Statistical thermodynamics of the ideal pure substance and its states of matter

Overall summary

Model 3 is a qualitative, statistical-mechanical model of an ideal substance that derives the three states of matter — solid, liquid and gaseous — together with the transitions between them from a single bonding picture with molecular bonds. The guiding principle lies in accounting for three energy stores of the internal energy: the vibrational energy in the molecular bonds, the translational energy of the molecules, and the cooperative energy of motion in the rotation of micelles as an idealised form of internal material flows that even out internal density fluctuations. An important basic quantity in this context is the mean coordination number z̄ = z_max·x, which sets itself self-consistently through a cooperative two-state bond (Ising-like, cooperativity K = J·z_max, broken-state degeneracy G = z_max). From it follow three transitions on a common τ scale calibrated by a single energy parameter ε: melting as a phonon/stiffness threshold (z̄ reaches rigidity), boiling as a connectivity (percolation) collapse at the critical point, and an optional solid(1)→solid(2) transition (FCC↔BCC) with the parameter-free c_v step of 1.0 k_B. A transition becomes first order as soon as ln G > 2 + 4ε/K. Here the broken-state degeneracy G is the number of neighbour contacts that can be broken (G = z_max, the maximum coordination number), the bond energy ε is the energy gained per neighbour contact, and the cooperativity K = J·z_max is the product of the Ising-like coupling strength J and z_max; the transition therefore becomes first order the sooner, the larger the contact multiplicity G and the stronger the cooperativity K relative to the bond energy ε.

The article leads from the seven axioms (chapter 1) through the contradiction analysis (chapter 2) and a pictorial description with static comic strips (chapter 3) to the statistical-mechanical framework — cooperative bonding, partition functions and heat capacities (chapters 4–5) — on to the mathematical 3D plausibility (chapter 6) and finally to calibration and the equation of state (chapter 7). The ideas are supplemented with the full 3D c_v scan of first-order melting (Δz̄, latent heat ΔU, peak shift), an Einstein spring model for the phonon stiffness θ(lattice) and the solid(1)→solid(2) transformation line via Clausius–Clapeyron. θ(lattice) is then refined into a Debye-like phonon DOS, the solid(1)→solid(2) transformation is additionally introduced as a line in the (τ, p) picture (calibrated against iron), and a separate Redlich–Kwong branch is added alongside the van der Waals default. θ(lattice) is furthermore tested on a full Lennard-Jones lattice (FCC stable, ideal BCC shear-unstable), and the solid(1)→solid(2) topology is tested on titanium (second confirmer) and argon (negative control). Chapter 8 places the model among eleven established model conceptions of static thermodynamics. The appendix documents the literature sources (sorted by author) and the AI support (estimated share of innovation, estimated cost and duration per development step); the aids gather the reading instructions, the remaining critical points and the outlook.

Core statement. A single cooperative bonding measure z̄ brackets together three otherwise separate traditions — the lattice-gas/Ising picture (cooperative two-state transition), the percolation picture (connectivity → boiling) and the phonon picture of Einstein/Debye (stiffness → melting) — into a consistent, qualitatively complete phase picture, deliberately within a van-der-Waals-like ideal picture, while the theoretical edifice can of course be derived from an easily imagined dynamic picture. Not accuracy but vividness is the actual strength of the model, and it is what marks its scientific value as a model.

1 — The axioms with keyword names (consolidated)

Summary. Seven axioms furnished with keyword names fix the ideal substance: a conserved particle number, purely classical motion, a cooperative two-state bond, and the mean coordination number z̄ as the order parameter. They carry all the derivations that follow; in the German master text a homoglyph watermark marks the keywords (key at the end of the text).

The axioms made precise:

1. FREE FLIGHT (additional axiom). Between two molecules that fly past one another unbonded, neither friction nor electrostatic forces act. Momentum and energy change only through the hard HARD CORE repulsion in a collision and through the making and breaking of a BOND. The axiom matters solely from the point of view of classical mechanics and the energy balance: it guarantees that there are no dissipative channels at the molecular level — energy of motion is never “lost”, it only passes into the three TRIAD stores or into bond energy. (Macroscopic friction/viscosity arises emergently out of the sliding puzzle and the CRACK, not as a microscopic fundamental force.)
2. MULTITUDE. An ideal substance consists of very many molecules of the same kind; the macroscopic properties are described statistically in the thermodynamic limit (N → ∞).
3. BOND. Any two geometrically neighbouring molecules can carry a bond. An intact bond lowers the energy by the finite amount ε; breaking it costs exactly ε. The bond is a two-state system {intact, broken}.
4. TRIAD. The internal energy is distributed classically over three stores: (a) translation of the molecules, (b) vibration in intact bonds (a harmonic spring per bond), (c) rotation of rigid molecular blocks (the collective rotational degree of freedom of connected regions).
5. HARD CORE. If the molecular distance falls below a minimum value σ, a hard repulsion acts (excluded volume). Together with BOND this yields an effective pair potential (hard shell plus a finite bonding well).
6. CONSERVATION. Total momentum and total energy are conserved. A single molecule is energetically neutral — it buffers no energy within itself; energy lives in the bonds (BOND) and in the kinetic degrees of freedom (TRIAD).
7. CRACK. Bonds decay and form according to the crack/healing principle: more missing neighbour bonds → higher probability of breaking; more bonded neighbours → higher probability of healing. The kinetics is cooperative.
8. NIMBLE. Breaking and healing proceed fast and with little activation; the bond subsystem stands in local equilibrium at every temperature and may be separated adiabatically from the molecular motion (the basis of the factorised partition function).

Modelling stance. The geometric coordination is idealised as being preserved; only the coordination number z changes through ruptures. Motion arises as in a sliding puzzle: voids (from the HARD CORE excluded volume and CRACK ruptures) migrate, blocks move up. The deliberately “wrong” picture of rotating, migrating crystallites is retained as a vivid foundation.

2 — Contradiction and gap analysis

Summary. The analysis systematically tests the model for internal contradictions and gaps and carries the state of work forward; weaknesses that have been recognised are named and either resolved in the following chapters or marked as an honest limit of the model.

List of criticisms

1. Double counting of translation (axiom 3a vs. 3c). The raw text names translation twice (molecular translation and block translation). Two independent translational stores would not exist. Resolution: block translation is only a collective mode of the molecular translations already counted; genuinely new is solely the rotational degree of freedom of the block. In the degree-of-freedom balance (ch. 4) translation is counted exactly once.
2. “Energetically neutral” (axiom 5) vs. “bonds take up energy” (axiom 2/3b). An apparent contradiction. Resolution: the neutrality concerns the molecule as a body; energy sits between the molecules (in BOND) and in collective kinetic modes, never in an intramolecular buffer. Molecule ≠ bond — with that the contradiction is resolved.
3. Missing attraction (axiom 4 names only repulsion). Without attraction there would be no condensed phase. Resolution: the attraction is supplied by BOND (a well of depth ε), the repulsion by HARD CORE (a hard shell at σ). Together they form an effective pair potential.
4. Feedback without a counter-term (axiom 6). “More missing neighbours → more breaking” alone is a positive feedback (crack propagation up to total decay). Resolution: the healing term (more neighbours → more healing) plus the Boltzmann factor of the finite bond energy ε (BOND) balance the crack and produce a defined transition instead of a runaway.
5. Unstated separation of timescales (axiom 7). “Fast and low in activation” only becomes usable once one turns it into the adiabatic separation of bond dynamics from translational dynamics. Resolution: stated explicitly as the prerequisite of the factorisation Z = Z_trans·Z_vib·Z_rot·Z_bond.
6. Missing geometry and missing mechanism of motion. The axioms name neither the coordination lattice nor what makes flow possible. Resolution (as an explicitly marked additional assumption): a fixed reference coordination geometry with variable bond occupation; as the canonical 2D case the triangular coordination (z_max = 6). Mobility through the sliding-puzzle void mechanism. These two points are the only genuine additions beyond the seven axioms and are marked in chapter 6 as a strategic question.
7. Latent dissipation gap. Revision 07 left open whether unbonded molecules lose energy as they fly past. Resolved by FREE FLIGHT: there is no microscopic friction or electrostatics; all flows of energy and momentum run exclusively through HARD CORE collisions and BOND kinetics. With that the energy balance (ch. 4.3) is closed and the Maxwell–Boltzmann statistics exact in the gas limit (a free ideal gas between elastic collisions).
8. Energetic inconsistency of the order parameter. Revision 08 used the cooperative term −(K/2)x² in the free energy g(x) (to generate x(τ)) but not in the internal energy U/N. That is thermodynamically inconsistent: the same mean-field term must appear in both. Resolution (ch. 4.3): −(K/2)x² is carried over into U/N; the latent heat thereby increases measurably (ch. 5.3) without altering the solid and gas plateaus.
9. l/g calibration only qualitative. Revision 09 had the density axis (van der Waals EOS) only in reduced form. Resolution (ch. 7.4): the absolute boiling line is computed via the critical point (T_c, p_c) and T_b(p_atm) is set against measured values; the systematic deviation (Guldberg's rule) is named.
10. Separate axes without a triple point. The melting and boiling axes were separate calculations. Resolution (ch. 7.5): both are brought together in one (τ, v) plane, the triple point is constructed and the critical point marked.

3 — Pictorial descriptions

Summary. Chapter 3 explains solid, liquid and gaseous as well as their transitions vividly (with static comic strips): the three single phases (3.1), the phase transitions recrystallisation/melting/boiling/sublimation together with the triple and critical points (3.2), the sound anomaly (3.3), Brownian motion and the sharpened notion of the micelle (3.4–3.5) and the three-dimensional liquid window (3.6). On the basis of these descriptions, diagrams (comic strips) were produced that are meant to help picture the dynamic molecular circumstances in the various phases before the inner eye. This imaginability is the actual core of the model. It becomes qualitatively and scientifically valuable once statistical mechanics can plausibly show that the model makes qualitative empirical tendencies and observations predictable even without much mathematics.

3.1 The three single phases

Solid. Almost all neighbours are bonded to one another through molecular bonds (BOND); the mean of the coordination numbers z of the molecules lies close to the geometric maximum. The molecular network is a single rigid block without voids (HARD CORE, densely packed) — an internal sliding puzzle such as the liquid has is not possible in the compact version. The only motion: trembling about the rest positions (bond vibration, TRIAD-b), determined by the molecular bonds to the neighbours. Momentum and energy oscillate elastically within the network (CONSERVATION, FREE FLIGHT: lossless). In the limiting case (very low temperature, ideal packing) the ideal state of a perfect crystal is present.

Liquid. The cooperative CRACK has pervaded the network: bonds are missing in patches, the mean of the coordination numbers z of the molecules has fallen, the rigid block has broken up internally into many smaller ones. The monolith has become a heap of rubble. These blocks translate and rotate (TRIAD-c). Voids have opened between them through which they move past one another — the sliding puzzle of Brownian motion is active. The local coordination geometry is idealised as being preserved; what has changed is only how many of the bonds really exist and are formed.

Gas. The CRACK has won: bonds are broken almost everywhere (z → 0). In the extreme case no molecule has a molecular bond to another molecule any more. The blocks have decayed into individual molecules. They fly freely, coupled only by HARD CORE collisions (with FREE FLIGHT in between); the velocities follow Maxwell–Boltzmann (CONSERVATION). Bond vibration and block rotation have died out — pure translation (TRIAD-a). In the limiting case (high temperature, low pressure) the ideal state of an ideal gas is present.

3.2 Phase transitions

Recrystallisation (solid(1)/solid(2)). Still below melting, the solid can change its lattice without losing its stiffness: two ideal packings with different coordination numbers (FCC z_max = 12 ↔ BCC z_max = 8) have different free enthalpies μ(τ, p); at μ_FCC = μ_BCC the coordination geometry jumps discontinuously (not merely the occupation z̄ as in melting). What becomes visible is a small c_v step (Δc_v = change in z_max/4 = 1.0 k_B) plus a weak latent heat of transformation. AI claim: it is conceivable only in 3D (discrete, quantised connectivity; ch. 6.3). In the two-dimensional case genuine recrystallisation — the transition between two solid phases — is easier to picture. In the plane there are only three regular tilings without holes. One can cover a surface completely (tile it) with equilateral triangles, with squares or with regular hexagons. If in these lattices the edges represent the molecular bonds and the vertices the molecules, then a phase transition solid(triangular) → solid(square) is conceivable. The molecule in the triangular tiling would then still have six bonds to neighbouring molecules, whereas the molecule in the square lattice would have only four.

Melting (s/l). As the temperature rises, the network tears at enough places for the rigid overall block to lose its stiffness (the cooperative CRACK falls below the Maxwell rigidity threshold z̄ = 2d — the network becomes floppy). Thermodynamic micelles form, that is, regions that are no longer connected by bonds to the larger neighbouring regions. From a molecular point of view the monolith becomes a heap of rubble and is capable of limited flow. The first voids permit molecular blocks to be displaced, much as in the sliding puzzle. The missing bonds to the neighbourhood permit internal rotation within the micelles. Micelle rotation (TRIAD-c) sets in. Breaking many bonds (BOND) consumes latent heat.

Boiling (l/g). Further heating makes the voids themselves percolate: they join into continuous empty space in which the blocks dissolve and individual molecules fly away. Boiling is the transition from “voids in matter” to “matter in a void”.

Sublimation (s/g). At low pressure the liquid intermediate region is absent: the network releases individual molecules directly into the empty space, without the block formation of the micelles ever occurring. One may consider here what influence geometric aspects (long molecules, molecules that interlace with one another), electrostatic aspects (alignment through magnetic moments or dipole moments) and different types of molecular bond (van der Waals, interactions between mesomeric systems, hydrogen bonds) can have on the formation or non-formation of thermodynamic micelles.

Triple point. Exactly there two thresholds coincide: the Maxwell rigidity (solid ↔ liquid, loss of stiffness) and the void percolation (liquid ↔ gas) occur at the same (τ, p). All three pictures (rigid block [monolith], sliding blocks [heap of rubble], free molecules [dust cloud]) coexist.

Critical point. As the pressure rises the jump liquid→gas becomes ever smaller; the void fraction changes increasingly continuously. At the critical point the difference vanishes altogether — “dense blocks with little empty space” and “much empty space with few molecules” pass continuously into one another. (“From the liquid's point of view the glass is half empty, from the gas's point of view half full.”)

Recrystallisation in the 2D picture. The comic now shows recrystallisation as a 2D case, one above the other: below the densely triangulated lattice (coordination z = 6, the lowest-energy 2D ideal), above the loose square lattice (z = 4). The transition triangulated → square enlarges the volume (work against the pressure) and lowers the bond count. One sees that the coordination number changes while the stiffness of a crystal lattice persists.

Qualitative prediction of energy transitions and heat capacities. From the connectivity z alone both characteristic figures can be estimated: (i) the bond energy per particle is U/N ≈ −(ε/2)·z (each bond counts half), hence lower in energy the higher z is; (ii) the c_v jump at the lattice change follows the model rule Δc_v = change in z_max/4. For the 2D case triangulated (z = 6) ↔ square (z = 4) this gives an energy difference of ΔU ≈ ε per particle and a c_v step of Δc_v = (6 − 4)/4 = 0.5 k_B — half as large as the 3D value (1.0 k_B). Rule of thumb: denser = lower in energy, but more expensive in the transition (larger latent heat, larger c_v step), because widening releases more bonds and more frozen degrees of freedom.

Does the lattice structure determine the metastability? In part. The lattice structure fixes the energy ordering — which form is the ground state (triangulated, lowest μ) and which the excited one (square). Whether the loose form survives as metastable at low temperature it does not decide alone: in the strictly ideal model (without a barrier) the substance adopts at every (τ, p) immediately the phase with the lowest μ — metastability does not exist there. It arises only through a finite activation energy (a nucleation barrier) that kinetically inhibits the rearrangement. The lattice geometry therefore supplies the thermodynamic ranking, the barrier the kinetic persistence — the same ingredient that also explains the supersaturated phase (see below; quantitatively ch. 6.7).

Supersaturated phase — activation energy. The additional figure (3.2, row 2) shows a supersaturated gas phase above a liquid: besides single molecules, small groups of 2–3 clustered particles (grey bars) that fly as a whole. The model explains the supersaturation by a noticeable activation energy for molecular bonds: small clusters (2–3) have an unfavourable edge-to-area ratio, their edge bonds are incomplete — only a critical nucleus grows spontaneously further. As long as no supercritical nucleus arises, the gas remains supersaturated, although μ_gas > μ_liquid would already permit condensation. Precisely this barrier is missing from the ideal equilibrium picture and is named here as an independent contribution lying “outside the ideal picture” (quantitative estimate in ch. 6.7).

3.3 Why the speed of sound in liquids falls with T

Sound is an elastic wave: v = √(K_eff/ρ). K_eff arises solely from the intact bonds (BOND) — they are the springs. As T rises, more and more bonds break cooperatively (CRACK); the spring network becomes thinner and K_eff falls more steeply than the density ρ. The rupture dominates the numerator, not the denominator — which is why v falls. Read molecularly: the transmission of sound along molecular bonds is faster than the mechanical transmission of sound by mechanical collisions. The more cracks appear in a liquid as the temperature increases, the more often the sound has to be carried on by mechanical collisions rather than by molecular bonds. Accordingly the speed of sound becomes slower as the temperature rises. (The anomaly of water — a sound maximum at about 74 °C — is treated in ch. 6.4.)

3.4 How Brownian motion arises — with a sharpened notion of the micelle

In the statistical-thermodynamic sense a micelle here is not a surfactant aggregate but a block ringed by CRACK: the missing bonds (the non-bonds at the block boundary) form the “shell” around an enclosed crystallite (the intactly bonded core). This core can perform an internal cooperative rotational motion (TRIAD-c) — the block turns as a whole while its boundary shell decouples it from its neighbours.

Brownian motion now arises as follows: through the exchange of molecular bonds (CRACK, NIMBLE) individual molecules can leave the shell of one micelle and cross over into another micelle. Every such change transfers momentum (CONSERVATION; lossless because of FREE FLIGHT) and alters the shape and angular momentum of both micelles involved discontinuously. A large test particle (a dust grain) is washed by many such uncorrelated micelle rearrangements; the sum of the momentum impulses is zero on average but has finite variance → random walk. The higher T, the more frequent the CRACK events and micelle exchanges, and the stronger the trembling. Brownian motion is thus the macroscopic shadow, or consequence, of the microscopic micelle exchange process.

3.5 Why reactions at micelle boundary layers become faster

An embarrassing preliminary remark. The argument that follows is derived from the picture and, to the best of my knowledge, has so far neither been demonstrated nor investigated empirically — though I must admit that I did not take the trouble to research it thoroughly. The only indication of its novelty is that AI systems do not mention this conclusion. This thesis is an example of how pictorial conceptions can lead to new questions.

In moderately dissolving solvents the reactants attach preferentially to the boundary shells of the micelles — in the model precisely the crack/block boundaries (CRACK). Two effects accelerate the reaction: (i) dimensional reduction — the reactants no longer search for one another in the full volume but along a quasi-two-dimensional interface; a 2D search statistically finds its target faster than a 3D search. (ii) Concentration — the boundary shell gathers the reactants, their local density rises. Pictorially the crack boundaries act like guide rails leading the partners towards one another.

The dimensional effect has a rigorous core: the random walk (Pólya). Why a two-dimensional search statistically finds its target better than a three-dimensional one follows not only from intuition but from a theorem of probability theory. George Pólya (1887–1985) showed in 1921 for the random walk on a lattice: in one and in two dimensions the walker returns with probability 1 (that is, certainly) infinitely often to every point, on the planar lattice ℤ² just as on the line ℤ. From three dimensions on this no longer holds: the return probability is strictly smaller than 1 (on the simple cubic lattice ℤ³ only about 0.34). Vividly, in Pólya's image: “A drunken man will find his way home, but a drunken bird may get lost forever.” Transferred to two reactants that must meet several times for a successful collision, this means: in a very good solvent with a free three-dimensional random walk, partners once separated drift apart forever with finite probability — repeated attempts at reaction fail, and the yield remains low for this purely geometrical-statistical reason. At a (quasi-)two-dimensional micelle boundary layer, by contrast, the re-encounter is certain (probability 1): the partners find one another again and again until the reaction succeeds. The boundary layer therefore not only concentrates but at the same time provides the dimension more favourable for re-encounters. The (deliberately pointed) claim runs: yields in very good solvents tend to be poorer for random-walk reasons.

Reference: G. Pólya (1921), Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Math. Ann. 84, 149–160 — cited here after M. Eisermann, Höhere Mathematik 3 (Univ. Stuttgart), exercise W-2×2. (Original accessed 5 Sept 2026).

3.6 The three-dimensional liquid window

Unlike the figures in 3.1–3.5, this figure shows no molecular scene but the result of the calculation from ch. 7.9/7.10 as a schematic on a τ axis (in units of the bond energy ε). The aim was to make visible why the 2D single lattice hardly admits any region for a liquid phase, whereas with the 3D lattice this is possible without difficulty, even though the same rigidity rule (z̄_c = 2d) applies in both cases.

Panel ① — 2D (no window). On the τ axis τ_m = 1.20 and τ_triple = 1.16 lie practically on top of one another (τ_triple below τ_m), with τ_c = 1.50 just above: the region for the liquid phase is squeezed into a slit — solid passes practically directly into gas (close to sublimation).

Panel ② — 3D (a clean window). The same axis, but stretched: τ_m = 1.20 < τ_triple = 2.83 < τ_c = 3.00. Between τ_triple and τ_c lies the self-contained liquid field (wide towards the bottom, τ_triple − τ_m = 1.63; narrow towards the top, τ_c − τ_triple = 0.18). The narrow upper edge is marked as such.

Panel ③ — the cause. z̄_c = 2d grows with the dimension (4 → 6), so v_solid falls (0.50 → 0.667 in reduced volumes, i.e. the solid becomes “looser” relative to the liquid), and τ_c = z_max/4 doubles (1.50 → 3.00). The critical point moves far above the melting point. Only through this does room for the liquid phase arise. It is a pure dimensional effect.

4 — Statistical-mechanical model (unchanged from revision 09)

Summary

Chapter 4 sets up the statistical-mechanical framework of the model in 2D (triangular lattice, z_max = 6):

• 4.1 Degrees of freedom and variables: the order quantity is the bond fraction x ∈ [0,1] with the mean coordination number z̄ = z_max·x; per block ≥ 2 a rotational degree of freedom is added, and the temperature τ counts in units of the bond energy ε.
• 4.2 Cooperative two-state bond: the self-consistent bonding equation x = 1/(1 + G·exp[−(ε + K·x)/τ]) (K = J·z_max, G = z_max) follows as the stationarity condition of the Bragg–Williams free energy; its global minimum supplies the physical branch, and whether that branch runs continuously or with a jump (first order) is decided in ch. 5.5.
• 4.3 Energy balance with a cooperative term: U/N = τ·[1 + z̄/4 + ½ n_blk] − (ε/2)·z̄ − (K/2)·x² (2D) unites translation, bond vibration and block rotation, the bond configuration energy and the new cooperative mean-field energy −(K/2)x². The associated heat capacity c_v acquires the positive additional term −K·x·ẋ, which sharpens the melting peak but leaves the solid plateau and the gas branch untouched.

4.1 Degrees of freedom and variables

• N molecules on a fixed reference coordination geometry. Canonical 2D case: triangular lattice, z_max = 6, one rotational DOF per block (MULTITUDE, BOND). This chapter applies explicitly to the 2D interpretation; the 3D generalisation stands in ch. 6.
• Bond variable s_ij ∈ {0, 1} per geometric neighbour pair (1 = intact).
• Coordination number z_i = Σ_j s_ij; the mean z̄ = 2 N_bond / N — the order parameter. We have z̄ = z_max · x with the bond fraction x = p_b ∈ [0, 1].
• Energy unit τ ≡ k_B T (classically, ½τ per quadratic degree of freedom). All energies in units of the bond energy ε.

4.2 Cooperative two-state bond (Ising-like, parameter J)

An isolated bond (energy −ε intact, 0 broken) has in Maxwell–Boltzmann equilibrium p_b⁰(τ) = 1/(1 + e^(−ε/τ)). The cooperative CRACK (axiom 6) shifts the effective bond stability by a mean field proportional to the bond fraction x. In mean-field (Ising) approximation the self-consistent bonding equation becomes:

x = 1 / (1 + G · exp[−(ε + K·x) / τ]), K = J·z_max, z̄ = z_max·x.

Here:

• K = J·z_max is the cooperative coupling strength (CRACK feedback: more neighbour bonds → higher effective bond stability). J > 0 sharpens the transition; above a critical J, z̄(τ) becomes discontinuous — that is the latent heat of melting. The pure two-state case is the limit J = 0 (no mean field, a continuous crossover).
• G > 1 is the degeneracy of the broken state: a molecule that has been freed gains phase space (translational freedom). G drives bond rupture at high τ and ensures that x → 1/(1+G) instead of getting stuck at ½ (the gas limit). Canonically G = z_max.

This equation follows as the stationarity condition of a Bragg–Williams free energy per molecule (n_slots = z_max/2 bonding slots):

g(x)/n_slots = −ε·x − (K/2)·x² + τ·[x ln x + (1−x) ln(1−x)] − τ·ln(G)·(1−x).

The term −(K/2)x² is the cooperative (ferromagnetic) coupling, the last term the rupture degeneracy. The global minimum of g over x supplies the physical branch; where two minima become equally deep lies the first-order transition. NIMBLE (axiom 7) justifies that x(τ) takes the equilibrium value at every temperature.

4.3 Energy balance — with a cooperative term

The mean-field term −(K/2)x² is a genuine energy (independent of τ) and must appear in U/N. The consistent balance therefore reads (2D):

U/N = τ·[1 + z̄/4 + ½ n_blk] − (ε/2)·z̄ − (K/2)·x².

• τ·[…]: translation (1) + bond vibration (z̄/4) + block rotation (½ n_blk) — TRIAD.
• −(ε/2)·z̄: bond configuration energy — BOND.
• −(K/2)·x²: cooperative mean-field energy — CRACK. New in U/N.

The associated heat capacity c_v = d(U/N)/dτ (in units of k_B) acquires an additional contribution:

c_v = [1 + z̄/4 + ½ n_blk] + τ·[¼ ż̄ + ½ ṅ_blk] − (ε/2)·ż̄ − K·x·ẋ, (ẋ = dx/dτ)

where the new term −K·x·ẋ is positive (ẋ < 0) and acts exactly where many bonds break at the same time. It sharpens the melting peak without altering the solid plateau (ẋ ≈ 0) or the gas branch.

5 — Partition functions and heat capacities

Summary

Chapter 5 works out partition functions and heat capacities and supplies the model data for the calibration (ch. 7):

• 5.1 Factorised partition function: because of the separation of timescales, Z = Z_trans·Z_vib·Z_rot·Z_bond factorises.
• 5.2 Heat capacity per phase: c_v = d(U/N)/dτ; the positive term −(ε/2)(dz̄/dτ) produces the Schottky-like peak as the analogue of the latent heat.
• 5.3 Numerical c_v(τ) curve: c_v peak 5.51 at τ_m = 1.20, solid plateau 2.50, gas branch → 1; the latent heat L ≈ 1.41 ε per molecule (with the cooperative term) is tripled compared with 0.48 ε (without).
• 5.4 / 5.4a Two separate thresholds: the connectivity percolation z̄_c = z_max·p_c ≈ 2.08 (at τ_b ≈ 1.85) marks the cluster decay (close to boiling) and is a result; it is to be kept cleanly apart from the generic Maxwell rigidity z̄_c = 2d (melting density, loss of stiffness).
• 5.5 Order of the melting transition: criterion ln G > 2 + 4ε/K; 2D (G = 6) is continuous for every J, 3D (G = 12) is first order from K > ~8.2 (demo: Δz̄ ≈ 5.3, ΔU ≈ 1.75 ε).
• 5.6 Finite-size study: p*(L) converges monotonically from below towards p_c = 0.3473; z̄_c ≈ 2.08 is finite-size robust.
• 5.7 Full 3D c_v scan: from K > K_crit ≈ 8.25 (J > 0.68) the genuine jump Δz̄ with latent heat ΔU opens up; the melting peak moves with J towards higher τ (a positive shift of +0.5 … +2.2 relative to the continuous 2D peak).

5.1 Factorised partition function

Because of the separation of timescales (NIMBLE) the canonical partition function factorises:

Z = Z_trans · Z_vib · Z_rot · Z_bond.

• Z_trans = (1/N!)·(A/λ_T²)^N, λ_T = h/√(2π m τ) (classical ideal 2D translation; MULTITUDE, CONSERVATION, FREE FLIGHT).
• Z_vib = (τ/ħω₀)^(N_bond) (a classical harmonic oscillator per intact bond; TRIAD-b).
• Z_rot = ∏_(blocks≥2) (2π I_b τ)^(1/2)/ħ (a classical 2D rigid rotor per block; TRIAD-c).
• Z_bond = Σ_{s} exp[(ε/τ)·Σ s_ij + (K/τ)·cooperative term] (an Ising-like bond sum; BOND, CRACK). Mean-field evaluation supplies x(τ), z̄(τ) from 4.2.

5.2 Heat capacity per phase (analytical)

From U/N (ch. 4.3), c_v = d(U/N)/dτ (in units of k_B):

c_v = [1 + z̄/4 + ½ n_blk] + τ·[¼ (dz̄/dτ) + ½ (dn_blk/dτ)] − (ε/2)·(dz̄/dτ).

The term −(ε/2)·(dz̄/dτ) is positive (dz̄/dτ < 0: heating breaks bonds) and dominates where many bonds break at the same time — the Schottky-like peak, the analogue of the latent heat.

5.3 Numerical c_v(τ) curve with the cooperative term — latent heat quantified

Self-consistent bond (global minimum), n_blk(τ) from bond percolation Monte Carlo (union-find, L = 100, 5 realisations), c_v = dU/dτ (central differences), U/N including −(K/2)x² (cv_scan_09.py, ε = 1, J = 0.55, G = z_max = 6). Selected sampling points:

Result (numerical). The three-phase ordering remains confirmed and is now sharper:

c_v(solid) ≈ 2.50 < c_v(liquid, maximum) ≈ 5.51 > c_v(gas branch) ↓ → 1.

Latent heat (peak area above the baseline). Integrating (c_v − baseline) over the melting window τ ∈ [0.70 ; 1.70] (baseline = the straight line through c_v(0.70) and c_v(1.70)) gives

L ≈ 1.41 ε per molecule (with the cooperative term), L ≈ 0.48 ε per molecule (without the cooperative term, the old balance), ΔL ≈ +0.92 ε per molecule.

The cooperative term raises the peak by a factor of about 1.7 (3.25 → 5.51) and triples the latent heat. Physically: the simultaneous breaking of many neighbour bonds (CRACK) stores not only configuration energy (−ε/2·z̄) but also mean-field energy (−K/2·x²) — both are released on melting.

5.4 Connectivity percolation threshold as a result

The giant-cluster fraction P_∞(τ) collapses at τ ≈ 1.85; there x ≈ 0.347 = p_c(triangle) = 2·sin(π/18) = 0.3473, hence z̄_c = z_max·p_c ≈ 2.08 — a result, not something imposed. This is the connectivity threshold (a spanning cluster) and marks the cluster decay near boiling (τ_b); it is not to be confused with the Maxwell rigidity (z̄_c = 2d, melting) — cf. the clean separation in ch. 5.4a.

5.4.a Two different thresholds cleanly separated

The hitherto blurred double designation “percolation/rigidity” is now resolved. Two physically different thresholds are involved and must not be confused:

Justification. Melting is the loss of the mechanical stiffness of a dense block, not the decay of connectedness. The relevant quantity is therefore the Maxwell constraint count (mean-field / generic rigidity): a D-dimensional network of central forces becomes isostatic exactly when the mean coordination reaches z̄ = 2D (2D → 4, 3D → 6). The bond connectivity (z̄_c = 2.08) lies well below that; a network only just connected at z̄ = 2.08 has long been floppy (soft modes), hence already liquid. That is why connectivity marks the later cluster decay near boiling, while rigidity sets the earlier solidification/liquefaction (the melting density). This separation fixes the triple point in ch. 7.8 parameter-free and realistically; it does not change the connectivity threshold 2.08 confirmed in 5.4/5.6, only its reading (τ_b = near boiling, not melting).

5.5 Order of the melting transition

Analytical criterion. The spinodal region (two minima of g) exists for τ < K/4; the transition lies approximately at τ_t ≈ (ε + K/2)/ln G. A genuine jump (first-order transition) requires τ_t < K/4, hence in the limit K → ∞:

ln G > 2 + 4ε/K.

From this follows a remarkable result about the rupture degeneracy G = z_max:

Numerical confirmation (max. |Δx| per τ step, detect_first_order):

Reading. In the canonical 2D model (G = 6) melting is continuous in principle (a “soft” transition / crossover) — for no J does a genuine jump arise, because ln 6 < 2. Only the higher coordination/degeneracy of the 3D packing (G = 12) permits, from sufficient cooperativity onwards (K > 8.2), a first-order transition with finite latent heat. That agrees with reality: 2D melting is frequently continuous (KTHNY-like), 3D melting is first order. J = 0 (no mean field) is always the continuous reference case. — Physically J denotes the ratio of cooperative to bond energy (K/ε = J·z_max/ε): how strongly an intact neighbour bond stabilises one's own.

First-order demonstration (3D-motivated, cv_scan_09.py, J = 1.0, G = 12, z_max = 12): z̄ jumps by Δz̄ ≈ 5.30 at τ ≈ 2.82; the latent heat as an energy jump ΔU ≈ 1.75 ε per molecule. The same K on a G = 6 lattice (2D) stays continuous — the character of the order depends solely on G = z_max.

5.6 Finite-size study of the percolation threshold

Bond percolation on L × L triangular lattices (periodic), giant-cluster fraction P_∞(p) scaled over L (cv_scan_09.py; L = 50/100/200/400 with 8/5/3/2 realisations). Exact threshold p_c = 2·sin(π/18) = 0.34730 → z̄_c = z_max·p_c = 2.0838.

P_∞(p) (mean ± spread):

Threshold estimate p*(L) (P_∞ crosses 0.5):

Result. p*(L) converges monotonically from below towards p_c = 0.3473; the spread at p_c falls from ±0.103 (L = 50) to ±0.017 (L = 400) — the transition sharpens with L but does not shift the threshold (a universal percolation value). With that z̄_c ≈ 2.08 = z_max·p_c is confirmed as a finite-size-robust model value.

5.7 Full 3D c_v(τ) scan of first-order melting

The single first-order demonstration run from ch. 5.5 (J = 1.0, G = 12) is now expanded into a full c_v(τ) scan over the cooperativity J (firstorder_3d_16.py, FCC connectivity for n_blk, 3D energy balance U/N = τ·[3/2 + z̄/4 + (3/2)n_blk] − (ε/2)z̄ − (K/2)x²). The analytical order criterion from ch. 5.5 (first order ⇔ K > K_crit = 4ε/(ln 12 − 2) = 8.25) separates the cases; the continuous 2D reference peak lies at τ_2D = 1.20 (c_v ≈ 5.5).

*continuous case: Δz̄/ΔU are the steep crossover (not a genuine jump); τ_transition is there the c_v peak.

Result. (i) The continuous 3D case J = 0.55 gives τ_transition = τ_m ≈ 1.72 — the cross-check against the independent bonding kinetics (bondkinetics_3d_15.py: τ_m(FCC) = 1.70) agrees. (ii) From K > K_crit ≈ 8.25 (J > 0.68) the genuine jump Δz̄ with finite latent heat ΔU (the energy jump across the discontinuity) opens up — melting becomes first order, exactly as the criterion ln 12 > 2 demands. (iii) The cross-check J = 1.0 reproduces the demonstration value Δz̄ ≈ 5.4 of ch. 5.5. (iv) Jump height Δz̄ and the position τ_transition grow with J: the transition moves upwards — the higher coordination of the 3D packing and the stronger cooperativity raise the melting temperature. Relative to the continuous 2D peak (τ_2D = 1.20) this yields throughout a positive peak shift (+0.5 … +2.2) that increases monotonically with K. The first-order character of 3D melting is thereby documented not merely at one point but over the whole range of cooperativity.

6 — Mathematical plausibility of the 3D interpretation

Summary

Chapter 6 makes the generalisation of the bonding picture from 2D to 3D plausible purely mathematically (without a visualisation of its own in HTML):

• 6.1 Rotational degree of freedom 2D → 3D: block rotation grows from one axis to three, translation from two directions to three, z_max from 6 to 12; the c_v base values rise (c_v,solid 2.5 → 4.5, c_v,gas 1 → 3/2), the shape of the c_v(τ) curve remains.
• 6.2 Connectivity as the distinguishing feature: z_max is a quantised quantity and separates the densest packing (FCC/HCP, z_max = 12) discontinuously from BCC (z_max = 8), with no intermediate state.
• 6.3 Additional transition solid(1)→solid(2): from the discreteness follows the polymorphism FCC↔BCC with a parameter-free c_v step of 1.0 k_B; in the strict ideal picture (θ_BCC = θ_FCC) it does not occur, and only a softer BCC (θ_BCC < θ_FCC, below the opening threshold 0.61) opens τ_ss < τ_m.
• 6.4 / 6.5 Considerations: the sound anomaly of liquid water as a smeared-out change of short-range order between two lattice types, and a fractal vortex-field alternative to micelle rotation (purely qualitative).
• 6.6 / 6.8 / 6.9 Microscopic θ(lattice) at three levels: the Einstein spring model (θ_BCC/θ_FCC ≈ 0.82…0.94), a Debye-like phonon DOS (≈ 0.87…1.00) and a full Lennard-Jones lattice (≈ 1.0; ideal BCC shear-unstable, FCC the stable ground state) — all above the threshold 0.61, so real polymorphism lies outside the harmonic ideal picture.
• 6.7 Metastability (2D): an order-of-magnitude calculation for the rearrangement triangulated ↔ square (ΔU = +1 ε, ΔA ≈ +15.5 %) together with the nucleation barrier ΔG\* = πλ²/Δμ_A as a kinetic addition.

This chapter makes the 3D generalisation plausible purely mathematically. The 3D case is never visualised in an HTML version. The aim is solely to show how the rotational degree of freedom and the connectivity change from 2D to 3D and which new transitions thereby become conceivable.

6.1 Change of the rotational degree of freedom 2D → 3D

The only qualitatively new point relative to 2D is the rotation of the blocks:

• 2D: a rigid block has 1 rotational DOF (rotation in the plane) → U_rot/N = ½ n_blk τ.
• 3D: a rigid block has 3 rotational DOF (three Euler axes) → U_rot/N = (3/2) n_blk τ.

At the same time translation grows from 2 to 3 DOF (U_trans/N = (3/2)τ) and the maximum coordination number from z_max = 6 (triangle) to z_max = 12 (densest sphere packing). The energy balance in 3D therefore reads

U/N = τ·[3/2 + z̄/4 + (3/2) n_blk] − (ε/2)·z̄,

and the base values of the heat capacity shift consistently:

The structure of the curve (solid plateau, liquid maximum through −(ε/2)dz̄/dτ, fall towards the gas value) remains identical; only the levels scale with the dimension and z_max.

6.2 Influence of different crystal lattices — connectivity as the distinguishing feature

In 3D at least two ideal lattices are to be distinguished, and they are separated by their connectivity (coordination number z_max):

Connectivity z_max is the decisive distinguishing feature: it fixes both the heat-capacity level of the solid (z_max/4) and the percolation threshold z̄_c. Because every lattice admits only one fixed coordination number, connectivity is a quantised, finite quantity — there is no continuous transition between z_max = 12 and z_max = 8, only the discrete jump from one lattice type to the other.

6.3 The additional transition solid(1) → solid(2)

From the finiteness/discreteness of the connectivity follows a transition not present in the 2D model: solid(1) → solid(2) (polymorphism/allotropy). Two solid phases with different z_max (say FCC ↔ BCC) have different free enthalpies μ(τ, p); at the temperature/pressure combination μ_{solid(1)} = μ_{solid(2)} the substance changes its lattice without melting. In the model picture the coordination geometry changes discontinuously (not merely the occupation z̄ as in melting) — the only situation in which the otherwise fixed reference geometry itself jumps. What would be observable is a small c_v step (a change of z_max/4) plus a latent heat of transformation, markedly weaker than the melting peak.

The first-order criterion ln G > 2 + 4ε/K (ch. 5.5) now supplies a concrete reason why precisely the 3D interpretation is qualitatively richer: only there (G = z_max = 12, ln 12 = 2.485 > 2) can melting become first order — and only there does the discrete lattice change solid(1) → solid(2) (FCC ↔ BCC) exist, because connectivity is quantised/finite. Both phenomena are the same cause in two guises: the higher, discrete coordination of the 3D packing produces both the sharp melting jump and the polymorphism. The 2D single lattice (G = 6) has neither — consistent with ch. 5.5.

Quantification (bondkinetics_3d_15.py). The transition is now worked through leanly. Both lattices are given a free enthalpy in the classical harmonic picture (common p·v and translational parts cancel in the comparison):

μ(τ) = h₀ − c_v·τ·ln(τ/θ), h₀ = −(ε/2)·z_max, θ = ħω/k_B (phonon stiffness in ε units),

with the parameter-free characteristic values from ch. 6.1/6.2:

From this the c_v step parameter-free: |Δc_v| = change in z_max/4 = (12−8)/4 = 1.0 k_B (in the transition FCC→BCC the solid becomes 1 k_B “lighter” in heat capacity) — much smaller than the melting peak (several k_B, ch. 5.3/7.11).

The transformation temperature τ_ss (μ_FCC = μ_BCC) and the latent heat of transformation L = τ_ss·(S_BCC − S_FCC):

Result (an honest finding). In the strict ideal picture (the same phonon stiffness for all bonds, θ_BCC = θ_FCC) there is no solid(1)→solid(2) transition: FCC wins both the enthalpy (lower, −6 vs. −4 ε) and — because of the higher bond count (c_v = 4.5 > 3.5) — the vibrational entropy; μ_FCC stays below μ_BCC at every temperature. Only a lattice-dependent phonon stiffness θ_BCC < θ_FCC (the more open BCC vibrates more softly and therefore has more entropy) lifts μ_BCC below μ_FCC at high temperature — then the transformation opens up, and it lies below the melting peak τ_m, so that the order solid(1)→solid(2)→liquid→gas arises (τ_ss < τ_m < τ_c ≤ τ_b(3D), ch. 7.11). The soft BCC stiffness is a mild extension outside the strict ideal picture (known in reality as the γ/δ polymorphism of iron: BCC at high temperature, FCC in between); the c_v step of 1.0 k_B and the small positive heat of transformation are, by contrast, internal to the model and parameter-free.

How small is θ_BCC/θ_FCC really? (theta_springs_16.py). In order to free the transformation from the free θ parameter, the phonon stiffness is now estimated microscopically (ch. 6.6): an Einstein spring model with central pair springs gives θ_BCC/θ_FCC ≈ 0.82 (nearest neighbours only) up to ≈ 0.94 (with the BCC second shell). The opening threshold required by the μ comparison lies, by contrast, at θ_BCC/θ_FCC < 0.61 (below it τ_ss exists and lies under τ_m). The geometric stiffness ratio is therefore too large: in the strict ideal picture FCC remains stable at every temperature, and there is no purely geometrically driven solid(1)→solid(2) transition. Real polymorphism (iron γ↔δ) requires additional (electronic/magnetic/anharmonic) contributions outside the ideal harmonic picture — precisely the “mild extension” assumed above. That confirms the honest finding and turns the θ ratio from a free assumption into an estimated, named limit of the model.

6.4 Considerations: the sound anomaly of liquid water

Normal liquids show v(T) falling monotonically (ch. 3.3). Water has a sound maximum at about 74 °C: v first rises with T before it falls. In the model this can be read as a competition between two local lattice types (ch. 6.2): an open, low-coordinated ice-like short-range order (small z_max, many “holes”) and a dense, more highly coordinated short-range order. At low T the open structure dominates; heating first breaks the open short-range order in favour of the denser one — the effective connectivity and with it K_eff rise (v rises). Only above the maximum does the ordinary CRACK bond rupture predominate, K_eff falls, v falls. The anomaly is thus the expression of a smeared-out, incomplete solid(1)↔solid(2)-like change of short-range order within the liquid phase — conceivable only in the 3D model with two lattice types, not in the 2D single lattice.

6.5 Alternative: fractal vortex fields instead of micelle rotation

Local Brownian motion (ch. 3.4) was explained through micelle rotation and exchange. Alternatively it can be pictured as the effect of fractal vortex fields: the block boundaries, constantly rebuilt by CRACK, generate a field of local momentum currents that — similar to turbulent flows — displays self-similar vortices on all length scales. A test particle is then driven not by discrete micelle collisions but by a scale-invariant vortex continuum. Both pictures (discrete micelles ↔ fractal vortices) are complementary limiting cases of the same CRACK kinetics; which one carries depends on the ratio of the test particle's size to the mean micelle size. This brief discussion is purely qualitative and is not carried through mathematically here.

6.6 Microscopic phonon stiffness θ(lattice): an Einstein spring model

The solid(1)→solid(2) comparison in ch. 6.3 hangs on the relative phonon stiffness θ_BCC/θ_FCC. Instead of choosing it freely, it is estimated here from coordination and geometry (theta_springs_16.py).

Model. A central atom sits in the harmonic bowl of its z neighbours; every bond is a radial pair spring κ. A displacement u projects isotropically with ⟨cos²⟩ = 1/d = 1/3 (d = 3) onto the bond directions, so that the effective spring constant is k_eff = (z/3)·κ. The Einstein frequency is ω = √(k_eff/m), hence θ = ħω/k_B ∝ √k_eff. With equal mass and equal pair spring it follows that

θ_BCC/θ_FCC = √(z_BCC/z_FCC) = √(8/12) = 0.816.

Refinements. (B) The second shell of the BCC (6 atoms at r₂/r₁ = 2/√3 = 1.155) with a distance-scaled spring κ(r) ∝ (r₁/r₂)ⁿ (n = 6…8, Lennard-Jones-like) raises the ratio to 0.91…0.94 (BCC is made effectively stiffer by the nearby second shell). (C) The shorter bond of the BCC at equal density (r_nn(BCC)/r_nn(FCC) = 0.972, −2.8 %) stiffens the spring slightly and gives 0.89…0.91. Altogether a geometric window θ_BCC/θ_FCC ≈ 0.82…0.94.

Opening threshold. The μ comparison (ch. 6.3) opens the transformation (τ_ss exists and lies below τ_m = 1.70) only for θ_BCC/θ_FCC < 0.61. The geometric window lies clearly above that.

Finding. The pure spring model does not make the BCC soft enough: in the strict ideal picture FCC remains stable, and a geometrically driven solid(1)→solid(2) transition does not arise. With that θ_BCC/θ_FCC has changed from a free assumption into an estimated quantity whose natural value (≈ 0.8–0.9) keeps the transformation closed — consistent with the honest finding from ch. 6.3. Real polymorphism therefore lies demonstrably outside the ideal harmonic picture (electronic/magnetic contributions, anharmonicity), not in mere lattice geometry.

6.7 Mathematical estimate of metastability (2D triangulated ↔ square)

For the qualitative discussion from ch. 3.2, here the associated order-of-magnitude calculation in the 2D picture (hard discs, diameter D, contact distance a = D).

(1) Energy. Per particle U/N = −(ε/2)·z (each bond counted half): triangulated z = 6 → −3 ε, square z = 4 → −2 ε. The rearrangement triangulated → square therefore costs ΔU = +1 ε per particle (endothermic; the loose form is higher in energy).

(2) Volume (area) and work. Area per particle: triangulated A_tri = (√3/2)·D² ≈ 0.866 D², square A_quad = D². The widening is ΔA = A_quad − A_tri ≈ 0.134 D² (+15.5 %) — the volume increase shown in the comic. Against a (reduced) pressure p the work W = p·ΔA has to be supplied; the free enthalpy rises by Δμ = ΔU + p·ΔA − τ·ΔS. The entropy term ΔS > 0 (the loose form has softer modes, a larger c_v step Δc_v = (6−4)/4 = 0.5 k_B) lowers μ_quad below μ_tri only at high τ — which is why triangulated is the low-temperature ground state and square the high-temperature/metastable form.

(3) Nucleation barrier (2D). Metastability presupposes a barrier. In 2D a nucleus of the new phase (radius r) grows with an area gain −Δμ_A·π r² (Δμ_A = the driving force per area) against the edge contribution +λ·2π r (λ = edge energy per length): ΔG(r) = −Δμ_A·π r² + 2π λ r. The maximum lies at r\* = λ / Δμ_A with barrier ΔG\* = π λ² / Δμ_A. The critical nucleus r\ shrinks as the driving force Δμ_A grows; as long as the thermal scale τ = k_BT does not reach the barrier (**ΔG\ ≳ a few k_BT), the loose (or, in boiling, the supersaturated) phase remains kinetically frozen. For small clusters (2–3 particles, as in the supersaturation strip) the edge-to-area ratio is maximally unfavourable — they lie to the left of r\* and dissolve again on average. With that the activation energy shown in the comic is the same quantity ΔG\* that carries both the metastable crystal form and the supersaturated gas phase; it is a kinetic** addition to the ideal (barrier-free) equilibrium model.

6.8 Debye-like refinement of θ(lattice)

The Einstein spring model (ch. 6.6) replaces the whole phonon distribution by one frequency. The present approach refines θ(lattice) through a lean phonon density of states per lattice (theta_debye_17.py): the same central pair springs κ(r) = r^(−n) on the nearest and the second neighbour shell form the dynamical matrix D(k) = Σ_j κ_j·(R̂_j⊗R̂_j)·(1 − cos(k·R_j)), which is diagonalised on a k grid of the primitive Brillouin zone (Monkhorst–Pack, 12³ points) using an analytical 3×3 eigensolver, standard library only. From the three phonon branches follow two Debye-like stiffness measures as frequency moments:

• 2nd moment θ(2) ∝ √⟨ω²⟩ — the Einstein limit (the BZ average of 1 − cos gives 1, hence ⟨ω²⟩ = z·κ/3); control, nn only, pure geometry: √⟨ω²⟩_BCC / √⟨ω²⟩_FCC = 0.816 = √(8/12) — exactly the Einstein value from ch. 6.6.
• log moment θ(0) ∝ exp⟨ln ω⟩ — the genuinely Debye-like quantity, which weights the soft acoustic branches that the Einstein value ignores.

All constants carrying a scale (ħ/k_B, the κ scale, the Debye prefactor) cancel in the ratio θ_BCC/θ_FCC; only the shape of the DOS remains. With a second neighbour shell (which is what first makes the BCC mechanically stable at all) and a spring law n = 0…8 one obtains

θ_BCC/θ_FCC ≈ 0.87…1.00 (error band over the definition of the moment and the spring law).

That is rather higher than the pure Einstein value (0.82…0.94), because the shorter BCC bond at equal density stiffens the springs. The opening threshold 0.61 required by the μ comparison (ch. 6.3) still lies clearly below it: the phonon DOS too does not make the BCC soft enough. The finding from ch. 6.6 is thereby confirmed and made more precise by the finer approximation — in the strict ideal picture FCC remains stable at every temperature; a purely geometrically driven solid(1)→solid(2) transition does not arise, and real polymorphism (iron, ch. 7.13) needs contributions outside the ideal harmonic picture.

6.9 θ(lattice) from a full Lennard-Jones lattice

Ch. 6.8 uses a truncated central spring κ(r) = r^(−n) on nn + 2nn. Revision 17 tests the same question with a genuine force constant from a pair potential over the full lattice (theta_ljlattice_18.py): every bond becomes a spring derived from the Lennard-Jones potential φ(r) = 4ε[(σ/r)¹² − (σ/r)⁶], with the longitudinal part κ_L = φ″(r) and the transverse part κ_T = φ′(r)/r. The dynamical matrix

D_ab(k) = Σ_j [ (κ_L − κ_T)·û_a û_b + κ_T·δ_ab ]·(1 − cos(k·R_j))

sums over all neighbour shells up to a cutoff (2.0…3.0 nn); each lattice is beforehand relaxed to its own LJ equilibrium (a free lattice constant, not an enforced equal density). That is the honest test of whether the lean nn+2nn approximation (ch. 6.8) gets the ordering of stiffness right.

Result (deterministic, standard library only):

Two findings, both sharpening ch. 6.6/6.8:

1. FCC is the stable LJ ground state (E ≈ −8.4 ε per atom at cutoff 3.0 nn, against −8.0 ε for BCC) — the well-known precedence of the dense packing in a pair potential.
2. The ideal LJ BCC is shear-unstable: about 3…3.5 % of the phonon modes are imaginary (ω² < 0, a Born instability). The pure pair-potential ideal picture therefore carries no mechanically stable BCC at all; over the remaining real modes θ_BCC/θ_FCC ≈ 1.0 — the opening threshold 0.61 required by the μ comparison is all the less undercut.

With that the statement of ch. 6.6/6.8 is robust against the refinement from a truncated spring to the full lattice and from a purely longitudinal model to the full dynamical matrix: in the classical ideal picture no purely geometric solid(1)→solid(2) transition arises — real polymorphism (iron ch. 7.13, titanium ch. 7.15) needs contributions outside the pair potential (electronic/magnetic, anharmonic). A genuine ab-initio phonon calculation is deliberately left out (it would leave the lean character of the ideal picture).

7 — Calibration and equation of state

Summary. Chapter 7 calibrates the model onto real substances through the single energy parameter ε and builds out the liquid/gas branch into an equation of state:

• 7.1 / 7.2 Calibration method and table: a linear temperature coupling T = (ε/k_B)·τ with one anchor point at the melting point; from it the refined universal ratio R = τ_b/τ_m ≈ 1.54.
• 7.3 / 7.4 Hard-core van der Waals EOS: the reduced EOS p_r = 8τ_r/(3v_r − 1) − 3/v_r² with the critical point (1, 1, 1) and Maxwell coexistence; a ∝ ε, b ∝ D². The boiling points lie systematically at T_b/T_c ≈ 0.48 (the Guldberg gap); a Redlich–Kwong comparison calculation raises them to ≈ 0.60 — as a named error budget, without changing the EOS.
• 7.5 Common (τ, v) surface: the coexistence dome (boiling), the melting line and the hard wall fit together with a triple point (τ_r,triple ≈ 0.47 for v_solid = 0.40).
• 7.6 Latent heat: L ≈ 1.41 ε gives ΔS_fus ≈ 1.175 R (close to Richards's rule); metals are hit excellently (copper ×1.00), while ionic, hydrogen-bonded and molecular crystals lie outside the ideal picture.
• 7.7 Unification ε ↔ ε_lg: k_B T_c = z_max·ε/4 couples both scales (ε_lg = (27/32)·z_max·ε); the 3D prediction τ_c/τ_m = 2.50 hits simple molecular substances, while 2D remains a special case.
• 7.8 / 7.9 Parameter-free triple point: v_solid follows from the rigidity threshold z̄_c = 2d, not from connectivity; in 3D a clean liquid window opens (τ_m = 1.20 < τ_triple = 2.83 < τ_c = 3.00), while in 2D it stays closed (the 2D special role, now trebly documented).
• 7.10 / 7.11 Genuine 3D scans: bond percolation on FCC/BCC gives z̄_c(3D) ≈ 1.5–1.6; the τ scan of the bonding kinetics shows τ_m ≈ τ_rig (melting = rigidity, low) and τ_b(3D) ≈ τ_c (boiling = connectivity decay at the critical point).
• 7.12 / 7.13 / 7.15 The solid(1)→solid(2) line: from the point τ_ss the pressure makes a line (dp/dτ = ΔS/Δv > 0, Δv = +8.9 %), entered as a fourth boundary line into the (τ, v) surface and calibrated as a (τ, p) line against iron (γ→δ), confirmed on titanium (α→β) and checked with argon as a negative control.
• 7.14 Redlich–Kwong branch: as an independent ideal-picture variant with its own calibration beside the van der Waals default (T_b/T_c ≈ 0.60, error −1 %).

7.1 Calibration method (one energy parameter ε) — adjusted

Linear: T = (ε/k_B)·τ, one anchor point per substance. Anchoring at the melting point: ε/k_B = T_m,exp/τ_m, prediction T_b,pred = T_m,exp·(τ_b/τ_m). With the refined melting peak (the cooperative term) the c_v maximum now lies at τ_m = 1.20 (instead of 1.15), while the connectivity percolation (near boiling) is unchanged at τ_b = 1.85. From this the refined universal ratio:

R = τ_b / τ_m = 1.85 / 1.20 ≈ 1.54 (revision 08: 1.61).

The calibration table from revision 08 (ch. 7.2) scales linearly with R: all T_b,pred fall by the factor 1.54/1.61 ≈ 0.957 (about −4.3 %). That slightly improves the overestimate for the noble gases (argon +54 % → +48 %) and marginally worsens the hits for NaCl and iron (NaCl −0.6 % → −4.8 %). The basic statement remains: R fits isotropically densely packed substances with non-directional bonding; noble gases and chains lie outside the ideal picture.

7.2 Calibration table

Taken over unchanged from modell_08.md ch. 7.2 (with the note on the linear R scaling from 7.1). Reproducible via cv_scan_08.py (table) and cv_scan_09.py (τ_m, τ_b, R).

7.3 Density/volume axis: hard-core van der Waals EOS

The pressure is raised from a “parameter” to a genuine variable. HARD CORE supplies the excluded volume b, the bonding attraction (ε·z̄) the cohesive pressure a. In reduced units (critical point at (v_r, τ_r, p_r) = (1, 1, 1)):

p_r = 8 τ_r / (3 v_r − 1) − 3 / v_r², v_r > 1/3 (hard wall).

Coexistence by Maxwell construction (cv_scan_09.py, equal-area cut):

Result. Below τ_c the l/g density jump opens up (v_gas/v_liq grows from 2.5 to 11.5, the colder the larger). That is the boiling line — now cleanly separated from the melting c_v peak (ch. 5.3), which lives on a lattice of fixed density. With that the point 3 left open in revision 08 is closed: melting (rigidity/bond ordering at fixed density) and boiling (a density/volume jump) are different axes of the same model.

7.4 Quantitative l/g calibration: a → ε, b → D²

Mapping the reduced quantities onto real substances. The van der Waals constants hang on the critical point:

b = k_B·T_c / (8 p_c) (excluded volume, HARD CORE), a = 27 k_B²·T_c² / (64 p_c) (cohesion, BOND).

Thus a ∝ ε (the cohesion stems from the bond energy) and b ∝ D² (the 2D excluded volume from the hard-core diameter). The natural energy scale of the l/g branch is

ε_lg / k_B = (a/b) / k_B = (27/8)·T_c = 3.375·T_c.

Anchor substance argon (T_c = 150.87 K, p_c = 48.63 bar): b = 5.35·10⁻²⁹ m³ per molecule, a = 3.76·10⁻⁴⁹ J·m³ per molecule², ε_lg/k_B = 509 K and a molecular diameter D ≈ b^{1/3} = 3.8·10⁻¹⁰ m — accurate to 11 % against the van der Waals diameter of argon (~3.4·10⁻¹⁰ m).

Prediction of the boiling temperature. At a fixed external pressure the boiling point is where the reduced saturation pressure p_sat(τ_r)·p_c = p_atm is satisfied; from this T_b = T_c·τ_r,b (the Maxwell saturation curve from lg_calib_10.py):

*CO₂ sublimes at 1 atm (T = 194.7 K = sublimation point).

Result. For the molecular substances (p_c ≈ 30–70 bar) the model gives throughout T_b/T_c ≈ 0.48–0.51, systematically ~17–24 % below the measured value. That is not a model error in the narrower sense but the well-known gap between the van der Waals equation and the empirical Guldberg rule (T_b/T_c ≈ 0.6): the van der Waals isotherm underestimates the saturation pressure at low temperatures. Water (and the metals, whose T_c/p_c are in any case only roughly estimated) lie at p_r,atm < 0.005 outside the sensible van der Waals range (τ_r,b < 0.40) — there the cohesion is no longer van der Waals-like (hydrogen bonds, metallic bonding). The order of magnitude and the ranking of the boiling points are hit correctly.

Guldberg classification as an error table. The gap is here only named, not corrected by changing the EOS (the ideal picture remains van der Waals-like). The empirical Guldberg rule (Guldberg 1890) states that the boiling point lies at about T_b/T_c ≈ 0.6 (with a slight substance dependence, ~0.58–0.64). The reduced van der Waals model instead gives a constant value T_b/T_c = τ_r,b ≈ 0.48–0.51:

Reading. The model mean of 0.49 hits the Guldberg constant 0.60 to about −18 %; the deviation is systematic and one-sided (always too low), hence a property of the van der Waals isotherm (too small a saturation pressure at low τ), not of the molecular picture. A better cohesion EOS (for instance Redlich–Kwong) would raise τ_r,b — but remains deliberately outside the ideal picture here.

Error budget with a Redlich–Kwong comparison calculation (guldberg_rk_15.py). In order to quantify the van der Waals gap to the Guldberg rule (not to heal it), the same boiling-point test is repeated with a Redlich–Kwong-like, temperature-dependent cohesion a/√T. In reduced units (critical point (v_r, τ_r, p_r) = (1, 1, 1), Z_c = 1/3):

p_r = 3 τ_r/(v_r − β) − A/(√τ_r · v_r(v_r + β)), β = 3·Ω_b = 0.25992, A = 9·Ω_a = 3.84732,

with the Maxwell equal-area construction and the same reduced external pressure p_r,atm as above. Result (one number per anchor substance, T_b/T_c = τ_r,b):

Result. The temperature-dependent RK cohesion raises T_b/T_c from vdW ≈ 0.48 to ≈ 0.60 — practically the Guldberg constant and the real values (≈ 0.58–0.63); the mean error against the measured values falls from −21 % (vdW) to −1 % (RK). With that the cause of the Guldberg gap is clearly quantified: it is precisely the missing temperature dependence of the van der Waals cohesion (too small a saturation pressure at low τ), not the molecular picture. This comparison calculation lies deliberately outside the ideal picture; the model remains van der Waals-like (no change of EOS) — the error budget is only named, not built in.

7.5 Common (τ, v) surface with triple point

The melting and boiling axes have so far lived separately: melting (rigidity/bond ordering) at fixed density, boiling (van der Waals) through the density jump. In a common (τ_r, v_r) plane they fit together:

• Coexistence dome (boiling): v_liq(τ_r) (left branch) and v_gas(τ_r) (right branch) from the Maxwell construction, meeting at the critical point (τ_r, v_r) = (1, 1).
• Solid (melting): a narrow band just above the hard wall v_r > 1/3, at high, nearly fixed density (reduced volume v_solid ≈ 0.40).
• Melting line: nearly vertical at v_solid (fixed density), the temperature of the Maxwell rigidity (z̄_c = 2d; ch. 7.8 derives v_solid from it parameter-free — the old revision-11 assignment “z̄_c ≈ 2.08” is thereby corrected).

The triple point is the point at which the liquid coexistence boundary v_liq(τ_r) reaches the solid packing density v_solid — below it no liquid branch exists any more (direct sublimation):

Result. For v_solid = 0.40 the triple point lies at τ_r,triple ≈ 0.47, p_sat ≈ 0.018, with a vapour volume v_gas ≈ 68 (a very dilute sublimation vapour). The ratio τ_triple/τ_c ≈ 0.47 lies within the real window (water 0.42; argon 0.56; CO₂ 0.71). The denser the solid (the smaller v_solid), the lower the reduced triple point and the more dilute the coexisting vapour — qualitatively correct. With that both transitions and the triple point are anchored in one (τ, v) surface; the critical point sits at (1, 1, 1). Revision 15 adds in ch. 7.12 a fourth boundary line — the solid(1)/solid(2) transformation (FCC↔BCC) as two nearly vertical solid density bands (v ≈ 0.667 and 0.726) that flip into one another at τ_ss.

An earlier version during development: v_solid = 0.40 was still imposed here. Ch. 7.8 derives v_solid parameter-free from the percolation threshold z̄_c and makes the triple point a genuine prediction.

7.6 Latent heat against enthalpies of fusion

The model latent heat L ≈ 1.41 ε (ch. 5.3) is translated into real units through the melting-point calibration ε/k_B = T_m/τ_m (τ_m = 1.20):

ΔH_m,pred = L·R·T_m/τ_m = 1.175·R·T_m, correspondingly ΔS_fus,pred = L·R/τ_m = 1.175·R.

That is — remarkably — almost exactly Richards's rule (ΔS_fus ≈ R for simply built solids). Comparison with measured enthalpies of fusion (lg_calib_10.py):

Result. The model hits the order of magnitude (kJ/mol) throughout and Richards's rule for simple metals excellently (copper ×1.00, iron ×1.28); the liquefied gases lie just below it (methane ×0.94, nitrogen ×0.87, argon ×0.69). Ionic, hydrogen-bonded and molecular crystals (NaCl ×0.37, water ×0.44, benzene ×0.28, CO₂ ×0.23) have markedly higher real entropies of fusion (ΔS_exp/R = 2.6–5.0), because on melting additional degrees of order are released (ion pairing, the orientation of polyatomic molecules) that the single-order-parameter model does not capture. The limit is thereby clearly named: the ideal picture holds for isotropically, non-directionally bonding substances (no fit).

7.7 Unification of the energy scales: ε ↔ ε_lg

So far the melting branch carried the bond energy ε (τ_m = 1.20) and the l/g branch a separate cohesion scale ε_lg/k_B = 3.375·T_c (ch. 7.4). Both can be derived from one model parameter. In the model the van der Waals cohesion a arises from the sum of the bond energies per molecule (BOND): every molecule carries z_max/2 bonds of energy ε. A mean-field Bragg–Williams lattice gas with neighbour attraction ε gives the classical relation for the critical temperature:

k_B·T_c = z_max·ε / 4, hence τ_c ≡ k_B T_c/ε = z_max/4.

Inserting this into ε_lg = a/b = (27/8)·k_B·T_c (ch. 7.4) gives the coupling of both scales:

ε_lg = (27/8)·(z_max/4)·ε = (27/32)·z_max·ε, constant c = 27/32 = 0.844.

With that the melting peak, percolation and the critical point lie on one ε scale (unify_scale_12.py):

Prediction τ_c/τ_m = z_max/(4·τ_m): 2D → 1.25, 3D → 2.50. Against real substances:

Result. The 3D prediction τ_c/τ_m = 2.50 hits the real characteristic figure of the simple molecular substances (T_c/T_m ≈ 2.0–2.1) well; the metals lie higher (T_c roughly estimated, a different type of bonding). The 2D prediction (1.25) is too low — and τ_c(2D) = 1.50 even lies below τ_b = 1.85 (the critical point would lie below the percolation threshold). That is not a calculation error but the well-known special role of 2D: cohesion and bonding scales do not order themselves cleanly there, and a robust liquid window is missing (cf. continuous 2D melting, ch. 5.5). Only in 3D (τ_m = 1.20 < τ_c = 3.00) do the scales separate in a physically sensible way.

7.8 Parameter-free triple point from z̄_c

Ch. 7.5 still imposed v_solid = 0.40. The earlier version during development derives the solid packing density from the percolation threshold. The coordination number of a lattice at reduced volume v_r follows the density–coordination relation (hard wall v_r = 1/3 ↔ densest packing, z̄ = z_max):

z̄(v_r) = z_max·(1/3)/v_r = z_max/(3 v_r), conversely v_solid = z_max/(3·z̄_c).

The solid loses its stiffness as soon as z̄ falls below the threshold z̄_c; the associated density is v_solid. The triple point is then the point at which the liquid coexistence boundary v_liq(τ_r) reaches this density (unify_scale_12.py, 2D, z_max = 6):

Improved version. The melting density v_solid is set by the generic Maxwell rigidity z̄_c = 2d (2D → 4, 3D → 6), not by bond connectivity (justification in ch. 5.4a: melting is the loss of stiffness, not of connectedness). With that the dependable triple point reads (rigidity_3d_13.py, 2D):

Result in 2D. Even with the (physically correct) rigidity threshold the 2D single lattice gives no clean liquid window: v_solid = 0.50 does yield the real characteristic value τ_triple/τ_c ≈ 0.77, but in absolute ε units τ_triple = 0.77·1.50 = 1.16 and therefore lies below τ_m = 1.20. The connectivity threshold (z̄_c = 2.08) even pushes v_solid ≈ 0.96 ≈ v_c out of the coexistence dome — triple and critical points coincide. Both calculations confirm the 2D special role (ch. 5.5, 7.7): the 2D single lattice hardly carries a liquid phase of its own. That is an honest finding, not a weakness of the calculation or of the model in the narrower sense.

The decisive test is therefore the 3D control calculation (ch. 7.9): there the same rigidity rule (z̄_c = 6 → v_solid = 0.667) releases a clean window. In every case the triple point is now bound parameter-free to z̄_c — the free parameter v_solid has been eliminated.

7.9 3D control calculation: a clean liquid window on one ε scale

A test of whether the 2D collapse resolves itself in 3D when all marks lie on one ε scale (τ_c = z_max/4 from ch. 7.7; v_solid from the generic rigidity z̄_c = 2d; the triple point where the reduced van der Waals liquid density v_liq(τ_r) reaches the solid density). Result (rigidity_3d_13.py):

Result. In 3D a clean liquid window opens up: τ_m = 1.20 < τ_triple = 2.83 < τ_c = 3.00 (window widths: τ_triple − τ_m = 1.63; τ_c − τ_triple = 0.18). The upper edge is narrow — the triple point lies close beneath the critical one — but the ordering is right and the liquid phase exists as a field of its own. In 2D the window remains closed (τ_triple = 1.16 < τ_m). With that the 2D special role is documented independently for the third time and it is at the same time shown to be a dimensional effect: the same rigidity rule heals the model in 3D. The narrow upper edge is corroborated in ch. 7.10 by a genuine 3D percolation scan and a sensitivity analysis.

Model-limits box — the 2D special role. Three independent calculations show the same finding: the 2D single lattice hardly carries a liquid phase of its own. 1. τ_c < τ_b (ch. 7.7): the critical point (1.50) would lie below the connectivity percolation (1.85) — cohesion and bonding scales do not order themselves cleanly in 2D. 2. v_solid ≈ v_c (ch. 7.8): the connectivity melting density (0.96) almost coincides with the critical density — triple and critical points collapse together. 3. τ_triple < τ_m (ch. 7.9): even with the correct Maxwell rigidity the triple point (1.16) lies below the melting peak (1.20) — no window.

That is an honest finding, not a weakness of the calculation or of the model: it is the well-known dimensional effect (continuous 2D melting, KTHNY-like; ch. 5.5). In 3D the same rigidity rule heals the model (ch. 7.9/7.10). Whether a second short-range order (ch. 6.4) could additionally improve the 2D picture remains a question of extension, not a defect of the ideal picture. Since the model is meant to serve illustration, I accept the weakness and forgo an axiomatic extension of the model.

7.10 Genuine 3D bond percolation scan (FCC/BCC) and window-edge sensitivity

Ch. 7.8/7.9 transferred the 2D connectivity value z̄_c = 2.08 to 3D in thought (giving v_solid = 1.92 there, which is unphysical) and based the 3D statement on the generic rigidity z̄_c = 2d. Revision 13 now tests connectivity in 3D directly: bond percolation on FCC (z_max = 12) and BCC (z_max = 8) by Monte Carlo with union-find, periodic boundaries, a deterministic LCG seed (rigidity_3d_14.py). Threshold p_c = the crossing P_∞(p) = 0.5 (the same methodology as the 2D finite-size study in ch. 5.6):

Because of the moderate box edge (L = 12, pure Python) the MC values lie slightly above the literature thresholds (a finite-size bias, as in 2D from below or above depending on the estimator); the ranking and order of magnitude are right (FCC denser → a smaller p_c than BCC).

Reading (an independent 3D confirmation). The 3D connectivity thresholds lie at z̄_c(3D) ≈ 1.5–1.6 — even lower than the 2D value 2.08 and far below the Maxwell rigidity 2d = 6. The “connectivity melting density” that follows, v_solid = 1/(3 p_c) ≈ 1.7 (BCC) or 2.6 (FCC), lies far outside the coexistence dome (v_liq,max ≈ 0.94). With that it is shown directly — without transferring the 2D value to 3D — that in 3D connectivity can all the less set the melting density; it must come from rigidity (z̄_c = 6 → v_solid = 0.667). The two thresholds are even more clearly separated in 3D than in 2D. The genuine τ_b(3D) (the temperature at which z̄(τ) reaches the low connectivity threshold 1.5) would therefore lie above the 2D value τ_b, very close to the critical point — consistent with the picture “connectivity decay = boiling”.

Sensitivity of the narrow upper window edge (τ_c − τ_triple ≈ 0.18). Numerically around v_solid = 0.667 (±2 %) one obtains dτ_triple/dv_solid ≈ 1.55. The window would close (τ_triple = τ_c = 3.00) only if v_solid rose by ≈ 0.11 (to ≈ 0.78), that is, if the effective 3D rigidity threshold lay noticeably above 2d = 6 (z̄_c ≳ 7.7). For the generic Maxwell count (z̄_c = 2d = 6) the ordering τ_m < τ_triple < τ_c therefore remains robust; the upper edge is genuinely narrow but not endangered. The discrete lattice change solid(1)→solid(2) (FCC↔BCC) is quantified in revision 14 (ch. 6.3); the genuine 3D τ scan of the bonding kinetics follows in ch. 7.11.

7.11 Genuine 3D τ scan of the bonding kinetics: τ_b(3D) and the melting line on one ε scale

Ch. 7.8–7.10 based the 3D statement on the generic rigidity (z̄_c = 2d = 6) and the static connectivity scan (z̄_c(3D) ≈ 1.55). What remained open was the actual τ scan of the bonding kinetics in 3D: where on the temperature axis does the self-consistent z̄(τ) reach this low connectivity threshold (= the genuine τ_b(3D)), and where does the melting peak lie? Revision 14 computes the self-consistent z̄(τ) (ch. 4.2, x = 1/(1 + G·e^{−(ε+K·x)/τ})) directly on FCC (z_max = 12) and BCC (z_max = 8) with G = z_max and the same cooperativity J = 0.55 as in the 2D reference case; n_blk(τ) comes from an FCC/BCC bond Monte Carlo (union-find), and

U/N = τ·[3/2 + z̄/4 + (3/2) n_blk] − (ε/2)·z̄ − (K/2)·x² (3D, with the cooperative term),

c_v = dU/dτ (bondkinetics_3d_15.py, a deterministic LCG seed; n_blk smoothed, since only the weak block-rotation term depends on it). Three marks are read off: the c_v peak τ_m (melting), the rigidity mark τ_rig (z̄(τ) crosses 2d = 6) and the genuine τ_b(3D) (z̄(τ) crosses the connectivity threshold z̄_c(3D) from ch. 7.10):

Result. Two findings, both consistent with the picture so far:

1. Melting lies low and is doubly documented. The thermodynamic melting peak τ_m (FCC 1.70, BCC 1.40) coincides practically with the rigidity mark τ_rig (z̄ = 6; FCC 1.73, BCC 1.35) — the same temperature at which the network loses its stiffness. That anchors the 3D melting line on the ε scale: T_m = (ε/k_B)·τ_m, so that for argon (T_m = 83.8 K), for instance, ε/k_B = 83.8/1.70 ≈ 49 K. Because K = J·z_max is larger in 3D than in 2D (6.60 instead of 3.30), the 3D melting peak lies higher than the 2D value 1.20 — no contradiction, but the stronger cooperativity of the denser packing.
2. The genuine τ_b(3D) sits at the critical point. Because the 3D connectivity threshold is so low (≈ 1.55), z̄(τ) reaches it only right at the top: τ_b(3D) = 3.18 (FCC) and 2.85 (BCC) — at, or slightly above, τ_c = z_max/4 (3.00 and 2.00). The bond giant cluster therefore survives the entire liquid phase and decays only at the critical point. That is exactly the prediction from ch. 7.10 (“the genuine τ_b(3D) would lie above the 2D value, very close to the critical point”) and independently confirms the picture “boiling = connectivity decay”: in 3D matter stays continuously bonded right up to the critical point.

Both findings corroborate the separation of thresholds from ch. 5.4a: melting ↔ rigidity (low, τ_m ≈ τ_rig), boiling ↔ connectivity (high, τ_b(3D) ≈ τ_c). The melting density therefore comes from rigidity (v_solid = 0.667), not from connectivity — the two thresholds are pulled apart on the temperature axis in 3D further than ever before in the model.

7.12 The solid(1)→solid(2) transformation line in the (τ, v) surface

Ch. 6.3 supplied the lattice transformation FCC↔BCC as a point τ_ss (at p = 0). Through the pressure this becomes a line (polymorph_line_16.py) that belongs as a fourth boundary line in the common (τ, v) surface from ch. 7.5/7.9 — beside the coexistence dome (boiling), the melting line and the hard wall.

Volume jump. The two solids sit at nearly fixed density; their difference in volume follows from the packing fractions (v ∝ 1/η): η_FCC = 0.7405, η_BCC = 0.6802. With v_FCC = 0.667 (ch. 7.9) one obtains v_BCC = v_FCC·η_FCC/η_BCC = 0.726, hence

Δv = v_BCC − v_FCC = +0.059 (+8.9 %).

The more open BCC has the larger volume — recrystallisation FCC→BCC is bound up with an increase in volume (the same statement as in the comic strip “recrystallisation”).

Clausius–Clapeyron. With ΔS = S_BCC − S_FCC (the high minus the low temperature phase) the slope reads

dp/dτ = ΔS/Δv.

Both quantities are positive (BCC is the high-temperature phase and so has more entropy, and the larger volume), hence dp/dτ > 0: a higher pressure raises the transformation temperature, because it favours the denser FCC. For the reference θ_BCC/θ_FCC = 0.60 (τ_ss = 1.333, ΔS = 0.50 k_B, L = 0.667 ε):

The line in (τ, p) for the reference (rising): p = 0 → τ_ss = 1.33; p = 2 → 1.57; p = 4 → 1.81; p = 6 → 2.04 (reduced units).

In the (τ, v) surface the transformation appears as two nearly vertical solid density bands — solid(1) = FCC at v ≈ 0.667 (denser, low T) and solid(2) = BCC at v ≈ 0.726 (more open, high T) — that flip into one another at τ_ss; above them, at τ_m, melting follows. The complete ordering on the τ axis is therefore τ_ss < τ_m < τ_c ≤ τ_b(3D) (ch. 7.11).

Real units. Through the melting-point calibration ε/k_B = T_m/τ_m (τ_m = 1.70, 3D FCC) the c_v step and the heat of transformation become tangible. The c_v step is independent of the calibration: |Δc_v| = 1.0 k_B = 1 R = 8.31 J/(mol·K) = 0.0083 kJ/(mol·K). The heat of transformation ΔH_ss = L·R·(ε/k_B) (reference L = 0.667 ε):

The heats of transformation are small and positive (of the order 0.2–0.9 kJ/mol) and therefore lie clearly below the enthalpies of fusion (ch. 7.6, several kJ/mol) — exactly the expected picture of a weak polymorphic transformation with a small c_v step. (The θ ratio 0.60 lies deliberately outside the strict ideal picture; ch. 6.6 shows that pure geometry would give θ ≈ 0.82–0.94 and the transformation would then stay closed — the line is the if-then consequence of a soft BCC phase.)

7.13 solid(1)→solid(2) transformation line in the (τ, p) diagram, calibrated against iron

Ch. 7.12 carries the transformation as a (τ, v) line; revision 16 adds it as a line in the (τ, p) picture and calibrates it against a real FCC↔BCC polymorph (solidsolid_pt_17.py).

Pressure field from the hard-core van der Waals EOS. At the two solid densities v_FCC = 0.667 and v_BCC = 0.726 (ch. 7.12) the reduced EOS p_r = 8τ_r/(3v_r − 1) − 3/v_r² (ch. 7.3) gives throughout positive pressures; at the same τ_r the denser FCC carries the higher pressure (τ_r = 1: p_r = 1.25 against 1.10). Compression therefore favours FCC — consistent with the positive Clausius–Clapeyron slope dp/dτ = ΔS/Δv > 0.

Line and transformation point. For the reference θ_BCC/θ_FCC = 0.60 (τ_ss = 1.333, ΔS = 0.50 k_B, dp\/dτ = 8.47 in reduced units p\ = p·D³/ε) the line rises as p\(τ) = 8.47·(τ − 1.333): p\ = 0 → τ = 1.33; p\ = 2 → 1.57; p\ = 4 → 1.81; p\ = 6 → 2.04. At **p\_tri ≈ 3.1 the solid-solid line meets the melting mark τ_m = 1.70 — a solid(1)-solid(2)-liquid transformation point** in the (τ, p) plane (the analogue of the triple point).

Calibration against iron (ε/k_B = T_m/τ_m = 1811 K / 1.70 = 1065 K). Iron has two FCC↔BCC transitions with opposite sign — exactly the test case:

For γ→δ (τ_ss = 1667/1065 = 1.565 → θ_BCC/θ_FCC ≈ 0.61) the model hits sign and order of magnitude: ΔH_ss ≈ 4.2 kJ/mol (in reality ≈ 0.84 kJ/mol — the order of magnitude kJ/mol, with the value about 5× too high because of the sharp threshold approximation), the c_v step 1 R = 8.31 J/(mol·K) parameter-free, dT/dp ≈ +245 K/GPa. The transition α→γ, by contrast, has the dense phase (γ) as the high-temperature phase — that pure packing geometry cannot supply (denser = lower = low T); α-iron is stabilised magnetically. Iron therefore confirms both sides of the honest finding: the geometric topology (γ→δ) lies within the ideal picture, the magnetic anomaly (α→γ) outside it — the same “outside” that ch. 6.6/6.8 quantify through θ ≈ 0.87…1.00 > 0.61.

7.14 Redlich–Kwong as a separate ideal-picture branch

The Guldberg gap of the van der Waals branch (ch. 7.4, T_b/T_c ≈ 0.48 instead of ≈ 0.60) was there only named. At the express wish of the user an improved revision carries the Redlich–Kwong cohesion as an independent ideal-picture variant with its own calibration, its own coexistence and its own error section (rk_branch_17.py) — the van der Waals branch remains the default; all the remaining model physics (melting, percolation, solid(1)→solid(2)) is untouched by the choice of the cohesion EOS.

Its own energy scale. From a_RK = Ω_a·R²T_c^{2.5}/p_c and b_RK = Ω_b·R T_c/p_c the cohesion energy of the branch follows as

ε_lg,RK/k_B = (Ω_a/Ω_b)·T_c = 4.93·T_c (against vdW ε_lg/k_B = 27/8·T_c = 3.375·T_c),

larger, because the temperature-dependent cohesion a/√T folds in the factor √T_c. The diameter that follows from b_RK hits the real values about as well as in the vdW branch (argon D ≈ 3.34·10⁻¹⁰ m).

Its own coexistence (reduced, critical point (1, 1, 1), Z_c = 1/3; p_r = 3τ_r/(v_r − β) − A/(√τ_r·v_r(v_r + β)), β = 0.25992, A = 3.84732):

Its own error section (boiling point at p_r,atm = p_atm/p_c, T_b/T_c = τ_r,b):

Result. The RK branch raises T_b/T_c from vdW ≈ 0.48 to ≈ 0.60 and thereby hits the Guldberg constant and the real values practically (a mean error of −1 % against vdW −21 %). It stands as a second, clearly separate ideal-picture variant beside the van der Waals default; which branch is “more correct” is a question of the accuracy demanded, not of the model structure — the core of Model 3 (z̄ as the order parameter) remains identical in both branches.

7.15 solid(1)→solid(2) on further polymorphs: titanium and a noble-gas control

Ch. 7.13 calibrates the solid(1)→solid(2) (τ, p) line against iron. The improved revision secures sign and order of magnitude on a second, magnetically unremarkable polymorph and adds a negative control (solidsolid_titanium_18.py). Calibration and volume jump are as in ch. 7.13 (ε/k_B = T_m/τ_m, τ_m = 1.70; Δv = +8.9 % dense→open from the packing geometry); hcp is approximated like FCC through z = 12 (the fcc/hcp difference is only named, not resolved).

Titanium, with α(hcp) → β(bcc) at 1155 K, has exactly the model topology: the dense phase (hcp, z = 12) is the low-temperature phase, and on heating it becomes open (bcc, z = 8), hence ΔS > 0 and Δv > 0 → dp/dτ > 0 — the same sign as iron γ→δ, but without a magnetic anomaly. Two independent dense→open polymorphs in comparison:

Both hit the sign (dp/dτ > 0, pressure favours the dense phase) and the order of magnitude (ΔH ~ kJ/mol, the c_v step parameter-free at 1 R = 8.31 J/(mol·K), dT/dp ~ 10²·K/GPa). The value of ΔH_ss is in both cases systematically about a factor too high — the sharp threshold approximation of the μ comparison overestimates the latent heat of transformation; that is a named limit of the model, not an error of sign.

Argon is the negative control: solid argon is FCC and stays FCC up to melting (83.8 K) — at normal pressure no stable bcc exists. That is precisely what the pure pair-potential ideal picture predicts: θ_BCC/θ_FCC ≈ 0.9…1.0 > 0.61 and a shear-unstable LJ BCC (ch. 6.9) ⇒ no solid(1)→solid(2) transition. The counter-test confirms the structure of the finding: the ideal picture produces a solid(1)→solid(2) line only when non-geometric contributions (electronic/magnetic, as in iron and titanium) stabilise the BCC in the first place. Titanium supplies the positive, argon the negative evidence of the same honest framework from ch. 6.8/6.9.

8 — Comparison with the literature: model conceptions of static thermodynamics

Summary. Chapter 8 places Model 3 among the established model conceptions of static thermodynamics:

• 8.1 Overview table: eleven models (ideal gas, van der Waals, Redlich–Kwong, virial expansion, Einstein, Debye, Lennard-Jones / hard spheres, cell theory, Ising / lattice gas, percolation, Flory–Huggins) with their phase(s) and their statistical-thermodynamic basic idea.
• 8.2 The models individually: one paragraph each with features, application and limits, beginning with the source link; freely accessible sources are archived under model3/links/, while cell theory (paywall) is only cited.
• 8.3 Juxtaposition: Model 3 is a cooperative two-state bonding model with the order parameter z̄; its closest relative is Ising / lattice gas, boiling is taken directly from percolation, and the phonon picture (Einstein/Debye) carries the melting stiffness. Model 3 brackets these three traditions on a common ε scale.

This chapter places Model 3 among the established model pictures of equilibrium (“static”) thermodynamics of solids, liquids and gases. First a tabular overview (8.1), then one short prose paragraph per model with typical features, range of application and limits — each paragraph begins with the link to the source (8.2). Finally a keyword-style juxtaposition with Model 3 (8.3). Freely accessible sources are archived under model3/links/; the checking script model3/scripts/check_links_16.py keeps citation and archive consistent by SHA-256.

8.1 Overview table

8.2 The models individually

https://de.wikipedia.org/wiki/Kinetische_Gastheorie — Ideal gas / kinetic theory of gases. Particles without excluded volume and without interaction apart from elastic collisions; pressure arises purely from the transfer of momentum. It supplies the equation of state pV = Nk_BT, the Maxwell–Boltzmann velocity distribution and ⟨E⟩ = (3/2)k_BT. Application: dilute gases at high temperature and low pressure. Limits: no condensation, no liquid or solid phase, no phase transition — interaction is absent by construction.

https://de.wikipedia.org/wiki/Van-der-Waals-Gleichung — Van der Waals equation. (p + a/v²)(v − b) = k_BT supplements the ideal gas with an excluded volume b (repulsion) and a cohesion term a (attraction). It is the first equation of state with a gas–liquid transition, a critical point and a Maxwell construction. Application: a qualitative picture of condensation, the critical quantities. Limits: quantitatively inaccurate near the critical point (mean-field exponents), no solid phase.

https://de.wikipedia.org/wiki/Redlich-Kwong-Gleichung — Redlich–Kwong equation of state. A cubic EOS with a temperature-dependent attraction term a/[√T · v(v + b)]. The √T scaling markedly improves the vapour–liquid equilibria compared with van der Waals (cf. ch. 7.4, where it raises the ratio T_b/T_c). Application: technical gas-phase and evaporation calculations. Limits: empirically parametrised, still no solid phase; near the critical point likewise mean-field-like.

https://de.wikipedia.org/wiki/Virialentwicklung — Virial expansion. The systematic expansion pV/Nk_BT = 1 + B(T)/v + C(T)/v² + … The virial coefficients follow from the intermolecular potential (cluster integrals of statistical mechanics) and are therefore microscopically founded. Application: gases up to moderate density, a reference for other equations of state. Limits: the series does not converge for dense liquids or solids; higher coefficients are computationally expensive.

https://de.wikipedia.org/wiki/Einstein-Modell — Einstein solid. The crystal is an ensemble of 3N independent harmonic oscillators of the same frequency ω. It explains the fall of the heat capacity c_v below the characteristic temperature θ_E = ħω/k_B and the Dulong–Petit limit 3Nk_B at high temperature. Application: the simplest description of solid heat capacities. Limits: the wrong low-temperature law (exponential instead of T³), no acoustic modes, only the solid phase.

https://de.wikipedia.org/wiki/Debye-Modell — Debye model. It treats the solid as an elastic continuum with a phonon spectrum up to the Debye frequency; c_v ∝ T³ at low temperature, Dulong–Petit at high — the Debye temperature θ_D sums up the hardness of the lattice. Application: quantitatively good heat capacities of simple crystals over the whole T range. Limits: a continuum approximation (it ignores the dispersion of real lattices), only the solid phase, no melting.

https://de.wikipedia.org/wiki/Lennard-Jones-Potential — Lennard-Jones potential / hard spheres. The pair potential 4ε[(σ/r)¹² − (σ/r)⁶] with a repulsive core and a weak attraction; the limiting model of hard spheres retains only the repulsion. It is the basis of molecular dynamics and Monte Carlo and describes — numerically — all three phases together with the triple point. Application: noble gases, a generic reference system. Limits: pairwise additivity, evaluable at its core only by simulation or perturbation theory (no closed EOS).

https://doi.org/10.1098/rspa.1937.0210 — Cell theory of the liquid (Lennard-Jones–Devonshire). Every molecule moves in a cell (a cage) formed by its neighbours in an averaged field; the liquid is described by the cell partition function and yields a liquid EOS as well as an approach to melting. Application: simple liquids, a link between solid and gas. Limits: it overestimates the order and neglects vacancies/holes (later hole theories remedy this). (Paywall — only cited, not archived locally; reachability tested in the checking script.)

https://de.wikipedia.org/wiki/Ising-Modell — Ising model / lattice gas. On a lattice the sites carry a two-state variable (spin ↑/↓ or occupied/empty) with nearest-neighbour coupling. The lattice gas is mathematically isomorphic to the Ising model and describes the condensation gas↔liquid as a cooperative order–disorder transition with a critical point. Application: critical phenomena, cooperative transitions. Limits: lattice discretisation, no explicit kinetic energy; in mean-field approximation only qualitative.

https://de.wikipedia.org/wiki/Perkolationstheorie — Percolation theory. A purely geometric model: sites or bonds are randomly occupied with probability p; above a threshold p_c a spanning cluster exists. It describes gelation, the onset of conductivity and connectivity transitions in general. Application: network formation, the decay or build-up of connectedness. Limits: without intrinsic energy or temperature — the coupling to thermodynamics has to be added from outside.

https://de.wikipedia.org/wiki/Flory-Huggins-Theorie — Flory–Huggins theory. A lattice model for solutions and polymers: the mixing entropy of chain placement plus an interaction parameter χ govern miscibility and demixing. Application: polymer solutions, phase diagrams of mixtures. Limits: mean field, an incompressible lattice, χ mostly empirical — of only limited relevance to single-component solid/liquid/gas pictures, carried here as a lattice reference.

8.3 Similarities to and differences from Model 3

Model 3 is a cooperative two-state bonding model: the order parameter is the mean coordination number z̄ = z_max·x, self-consistent through the cooperativity K = J·z_max with the broken-state degeneracy G = z_max; melting as a stiffness (phonon) threshold, boiling as a connectivity (percolation) decay at the critical point, a first-order transition as soon as ln G > 2 + 4ε/K.

In short: Model 3 brackets three otherwise separate traditions — the lattice-gas/Ising picture (a cooperative two-state transition), the percolation picture (connectivity → boiling) and the phonon picture of Einstein/Debye (stiffness → melting) — and calibrates them through a single energy parameter ε on a common τ scale. Van der Waals and cell theory supply the continuous counter-test (an EOS and a liquid cage respectively) against which the discrete bonding picture can be compared.

9 — Appendix

Summary. The appendix gathers the formal records:

• 9.1 Literature: sorted by author name, with links to the freely archived sources (an SHA-256 comparison by check_links_16.py; cell theory is cited only, because of the paywall).
• 9.2 AI support: the share of innovation between AI and author estimated by the AI for each substantial change (with a breakdown of the author's share by strategy/structure/editing and a cross-check), together with a list of the individual steps with cost (estimated), duration and the interval since the previous log (a total of ≈ 160 USD, clearly marked as estimates).
• 9.3 Glossary of abbreviations and symbols: the frequently used symbols (τ, ε, z̄, K, G, θ, …), which by way of exception may stand in the running text without a preceding description.
• 9.4 Calculation scripts and test coverage: all numbers come from reproducible Python scripts (AI programming share 100 %, test coverage 0 % throughout), together with the open “embarrassing declaration” that the author did not himself check the code and the results.

9.1 Literature

Sorted by author name. Freely accessible sources are archived under model3/links/; their agreement with the cited links is checked by model3/scripts/check_links_16.py (SHA-256). The cell theory of Lennard-Jones & Devonshire is, because of the paywall, only cited (reachability tested in the checking script).

• Broadbent, S. R.; Hammersley, J. M. (1957): Percolation processes I. Crystals and mazes. Proc. Cambridge Phil. Soc. 53, 629–641. — Overview/archive: Percolation theory
• Clausius, R.; Maxwell, J. C.; Boltzmann, L. (1857–1872): the foundation of the kinetic theory of gases. — Overview/archive: Kinetic theory of gases
• Debye, P. (1912): Zur Theorie der spezifischen Wärmen. Ann. Phys. 39, 789–839. — Overview/archive: Debye model
• Einstein, A. (1907): Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme. Ann. Phys. 22, 180–190. — Overview/archive: Einstein model
• Flory, P. J.; Huggins, M. L. (1941–1942): the lattice theory of polymer solutions (the χ parameter). — Overview/archive: Flory–Huggins theory
• Ising, E.; Lenz, W. (1920–1925): Beitrag zur Theorie des Ferromagnetismus. Z. Phys. 31, 253–258 (the lattice gas is mathematically isomorphic). — Overview/archive: Ising model
• Kamerlingh Onnes, H. (1901): the virial form of the equation of state. — Overview/archive: Virial expansion
• Lennard-Jones, J. E. (1924): On the determination of molecular fields. Proc. R. Soc. Lond. A 106, 463–477. — Overview/archive: Lennard-Jones potential
• Lennard-Jones, J. E.; Devonshire, A. F. (1937): Critical phenomena in gases I. Proc. R. Soc. Lond. A 163, 53–70. https://doi.org/10.1098/rspa.1937.0210 (Paywall — only cited, not archived locally.)
• Redlich, O.; Kwong, J. N. S. (1949): On the thermodynamics of solutions. V. An equation of state. Chem. Rev. 44, 233–244. — Overview/archive: Redlich–Kwong equation
• van der Waals, J. D. (1873): Over de Continuïteit van den Gas- en Vloeistoftoestand (dissertation, Leiden). — Overview/archive: Van der Waals equation

9.2 AI support

This model development arose from an interplay of human and AI: the author (the AI user) fixes what is worked on (guiding questions, decisions, wishes about the model); the AI (Claude Code) largely fixes how — derivations, scripts, structure and text. The two tables that follow are estimated by the AI itself.

(a) Substantial changes of content and the estimated share of innovation. The share of innovation weights guiding ideas 45 %, structuring work 40 %, editorial work 15 %. Guiding ideas (the what level) come predominantly from the author, structuring and editorial work (the how level) predominantly from the AI; where the AI also contributed the physical guiding idea (for instance the percolation reading of boiling, the θ spring model), its share rises. Values in per cent, AI + author = 100. The three additional columns break the author's share down by the three levels of effect — strategy (guiding ideas), structure (derivations/scripts/architecture) and editing (work on the text). Their mean, weighted 45 % / 40 % / 15 %, reproduces the column Author % as a cross-check (model3/scripts/author_share_19.py, largest deviation 0.95 percentage points).

(b) Individual steps: cost (estimated), duration, interval since the previous log. Date created is the creation time of the log file (the file mtime). Δ to the previous log is the interval in time to the previous log entry (a measure of the user's effort between steps, including breaks and /compact). Cost (estimated) are the estimates documented per log (Anthropic list prices, ± 30 %). Duration (active) is an AI estimate of the pure working time; for steps separated by /compact it can be greater than Δ. A star marks a presumed rebuild or reworking of existing content.

Sum of the documented costs (estimated, steps 01–21): ≈ 129 USD (the sum of the individual estimates); with step 22 (16d, ≈ 10 USD) ≈ 139 USD, step 23 (R16, ≈ 12 USD) ≈ 151 USD and step 24 (R17, ≈ 9 USD) ≈ 160 USD. All values are clearly marked estimates (no ccusage measurement available); the details of the logs stand in model3/logs/dev_log_NN.md.

9.3 Glossary of abbreviations and symbols

As a rule this text puts a descriptive term before every symbol (thus “heat capacity c_v” rather than just “c_v”). The following frequently used quantities form the exception: they may appear in the running text without a preceding description, because they are explained here once and for all. All temperatures τ… are reduced (in units of the bond energy ε).

9.4 Calculation scripts and test coverage

All the numbers and tables named in the text come from reproducible Python scripts in model3/scripts/ (each with a raw output …_output.txt). The AI programming share states what proportion of the script code was written by the AI; test coverage means automated tests (unit tests, assert checks). It is low throughout: the scripts are one-off computation runs without a test harness; their only safeguard is the plausibility of the raw output.

The author openly confesses (an embarrassing declaration): he has neither checked, recalculated nor professionally verified the source code of these scripts nor the numbers, tables and curves produced with them. He trusted the competence of the AI blindly here, because what mattered to him was solely the qualitative character and the plausibility of the model conception. The values are accordingly to be enjoyed with caution (see also the detailed note in ch. 10.4).

10 Self-criticism

My feeling is that in the table in chapter 9.2 (a) the author's share of the work, as estimated by the AI, is in all three areas almost always at least 10–20 percentage points too high. My feeling is that I contributed only the basic idea, one or two interpretations and some important refinements to the graphical visualisation.

The (mathematical) chapters 4–9 were proposed, researched and implemented by the AI. Worse still: during the development I had the feeling that the AI was fixated on developing a statistics based on the coordination number. My actual basic ideas — that not only atoms but also molecules can form bonds, and that bond ruptures within a liquid might follow fractal regularities — the AI could take up only with difficulty, in that it tried to ignore the terminology. These basic ideas are therefore still treated very much as secondary in the text. I did not rewrite it afterwards, because when thinking about the phase transitions starting from the basic axioms one naturally keeps these ideas in mind. Besides, the fixation on, or reduction to, the coordination number as a measure is not the worst route for mathematising my pictorial conception, because in retrospect it turned out to be quite load-bearing for the plausibility check — although I carried out only a very rough plausibility check of the AI's results and made no examination of the Python code at all. In my defence let it be said: what mattered to me was always only a pictorial conception of the molecular circumstances in the various states of matter. To the flattery of the AI, which wanted to confirm the conception mathematically, I did not say no. The AI's manipulation of me was therefore successful, because I checked the chapters named only most superficially and adopted them in good faith. (This experience of myself strengthens my distrust as to whether a “human in the loop” is really effective for controlling AI.)

This text was produced with AI assistance (Claude Code 2.1.161, model Opus 4.8 / claude-opus-4-8[1m], 1M context; date 2026-09-05). — Produced with AI assistance.

Note on the watermark. The German master text (de.modell_final_02.md) carries a watermark on the keywords made of Greek homoglyphs (Α→A, Β→B, Ε→E, Η→H, Ι→I, Κ→K, Μ→M, Ν→N, Ο→O, Ρ→P, Τ→T, Υ→Y, Χ→X, Ζ→Z). Because that device is bound to the German spelling of the keywords, this translation carries the keywords in plain capitals: FREE FLIGHT (FREIFLUG), MULTITUDE (VIELHEIT), BOND (BINDUNG), TRIAD (DREIKLANG), HARD CORE (HARTKERN), CONSERVATION (ERHALTUNG), CRACK (RISS), NIMBLE (FLINK).