exponent) = 0.61
    Fitted by least squares; contributes to the FoR through projectile mass beyond its role in kinetic energy.
  • alpha4 (rate of fire exponent) = 0.39
    Fitted by least squares; determines the contribution of maximum rate of fire to the FoR.
  • theta2 (quadratic temporal coefficient) = 8.27e-6
    Fitted coefficient in the selected temporal model log(FoR) = 1.0 + theta2 (t - 1200)^2, with normalization at 1200 CE.
assumptions (5)
  • domain assumption The four attributes V, D, M, and R are the significant and sufficient characteristics for a small-arms FoR across 800 years.
    Adopted from domain expertise without a formal derivation; Section 4.2 states these are considered important.
  • ad hoc to paper The FoR has a multiplicative power-law form f(x) = k product x_i^alpha_i.
    Inherited from Alexander-Nelson regression and assumed without a derivation from weapon physics; Section 4.2.
  • domain assumption The historical data in Kott [2019] are accurate and consistently defined across weapon families.
    The entire empirical basis comes from the author's technical note; Section 4.1.
  • domain assumption The set of five candidate temporal models contains the true growth form.
    Model selection is restricted to these five curves, so if the true dynamics are outside this set, the selected model is misspecified; Section 4.3.
  • standard math Gaussian error model is appropriate for BIC calculations.
    The paper uses BIC under a Gaussian assumption following Friedman et al. 2001; Section 2.
invented entities (1)
  • Figure of Regularity (FoR)
    purpose: A composite measure serving as a substitute for a performance measure, intended to exhibit a regular temporal trend for technologies without an agreed MoP.
    The FoR is defined and fitted entirely within this paper, with no external dataset or experimental validation. Its exponents are chosen to fit historical time series, so it is an invented index rather than a discovered physical quantity.

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Cite this review

Pith. "Pith review of Discovering a Regularity: the Case of An 800-year Law of Advances in Small-Arms Technologies." pith.science (2026). https://pith.science/paper/ZGOMQSKF

@misc{pith2026190803435,
  author       = {Pith},
  title        = {Pith review of: Discovering a Regularity: the Case of An 800-year Law of Advances in Small-Arms Technologies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZGOMQSKF}},
  note         = {Machine review of arXiv:1908.03435}
}
read the original abstract

Considering a broad family of technologies where a measure of performance (MoP) is difficult or impossible to formulate, we seek an alternative measure that exhibits a regular pattern of evolution over time, similar to how a MoP may follow a Moore's law. In an empirical case study, we explore an approach to identifying such a composite measure called a Figure of Regularity (FoR). We use the proposed approach to identify a novel FoR for diverse classes of small arms - bows, crossbows, harquebuses, muskets, rifles, repeaters, and assault rifles - and show that this FoR agrees well with the empirical data. We identify a previously unreported regular trend in the FoR of an exceptionally long duration - from approximately 1200 CE to the present - and discuss how research managers can analyze long-term trends in conjunction with a portfolio of research directions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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