The g-series

The stages of one phase change, the residue it leaves behind, and the new level that residue seeds.

Layer 1 · the stages of one phase change

The shape underneath

Take the potential W(q) = q⁴/4 − 3q²/2 + a·q, with a a slowly drifting control. Its resting states satisfy q³ − 3q = −a. Plotted against the control, they form an S:

• Folds at (a, q) = (2, 1) and (−2, −1), where the stable and unstable branches meet.
• The jump. Raise a past 2 and the upper branch ends. The only resting states at a = 2 are q = 1 (the fold) and q = −2, because q³ − 3q + 2 = (q − 1)²(q + 2). The system lands at q = −2, which is stable.
• The residue. Lower a again and the system stays on the lower branch until a = −2. For |a| < 2 both stable states exist, and which one the system occupies depends on its history.

The curve q³ − 3q is the V(q) of Principia Orthogona Vol. I. Its facts V′(1) = 0, V″(1) = 6, V(1) = −2 are exactly the fold conditions at (1, 2), once V is read as this equilibrium curve rather than as the energy.

The same fold, in time: a market

The S-curve is the fold as a shape. A market meets it as a path in time. Let the state (think log price) push back on the control: while the price is low the control drifts one way, while it is high it drifts back. E. C. Zeeman used the cusp this way for stock exchanges in 1974. Simulated:

• g⁶ accumulation. The price climbs slowly along the lower branch.
• g³³ accelerating — the parabolic phase. Near a fold the branch steepens by itself: the distance to the fold point shrinks like the square root of the distance in the control, so the price moves faster and faster with no extra push. The softening shows as larger, slower wobbles.
• g⁶⁴ blow-off. The branch ends and the price goes vertical to the upper branch.
• Crash. The top is short. The control reverses, the upper branch erodes, and its own fold is the crash.
• Residue. The price lands on a floor. In this simulation the floors rise only because a slow trend was added by hand; the fold alone would return to the same floor each cycle.

Whether Bitcoin, or any market, follows this model is an empirical question. Fitting it needs that market's own data.

The same fold, as a shape: geology

In rock the fold can be seen. A layer squeezed from the sides buckles. For a perfectly straight layer that is Euler buckling; any imperfection in the layer turns it into exactly this cusp: past a critical load the layer snaps into its bent state, and easing the load does not straighten it. Drawn as a cross-section, the overturned S above has the shape of a recumbent fold. Rock keeps its folds for a further reason — it flows under long stress and does not spring back — so there the residue is permanent deformation, recorded in the strata.

Layer 2 · levels

The residue of one passage is the seed of the next: the system now runs at a new stable point, often at a new scale or in a new domain, and the stages g⁰ … g⁶⁴ can run again there. Levels are numbered separately — level 1, level 2, level 3 — so that a g-label always names a stage, never a level. The numbers 12, 24, 48 and 96, used as rungs in earlier chapters, are retired as g-indices; those rungs are levels.

That one fold's landing seeds a further fold is a modelling choice made for each system. It is not a theorem.

Translation and reintegration

The idea behind the whole series is about systems and reintegration. A system meets a threshold, changes state, and reintegrates at a new stable point, carrying the change with it into the next level, often at a new scale or in a new domain. Urie Bronfenbrenner described people in the same way, inside nested systems (family and classroom, the links between them, the wider community and culture, and change over time), where a change at one level carries into the others.

The series works as translational research in two senses. In the sense used by the Bronfenbrenner Center for Translational Research at Cornell, research and practice inform each other in a continuing loop, and every finding carried into practice is evaluated there to see whether it holds. The chapters also translate between fields: a scientist who knows a transition in one field is shown how it reads in another. What carries across is the shape of the fold, its exponents and the operators. The constants (where the fold sits, in what units, how fast) and the mechanism do not carry across.

The same move works for vocabulary. Book 8 names the fourth face of its topology–topography–holography grid holology, the logic by which a whole is coherent as a whole, and uses the word as a translation slot: a mathematician finds topos theory there, a psychologist finds Gestalt, a neuroscientist finds the binding problem (Book 8 · Holology).

Why it recurs. The fold is the generic way a stable state is lost when one control drifts, so different fields meet it independently. Near the fold every system reduces to the same normal form, and that form has no preferred scale: rescaling space by b, the control by b² and time by 1/b gives the same equation back (FoldScaling.lean). The exponents are therefore shared: resting states sit √d from the fold at distance d, and recovery slows like 1/√d.

The evaluation step. Each operator comes with a test that can be run on a field's own data:

Each field then carries a status: established (its own literature shows the fold), model (a fold model exists, evidence mixed), or analogy (the shape only, untested). A field moves up only on its own data or a derivation, never on resemblance alone.

What is proved, and what is not

The earlier formal model

The g-series was first formalised in GCTC.Operators.Chain (GTCT repository). That file is superseded. Its g33_stability_index holds by definition; in poincare_collatz_contracting the 33 plays no role (the theorem holds for any threshold); and two of its axioms are false — GSeries.lean refutes poincare_collatz with a counterexample and inner_basin_is_asymmetric outright. Because a false axiom makes every statement derivable, results from that file should not be cited as proofs.

Other versions, and where they now sit