Papers deposited on Zenodo. 327 at present. Each title links to its Zenodo record (Concept DOI).
The corpus keeps three layers apart: established results, structural re-readings of them, and private speculation that is not science. Every paper states at its head what it does and does not claim.
Jump to: 001–020 ｜ 021–040 ｜ 041–060 ｜ 061–080 ｜ 081–100 ｜ 101–120 ｜ 121–140 ｜ 141–160 ｜ 161–180 ｜ 181–200 ｜ 201–220 ｜ 221–240 ｜ 241–260 ｜ 261–280 ｜ 281–300 ｜ 301–320 ｜ 321–328
Papers 001–020
- 001 論文1
In 3D Euclidean space R³, we prove that the necessary and sufficient condition for the spherical-shell mass growth rate dm/dr of a spherical… article - 002 論文2
In three-dimensional Euclidean space, when integrating over a spherical region whose density follows the inverse-square law ρ(r) = 1/r², the… article - 003 論文3
[4th edition (August 2026)] No conclusion has changed. article - 004 論文4
A brute-force search that actually scans 2^n candidates is bounded below by Landauer’s principle. article - 005 論文5
Classical Yang–Mills theory is scale invariant: its action contains not a single parameter of mass dimension. article - 006 論文6
The critical line Re(s)=1/2 is the fixed-point set of the involution s↦1−s defined by the functional equation ξ(s)=ξ(1−s) of the completed z… article - 007 論文7
We present an information-geometric reformulation of the 3D Poincaré Conjecture (Perelman’s Theorem). article - 008 論文8
For the four-dimensional topological Poincaré conjecture (Freedman’s theorem) and the exotic differentiable structures of four-dimensional… article - 009 論文9
This paper states the geometric origin of the 16π² appearing in the quadratic divergence of the Higgs mass, δm_H² = (λ/16π²)Λ², and examines… article - 010 Only Dimensionless Numbers Can Be Fine-Tuned
This paper called the divergence of dm/dr = 4π r^(2−n) for n≠2 a "system crash theorem" and approached the fine-tuning problem from it. article - 011 Factors That Cancel in Normalisation, and the Exponent 2 of Born’s Rule
This is the first paper this series wrote about quantum measurement. article - 012 From Boundary Energy to Volume Energy: The Einstein-Yamagishi Volume Restoration Equation and the Overclock Pythagoras Theorem [Paper 12]
The central observation of this paper is a change of reading, not a calculation. article - 013 Physical Law as a Description Language of the Universe OS
Modern physics has offered many frameworks toward a Grand Unified Theory (GUT), yet an interpretational gap remains between General Relativi… article - 014 The Two Patches Are Not the Same Kind of Thing
This paper introduced ε₀=r (the sync patch) and ε₀=c (the overclock patch), and the second edition organised them as a mode-switching specif… article - 015 A Convention Is a Number, Not a Function of the Configuration
This paper applied two "patches", m=4π and ε₀=r, to the balance equation of the Bohr model and obtained the form 16π²v²r²=e², that… article - 016 The Equals Sign Has Distinct Roots Too
The first edition of this paper presented the one-line chain p=4πv=e/r=c√n_opt and claimed that four domains of physics were thereby "f… article - 017 The 8π Is an Artefact of the Units
This paper showed that substituting m=4π into the kinetic term of the Schrödinger equation gives −(1/8π)∇²ψ + Vψ = Eψ, and the first edition… article - 018 The Yamagishi Kinetic Energy Equation
Substituting the Yamagishi patch m = 4π = Vol(S²), supplied by the n = 2 uniqueness theorem, into the classical kinetic energy E_k = ½ m v² … article - 019 The Yamagishi Spherical Compression Pressure Equation
Substituting m=4π into the kinetic-theory pressure P = (1)/(3) n m v² gives P = (4)/(3)π n v² , in which the coefficient (4)/(3)π equals the… article - 020 The Yamagishi Overclock Temperature Equation: Derivation of kT = frac43pi v^2 and Consistency with the Standard Equation of State
We examine the geometric expression of the temperature kT. article
Papers 021–040
- 021 Zero-Tension Does Not Act on an Identity
This paper substituted m₀=4π into the relativistic energy–momentum relation to write E²=(pc)²+(4πc²)², and the first edition called the rear… article - 022 The Yamagishi Zero-Tension Plasma Equation
We give a geometric reformulation of the plasma frequency _p = √((n e²)/(m ₀)) by applying the Yamagishi patch m=4π (from the n=2 uniqueness… article - 023 Making the Curvature Constant Does Not Fix the Density
Making the curvature constant does not fix the density – what Yamabe’s theorem can and cannot do to the component light leaves behind. article - 024 The One-Way Speed of Light Cannot Be Measured
The one-way speed of light cannot be measured – the convention of simultaneity, and its fence. article - 025 “Unsolvable” Has Three Distinct Roots
Unsolvable has three distinct roots – the separator is which tools were allowed. article - 026 The Yamagishi Classical Mechanics Reformulation
We substitute the Yamagishi mass patch m=4π into the basic equations of Newtonian mechanics. article - 027 “We Measured the Curvature”
We measured the curvature – of which manifold? Gauss’s triangle saw the ground. article - 028 The Yamagishi-Kepler Equation
Substituting the Yamagishi patches M = 4π and G = 4π into Kepler’s third law T² = (4π²/GM)a³ cancels the 4π² of the numerator against the 16… article - 029 The Yamagishi
The Adler–Bardeen theorem (1969) states that the coefficient of the chiral anomaly is fixed at one loop and receives no corrections from higher orders. article - 030 The Yamagishi Double-Torus Stellar Fortress Equation
We rewrite the self-gravitational potential energy of a uniform sphere, U = -(3GM²)/(5R), using M = (4)/(3)π R³ ρ and the Yamagishi mass pat… article - 031 The π That Carries No Dimension
The pi that carries no dimension – the pi on Ramanujan’s page did not come from a circle. article - 032 The Yamagishi Unified Budget Canvas Theory
This paper examines the geometric constant 32π² common to the instanton topological charge density q(x) of quantum chromodynamics and the Po… article - 033 The Wigner Angle Is an Area
The Wigner angle is an area – velocities do not add plainly because the place they live in is curved. article - 034 A Census of the c⁵s
This paper formally integrated E²=(4πc²)²+(4πvc)² with respect to the speed of light c and, on the strength of the resulting (16π²/5)c⁵, cla… article - 035 The Yamagishi Gravitational Constant Unpacking Theory
Substituting the Yamagishi gravitational patch G_geom = 4π = Vol(S²), obtained from the n = 2 uniqueness theorem, into Newtonian gravity and… article - 036 The π in Stokes’ 6π Is a Solid Angle
The pi in Stokes’ 6 pi is a solid angle – the 6 does not cross dimensions. article - 037 Is 6π Really 4π + 2π?
Is 6 pi really 4 pi plus 2 pi? The same number, and the supports do not meet. article - 038 The Yamagishi Master Table of Classical-Physics Geometric Unpacking
This paper is a comparison table listing how representative formulas of mechanics and astrophysics look once the Yamagishi patches (m = 4π, … article - 039 The Yamagishi Planck Length Unpacking Theory
[4th edition (August 2026)] No conclusion has changed. article - 040 The Four Planck Quantities Are Four Instances of One Rule (Third Edition)
The Four Planck Quantities Are Four Instances of One Rule: Mass Dimension Fixes the Power of 4pi, and l_p/t_p = c Holds in Every Unit System. article
Papers 041–060
- 041 The Fringes Do Not Remember the Exponent (Third Edition)
The Fringes Do Not Remember the Exponent: Flux Conservation Forces n = d-1, and the Fringe Spacing Comes from Phase – Two Distinct Roots. article - 042 The G and the c⁵ of the Quadrupole Formula Cannot Be Separated (Third Edition)
The G and the c^5 of the Quadrupole Formula Cannot Be Separated: G/c^5 Is the Reciprocal of the Planck Power, and the Two 5s Have Different Origins. article - 043 Keep the Patch Constant as a Letter and the Ambiguity Dissolves (Third Edition)
Keep the Patch Constant as a Letter and the Ambiguity Dissolves: 15360 pi Is Physics, a^3 Is Convention, and the Power of a Is the Mass Dimension. article - 044 The Power of 4π Is the Mass Dimension (Third Edition)
The Power of 4pi Is the Mass Dimension: “Zero or One” Is a Property of the List, and What Fixes a = 4pi Is an Electromagnetic Unit Convention. article - 045 An Area Is Not a Solid Angle (Third Edition)
An Area Is Not a Solid Angle: The Hopf Factorisation Holds at Every Radius, and R = 2 Is a Choice of Unit. article - 046 Of 16π², Only Ω₃ = 2π² Is Geometry (Third Edition)
Of 16 pi^2, Only Omega_3 = 2 pi^2 Is Geometry: The (2 pi)^4 Is a Fourier Convention, and the 4 pi of 4 pi v Is Not a Solid Angle. article - 048 The Pfaffian’s 2ⁿ n! Is One Block (Third Edition)
The Pfaffian’s 2^n n! Is One Block: Where the Yamagishi Euler-Characteristic Unpacking Cut (4 pi)^n n!. article - 049 2.5PN Is the First Dissipative Order Because Mass Conservation Kills n=1 (Third Edition)
2.5PN Is the First Dissipative Order Because Mass Conservation Kills n = 1: Where the Even-Order Rule of the Yamagishi Higher-Order Delay In… article - 050 The Internal Structure of the Exponent 7 Is Not Unique (Third Edition)
The Internal Structure of the Exponent 7 Is Not Unique: The 5/2 + 9/2 of the Yamagishi Planck-Pressure Limit Names the Route Taken. article - 051 The Yamagishi Gravitational Multipole Aggregation Theory (Fourth Edition)
The Yamagishi Gravitational Multipole Aggregation Theory — fourth edition: the licence is the exponent two and four pi does not survive into… article - 052 The Yamagishi Absolute-Zero Standby and Hawking Phase-Transition Theory (Third Edition)
The Yamagishi Absolute-Zero Standby and Hawking Phase-Transition Theory: the critical radius contains no G, and the 8π is not a solid angle. article - 053 The Yamagishi Born-Rule Geometric Structure Theory (Third Edition)
The Yamagishi Born-Rule Geometric Structure Theory: the exponent 2 comes from two distinct roots, and here the 4π survives. article - 054 The Yamagishi Time-Operator and Clock-Subsystem Theory (Third Edition)
The Yamagishi Time-Operator and Clock-Subsystem Theory: Pauli’s theorem needs the Weyl relation, and the internal clock only moves the obstruction. article - 055 The Yamagishi Macroscopic Superposition Limit Theory (Third Edition)
The Yamagishi Macroscopic Superposition Limit Theory: the density correction never reached the table of radii, and E_G=Gm²/R hides the separation. article - 056 A Structural Study of the 4π Patch (Second Edition)
A Structural Study of the 4π Patch: of the four claims about the Yamagishi patch, two have been overtaken by their own author. article - 057 A Kaluza-Klein Re-entry (Second Edition)
A Kaluza-Klein Re-entry: KK charge seen through the Yamagishi patch. article - 058 The Agreement of the Factors 2 Is a Case of Distinct Roots (Second Edition)
The Agreement of the Factors 2 Is a Case of Distinct Roots — second edition: the first edition called to the stand the very coincidence it h… article - 059 The Volume of the Unit Ball Is Greatest in Five Dimensions (Second Edition)
The Volume of the Unit Ball Is Greatest in Five Dimensions — second edition: that five is a consequence of having set the radius to one. article - 060 c as the Ball of Velocity Space (Second Edition)
c as the Ball of Velocity Space — second edition: this was a result about c, not about four pi. article
Papers 061–080
- 061 c as the Light Sphere
c as the Light Sphere — The Causal Sphere r=ct, Volume Restoration, and the Origin of the Inverse-Square Law. article - 062 +C and Λ: A Duality on the Volume Degree of Freedom
+C and Λ: A Duality on the Volume Degree of Freedom — A Structural Interpretation with Unimodular Gravity as the Lens. article - 063 The Velocity–Spacetime Curvature Duality
The Velocity–Spacetime Curvature Duality: Energy as the Hinge between E = 4π/|K| and G_μν = κ T_μν — Thomas Precession and Gravitational Holonomy. article - 064 The Special-Dimension Distribution Theorem
The Special-Dimension Distribution Theorem: Why There Is No Single “Magic Dimension” — An Extension of the Function-Comparison Method of Paper 58. article - 065 The Bekenstein–Hawking Entropy of the Causal Sphere and c⁵
The Bekenstein–Hawking Entropy of the Causal Sphere and c⁵: The Dimensional Ladder of Information Capacity — An Exponent-Structure Observati… article - 066 Velocity-Space Curvature and Gravitational Redshift
Velocity-Space Curvature and Gravitational Redshift: Why “Local c” and “Coordinate c” Are Distinct Roots. article - 067 The Euler-Characteristic Origin of the 4π Patch
The Euler-Characteristic Origin of the 4π Patch: Reading m = 4π as 2π·χ(S²) via Poincaré–Hopf. article - 068 The Qubit as a Hopf Fibration
The Qubit as a Hopf Fibration: Is Born’s “2” the Same Root as the Hopf Map’s “2”? — A Disciplined Census of the Twos. article - 069 The Dimension Census
The Dimension Census: Why 3+1 Dimensions, as an Over-Determination by Independent Roots. article - 070 The Home of the Constants
The Home of the Constants: 4π and 32π² as Index-Theorem Normalizations — Placing the Yamagishi Patch in the Atiyah–Singer Family, Without Me… article - 071 Born’s Exponent and Gleason’s Theorem
Born’s Exponent and Gleason’s Theorem: Placing the Geometric Reading Beneath the Stronger Derivation — and the Qubit, Where Gleason Is Silent. article - 072 Reframing the Cosmological Constant
Reframing the Cosmological Constant: Λ as an Integration Constant, Not a Vacuum Energy — Unimodular Gravity and the +C Duality of Paper 62. article - 073 The “1/4” in Black-Hole Entropy
The “1/4” in Black-Hole Entropy: Anchoring It via the 4π Solid Angle, the Euler Characteristic, and Equipartition — and Placing the Causal S… article - 074 The Yamagishi Patch within Emergent Gravity
The Yamagishi Patch within Emergent Gravity: Reading G=4π as the Elastic Constant of Spacetime — The Umbrella over the Thermodynamic Arc (Pa… article - 075 The Arrow of Time and the Growing Causal Sphere
The Arrow of Time and the Growing Causal Sphere: Reading Penrose’s Weyl Curvature Hypothesis as the Growth of the Light Sphere’s Entropy — T… article - 076 A Covariant Symplectic Structure for Velocity-Space Curvature and the Spacetime Metric on Phase Space T*M
A Covariant Symplectic Structure for Velocity-Space Curvature and the Spacetime Metric on Phase Space T*M: An Open Framework for the Open Pr… article - 077 Liouville’s Theorem and the Arrow of Time: Fine-Grained Conservation and Coarse-Grained Growth
Liouville’s Theorem and the Arrow of Time: Fine-Grained Conservation and Coarse-Grained Growth — Closing the Arc of Paper 75 on the Hamilton… article - 078 Friction and Dissipation as Coarse-Graining
Friction and Dissipation as Coarse-Graining: Why Energy Conserved in the Fine-Grained Appears “Lost” — The Fluctuation–Dissipation Theorem a… article - 079 The Hidden Symmetry Unique to n=2
The Hidden Symmetry Unique to n=2: Why Inverse-Square Orbits Close — The Laplace–Runge–Lenz Vector and the Dynamical Twin of Paper 01. article - 080 Conservation Laws Are the Shadow of Symmetry
Conservation Laws Are the Shadow of Symmetry: Reading Noether’s Theorem on Phase Space T*M — Why Things Are “Conserved”. article
Papers 081–100
- 081 The Poincaré Recurrence Theorem and the Arrow of Time
The Poincaré Recurrence Theorem and the Arrow of Time: The Microscopic Always Returns, but the Universe Ends First — The Companion to Paper 77. article - 082 Renko Time
Time as a Renko Chart: Time as the Accumulation of Events — An Open Framework across Proper Time, Coarse-Graining, and Relational Time. article - 083 Why Does a “Maximum Speed” Exist?
Why Does a Maximum Speed Exist? — The Two Relativities (Galilean and Lorentzian) and the Theorem of Ignatowski. article - 084 The Principle of Least Action
The Principle of Least Action — The Foundation Beneath the Phase-Space Trilogy (Euler), yet Not Teleology. article - 085 From Galois to Noether
From Galois to Noether — Symmetry as the Arbiter of the Possible: The Lineage of Papers 79 and 80, yet Discrete and Continuous Are Distinct Roots. article - 086 The Four Roots of Necessity
The Four Roots of Necessity: Why a Quantity Is “Fixed” — A Census of Distinct Roots across Topology, Symmetry, Algebra, and Dimension. article - 087 The Action Principle as the Description Language of Physical Law
The Action Principle as the Description Language of Physical Law: From Particle to Field — One Grammar, Not One Equation. article - 088 Two Entropies, Not to Be Conflated
Two Entropies, Not to Be Conflated: A Structural Contrast of 3D Coarse-Graining Entropy (the Arrow of Time) and the Black-Hole Horizon Area … article - 089 Zero-Tension and the Inverse Problem
Zero-Tension and the Inverse Problem: Reading the Hidden Cause from a Perfect Null — The Reversible Reference Layer and Inverse Scattering. article - 090 Continuous Symmetry Emerges from Weak Assumptions
Continuous Symmetry Emerges from Weak Assumptions: Hidehiko Yamabe and Hilbert’s Fifth Problem (The Deep Cousin of Ignatowski). article - 091 Curvature Is Intrinsic
Curvature Is Intrinsic: Riemann, and the Language for Not Conflating the Corpus’s Many “Curvatures”. article - 092 The Shared Alphabet
The Shared Alphabet: Cartan’s Differential Forms and the Language Beneath the Corpus’s “Grammar”. article - 093 The Discovery of the Field
The Discovery of the Field: Faraday’s Lines of Force and the Physical Intuition That Came Before the Mathematics. article - 094 “Dimension” Is Not One Thing
“Dimension” Is Not One Thing: Mandelbrot’s Fractal Dimension and Not Conflating the Corpus’s Special Dimensions (From the Roughness of Markets). article - 095 Convention and Fact
Convention and Fact: Poincaré’s Conventionalism and the Corpus’s “Scaffolding” (the Defined Value of c, Unit Conventions). article - 096 The Gauge Principle
The Gauge Principle: Not Conflating Weyl’s Two “Gauges” (Scale and Phase). article - 097 Time and Space as Relation
Time and Space as Relation: Leibniz’s Relationalism, and Not Confusing the Teleology of “the Best” with Least Action. article - 098 The Past Hypothesis
The Past Hypothesis: Boltzmann’s H-Theorem, and Honestly Admitting the Arrow of Time Is Not Yet Solved. article - 099 Reading the Interior from the Boundary
Reading the Interior from the Boundary: the Calderón Problem, Electrical Impedance Tomography (EIT), and the Inverse-Problem Stance. article - 100 The Crossing of Royal Roads
The Crossing of Royal Roads: a Census of Structural Interpretation (Papers 63-99), and the One Discipline of “One Word, Distinct Roots”. article
Papers 101–120
- 101 Admiration and Caution
Admiration and Caution: Eddington’s Fundamental Theory and the Discipline for Not Falling into Numerology (Not Mistaking Derivation for Selection). article - 102 Hearing Shape from Spectrum
Hearing Shape from Spectrum: Connes’ Non-Commutative Geometry, Kac’s Inverse Spectral Problem, and the Highway Linking Geometry and Inverse Problems. article - 103 Flowing Geometry to Canonical Form
Flowing Geometry to Canonical Form: Hamilton’s Ricci Flow, Perelman, and “Dynamical Uniqueness”. article - 104 The Algebra That Holds the Metric
The Algebra That Holds the Metric: Clifford’s Geometric Algebra, Spinors, and Not Conflating It with Cartan’s Exterior Algebra. article - 105 The Exception Only in Four Dimensions
The Exception Only in Four Dimensions: Donaldson’s Exotic R^4 and One More Entry in the Overdetermination of “Why 3+1 Dimensions”. article - 106 Extracting Signal from Noise
Extracting Signal from Noise: Wiener’s Filter, Cybernetics, and the Statistical Face of the Inverse Problem (the Origin in Markets). article - 107 The Two Noethers
The Two Noethers: Not Conflating the First Theorem (Conservation) with the Second Theorem (Identities and Constraints). article - 108 The Geometry Where the Invariant Lives
The Geometry Where the Invariant Lives: Minkowski’s Spacetime, and Not Conflating the “Geometries of Relativity”. article - 109 The Geometry of Dimensional Lift
The Geometry of Dimensional Lift: Why the Power of pi Rises with Dimension, a Census of Royal-Road Topology. article - 110 Ricci Flow IS a Renormalization Group Flow
Ricci Flow IS a Renormalization Group Flow: Where Two Lineages Shake Hands over Taming the Infinite, a Census of a Royal-Road Bridge. article - 111 How Far Does Light Determine Geometry?
How Far Does Light Determine Geometry? Two Roads That Stop at Conformal Structure (Differential Geometry and Order Theory), and the Distinct… article - 112 The Allotted Region and the Freedom Within
The Allotted Region and the Freedom Within: A Census of Six Distinct Roots Sharing One Rhyme – Prohibition, Resolution, Postulate, Theorem, … article - 113 Modules Are Not Walls but Limits
Modules Are Not Walls but Limits: A Distinct-Roots Census of Contraction, Deformation and Decoupling, and the Beautiful Temptation of “Stability”. article - 114 Which Objects Know They Are in Four Dimensions?
Which Objects Know They Are in Four Dimensions? A Census of Dimension-Selective and Dimension-Agnostic Structures: Why the Tesseract Became … article - 115 Two Staircases
Two Staircases: Separating Value from Generator – The Sphere-Volume Recursion, Its Parity Split, and Where This Series’ Constants Live. article - 116 What Does the Exponent of c Count?
What Does the Exponent of c Count? – The Exponent Formula of the Planck Reduction, epsilon as Mass Dimension, the Non-Uniqueness of 4pi, and… article - 117 Scale Invariance Fixes an Exponent
Scale Invariance Fixes an Exponent – in Four Different Ways: Separating Identity, Dimension Matrix, Measurement and Fixed Point; the Conditi… article - 118 Two Clocks and One Scale
Two Clocks and One Scale: Coordinate Time dt, Proper Time dtau and Entropy dS Are Three Different Logical Types. article - 119 The Fourth Clock
The Fourth Clock: A Flow Parameter Advances Not Inside a Manifold but Through the Space of Manifolds. article - 120 Dimensional Analysis Fails in Two Ways
Dimensional Analysis Fails in Two Ways: beta (Dimensional Transmutation) and gamma (Anomalous Dimension) Are Independent. article
Papers 121–140
- 121 The Space of Instruments Carries a Metric of Its Own
The Space of Instruments Carries a Metric of Its Own: A Trajectory Has a Length, but Not a Proper Time – Two Geometries on Met(M). article - 122 How Much Do the Four Clocks Know About the Dimension They Are In
How Much Do the Four Clocks Know About the Dimension They Are In: putting the exponent formula of the Yamagishi patch to the Einstein-Yamagi… article - 123 What Differs Between the Twins Is Not a Lagging Clock but the Length of a Worldline
What Differs Between the Twins Is Not a Lagging Clock but the Length of a Worldline: time dilation read through the Yamagishi zero-tension, … article - 124 Three Distinct Roots of Aggregation
Three Distinct Roots of Aggregation — the shell theorem, Barnes-Hut and the fast multipole method drop terms of the same expansion for three… article - 125 There Are Three Hopfs
There Are Three Hopfs — a proper name also needs an address: applying one word, distinct roots to people. article - 126 The Existence of an Integrating Factor Does Not Give You a Temperature
The Existence of an Integrating Factor Does Not Give You a Temperature — Caratheodory’s axiomatisation in three steps. article - 127 One Phrase, “the Canonical Commutation Relation”, Covers Three Different Situations
One phrase, the canonical commutation relation, covers three different situations – where the Stone-von Neumann uniqueness appears, and wher… article - 128 4π=2πχ Holds in Three Dimensions and Nowhere Else
4 pi = 2 pi chi holds in three dimensions and nowhere else – Poincare-Hopf delivers only one direction. article - 129 ℏ Is Not on the Mathematical Side of the Inequality
hbar is not on the mathematical side of the inequality – the uncertainty principle is not one theorem. article - 130 A Vortex Line Cannot End Because of an Identity, Not a Law
A vortex line cannot end because of an identity, not a law – Helmholtz’s three theorems do not cost the same. article - 131 Of k_B T ln2, Only the T Is Physics
Of k_B T ln2, only the T is physics – taking the number in Landauer’s bound apart. article - 132 The 2 in Ito’s Lemma Is Not a Dimension
The 2 in Ito’s lemma is not a dimension – a distinct-root census of the tools of stochastic calculus. article - 133 The Fisher Metric Is Unique Only on the Classical Side
The Fisher metric is unique only on the classical side – a census of information geometry. article - 134 The 4 in the Light-Bending Angle Is Not 4π but 2(1+γ)
The 4 in the light-bending angle is not 4 pi but 2 times (1 + gamma) – a census of lensing. article - 135 Volume or Energy: You Cannot Have Both
Volume or energy: you cannot have both. article - 136 The Equality Is Exact, and the Average Is Held by the Tail
The equality is exact, and the average is held by the tail – a census of non-equilibrium. article - 137 The KL Divergence Is Not a Distance, but Its Second Derivative Is a Metric
The KL divergence is not a distance, but its second derivative is a metric. article - 138 Which Part Is the Hamiltonian Is a Convention
Which part is the Hamiltonian is a convention – a census of the Lindblad equation. article - 139 Measuring Overlap, or Measuring Displacement
Measuring overlap, or measuring displacement – KL and Wasserstein are not competitors. article - 140 Symmetry Fixes the Ratios, Dynamics Fixes the Scale
Symmetry fixes the ratios, dynamics fixes the scale – a census of the Wigner-Eckart theorem. article
Papers 141–160
- 141 A Limit Exists Because Two Forces Carry the Same Exponent
A limit exists because two forces carry the same exponent – a census of the Chandrasekhar mass. article - 142 Move It, and Only the 2π Survives
Move it, and only the 2 pi survives – a census of Unruh and Hawking. article - 143 Minimum Entropy Production Is Not a Principle but a Theorem with a Fence
Minimum entropy production is not a principle but a theorem with a fence. article - 144 The Exponent Is the Signature of What Is Conserved
The exponent is the signature of what is conserved – a census of the Sedov-Taylor solution. article - 145 Pressure Pulls Too, and That Is Why There Is a Bound
Pressure pulls too, and that is why there is a bound – a census of Buchdahl’s limit. article - 146 The Dirac String Is the Mark of Trying to Write It on One Sheet
[2nd edition (August 2026)] No conclusion has changed. article - 147 The Same H^2, and with Different Coefficients Nothing Is Quantised
The same H^2, and with different coefficients nothing is quantised – a census of the obstructions from local to global. article - 148 eps^2=0 Is a Definition of a Ring and (dW)^2=dt Is a Fact About a Path
eps^2 = 0 is a definition of a ring and (dW)^2 = dt is a fact about a path – a census of the word point. article - 149 What Fixes the Denominator Is Not the Determinant but the Largest Invariant Factor
What fixes the denominator is not the determinant but the largest invariant factor – this paper applies a brake to the corpus itself. article - 150 What Light Cannot Measure Is a Density
What light cannot measure is a density – one conformal rescaling separates the four clocks by weight. article - 151 A Two-Set Cover Returns the Wrong Answer for the Circle
A two-set cover returns the wrong answer for the circle – this paper applies a brake to Paper 146, without withdrawing its conclusion. article - 152 The Same Character Table, and Still Not Isomorphic
The same character table, and still not isomorphic – the same has four strengths, and there is one more rung below them. article - 153 What Was Forgotten Can Be Read Off by Counting Maps
What was forgotten can be read off by counting maps – forgetting comes in three rungs: property, structure, stuff. article - 154 A Limit Without Its Order Is Not a Quantity
A limit without its order is not a quantity – giving the singular limit an address. article - 155 There Are Spaces With No Points at All
There are spaces with no points at all – placing a sixth root for the word point. article - 156 The Inside of a Shell Is Not Flat Because of the Mean Value Theorem
The inside of a shell is not flat because of the mean value theorem – repair the reason, not the theorem. article - 157 A Census of the Twos
A census of the twos – one shape of exponent, arriving from this many different places. article - 158 Whatever Can Be Measured from Inside Has a Home Somewhere
Whatever can be measured from inside has a home – but the address is chosen by our own demand. article - 159 Ill Posed Is Not One Word
Ill posed is not one word – what separates them is not uniqueness but the rate of decay. article - 160 Three Ceilings over One Inequality
Three ceilings over one inequality – and no two are held down by the same thing. article
Papers 161–180
- 161 A Fixed Point Is Not One Theorem
A fixed point is not one theorem – sharing a name does not mean sharing a promise. article - 162 The Whole Can Be Better Known Than Its Parts
There are states about which the whole is completely known and the parts are not known at all. article - 163 A Good Code Exists and Here Is a Good Code Are Different Sentences
A good code exists and here is a good code are different sentences – some sixty years lay between them. article - 164 Probability Has Four Addresses
Probability has four addresses – sharing the axioms does not mean sharing the answers. article - 165 There Are Three Equivalence Principles
The single phrase equivalence principle names three claims that differ in what they assume and in how they are tested. article - 166 The Definition Divides d by p; the Theorem Divides p by the Discriminant
The definition divides d by p; the theorem divides p by the discriminant – the direction of division has changed hands. article - 167 Uniformization Gives Not One Shape but One Model and One Parameter
A split into three is not a statement of topology – the plane and the disk are homeomorphic. article - 168 What Lebesgue Bought Was Not a Bigger Class but Closure under Limits
One word, six integrals – and Lebesgue does not contain improper Riemann. article - 169 Sciama’s Famous Coincidence Is an Identity
When a number agrees exactly, the first thing to suspect is not nature but the definitions. article - 170 Cancelling the Differentials Gives -1, Not +1
A notation moves no theorem, and it does move which mistake looks natural. article - 171 The Fence around Least Squares Is Linearity, Not Normality
When told that nothing has been assumed, look for the assumption that was not written down. article - 172 Zeros Do Not Determine a Function
That the answer comes out right is no evidence that the middle was. article - 173 Kodaira’s Fence Is Integrality, Not Positivity
The condition being paid is not size and not smoothness but being an integer. article - 174 Whose Singularity Is It?
One is mending an injury the world left behind, and the other one it inflicted itself. article - 175 Symmetry Breaking Is Not One Word
The action is symmetric and the answer is not – and that same appearance arrives from three separate places. article - 176 Small Is Two Words, and They Do Not Agree
Almost every carries two meanings, and on one pair they give opposite verdicts. article - 177 The Order of the Quantifiers Is the Whole Content
Pointwise and uniform continuity become different properties by exchanging a for-all and a there-exists. article - 178 The Delta Is Not a Function but a Map
What to look at is neither the height nor the thinness but what it returns to a test function. article - 179 “Cannot Be Constructed” Has Distinct Roots
When something is called non-constructive, the next question is whether searching would find it. article - 180 “The Classical Limit” Is Not One Limit
When someone says take the classical limit, the next question is what is being asked to become classical. article
Papers 181–200
- 181 The Multiplication Does Not Decide the Second System
Knowing the symmetry group does not settle how two systems are put together. article - 182 “The Curse of Dimensionality” Is Not One Curse
When someone says the curse of dimensionality, the next question is what it is a statement about. article - 183 One Table, Two Correct Answers
It is not that there are too few numbers; it is that numbers are not enough. article - 184 Incompatibility Begins with the Third Option
Mathematics can say only that something must be given up, not which. article - 185 What Classifies a Defect Is the Target Space
When something is called topologically protected, the next question is the topology of which space. article - 186 When a Proxy Becomes the Target It Stops Being a Proxy
When a measure is said to have broken the moment it became a target, the next question is what broke the link. article - 187 The Gaussian Is Special for More Than One Reason
When someone says they assume a gaussian, it is not one assumption. article - 188 Fewer Measurements Than Unknowns, and Still Exact
When something is said to be determined by fewer measurements than unknowns, the next question is where the missing information came from. article - 189 “Unprovable” Is Not a Property of a Sentence
When a proposition is said to be unprovable, the next question is under which axioms. article - 190 “Rare” Is Measured on a Logarithmic Scale
When an event is called extremely rare, the next question is on what scale it was measured. article - 191 The Thing That Smooths Is What Makes the Pattern
When a pattern is said to have a definite size, the next question is where that length came from. article - 192 “A Small World” Is Three Separate Claims
When a network is called a small world, the next question is in which sense. article - 193 Two Coupled Costs Cannot Be Added
When told that a behaviour switches where one cost balances another saving, the next question is whether the two are independent. article - 194 The Four Means Are One Family, and There Is Only One Equality Fence
When told to take an average, the next question is which quantity is not allowed to move. article - 195 “Stable” Is Six Separate Words, and What Separates Them Is What You Perturb
When told that a fixed point is stable, the next question is: stable against what? The initial state, the equation itself, the configuration… article - 196 “Pressure” Is Four Separate Quantities, and Sharing a Unit Guarantees Nothing
When told about a pressure, the next question is what must be moved for that pressure to move. article - 197 “Converges” Is Four Separate Claims, and What Separates Them Is What You May Discard
When told that a sequence converges, the next question is in which sense. article - 198 “Straight” Does Not Mean “Shortest”, and There Are Two Fences
When told that a geodesic is the shortest path, the next question is which shortest. article - 199 What Decides Whether You Come Home Is the Exponent in the Denominator
Whether a point wandering at random gets home does not depend on the terrain, the stride, or luck. article - 200 The Two Hundredth Census: Measuring the Corpus as Its Own Data
A census earns its place only when setting things in a row costs one assumption held before the row was made. article
Papers 201–220
- 201 “Complete” Is Four Separate Statements, and What Ties Them Into One Is That the Metric Is Positive Definite
When told that a space is complete and therefore safe, the next questions are complete in which sense, and whether one is inside the fence o… article - 202 A Connection Is Not Derived from the Metric: It Is Pinned Only Once Two Demands Are Chosen
Fix the metric and the geometry follows is half right. article - 203 The Counts of Critical Points Move a Great Deal, and Their Alternating Sum Never Moves at All
When something is counted, the way to separate what belongs to the object from what belongs to the instrument is to change the instrument an… article - 204 The Power in the Eigenvalue Count Is Half the Dimension
From nothing but counting eigenvalues from the bottom up, out comes first the dimension, then the volume, and then the size of the boundary. article - 205 The Shortest Enclosure Is Not Always a Disc
Where the answer switches is also where descent no longer tells you anything. article - 206 The Spaces Converge and the Dimension Does Not
The question what is the dimension of this set has no answer until the scale is named. article - 207 There Are Three Tools for Writing Polarization, and One of Them Cannot Write Unpolarized Light
Whether a tool is adequate cannot be settled by checking whether the answers it gives are right. article - 208 Curvature Does the Same Work as Dimension
The same crowding is sold at two completely different prices. article - 209 Equal Volume Is Not Enough to Cut and Reassemble
What separated “not yet found” from “does not exist” was not precision but a remainder. article - 210 The Peak of a Pulse Can Leave Before It Arrives
What is early is the peak, not the news. article - 211 Adding More Terms Takes You Further from the Answer
What was wrong was not the series but the assumption that adding makes it better. article - 212 Every Map Must Distort Something
A map turns into a lie only when the absence of a bill is read as the absence of a debt. article - 213 Coming Back Has Distinct Roots
It does not return and it returns are both true at once of one motion. article - 214 A Lens Can Brighten and Cannot Increase Brightness
To brighten can be done. article - 215 One Sequence Goes Both to Zero and to Infinity
The theorem was not broken: we walked out of the ground it was guarding. article - 216 It Approaches and Has Not Come One Step Closer
What vanished was not the wave but only the wave’s engagement with a smooth partner. article - 217 Ask Only That Distances Be Kept, and Straightness Is Kept Too
What made the theorem work was not distance but having nowhere to escape to. article - 218 The Fringes Die and Come Back
They went out not because information was lost but because the information had moved over to the side of length. article - 219 A Flat Slab Brings Light to a Focus
The perfect instrument stands at a single point in the space of materials. article - 220 There Are Four Independences, and They Say Nothing to One Another
That a test was passed is often not news about the object but news about the test. article
Papers 221–240
- 221 Change the Last Bit and an Eigenvalue Moves by 0.16
The answer that a matrix is diagonalisable contained nothing at all of the meaning that it will survive computation. article - 222 The Sun Peaks in the Green, in the Orange and in the Infrared
What moves is not the Sun but the unit of counting. article - 223 It Passes Every Test, and the Points Lie on Fifteen Planes
A test sees only what it was built to see. article - 224 Flat Wherever You Touch It, and Four Pi in Total
Curvature is not spread thinly somewhere: it is set down all at once on the vertices, a set of measure zero. article - 225 One Ball Becomes Two of the Same Size
Only one thing is happening: part-way through, the quantity called volume ceases to exist. article - 226 A Surface That Comes Back Inside Out Still Has an Area
What broke was not the area but the machinery that would put a sign on it. article - 227 The Same Inequality Sign Points at a Wall You Can Touch and at One You Cannot
Writing the equality is not an assertion but a way of putting a question: the form in which one asks whether this wall can be touched. article - 228 There Are Eight Plus Signs, and Seven Become Real Addition in the Right Coordinate
Before asking whether things can be added, one must ask in which variable. article - 229 There Are Five Divisions, and Each One Fails for a Different Reason
Before asking whether something divides, one must ask what is being undone. article - 230 There Are Seven Minus Signs, and One of Them Undoes No Addition
Before subtracting, one must ask what is supposed to come back. article - 231 Every Doubling Costs the Product One Law
Before asking whether things can be multiplied, one must ask which law is being relied on. article - 232 Learn from a Book with the Proofs Removed and the Equals Sign Loses Its Type
A proof is both a warrant for a result and the vessel that carries its fence. article - 233 “Differentiable” Is Six Things, and Two of Them Take No Limit
After giving a step function a derivative, Paper 178 wrote that both cases are a question of in which sense one says derivative, and stopped there. article - 234 “Distance” Is Six Things, and One of Them Satisfies the Triangle Inequality Backwards
A metric has only four axioms. article - 235 “Boundary Condition” Is Four Things, and Every Eigenvalue Moves Without a Letter of the Equation Changing
The domain is not changed, the equation is not changed by one letter, and only one line at the ends is varied four ways. article - 236 One and the Same “= 0” Returns Five Different Kinds of Collection
When one says solve, the usual first question is how many answers there are. article - 237 “Infinitesimal” Is Five Things, and Squaring Them Gives Three Different Answers
Paper 148 wrote that the square of epsilon being zero is the definition of a ring while the square of a Brownian increment being dt is a fac… article - 238 “Regular” Is Three Things, and in the Complex Plane One Derivative Gives Infinitely Many
Paper 233 showed that a derivative in every direction does not give a total derivative, which is what happens when the demand about directio… article - 239 “Diverges” Is Four Things, and One of Them Converges in Another Metric
A series is said to diverge when its partial sums do not converge. article - 240 “Mass” Is Seven Things, One of Which Goes Negative and One of Which Depends on the Direction of the Push
This body of work uses the word mass in seventy-five papers, five hundred and forty times, and has never once counted it; Paper 165 counted … article
Papers 241–260
- 241 Before Asking Whether It Can Be Made into Addition, Ask Whether There Is a Binary Operation at All
Paper 228 asked whether a given plus sign becomes real addition after a change of coordinate, and found exactly one case that cannot be repa… article - 242 Do Not Divide by Zero Is Not One Prohibition
Paper 229 counted five obstructions that keep division from undoing multiplication. article - 243 Bounded by a Sum Is Not One Property
Paper 241 asked whether the whole is a function of the parts and listed the quantities for which it is not. article - 244 To Determine an Object You Need Every Other One
The Yoneda lemma says that an object is completely determined by its relations to every other object. article - 245 Two Limits Have No Answer Until the Order Is Written
Paper 154 showed that a limit written without an order is not a quantity. article - 246 Free and Forgetful Come in Pairs
Operations that add something and operations that throw something away appear again and again: free constructions, completions, localisation… article - 247 Splitting by Parity Is Not One Phenomenon
Splitting by parity appears repeatedly in this body of work: the parity separation of ball volumes in Paper 115, and the different standing … article - 248 Whose Cost Is the Demon’s? Five People Put It in Five Places
For more than a century Maxwell’s demon has been asked where the cost lies. article - 249 The Four Pi That Catches Molecules Is the Four Pi That Spreads a Field
When 4pi turns up in a biological setting, is it a coincidence of numbers or the same root? This paper treats exactly one case──the 4pi in t… article - 250 Two Instruments, Opposite Signs
How far apart can two operationalisations of one phrase land? This paper gives one instance with numbers: measuring the single phrase named … article - 251 Special in Two Dimensions Is Not One Phenomenon
Special in two dimensions appears again and again in this body of work: the absence of a steady absorber in Paper 249, the Huygens principle… article - 252 Information Is Four Quantities, and They Need Different Things
One word, information, names at least four different quantities: Shannon entropy, von Neumann entropy, Fisher information and Kolmogorov complexity. article - 253 Time Is One-Dimensional Because Prediction Demands It, Not Because a Law Says So
Why is time one-dimensional? The answer offered here is that no law decrees it; the requirement that one be able to predict allows nothing else. article - 254 That Knots Come Undone in Four Dimensions Is Not One Phenomenon
In four dimensions every knot in a string comes undone. article - 255 “Accuracy” Is Two Things, and Calibration Removes Only One
The single phrase that an instrument is accurate names two different quantities: the offset from the true value, and the scatter. article - 256 The Only Scale at Which Dynamic Similarity Is Exact Is One
Paper 117 showed that the condition under which dimensional analysis fixes an exponent is that there be exactly one Pi group, and that addin… article - 257 Changing the Speed Does Not Change the Destination
A catalyst makes a reaction faster. article - 258 4 Appears Only When the Source Is a Point
Paper 02 showed that a pure solid angle of four pi appears in an inverse-square field. article - 259 There Are Two Roads to Independence: One Narrows the Universe and the Other Widens It
That the continuum hypothesis is independent cannot be said by one proof. article - 260 The Equivalence Is a Theorem and the Thesis Is Not
Two claims of different standing hang from the phrase “can be computed”. article
Papers 261–280
- 261 What Fences In Poisson’s Ratio Is Not Geometry
An isotropic solid has only 2 elastic constants, and the two are not free to move. article - 262 There Are Three Ways to Break, and Strength Does Not Decide Which
The sentence “the strength of this material is 250 MPa” does not settle when it breaks. article - 263 Stationary Is Not Equilibrium, and Detailed Balance Is Only Sufficient
When a distribution has stopped moving, the system is said to have settled. article - 264 Inside One Phenomenon Live a Number That Moves with the Lattice and a Number That Does Not
Papers 117 and 120 treated the universality of exponents. article - 265 What Separated the Two Low-Temperature Models Is Not Quantisation but the Density of States
Einstein (1907) and Debye (1912) give entirely different answers for the specific heat at low temperature. article - 266 The Sampling Theorem Is Correct and Its Premise Is Never Met
The sampling theorem is a theorem. article - 267 The Displacement Current Is Not a Current
Between the plates of a charging capacitor, not one charge crosses. article - 268 That the Triple Point Is a Point Is Not an Experimental Fact
On a phase diagram the triple point is a point. article - 269 What a Crystal Forbids Is Not the Symmetry but the Periodicity
Fivefold symmetry, it is said, does not occur in crystals. article - 270 The Two in the Flux Quantum Is Neither a Dimension nor an Exponent but a Count of Carriers
Paper 157 called over the twos appearing across this series. article - 271 Mean Field Does Not See the Dimension, and Yet the Dimension Alone Settles How Far It Errs
The mean-field approximation predicts a phase transition in every dimension. article - 272 One and the Same “Twice as Strong” Opens by a Factor of 6.70 on the Stimulus Side
The sentence “the sensation has doubled” does not settle how much the stimulus was raised. article - 273 Four Different Quantities, All Written in J/kg
Absorbed dose, equivalent dose and effective dose all carry the unit J/kg. article - 274 Drude Was Right Because Two Errors Cancelled
Drude (1900) obtained the ratio of thermal to electrical conductivity in a metal from classical theory alone, and got a value close to measurement. article - 275 There Is No Natural Boundary between a Semiconductor and an Insulator
Materials are divided into semiconductors and insulators. article - 276 The Decibel Is a Ratio and Not a Quantity
The sentence “it went up by 3 dB” does not settle what became how many times larger. article - 277 A Solid Angle Is Hidden inside the 0.161 of the Reverberation Time
Sabine’s formula T_60=0.161 V/(Salpha) appears in every textbook of acoustics. article - 278 The Classes of Chemical Bond Are a Continuum of Strength Cut by a Temperature
Ionic, covalent, metallic and hydrogen bonds and the van der Waals force go by separate names. article - 279 What Changed the Exponent of a Chain Was Self-Avoidance Alone, and above Four Dimensions That Cost Disappears
The spread of a polymer chain is fixed by a power of the number of units N. article - 280 With the Same Magnetic Field, the Alfven Speed Differs by a Factor of 7.7410^7 Between the Corona and the Earth’s Core
The speed of a wave travelling along a magnetic field line is set by the Alfven speed v_A=B/sqrt(mu_0rho). article
Papers 281–300
- 281 It Cannot Be Copied, Yet It Can Be Corrected
A quantum state cannot be copied (the no-cloning theorem). article - 282 What Saturates Is Not the Earthquake but the Scale
“Magnitude” is not one thing. article - 283 One and the Same “Double the Q” Becomes 5.6910^18, 32, and 8
“Half-life” is not one mechanism. article - 284 The Langmuir Isotherm and the Fermi Distribution Are One and the Same Expression
The Langmuir adsorption isotherm of surface chemistry theta=Kp/(1+Kp), the Fermi distribution of solid-state physics f=1/1+e^(E-mu)/kT, and … article - 285 “Forbidden” Means “Slower by a Factor of 7.6210^7”
A selection rule is written as “this transition is forbidden.”Yet forbidden transitions are actually observed. article - 286 99.04% of the Proton’s Mass Is Not the Higgs
It is said that “the Higgs particle gives things their mass.”This paper asks what fraction of “things”──the answer, for the proton, is 0.96%. article - 287 What Discretises a Continuum of Errors Is Not the Code but the Measurement
The errors that can enter a qubit form a continuum. article - 288 “Which Is Fastest” Has Three Answers, and a Different Method Wins Each
Equations are solved numerically by Newton’s method, the secant method, or bisection. article - 289 One Number Sets the Limit of Forecasting
That a weather forecast cannot reach beyond a certain horizon is due neither to missing equations nor to slow computers. article - 290 That the Earth’s Field Begins at the Dipole Is Not a Property of the Earth
Expanded in spherical harmonics, the Earth’s magnetic field begins at the dipole (l=1). article - 291 One Sea Holds Three Lengths, Spanning a Factor of 4.3910^4
A rotating ocean has several characteristic lengths. article - 292 Only the Logarithm Adds
A “rate of return” looks like a quantity that can be added. article - 293 The Magic Numbers Come from Two Mechanisms
The nuclear magic numbers 2,8,20,28,50,82,126 look as though they follow from one rule. article - 294 Separation Improves Only as the Square Root of the Length
Chromatographic resolution goes as R_s proportional to sqrt(N) proportional to sqrt(L). article - 295 The Molar Mass of Carbon-12 Is No Longer Exactly 12
“One mole is the amount in 12 g of carbon-12” was for a long time a definition. article - 296 The Hierarchy of Languages Is Set by the Type of Memory, Not Its Amount
The Chomsky hierarchy divides languages into four levels. article - 297 Cosmic Distance Is a Ladder of Seven Rungs, and Only the First Is Measured Directly
When one says “that galaxy is a billion light-years away,” it is a measured quantity. article - 298 A Strategy That Wins 99.9999% of the Time Has Expectation Exactly 0
The doubling strategy──stake twice as much after every loss──can push the win rate arbitrarily close to 1. article - 299 A Drum Has No Pitch Because of Its Boundary, Not Its Equation
A guitar string has a pitch; a drum does not. article - 300 The Three Hundredth Census: Whether Two Things Share a Root Is Decidable
Paper 100 drew the method of papers 63 to 99 by hand. article
Papers 301–320
- 301 Grouping by Value Gives the Reverse of Grouping by Root
The golden ratio has appeared three times in this corpus──as the reason a crystal forbids five-fold symmetry (Paper 269), as the order of co… article - 302 “It Must Appear” Does Not Tell You When
Ramsey’s theorem says that if it is large enough, order must appear. article - 303 The Formula Returns 4, and at That One Point the Proof Does Not Reach
Heawood’s formula chi(g)=floor((7+sqrt(1+48g))/2) gives the chromatic number of a surface of genus g. article - 304 “Capacity” Is a Number Attainable Only at Infinite Length
Shannon’s coding theorem says that below the channel capacity C the error can be made arbitrarily small. article - 305 Security Rests on Three Grounds, and a Quantum Computer Breaks Only One
The sentence “this cipher is secure” rests on one of three different grounds. article - 306 How Many “Base” Units There Are Is a Promise
The SI has seven base units. article - 307 “Fitness” Is Four Quantities, and One of Them Changes Sign When the Population Changes
The sentence “its fitness is high” points at one of four different quantities. article - 308 The Gap between the Buses You Board Is Twice the Average
Suppose buses come at an average interval of 10 minutes. article - 309 The Length That Divides Gravity from Surface Tension Is Nobody’s Size
A liquid carries one length, l_c=sqrt(gamma/rho g). article - 310 “Conserved” Has Two Distinct Roots, and Noether Explains Only One
This corpus has cited Noether’s theorem in 27 papers. article - 311 The Correct Bound Is Not the Predictive One
The Carnot efficiency is correct as an upper bound on any heat engine. article - 312 Some Noise Falls When You Measure Longer, and Some Does Not
The phrase “too much noise” names several quantities with different mechanisms. article - 313 This Corpus Has Used “Synchrony” in Four Senses, and Only the Fifth Has a Threshold
This corpus has used the word “synchrony” in 20 papers. article - 314 There Is Only One Kind of Constant You Can Ask “Has It Changed?” About
Paper 10 established that only dimensionless numbers can be fine-tuned. article - 315 Which Dimension Is Special Depends on the Question
Paper 59 showed that the volume of the unit ball is greatest in 5 dimensions, and Paper 64 concluded that no single magic dimension exists. article - 316 The Same Integer Comes Back from Three Different Counts
Paper 67 read m=4pi=2pichi(S^2) as Poincare–Hopf. article - 317 A Significance Level Is a Promise per Test, Not per Study
“Significance at 5%” promises to hold false positives to 5% for one test. article - 318 A Key Halves for Two Different Reasons, and One of Them Is Not Waiting for a Quantum Computer
Paper 305 showed that a quantum computer halves the effective length of a symmetric key (Grover). article - 319 A Calendar Is Three Approximations Stacked
A year is not 365 days. article - 320 The Road into Chaos Carries a Number the Map Does Not
Paper 289 showed that one number, the error-doubling time, sets the limit of forecasting. article
Papers 321–328
- 321 The Length That Sets How Thick an Atmosphere Is Also Sets Whether It Stays
An atmosphere has one length in it, H=kT/(mg). article - 322 One and the Same Sensor Error Reaches Position by Three Different Powers
Paper 312 showed that averaging for longer stops helping past a certain point. article - 323 The “Three” in Three Primaries Is Not a Property of Light
Every device that reproduces colour uses three primaries. article - 324 That No Search Is Universal Follows from How the Problems Are Counted
The no-free-lunch theorem says that averaged over all problems, every search performs alike. article - 325 The Size of the Drop Is Not in the Angle of the Rainbow
A rainbow always appears at the same angle. article - 326 The Square-Cube “Law” Is Not a Law but an Identity
The square-cube law says that on enlargement area grows as the square and volume as the cube. article - 327 Two Curves Agree to Within 0.083 Per Cent and Are Not the Same Curve
A hanging chain and a suspension-bridge cable cannot be told apart by eye. article - 328 That the Same Ellipse Was Three Months Out Was the Evidence of a Different Cause
Over a year a star traces a small ellipse on the sky. article
Last updated 2026-09-01. 327 papers. Number 047 is permanently retired, having been merged into Paper 046.