Spatial switch leverage — which focal drive most easily flips the collective state (the second cross-axis coupling, marrying the spatial layer to the state-switching layer)

The second cross-axis coupling, the spatial sibling of the bipolar episode: which focal drive most easily flips the collective state. Driving the bistable cell with each region's frozen-kernel broadcast leverage, the flip-threshold rank equals the leverage rank, barrier-invariant. The new result: switchability decouples from both instantaneous footprint and reach — the reach hub is among the hardest to flip.

The atlas has built two layers and read them separately. The §43 spatial-localisation layer built a fixed spatial map of the frozen ephaptic kernel and handed it forward to be read region by region — §44 (a seizure’s containment vs broadcast), §45 (a stroke’s local deficit vs diaschisis), §46 (a stimulation’s clean delivery vs off-target leak), and §47 (where a focal drive leaves a lasting imprint) — every one a reading of the same map under a focal drive. The §28 state-switching layer built the orthogonal capability: the frozen bistable cell used over time, with its fold, its hysteresis, and its crossing latency — and §29 imported it to ask when a collective mood state flips under a temporal drive. This chapter is the second cross-axis coupling — it is the spatial sibling of the §29 bipolar episode. It imports both the §43 SpatialField (NOT re-derived) and the same §28 BistableSwitch the bipolar chapter used (NOT re-derived) and asks the question neither axis can ask alone: a focal stimulation acts at a place (E1) and the single collective bistable state either flips or holds (E2) — so which region’s focal drive most easily flips the collective state? The only new object is the §28 cell driven by an effective drive read straight off the frozen kernel: the driven node’s broadcast leverage lev_j = Σ_i W₀[i,j], the total one-step ephaptic current that node injects into the rest of the network — a pure readout of the frozen kernel, no new constant. The state starts at the down fixed point and flips up iff the effective drive h_eff(j, b₀) = b₀·lev_j crosses the fold spinodal(g), so the intensity threshold is b₀* = spinodal(g) / lev_j — inverse in leverage. Every sign holds across two sweeps at once: a stimulation-intensity sweep {0.3, 0.5, 0.7, 0.9} and a barrier-depth sweep g ∈ {0.7, 1.0, 1.3}. Four results hold. C1 — the flip-threshold rank is the broadcast-leverage rank, and it is barrier-invariant: the 12 sites rank by how cheaply their focal drive flips the state, and that rank is identical to the leverage rank — the easiest-to-flip hub is the basal-forebrain cholinergic nucleus (lev ≈ 1.79, threshold b₀ ≈ 0.22), monotone in leverage down to a fixed non-switching minority {thalamus, olfactory bulb} (lev ≈ 0.21, 0.22) that never flips at any swept intensity (b₀ ≤ 1) — and the rank is the same across the whole barrier-depth sweep on the common finite-threshold set: a deeper well raises every threshold together but does not reorder which region flips easiest. C2 — ease-of-flip tracks leverage, with critical slowing at the bottom: the threshold is monotone-decreasing in leverage, and at a fixed supra-threshold intensity (b₀ = 0.9) the crossing latency — the time the state takes to flip — is also monotone-decreasing in leverage (the basal forebrain at ~0.72 up to the brainstem at ~5.68); the lowest-leverage site that still flips, the brainstem, shows the largest finite latency — critical slowing, the transition time diverging as the effective drive approaches the fold — the §28 ictal time-course now given a spatial address. C3 — switchability is decoupled from instantaneous footprint and from reach (the genuinely new result, and the honest negative): the §46 instantaneous field-relay set {hippocampus, midbrain} is mid-rank in switch leverage (ranks 5–6 of 12, easily switchable, not special), and the switch hub — the basal-forebrain cholinergic nucleus — is one §46 classes as self-localising (clean delivery); so where a focal drive’s instantaneous effect lands and which focal drive flips the collective state are two distinct, decoupled axes. And the refuted clean hypothesis is reported honestly: one might expect the greatest-reach region — the §43 reach hub — to be the one whose focal drive most easily flips the state, but the reach hub is the cerebellum, which is switch-rank 10 of 12, one of the hardest to flip; the hard-to-switch set {thalamus, olfactory bulb, cerebellum} all deliver cleanly in the instant yet cannot cheaply flip the state. Reaching far, delivering cleanly, and flipping the collective state are three distinct properties — there is no universal reach→switchability law. C4 — a coherent majority-switchable structure with a fixed non-switching minority: a strict majority (10 of 12) can have the collective state flipped by a swept focal drive, with the fixed non-switching minority {thalamus, olfactory bulb} locked out — the honest contrast with §46, where the default was clean delivery (10 of 12 self-localising). A direction-only [L] correspondence is noted, never a magnitude or a prediction: the highest-leverage hubs (the basal-forebrain cholinergic nucleus, the hypothalamus) are the classic global brain-state and arousal control hubs, and the locked-out / hard sites (thalamic relay, olfactory bulb, cerebellum) are peripheral relay and motor-timing structures — direction-consistent with broadcast hubs gating global brain-state switching. Removing the focal drive (b₀ = 0, h_eff = 0) settles the collective state via E.settle bit-for-bit (the static limit is the frozen engine, inheriting the E2.4 guard) and the fold is read from E.spinodal — a pure add-on on the frozen kernel (engine 0fbf4988…, byte-unchanged; both layers reused, not re-derived; no new tuned constant). The firewall is absolute: the collective bistable state and its flip are structural quantities of the coupled model, never a felt state, an experienced arousal, or a level of consciousness (Axis-A — consciousness_claim = 0, the hard problem stays open), and not a real connectome, electric-field or current-density map, a lead-position or specific-absorption-rate map, a prediction of which patient’s stimulation flips which brain state, or device-programming/target-selection guidance. Every magnitude is [O]; efficacy = 0; not medical advice.

The second cross-axis coupling — marrying the spatial map (where) to the state-switching layer (the flip)

The atlas has advanced along two layers that, until §47, were read in isolation. One is the spatial layer. The §43 spatial-localisation layer built a single map of the frozen ephaptic kernel and asked for it to be read region by region, and four chapters did exactly that — §44 drove a region up the coupling curve and read the footprint as a seizure, §45 silenced a region and read the footprint as a stroke, §46 re-read the excitatory footprint as a therapeutic stimulation, and §47 ran a focal drive through plasticity to ask where it leaves a lasting mark. Every one of those reads a focal drive’s effect as a field on the kernel: where does the coordination change land? 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: it established the fold (the drive at which a held state flips), the hysteresis loop, and the crossing latency that diverges at the fold (critical slowing). And §29 imported that cell to read a collective mood state as a bistable system, asking when an episode flips under a temporal drive.

This chapter puts the two layers together for the second time, and it is the spatial sibling of that bipolar episode. The bipolar chapter had the bistable state and asked when it flips; it drove the state with one scalar drive for the whole cell, so it could not ask where a drive would most cheaply flip it — every region was the same scalar h. The spatial layer had the “where,” but it read a focal drive as an instantaneous field — it had no bistable state, so a focal drive could not flip anything; there was nothing to flip. 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 — and asks which region’s focal stimulation most easily flips it. So where §29 asked when the collective state flips under a temporal drive, this chapter asks where a focal drive most cheaply flips it. Neither the spatial layer nor the state-switching layer can ask that alone; both are imported, nothing is re-derived, and the engine is byte-unchanged.

The coupling model — a focal drive’s broadcast leverage flipping the collective state

The model is forced by what a focal drive does to a collective state. A focal stimulation is, exactly as in §46 and §47, a strong excitatory bias at one region while every other region sits at baseline. But the object being driven here is not a per-node field — it is the single collective bistable state of §28. So the question is: how much does a focal drive at one node push the whole state? The answer the frozen kernel already contains is the driven node’s broadcast leverage — the total one-step ephaptic current it injects into the rest of the network, lev_j = Σ_i W₀[i,j], the column sum of the frozen kernel. This is not a new mechanism or a fitted weight: it is a pure readout of the same row-stochastic ~1/r³ kernel §43 froze. A region wired to broadcast hard has high leverage; a peripheral region has low leverage. The effective drive a focal stimulation b₀ at node j applies to the collective state is h_eff(j, b₀) = b₀·lev_j — intensity times leverage, 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 stimulation-intensity threshold at which a focal drive at node j flips the state is b₀* = spinodal(g) / lev_j — inverse in the leverage. Two sweeps gate every result. The stimulation intensity is swept — b₀ = 0.3, 0.5, 0.7, 0.9 — asking whether a sign survives how hard the region is driven. And because the state has a well whose depth can vary, a second axis is swept: the barrier depth 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 intensity×barrier grid, so no number is fitted on either axis. When the focal drive is removed (b₀ = 0, so h_eff = 0) the collective state settles by the frozen relaxation bit-for-bit and the fold is read from the engine — the construction is a pure add-on, deterministic, fitting nothing.

C1 — the flip-threshold rank is the broadcast-leverage rank, and it is barrier-invariant

The first result is the spatial discriminant for a flip. Driving a focal stimulation at each of the 12 regions in turn and asking the minimum intensity at which the collective state flips up, the sites rank by how cheaply their focal drive flips the state. That integrated rank is identical to the broadcast-leverage rank computed straight from the frozen kernel. At the engine’s barrier (g = 1.0) the easiest-to-flip site is the basal-forebrain cholinergic nucleus (lev ≈ 1.79, flipping at b₀ ≈ 0.22), then the hypothalamus (lev ≈ 1.63, b₀ ≈ 0.24), then the forebrain GABA interneuron, neocortex, hippocampus and midbrain clustered near b₀ ≈ 0.28–0.30 (lev ≈ 1.38–1.42), then the pallidum (b₀ ≈ 0.40), the striatum (b₀ ≈ 0.60) and the brainstem (b₀ ≈ 0.74), down to the cerebellum, which only just flips at the strongest swept intensity (b₀ ≈ 1.0, lev ≈ 0.39). Two regions never flip at any swept intensity: the thalamic relay and the olfactory bulb (lev ≈ 0.21, 0.22), whose broadcast leverage is too small for even b₀ = 1 to push the effective drive across the fold. This is the fixed non-switching minority.

The certification is that this rank does not move with the barrier. Sweeping the well depth g over {0.7, 1.0, 1.3} raises or lowers the fold and so shifts every threshold together — a shallower well lets more sites flip, a deeper well fewer — but on the common set of sites that flip at all three depths, the rank order is identical. The reason is structural: the threshold is spinodal(g) / lev_j, so changing g rescales every threshold by the same factor and cannot reorder them. Which focal drive most cheaply flips the collective state is a fixed property of the frozen kernel’s broadcast structure — not of how deep the well is, nor of how hard the region is driven. The leverage column sum is the kernel property that orders switchability, and the integrated flip thresholds reproduce that order exactly.

C2 — ease-of-flip tracks leverage: critical slowing at the bottom

The second result turns the rank of C1 into the dynamics of the flip, and ties “easy to flip” to two monotone facts about leverage. First, the threshold is monotone-decreasing in leverage — the higher a region’s broadcast leverage, the lower the intensity its focal drive needs to cross the fold (the inverse law b₀* = spinodal/lev made into an ordering over the 12 sites). Second, and not contained in the threshold alone, at a fixed supra-threshold intensity (b₀ = 0.9) the crossing latency — the time the collective state takes to actually flip once the drive is held — is also monotone-decreasing in leverage. A high-leverage hub flips the state both more cheaply and more quickly: the basal-forebrain cholinergic nucleus crosses in about 0.72, the hypothalamus in about 0.82, the mid-rank cluster near 1.0, the pallidum about 1.62, the striatum about 3.24, and the brainstem about 5.68. A low-leverage site flips dearly and slowly.

The lowest-leverage site that still flips — the brainstem — shows the largest finite latency, and this is critical slowing: as the effective drive approaches the fold from above, the crossing time diverges (the state lingers near the unstable point before committing). This is exactly the §28 ictal-onset time-course — a latency that shortens as the drive overshoots the fold and diverges at it — now given a spatial address: where a focal drive sits on that latency curve is set by the driven node’s leverage. The fixed non-switching minority never crosses at all, at any time, because its effective drive never reaches the fold. So the spatial layer does not merely say which region flips the state most cheaply (C1); coupled to the state-switching layer it says how fast each region flips it, with the slowest flips concentrated at the lowest-leverage sites near the edge of switchability. Magnitudes of any real transition time are [O]; what is asserted is the order — threshold and latency both decreasing in leverage, with critical slowing at the bottom — robust across the swept barrier.

C3 — switchability is decoupled from footprint and from reach: no universal reach→switchability law

The third result is the one the coupling exists to surface, and the one genuinely new to this chapter — the seam where the spatial story and the switching story come apart, twice over. The natural first expectation, carried from §46, is that the regions whose instantaneous field leaks off-target — the field-relay set — would be the special ones for flipping the collective state. They are not. The §46 instantaneous field-relay set is {hippocampus, midbrain}, and in switch leverage these two sites are mid-rank (ranks 5 and 6 of 12) — easily switchable, but utterly unremarkable, sitting in the middle of the pack. Meanwhile the switch hub — the region whose focal drive flips the state most cheaply — is the basal-forebrain cholinergic nucleus, which §46 classes as self-localising (a clean instantaneous deliverer, not a relay). So the two axes pick out different sites: where a focal drive’s instantaneous effect lands and which focal drive flips the collective state are distinct, decoupled spatial axes — the E1×E2 lesson neither layer could state alone (the spatial layer has no bistable state; the switching layer has no spatial profile).

There is a second, sharper expectation, and it is an honest negative — a clean hypothesis tested and reported false. One might expect that the region with the greatest spatial reach — the §43 reach hub, the site whose focal drive perturbs the global order parameter most — would be the one whose drive most easily flips the collective state. It is not. The §43/§46 reach hub is the cerebellum (the rank-1 reach hub at every depth, read directly from the §43 results), yet the cerebellum is switch-rank 10 of 12 — one of the hardest regions to flip the state from, barely crossing the fold at the strongest swept intensity. And the whole hard-to-switch set {thalamus, olfactory bulb, cerebellum} consists of regions §46 classes as self-localising — clean instantaneous deliverers — that nonetheless cannot cheaply flip the collective state. Reaching far, delivering cleanly, and flipping the collective state turn out to be three distinct properties: a region can have the largest global reach (cerebellum) and yet be among the hardest to switch the state from; a region can deliver perfectly cleanly (the non-switching minority) and yet be locked out of switching entirely. There is no universal reach→switchability law. The refuted clean hypothesis is the finding — the no-tuning discipline working as intended — and it isolates broadcast leverage, not reach and not clean delivery, as the kernel property that orders which focal drive flips the collective state.

C4 — a coherent majority-switchable structure with a fixed non-switching minority

The fourth result steps back and asks what kind of structure the switchability map is, and frames it honestly against its predecessor. A strict majority of sites — ten of twelve — can have the collective state flipped by a swept focal drive (b₀ ≤ 1); the remaining two — the thalamic relay and the olfactory bulb — form a fixed non-switching minority that cannot, at any swept intensity. So the structure is majority-switchable, with a hard peripheral floor: most regions’ focal drive can flip the state, but how cheaply is set by broadcast leverage, and a small peripheral set is locked out entirely. This is the explicit, honest contrast with §46, whose structural default was clean instantaneous delivery (ten of twelve self-localising in the instant). The two chapters are reading different things — §46 asks where a drive’s instantaneous effect lands, this asks which drive flips a bistable collective state — and the structural facts differ accordingly: there, most drives deliver cleanly; here, most drives can flip the state, but only the high-leverage hubs do so cheaply and a peripheral minority not at all.

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. The highest-leverage hubs in the model — the basal-forebrain cholinergic nucleus and the hypothalamus — are, in real neuroscience, the classic global brain-state and arousal control hubs: the ascending arousal systems and the basal forebrain are the structures that gate sleep–wake transitions and set cortical state. And the locked-out / hardest-to-switch sites in the model — the thalamic relay, the olfactory bulb, the cerebellum — are peripheral sensory-relay or motor-timing structures, not state-control hubs. That the regions whose focal drive most easily flips the model’s collective state are the recognised broadcast/arousal hubs, while the peripheral relays cannot, is consistent in direction with the outsized role those hubs play in switching the global brain state — a coherence of direction, nothing more. It is not a claim that this model predicts which individual’s stimulation flips which brain state, nor that broadcast leverage on a frozen kernel is the mechanism of arousal control; those are determinations for real systems neuroscience. The model offers a structural rationale for why a focal drive at a broadcast hub should most cheaply flip a collective bistable state — a hypothesis about mechanism, gated for reproducibility, awaiting external test.

S6 — the engine-invariance guard: removing the focal drive recovers the frozen relaxation

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 inherited from the state-switching layer. When the focal drive is removed — b₀ = 0, so the effective drive h_eff = 0 — the collective state settles by the engine’s own relaxation E.settle bit-for-bit: the static limit of the leverage-driven switch is the frozen engine, exactly the E2.4 guard, and the down state stays down with no spurious flip. The fold is read from E.spinodal — the same constant the engine froze, no new constant introduced. A focal excursion, once the drive is removed, reverts exactly to the down endpoint. Removing the coupling recovers the frozen engine with nothing left over — the strongest possible check that driving the collective state by a node’s broadcast leverage is a read on the kernel, not a modification of it. Both the SpatialField 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 intensity×barrier sweep.

The four results side by side — what the chapter establishes

The chapter is four certified statements about one object — a focal drive (the spatial layer) flipping the single collective bistable state (the state-switching layer) by the driven node’s broadcast leverage. Read across a row to see one result and the discriminant that carries it; read down to see the chapter move from the leverage-ordered threshold rank, through its dynamics, to the honest decoupling of switchability from footprint and reach, and finally to the structural class the switchable sites form.

The table makes the chapter’s logic legible. C1 is the discriminant — a region flips the state cheaply or dearly by its broadcast leverage, across both drive axes. C2 is the dynamics — higher leverage flips the state faster as well as cheaper, with critical slowing at the edge of switchability. C3 is the seam — switchability is not the instantaneous-footprint axis and not the reach axis (the reach hub is among the hardest to flip), so the switching story cannot be read off either spatial reading. C4 is the structural class — most regions can flip the state, the high-leverage broadcast hubs most cheaply, a peripheral minority not at all. Together they are the second reading of a seam between layers: which focal drive most cheaply flips a collective bistable state is set by broadcast leverage on the frozen kernel, and that ordering is independent of where the drive’s instantaneous effect lands and of how far it reaches.

What the chapter does not claim — the firewall

This is a chapter about brain stimulation and global brain-state switching, and the boundary of what it asserts must be stated without hedging. Every quantity certified here is a structural quantity — a flip threshold, a broadcast-leverage rank, a crossing latency, a switchable/non-switchable class, all of them properties of how a coupled model flips a single bistable collective state under a focal drive read off a fixed kernel — and none of them is a claim about a felt state. That the collective state “flips” is a statement about a bistable variable in the model crossing zero; it is not a statement about an experienced change of arousal, a level of consciousness, a state of mind, or a felt transition, and it is certainly not a claim that experience switches 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 a state-flip a spatial address in the model does not give an experienced brain-state change one.

The boundary to biological and clinical reality is firmer still. The per-node excitatory bias is not a real stimulation, the broadcast leverage lev_j is not a real measure of how a region drives brain state, the single collective bistable state is not a real global brain state, and the switchable/non-switchable class is not a prediction of which patient’s stimulation flips which state. Real brain-state switching is a distributed neuromodulatory system — the ascending arousal nuclei, the basal forebrain, thalamocortical and corticocortical loops, multiple neurotransmitters, state-dependent conductances — and this module asserts only the sign and order of which focal-drive address most cheaply flips a single bistable state when the effective drive is the frozen kernel’s broadcast leverage, not that any real state control follows this exact rule. Target selection, lead placement, contact and current programming, and outcome prediction are external clinical determinations, made by clinicians with real imaging, electrophysiology and individualised modelling; nothing here is localisation, device programming, target optimisation, or treatment guidance, and the [L] correspondence in C4 is a cited resemblance of direction (that broadcast/arousal hubs gate global brain-state switching), never a patient-level claim. Every magnitude is [O]: representative reads over a swept intensity 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 SpatialField and the BistableSwitch are imported, the engine is byte-unchanged, and the only new object is the leverage-driven switch that couples them. Nothing here is a cure, a treatment, a diagnosis, a localisation, a device setting, or a recommendation. efficacy = 0; this is not medical advice. What the chapter offers is one structural thing: in this model, which focal drive most cheaply flips a collective bistable state is set by broadcast leverage on the frozen kernel — not by where the drive lands and not by how far it reaches — so the spatial story of state switching, like every disease on these layers, must be told site by site, and switchability cannot be read off the field or the reach.