Thinking In Structure

Tag: algorithms

  • Invariant Selection and the Problem of Novelty

    Why good work disappears in stable systems — and when systems quietly outlive their legitimacy

    If you publish a good Substack, write a strong novel, or ship a thoughtful indie game, the dominant experience is rarely rejection. More often, it is non-selection. The work does not fail. It simply never enters the flow.

    This is usually explained away psychologically: bad timing, weak marketing, the wrong audience. But that explanation is unsatisfying, because the same pattern repeats across domains. Writing, games, research, startups — different surfaces, same outcome.

    The deeper reason is not cultural.
    It is dynamical.

    The hidden rule of modern ranking systems

    Most large-scale discovery systems — search engines, recommendation feeds, citation graphs, storefronts — are not designed to find what is new. They are designed to identify what is stable.

    They rank according to invariant structure: patterns of attention that persist under repeated mixing.

    This family of effects is well known. Preferential attachment, Matthew effects, popularity bias, exposure concentration — these have been documented repeatedly in networks ranging from scientific citations to music streaming (Barabási & Albert, 1999; Merton, 1968; Salganik et al., 2006).

    The claim here is not novelty of diagnosis, but precision of mechanism: many systems do not merely reward popularity; they reward self-reproducing patterns of flow.

    What is being selected is not “what many people liked once,” but “what keeps being encountered after the system updates itself.”

    Not just “rich get richer”

    This distinction matters because many popular things do not persist.

    Most viral content decays rapidly. In citation networks, the median paper receives the majority of its citations within 2–5 years and then effectively disappears from the flow. In app stores, industry analyses routinely show that well over 90% of indie releases receive negligible long-term visibility.

    Popularity spikes are common.
    Persistence is rare.

    What systems converge on is not raw popularity, but configurations that survive repeated redistribution of attention.

    The mathematical core (as approximation, not dogma)

    To capture this idea cleanly, it helps to use a simplified model.

    Let the discovery process be represented by a linear operator PPP, describing how attention, citations, or visibility move from one node to another.

    Invariant ranking means finding a vector v\*v^\*v\* such that:Pv\*=v\*P v^\* = v^\*Pv\*=v\*

    This says: once attention settles into this pattern, the system’s own dynamics keep it there.

    Any component not aligned with v\*v^\*v\* decays under repeated application of PPP.

    So:

    Novelty is structurally transient.

    This model is deliberately reductive. Real systems are not purely linear. They include nonlinear feedback, external shocks, human editorial interventions, and rule changes. But over long horizons — and between shocks — linear flow models often describe the dominant tendency of attention remarkably well.

    Think of this not as a law of nature, but as a local approximation, like frictionless planes in physics: wrong in detail, useful in structure.

    Why platform “fixes” only partially work

    Platforms know invariance is a problem. They add freshness boosts, exploration noise, personalization, decay of old signals.

    These interventions matter. They create eddies and side currents.

    But they rarely change the shape of the riverbed.

    Once the perturbation fades, attention flows back into the same channels.

    Local exploration does not rewrite global invariants.

    TikTok: novelty through instability

    TikTok is often cited as a counterexample — and rightly so.

    It differs in two key ways:

    1. The operator is local and conditional
      The For You Page is not one global ranking, but millions of short-horizon, behaviour-conditioned ones.
    2. The time constant is short
      Signals decay aggressively. What worked last week may vanish tomorrow.

    The result is not the absence of invariants, but rapid cycling between them.

    TikTok surfaces novelty — at the cost of persistence. Volatility replaces obscurity; burnout replaces invisibility.

    This confirms the trade-off rather than escaping it:
    stability suppresses novelty, novelty requires instability.

    Why invariant selection is not a bug

    Invariant selection often serves users well.

    Stable ranking systems:

    • reduce cognitive load
    • surface vetted material
    • suppress spam and adversarial gaming
    • converge quickly to “good enough” outcomes

    The cost is conservatism, not inefficiency.

    The problem is not that invariant systems exist.
    It is that they increasingly dominate every discovery context.

    Regime exhaustion: when the river keeps flowing but no longer convinces

    Here is the crucial transition:

    Some systems continue to function long after they have lost legitimacy.

    This is regime exhaustion.

    The rankings still converge. The metrics still update. The pipelines still run. But users feel that outcomes no longer reflect quality, relevance, or fairness.

    At that point, the problem is no longer optimisation.

    It is operator replacement — changing the rules by which attention flows at all.

    Operator replacement at scale (made concrete)

    Operator replacement rarely looks like collapse. More often it looks like attention routing around the official channels.

    Academic publishing is a clean example.

    Citation networks preserve canonical work extremely well, but integrate novelty poorly. Over time, legitimacy leaked elsewhere:

    • preprints (arXiv)
    • conferences overtaking journals in CS
    • blogs, talks, and open-source code becoming reputation carriers

    The old system continued to function.
    It simply stopped being where meaning accumulated.

    That is operator replacement.

    K-pop, briefly, as circulation physics

    K-pop illustrates the same structure in culture.

    Its success rests on an engineered circulation system: training pipelines, synchronized releases, fan mobilisation, platform-native artefacts.

    Attention recirculates efficiently. That efficiency is the strength — and the limit.

    Saturation occurs when the system becomes too good at reproducing itself. Novelty survives mainly as surface variation.

    The river flows.
    Surprise dries up.

    Local rewiring: Japanese indie devs and graph shaping

    At smaller scales, creators sometimes intervene directly.

    Japanese indie developers on Twitter/X form dense clusters of mutual review, retweeting, and visible interaction. This increases internal connectivity, creating a slow-mixing subgraph where attention lingers before leaking out.

    They are not changing the algorithm.
    They are reshaping the plumbing the algorithm operates on.

    This is not marketing.
    It is structural.

    Beyond individual levers: systemic alternatives

    The earlier “three levers” (legibility, local recirculation, graph shaping) describe individual agency. They matter — but they are not the whole story.

    Systemic responses also exist:

    • decentralised networks (e.g. federated social media) that weaken global invariants
    • public-interest discovery systems that privilege diversity over convergence
    • regulatory pressure on monopolistic ranking power

    None of these are panaceas. Each introduces new trade-offs. But they recognise the same underlying issue: when one operator governs too much of cultural flow, novelty suffocates.

    Closing

    Ranking systems based on invariant flow are not wrong. They are incomplete by design.

    They explain where attention stays, not where it should go. They preserve what already works, not what might work under different conditions.

    Understanding this does not guarantee success.
    It does something quieter and more honest:

    It tells you when the problem is you —
    and when it is the riverbed.

    And when a river keeps flowing long after it has stopped nourishing the land, the question is no longer how to swim better.

    It is whether the course itself needs to change.

    Ironically, as this essay itself predicts, its visibility may depend on whether it manages to route around the very invariants it describes.

    https://thinkinginstructure.substack.com/p/invariant-selection-and-the-problem

  • THE ANALYTIC STRUCTURE OF CONSTANTS

    THE ANALYTIC STRUCTURE OF CONSTANTS

    How singularities and symmetry determine the speed of numerical approximation

    Some mathematical constants are easy to approximate. Others converge painfully slowly. A few remain stubborn even after centuries of work. This variation is not random. It reflects the analytic structure of the functions that define the constants.

    The central idea of this article is simple:

    The ability of a function to continue analytically beyond the real line determines how fast any basic approximation method can converge. The location of singularities and the presence of global symmetries influence the decay of coefficients in Taylor, Fourier, or related expansions, and that decay controls the speed of computation.

    This gives us a clear way to understand why certain constants are intrinsically slow and why others allow rapid algorithms once the right structure is identified.

    1. Local and Global Analytic Structure

    Constants inherit their computational difficulty from the analytic behaviour of the functions behind them.

    Local structure

    Some functions have singularities very close to the real axis. For example:

    • arctan has singularities at ±i

    • 1/x has a pole at 0

    • algebraic functions have branch points near their roots

    Such functions have a limited radius of convergence for their power series. Their coefficients decay only at a polynomial rate, and this restricts how fast any elementary approximation can converge. By “elementary,” we mean methods that use:

    • Taylor expansions

    • Euler–Maclaurin corrections

    • Riemann sums and trapezoidal rules

    • simple algebraic transformations

    • Machin-type arctan decompositions

    These methods rely solely on real-line information and do not use any global structures such as periodicity or modular symmetry.

    A brief historical aside

    The contrast between “local” and “global” structure is not just a theoretical classification. When modular-form formulas for π were discovered and refined, the speed was so extraordinary that the Chudnovsky brothers built a home-made supercomputer in their New York apartment in the 1990s specifically to exploit them. The machine, assembled from spare parts and cooled with improvised plumbing, set world records for digits of π. It remains one of the clearest demonstrations of how global analytic structure can translate directly into raw computational power.

    Global structure

    Other functions behave nicely over large regions of the complex plane. Examples include:

    • sin(πx), which is entire and periodic

    • modular forms, which are analytic on the upper half-plane and satisfy transformation laws

    • elliptic functions, which are doubly periodic

    Their Fourier or spectral coefficients decay exponentially or faster, and this creates the possibility of very rapid convergence. Algorithms that use these structures are not elementary in the sense defined above. They rely on analytic continuation and global symmetry.

    2. Why Analytic Structure Determines Convergence

    The mechanism behind the phenomenon is classical. If a function is analytic inside a disk of radius R, then its Taylor coefficients are bounded by M divided by R to the power n. This means:

    • a nearby singularity (small R) leads to slow coefficient decay

    • entire behaviour (large R) gives exponential decay

    • modular or elliptic symmetries can create even faster decay

    Since all basic approximation schemes ultimately depend on expansions of this sort, the rate of coefficient decay sets a hard limit on the speed of convergence.

    This is a precise mathematical fact, not a heuristic.

    3. Constants Limited by Local Singularities

    These constants can only be reached slowly with elementary methods.

    π through arctan

    The singularities of arctan at ±i are at distance 1 from the real axis. Its Taylor coefficients behave like 1/n, which gives convergence of order 1/n for the usual Gregory series. This proves that real-line Taylor methods for π must be slow.

    Machin-type formulas help only because arctan(1/q) moves the singularities farther away, but the convergence is still polynomial.

    e and the logarithm

    The standard definitions through integrals or ODEs involve local behaviour. Any Riemann-sum or Euler–Maclaurin approach remains slow for the same analytic reason.

    γ (Euler–Mascheroni)

    The constant γ is the limit of Hₙ minus ln n. The defining function 1/x has a singularity at 0, so any elementary method that uses derivative information of 1/x, including Euler–Maclaurin, can only achieve polynomial convergence. There is no known elementary method that gives exponential decay of coefficients.

    4. Constants that Become Fast Once Their Global Structure Is Recognized

    ζ(2)

    The naive series 1 + 1/2² + 1/3² + … converges slowly. This is exactly what the coefficient-decay principle predicts.

    The situation changes completely once ζ(2) is linked to the sine function. The infinite product for sin(πx) is entire and periodic, so its associated coefficients decay exponentially. Fourier expansions and spectral methods then provide rapid convergence and lead directly to the closed form π²/6.

    This is the clearest example of how identifying the right global structure can transform a slow constant into a fast one.

    The Analytic Speed Limit

    Bars show digits gained per iteration. Local singularities (red) cap progress; global symmetries (green) accelerate it.
    Current Iteration
    0
    Step Size
    100
    Local (polynomial)
    Global (exponential)
    Click Run 100 repeatedly to see divergence.

    5. Constants With No Known Usable Global Structure

    ζ(3)

    The constant ζ(3) is analytically well-defined, and many series exist for it, but none of the known representations produce exponentially decaying coefficients using elementary constructions. At present there is no known periodic expansion, no simple entire product, and no modular-form identity that generates a rapidly convergent expression. Some series converge reasonably well, but never in a truly exponential way without heavy analytic work.

    Catalan and elliptic constants

    These constants are connected to functions with branch cuts and deep symmetries that are difficult to exploit. No simple representation with rapid coefficient decay is known.

    6. The Mechanistic Pattern

    The behaviour of constants now follows a very simple pattern:

    Local singularities produce polynomial convergence. Examples include π via arctan, e, the logarithm, γ, and the naive series for ζ(2) and ζ(3).

    Global periodicity or entire behaviour produces exponential convergence once the structure is used. Examples include ζ(2) through the sine product, and fast π algorithms based on modular forms.

    Deep analytic structure without accessible symmetry produces no known fast elementary convergence. Examples include ζ(3), Catalan’s constant, and elliptic integrals.

    The pattern is not historical. It is a direct consequence of standard complex analysis.

    7. Why Modular Forms Create Fast Algorithms for π

    Modular forms satisfy transformation laws that relate values at different points in the upper half-plane. By moving to regions where q = exp(2πiτ) is extremely small, one obtains series whose coefficients fall away at a superexponential rate. This behaviour is the reason the Chudnovsky and Ramanujan series converge so quickly. They harness global symmetry that elementary methods cannot access.

    This explains why polygon-based approximations are slow and why modular methods are exceptionally fast. The analytic behaviour is fundamentally different.

    Chudnovsky π Calculator

    Ready.
    
    
    
    




    




    8. Counterexamples and Edge Cases
    




    BBP formulas for π
    




    Although the BBP series looks elementary, its derivation relies on analytic continuation of polylogarithms and special algebraic identities. It does not fall under the elementary methods described here.
    




    Euler–Maclaurin for γ
    




    The method improves constants but not the overall rate. It remains polynomial.
    




    Continued fractions
    




    Some continued fractions converge quickly for algebraic constants, but analytic limitations prevent them from giving exponential speed for transcendental constants like π or γ without global structure.
    




    Nothing here contradicts the mechanism.
    




    




    9. Why These Ideas Matter
    




    The analytic structure of a constant provides a practical guide to its computational difficulty. It tells us:
    




    • no simple fast algorithm for γ exists unless new global structure is found • ζ(3) will not yield rapid convergence without discovering symmetry now unknown • every fast algorithm for π must rely on entire or modular behaviour
    




    These are clear predictions grounded in complex analysis.
    




    The principle is concise. The decay of coefficients controls convergence. The analytic continuation of a function controls the decay of its coefficients.
    




    Local structure gives slow convergence. Global structure gives fast convergence. Deep structure remains inaccessible without heavy machinery.
    




    This is why some constants are easy and others are not, and why the discovery of global analytic structure has such dramatic computational consequences.
    




    https://thinkinginstructure.substack.com/p/the-analytic-structure-of-constants