Visceral afferent gain (IBS subtypes, functional abdominal pain)

Irritable bowel syndrome rides a new Tier-2 primitive: a visceral afferent gain, the R19 element's susceptibility χ = 1/k. IBS is a §14 motility subtype (the transport bias orders IBS-C → IBS-M → IBS-D by retained fraction) plus a raised afferent gain (visceral hypersensitivity — allodynia, then spontaneous firing past the R19 spinodal). Functional abdominal pain is the same raised gain at normal motility.

The third Tier-2 primitive is a visceral afferent gain: a visceral afferent is the §2 R19 switch ds/dt = g·s − s³ + h resting quiescent, and its static susceptibility to a wall-distension input is the restoring-curvature inverse χ = ds*/dh = 1/k = 1/(3s² − g) — the IDENTICAL quantity the §17 reservoir reads as fundic compliance (one curvature, two readings). IBS is two things at once: a §14 motility subtype (the transport bias orders IBS-C → IBS-M → IBS-D by retained proximal fraction; the diarrhoea-side absolute magnitude hits the §15 conserved-bolus ceiling, an honest [O]) plus a raised afferent gain. A peripheral sensitization bias slides the afferent toward yield, so the gain rises and a fixed normal distension yields a larger signal (allodynia); past the R19 spinodal a normal distension flips the element discontinuously into the firing basin (spontaneous activity). Functional abdominal pain is the same raised gain at normal motility, no structural lesion. The gain diverges exactly at the R19 spinodal — the same marginal saddle-node as the §17 reservoir yield. Subtype ordering and the gain rise / allodynia are [V], the divergence-at-spinodal is the exact [F] identity; the felt pain (mind, behind the firewall) and the absolute rapid-transit magnitude / stool frequency are [O] with stated obstacles.

Irritable bowel syndrome and functional abdominal pain need one element the motility, gate and reservoir modules do not contain: an afferent gain — a peripheral sensitivity on the visceral afferent signal. This is the third Tier-2 primitive, and like the gate and the reservoir it is not new dynamics: a visceral afferent is the §2 R19 switch ds/dt = g·s − s³ + h resting quiescent in its contracted basin, and its static susceptibility to a wall-distension input is the restoring-curvature inverse χ = ds*/dh = 1/k = 1/(3s² − g). A peripheral sensitization bias slides the operating point toward yield, raising the gain; nothing is fitted.

IBS is two things at once: an §14 motility subtype plus a raised afferent gain. The subtype is set by the §14 transport bias — the retained proximal fraction (the §15 stasis reader) falls monotonically as the propulsive drive rises, so one bias parameter orders the subtypes: IBS-C (low drive, retained 0.64 — slow transit, constipation) → IBS-M (intermediate, 0.30) → IBS-D (cleared, 0.00 — rapid transit, diarrhoea).

The constipation side is the §14 slow-transit mechanism [V]; the diarrhoea side hits the same conserved-bolus ceiling as §15 dumping — the travelling occlusion wave cannot express transit faster than normal (the rapid side overshoots by only 0.0%), so the absolute rapid-transit magnitude is an honest [O] while the subtype assignment and the C→D ordering are [V].

The hypersensitivity that, layered on this motility subtype, makes it IBS rather than plain altered transit is a raised afferent gain. A peripheral sensitization bias (inflammation, mediators, peripheral facilitation) slides the quiescent afferent toward yield: the gain χ = 1/k rises from its baseline 0.500 and the afferent signal for a fixed normal wall-distension rises with it — allodynia, the same distension amplified ×1.99. Past the R19 spinodal that same normal distension flips the element discontinuously into the firing basin: spontaneous, un-provoked afferent activity.

Functional abdominal pain is the same raised gain at normal motility, with no structural lesion: at a normal transport bias (where the §14 retained fraction is 0.000, i.e. transit is normal) the afferent-signal proxy still rises with the sensitization bias. That is exactly what distinguishes functional abdominal pain (pure afferent gain) from IBS (afferent gain plus a §14 motility subtype).

The gain is not an independent new quantity: it is exactly the §17 fundic compliance read a second way. The afferent gain χ = 1/k equals the reservoir compliance C = 1/k for every operating point (maximum absolute difference 0e+00), the analytic susceptibility matches the simulated small-signal response (relative error 1.8e-04), and the gain diverges at the R19 spinodal — the same marginal saddle-node as the §17 reservoir yield (k = 1e-16 at s = −√(g/3)). One R19 marginal point, three readings: critical sensory gain here, maximal mechanical compliance at §17, the discontinuous flip of the bare switch.

Treatment (model reading). IBS -- subtype-directed motility correction PLUS lowering the afferent gain. The motility target is the s14 transport bias: IBS-C raises the propulsive drive (prokinetics / secretagogues), IBS-D lowers it (antimotility agents) -- the same knob, opposite sign by subtype. The hypersensitivity target is the afferent gain: neuromodulators lower the sensitization bias, sliding the operating point away from yield so the same distension produces a smaller signal (the model lowers the afferent signal monotonically as the gain is lowered). Functional abdominal pain -- lower the afferent gain alone (central / peripheral neuromodulation), since motility is normal. The brain-gut / felt / affective interpretation stays in `mind` behind the firewall; only the peripheral afferent term is moved here. Target directions [V]; absolute efficacy and the felt symptom mapping [O]. The complementary question — which pharmacological lever raises this afferent's firing threshold, lowering this exact gain χ = 1/k back toward its baseline, and toward which validated drug class the mechanism points — is answered in §28, which inherits a DNA-grounded 27-target non-opioid analgesic map and reads it on this very §18 spinodal (the analgesic firing-threshold scale |h_sp| = spinodal(γ) is identically the gain-divergence point established here).

The IBS motility-subtype ordering and the visceral-hypersensitivity gain rise (with allodynia) are forced by the substrate [V], and the divergence of the gain at the spinodal is the exact R19 identity [F]. What stays open [O], each with its obstacle: the absolute rapid-transit magnitude and stool frequency (the §15 conserved-bolus ceiling, needing the reservoir / pyloric-brake element and clinical transit calibration), and — the firewall — the felt visceral pain and affective experience, which is mind's; this section moves only the peripheral afferent-gain term, never the felt interpretation, and the HPA axis is not re-emerged here. This mind felt-symptom seam is now wired in §27 as a one-way forward-defer pointer (the peripheral afferent signal crosses OUT to mind’s interoception route; nothing returns).