Kindling — do repeated state-flips become easier? (the third cross-axis coupling, marrying the plasticity layer to the state-switching layer)

The third cross-axis coupling, closing the trace-to-threshold link the bipolar episode left open: do repeated state-flips become easier? Driving the bistable cell through the evolving plastic connectome, repeated flips deepen the trace and lower the flip threshold (kindling), barrier-invariant. The honest finding: kindling is consolidative, not degradative — coordination rises, it does not erode.

The atlas has built two temporal layers and read them apart. The §26 plasticity layer added a slow phase-correlation Hebbian update to the frozen ephaptic kernel, so the connectome can evolve and retain a structural trace; the §28 state-switching layer took the engine’s frozen bistable cell — the cubic ds/dt = g·s − s³ + h — and used it over time, establishing the fold, the hysteresis loop, and the crossing latency. The §29 bipolar episode imported the bistable cell and, in its kindling result, showed that alternating episodes deepen the connectome trace ‖dW‖, and separately noted that a lower barrier flips more cheaply — but it left the actual trace→threshold coupling, whether and how the accumulated trace lowers the threshold, explicitly open. This chapter is the third cross-axis coupling, and it is the module that closes that open link. It imports both the §26 PlasticConnectome (not re-derived) and the same §28 BistableSwitch the bipolar chapter used (not re-derived) and asks the question neither layer can pose alone: a plastic connectome has no bistable flip, and a bistable flip on a frozen connectome cannot accumulate — so do repeated flips of the single collective bistable state become easier over time? The only new object is the §28 cell driven through the §26 connectome: each flip episode is a sustained excursion that drives the phase-Hebbian update, so the connectome consolidates a retained trace, and the effective drive of an external push p on the collective state is scaled by the connectome’s coordination gain L(W) = R(W)/R_anchor — the order parameter of the evolving connectome over the frozen M9 anchor R(W₀) = 0.3896, a pure readout that is 1 at the frozen kernel and generalises the §48 broadcast leverage to the global coherence gain. The state flips up iff h_eff = p·L(W) crosses the fold spinodal(g), so the flip threshold in external push is p* = spinodal(g) / L(W) — and as the connectome consolidates and L grows, p* drops. Every sign holds across two sweeps at once: a plasticity-rate sweep η ∈ {0.03, 0.05, 0.08} and a barrier-depth sweep g ∈ {0.7, 1.0, 1.3}. Four results hold. K1 — repeated flips lower the flip threshold, and it is barrier-invariant: the retained trace ‖dW‖ deepens strictly monotonically (0 → 0.06 → … → 0.43 over eight episodes), and the flip threshold ends below the un-kindled fold (at g = 1.0, from 0.3849 down to 0.3783) — repeated flips become easier, and the kindling holds at every well depth because p* = spinodal(g)/L scales the fold by the same gain. This is the trace→threshold coupling the bipolar episode named but left open. K2 — a learned change of hysteresis, and a faster onset: the push-space hysteresis loop narrows (width 2·spinodal(g)/L, 0.770 → 0.757) and the crossing latency shortens at a fixed push (4.96 → 4.72). K3 — kindling is consolidative, not degradative (the genuinely new result, and the honest negative): the clean hypothesis that kindling erodes coordination is refuted — the threshold drops while the coordination R(W) rises (0.3896 → 0.3964) and the kindled connectome ends at or above the anchor, so easier-to-flip and erosion are decoupled: the threshold falls through consolidation, not degradation. K4 — a single coherent consolidative seam: every flip leaves a retained trace (E0) that lowers the next flip’s threshold (E2), and the threshold is monotone-decreasing in the accumulated trace, so the trace itself is the kindling variable. A direction-only [L] correspondence is noted, never a magnitude or a prediction: clinical kindling (Goddard) and bipolar cycle acceleration are the recognised phenomena in which repetition lowers the next transition’s threshold. With η = 0 the connectome stays frozen, R equals the frozen anchor bit-for-bit so the threshold is the un-kindled fold; with a zero push the collective state settles by the frozen relaxation bit-for-bit — a pure add-on (engine 0fbf4988…, byte-unchanged; both layers reused, not re-derived; no new tuned constant, no new rule). The firewall is absolute: the collective bistable state, its flip threshold, the retained trace and the coordination gain are structural quantities of the coupled model, never a felt state, an experienced mood, a level of consciousness, or an experienced ease of relapse (Axis-A — consciousness_claim = 0, the hard problem stays open), and not a real connectome, a synaptic-weight matrix, a measure of kindling or seizure threshold, or a prediction of whether any patient’s episodes will accelerate. Every magnitude is [O]; efficacy = 0; not medical advice.

The third cross-axis coupling — marrying the evolving connectome (plasticity) to the bistable flip (state switching), and closing the bipolar episode’s open link

The atlas has advanced along two temporal layers that, until this chapter, were read in isolation. One is plasticity. The §26 plasticity layer added the variable the structural atlas never had: a slow, activity-dependent, phase-correlation Hebbian update to the connectivity, W ← max(0, W·(1 + η·C)) row-renormalised, so a driven network consolidates a retained structural trace ‖W − W₀‖ the frozen engine could not represent. The other layer is state switching. The §28 state-switching layer took the engine’s frozen bistable cell — the same cubic ds/dt = g·s − s³ + h the engine froze — and used it over time, establishing the fold (the drive at which a held state flips), the hysteresis loop whose width is twice the fold, and the crossing latency that diverges at the fold (critical slowing). And the §29 bipolar episode imported that cell to read a collective mood state as a bistable system — and in its kindling result it showed that alternating episodes, driven through the §26 plasticity, deepen the connectome trace ‖dW‖, and it noted separately that a lower barrier lowers the fold and so flips more cheaply.

But the bipolar chapter stopped at a conditional. It said: episodes deepen the trace, and a lower barrier flips more cheaply, so — if accumulated episodes lower the barrier — cycling would accelerate. The crucial link, whether and how the accumulated trace actually lowers the threshold, it left explicitly open [O]: the magnitude of the trace→barrier coupling was owed to a later module. This chapter is that module. It puts the two layers together for the third time, and the question it asks is precisely the one the bipolar conditional could not close: do repeated flips of the collective bistable state make the next flip easier? Neither layer can ask it alone. The §26 plastic connectome has a retained trace, but it has no bistable state — there is nothing to flip, so “does the flip get easier” is not a question it can pose. The §28 bistable cell has the flip and its fold, but its connectome is frozen — nothing it does changes the threshold, so repetition cannot accumulate. The coupling this chapter makes is the smallest one that joins them: it keeps the single collective bistable state — the very same BistableSwitch object the bipolar chapter used — lets repeated flips drive the §26 plasticity so the connectome evolves, and reads how the evolving connectome changes the flip threshold. Both layers are imported, nothing is re-derived, and the engine is byte-unchanged.

The coupling model — repeated flips through the plastic connectome, the coordination gain lowering the threshold

The model is forced by what couples a plastic connectome to a bistable flip. A flip episode is a sustained excursion of the collective state — modelled, exactly as the bipolar episode modelled an episode, as a strong excitatory (or, on the return, inhibitory) bias driven through the network — and because the §26 plasticity is running, each episode consolidates: the phase-correlation Hebbian update writes a retained structural trace ‖dW‖ into the connectome. Repeated flips (alternating up and down, the repeated transitions) accumulate that trace. So far this is just the §26 layer driven by the §28 episode — the §29 trace-deepening. The coupling is in how the evolved connectome feeds back on the flip.

The object being flipped is the single collective bistable state of §28, so the question is: how much does an external push p on that collective state actually move it, given the connectome that the repeated flips have built? The answer the frozen framework already contains is the connectome’s coordination gain. The §48 coupling scaled a focal drive by a node’s broadcast leverage — the column sum of the kernel, how much that one node injects into the whole. The global generalisation, for a push that acts on the collective coordination mode, is the network’s overall coherence gain: the order parameter R(W) at the measured coupling, the same order parameter the engine froze, normalised by the frozen M9 anchor R(W₀) = 0.3896. We write it L(W) = R(W)/R_anchor — a pure readout of the evolving connectome, not a fitted weight, and L = 1 exactly at W = W₀ (the un-kindled limit recovers the engine). A more strongly coordinated connectome couples a coherent push to the collective mode with more gain; a less coordinated one with less. The effective drive a push p applies to the collective state is h_eff(p) = p·L(W) — push times coordination gain, with no new constant.

The collective state is the §28 cell started at its down fixed point s₀ = −√g. Under the held effective drive it relaxes, and it flips up iff h_eff crosses the fold spinodal(g) — the same fold (spinodal(1.0) ≈ 0.3849) as the M11 ignition threshold, the theta-cap, the epilepsy over-sync pole and §28 itself. So the flip threshold in external push is p* = spinodal(g) / L(W) — and as the connectome consolidates and L grows, the threshold drops: a push that did not flip the un-kindled state flips the kindled one. Two sweeps gate every result. The plasticity rate is swept — η = 0.03, 0.05, 0.08 — asking whether a sign survives how fast the connectome learns. And the barrier depth is swept — g = 0.7, 1.0, 1.3 (with g = 1.0 the engine’s universal R19 scale) — asking whether a sign survives how deep the well is. Every sign below is required to hold across the entire rate×barrier grid, so no number is fitted on either axis. When the plasticity is switched off (η = 0) the connectome stays frozen and the threshold stays the un-kindled fold bit-for-bit; when the push is removed (p = 0) the collective state settles by the frozen relaxation bit-for-bit — the construction is a pure add-on, deterministic, fitting nothing.

K1 — repeated flips lower the flip threshold, and it is barrier-invariant

The first result is the discriminant that closes the bipolar conditional. Driving repeated flip episodes through the plastic connectome and tracking two things after each one, the retained trace ‖dW‖ deepens strictly monotonically — 0 → 0.059 → 0.112 → 0.168 → 0.221 → 0.275 → 0.327 → 0.376 → 0.425 over eight episodes — which is the §26 consolidation signal and exactly the §29 trace-deepening. And the collective state’s flip threshold in external push, p* = spinodal(g)/L(W), ends below the un-kindled fold: at the engine’s barrier g = 1.0, where the un-kindled fold is 0.3849, the kindled threshold falls episode by episode to 0.3783 — a net drop of about 0.0066. Repeated flips become easier. This is kindling, on the same bistable cell the bipolar chapter used.

The certification has two parts. First, the kindling (the net threshold-decrease) holds across the entire rate sweep — faster or slower learning, the threshold ends below the un-kindled fold. Second, and the sharper claim, it is barrier-invariant: at every well depth the kindled threshold ends below that depth’s own un-kindled fold. At g = 0.7 the fold is 0.2254 and the kindled threshold ends at 0.2216; at g = 1.3 the fold is 0.5705 and the kindled threshold ends at 0.5608. The reason is structural: the threshold is spinodal(g)/L, so consolidation rescales the fold by the same coordination gain at every depth — a deeper well needs a larger drive, but the kindling lowers the requirement by the same factor everywhere. Whether repeated flips lower the threshold is a property of the consolidating connectome, not of how deep the well is. This is the trace→threshold coupling the bipolar episode named but left open: the accumulated trace lowers the threshold, and now we can say so — with the sign certified, the magnitude graded [O], across both sweeps.

K2 — a learned change of hysteresis, and a faster onset

The second result turns the kindling of K1 into the dynamics of the flip, and ties it to two consequences for the §28 hysteresis. The §28 layer established that the inter-state interval is bistable, with a hysteresis loop whose width is twice the fold — where the state sits depends on where it came from. Here that loop is reshaped by repetition. In push space the up- and down-transitions sit at ±spinodal(g)/L, so the loop width is 2·spinodal(g)/L — and as the connectome consolidates and L grows, the loop narrows: from 0.770 at the un-kindled connectome down to 0.757 after eight episodes. The bistable inter-state interval the §28 hysteresis carved out is not fixed; repeated flips learn a narrower one. This is a learned change of the state-switching hysteresis — the temporal seam where the plasticity layer (which carries a learned trace) and the state-switching layer (which carries a hysteresis) genuinely meet.

The second consequence is in the crossing time. The §28 layer established that past the fold the state crosses in finite time, with a latency that shortens as the drive overshoots the fold. Here, at a fixed supra-threshold push, the crossing latency shortens as kindling proceeds — from about 4.96 at the un-kindled connectome to about 4.72 after eight episodes. The mechanism is exactly the §28 one, now driven by consolidation rather than by a larger external push: as L grows, the same push p produces a larger effective drive p·L, which overshoots the fold further, so the onset is faster. The §28 ictal-onset time-course is shortened by repetition: a kindled state not only flips more cheaply (K1) but flips more quickly at a held drive. Both facts — the narrowed loop and the shortened latency — are net-monotone with episode count across the rate sweep; their magnitudes are [O], only their direction is asserted.

K3 — kindling is consolidative, not degradative: the genuinely new coupling result, and the honest negative

The third result is the one the coupling exists to surface, and the one genuinely new to this chapter — and it is an honest negative, a clean hypothesis tested and reported false. The intuitive story of kindling is a story of erosion: repeated violent flips wear the network down, degrade its stability, push it toward incoherence, and that is why the next flip is easier — a more-kindled connectome would be a less-coordinated one, a worn track down which the state slides more readily. It is a tidy picture, and the coupling refutes it. As repeated flips lower the threshold (K1), the connectome’s coordination R(W) at the measured coupling does not fall toward incoherence — it rises, from the frozen anchor 0.3896 up to 0.3964 over eight episodes, and the kindled connectome ends at or above the anchor. The connectome that flips more easily is more strongly coordinated, not less. (Honestly disclosed: at the very lowest swept rate, the first single excursion nudges the coordination a negligible amount below the anchor before consolidation takes over and it rises net; at the higher rates it never dips. The refutation is the net sign — the kindled connectome ends more coordinated — not a step-by-step claim.)

So easier-to-flip and degraded-coordination, which the intuitive picture welds together, are in fact decoupled. The threshold falls through consolidation, not through erosion: the §26 phase-Hebbian update writes the repeated transition into the connectome’s structure, strengthening the very coordination mode the external push acts on, so the same push couples to the collective state with more gain and crosses the fold more cheaply. The network does not learn to fall apart; it learns the transition, and learning it makes the transition easier. This is why the result needs the coupling to state it: the §26 layer alone shows consolidation (coordination rises with a retained trace) but has no flip to make easier; the §28 layer alone shows a flip but a frozen connectome whose coordination cannot change; only the two together can show that the same consolidation that raises coordination also lowers the flip threshold. The clean “kindling erodes stability” story is not forced onto the data — it is tested and reported false, the no-tuning discipline working as intended, and what it isolates is that the kindling mechanism here is constructive: a consolidative trace, not a degradative one.

K4 — a single coherent consolidative seam: the trace as the kindling variable

The fourth result steps back and asks what the three preceding facts are, together, and frames the coupling’s structural character. They are one phenomenon, not three. Every flip episode leaves a retained structural trace (the §26 plasticity), and that retained trace lowers the next flip’s threshold (through the §28 fold) — the kindling (K1), the narrowed hysteresis and faster onset (K2), and the consolidative-not-degradative character (K3) are all expressions of the same accumulating trace. The clean way to see this is to plot the threshold not against episode count but against the accumulated trace ‖dW‖ itself: the threshold is monotone-decreasing in the trace. The trace is the kindling variable. This is the explicit form of the trace→threshold law the bipolar episode (B3) named but left open: it is not merely that repetition lowers the threshold, but that the threshold is a (decreasing) function of the structural trace the repetition writes — the deeper the trace, the lower the threshold — and the whole seam is consolidative. Across the rate sweep the three signs co-hold (trace strictly up, threshold net down, coordination net up), so the seam is a single coherent object.

There is a direction-only correspondence to neuroscience worth naming — carefully, and graded [L], because it is a cited resemblance and not a derived or predicted quantity. Kindling is a real and named phenomenon: in Goddard’s classic electrical kindling, repeated sub-threshold stimulation progressively lowers the seizure threshold until a stimulus that once did nothing reliably triggers a seizure — repetition makes the transition easier. And in mood disorders, cycle acceleration — episodes recurring more readily, and on lesser provocation, over the course of an illness — is a recognised clinical pattern, and the kindling analogy for it is long-standing. That the model’s repeated flips lower the next flip’s threshold through a retained trace is consistent in direction with these phenomena, in which repetition lowers the threshold for the next transition — a coherence of direction, nothing more. It is not a claim that this model predicts whether any individual’s episodes will accelerate, nor that a phase-Hebbian coordination gain is the mechanism of clinical kindling; those are determinations for real systems and clinical neuroscience. The model offers a structural rationale for why repeated flips of a bistable collective state should lower its own threshold — and the perhaps surprising structural fact that they do so by consolidating coordination rather than eroding it — a hypothesis about mechanism, gated for reproducibility, awaiting external test.

S5 — the engine-invariance guard: removing the plasticity, and removing the push, both recover the frozen engine

As with every module in the series, the coupling is certified to be a pure structural read that adds nothing to the engine, and here the guard is doubly inherited — once from each layer. From the plasticity layer (E0): when the plasticity is switched off (η = 0), the connectome stays frozen through every episode, the order parameter R returns to the frozen M9 anchor 0.3896145516 bit-for-bit, so the coordination gain L = 1 exactly and the flip threshold is the un-kindled fold spinodal(g) exactly — with no plasticity there is no kindling, which is precisely why the bipolar chapter’s instant levers could not reach it: the kindling lives on the plasticity axis. From the state-switching layer (E2): when the push is removed (p = 0, so the effective drive h_eff = 0), the collective state settles by the engine’s own relaxation E.settle bit-for-bit, the down state stays down with no spurious flip, and the fold is read from E.spinodal — the same constant the engine froze, no new constant introduced.

Both limits recover the frozen engine with nothing left over — the strongest possible check that driving the collective bistable state through an evolving phase-Hebbian connectome is a read on two frozen layers, not a modification of either. Both the PlasticConnectome and the BistableSwitch are imported and the engine emerged read-only, confirmed byte-unchanged against the frozen tree hash (0fbf4988fc83…) with the M0–16 subtree identical. There is no new mechanism, no new measurement, and no new tuned constant: this chapter couples two frozen layers and reads four signs off the seam, all of which survive the rate×barrier sweep.

The four results side by side — what the chapter establishes

The chapter is four certified statements about one object — repeated flips of the single collective bistable state (the state-switching layer) driving an evolving phase-Hebbian connectome (the plasticity layer), whose consolidation lowers the flip threshold. Read across a row to see one result and the discriminant that carries it; read down to see the chapter move from the kindling itself, through its dynamics, to the honest decoupling of kindling from erosion, and finally to the structural class: the trace as the kindling variable.

The table makes the chapter’s logic legible. K1 is the discriminant — repeated flips lower the threshold, across both axes, closing the bipolar episode’s open link. K2 is the dynamics — the hysteresis loop narrows and the onset quickens as kindling proceeds. K3 is the seam, and the surprise — the kindling is not erosion; the connectome that flips more easily is more coordinated, not less, so easier-to-flip and degraded-stability are decoupled. K4 is the structural class — one consolidative phenomenon, the threshold a decreasing function of the accumulated trace, the trace the kindling variable. Together they are the third reading of a seam between layers: repeated flips of a bistable collective state lower its own threshold by consolidating the connectome, the deeper the trace the lower the threshold, and the easing comes from strengthened coordination rather than from erosion.

What the chapter does not claim — the firewall

This is a chapter about kindling, mood-episode recurrence and seizure thresholds, and the boundary of what it asserts must be stated without hedging. Every quantity certified here is a structural quantity — a flip threshold, a retained structural trace, a coordination gain, a hysteresis-loop width, a crossing latency, all of them properties of how a coupled model lowers the threshold of a single bistable collective state as repeated flips drive a phase-Hebbian connectome — and none of them is a claim about a felt state. That the collective state “flips” more easily over repetition is a statement about a bistable variable in the model crossing zero at a lower drive; it is not a statement about an experienced mood, a felt episode, a level of consciousness, or an experienced ease of relapse, and it is certainly not a claim that experience kindles when this variable does. This is the Axis-A firewall, held exactly as in every chapter of the series: consciousness_claim = 0, and the hard problem of experience stays open. Giving kindling a structural mechanism in the model does not give experienced recurrence one.

The boundary to biological and clinical reality is firmer still. The flip episode is not a real mood episode or a real seizure, the retained trace ‖dW‖ is not a real synaptic-weight change, the coordination gain L(W) is not a real measure of how a brain’s coordination scales a drive, and the flip threshold is not a real seizure or episode threshold. Real kindling and real cycle acceleration are heterogeneous processes — altered gene expression, mossy-fibre sprouting, receptor trafficking, network reorganisation, allostatic load — and this module asserts only the sign and direction of whether repeated flips lower the collective flip threshold when the connectome evolves by the imported phase-Hebbian rule and the effective drive is scaled by the imported coordination gain, not that any real kindling follows this exact rule. Whether any individual’s episodes will accelerate, and by how much, and what (if anything) should be done about it, are external clinical determinations, made by clinicians with real histories, electrophysiology and individualised assessment; nothing here is a prognosis, a treatment, a localisation, or a device setting, and the [L] correspondence in K4 is a cited resemblance of direction (that clinical kindling and cycle acceleration are phenomena in which repetition lowers the next transition’s threshold), never a patient-level claim. Every magnitude is [O]: representative reads over a swept rate and barrier, never quantities fitted to a target. There is no new mechanism, no new measurement beyond the read-only layers, and no new tuned constant in this chapter; the PlasticConnectome and the BistableSwitch are imported, the engine is byte-unchanged, and the only new object is the kindling switch that couples them. Nothing here is a cure, a treatment, a diagnosis, a prognosis, a device setting, or a recommendation. efficacy = 0; this is not medical advice. What the chapter offers is one structural thing: in this model, repeated flips of a bistable collective state lower its own threshold — the deeper the trace the lower the threshold — and they do so by consolidating the connectome rather than eroding it, so the trace→threshold link the bipolar episode left open is supplied with its sign, and the kindling story, like every disease on these layers, turns out to be one of structure being written, not worn away.