Clonal Selection as a Saddle-Node Threshold

Clonal selection is a saddle-node crossing of the R19 switch. Rather than assume the threshold, this volume simulates a rising antigen-affinity signal under noise and measures it: the commit-drive equals each organ's spinodal |h|=2(γ/3)^1.5 and orders by γ, while sub-spinodal drive stays tolerant. Immunodominance, maturation, imprinting, tolerance and exhaustion follow from the same shared-pool competition. Grade [V].

The activation threshold is measured, not posited: a rising-affinity simulation commits at the spinodal organ-by-organ (marrow ratio 0.999×, lymphoid 0.996×, approached from below by thermal activation), and a sub-spinodal drive stays tolerant (P(commit)=0.010). Immunodominance emerges from competition for a shared antigen pool: a subdominant clone that commits with probability 1.00 alone is competitively excluded to 0.22 while the dominant clone commits first (P=0.89).

The switch, not a dial

Clonal selection is discontinuous by construction. A clone is a single R19 switch whose drive is set by antigen affinity; the saddle-node at the spinodal means there is no graded half-activated clone, only a committed crossing.

Self-tolerance is the same statement read backwards: any clone whose affinity drive stays below the spinodal never leaves the OFF basin, so it cannot be activated against self. No separate tolerance rule is needed — it is the sub-spinodal region of the one switch.

The activation threshold is measured, not assumed

Rather than read the threshold off the closed-form spinodal, the volume simulates it. A resting clone is driven by a slowly rising antigen-affinity signal under independent cellular noise, and the drive at which it first crosses the ridge is measured. The commit-drive that comes out equals each organ's own spinodal (commit-drive ÷ spinodal = 0.999, 1.000, 1.001, 0.996 for marrow, spleen, thymus, lymphoid), so the saddle-node threshold is confirmed dynamically. It is approached from just below the spinodal because thermal activation lets a clone cross while the shrinking barrier still exists; the ratio rises toward 1 as the noise falls.

The measured thresholds order by γ — a higher-γ (deeper-well) compartment demands a stronger affinity signal to activate. And the bracket straddles the spinodal: a drive clearly below it essentially never commits however long it is held (self-tolerance, P(commit)=0.010 on marrow), while a drive clearly above it commits with probability 1.000. Only the absolute noise scale D is left [O].

Immunodominance emerges from clonal competition

A real response is not one clone in isolation but many clones of the adaptive compartment racing for one finite antigen pool. The volume simulates two clones of the same γ, differing only in affinity, sharing one pool that committed clones deplete. Immunodominance is then measured, not imposed: the higher-affinity clone reaches its spinodal first, commits, and consumes the shared antigen, which starves the lower-affinity clone. A subdominant clone that would commit with probability 1.00 on its own is competitively excluded to 0.22, and the dominant clone commits first with probability 0.89.

The hierarchy sharpens monotonically with the affinity gap and — honestly — vanishes when the gap does: at equal affinity the contest is a symmetric coin-flip (P(first)≈0.47), with no spurious winner. The depth of the hierarchy, set by the consumption rate and the noise scale, is [O]; its existence, direction, and monotonicity are measured.

A repertoire concentrates on a few clones

Scaling the two-clone contest to a full repertoire — N clones of the adaptive compartment with a spread of affinities sharing one pool — turns dominance into a measured concentration. With N=12 clones the response collapses onto an effective 5.8 clones (the participation ratio of the measured commit probabilities): the high-affinity clones capture it and the rest are competitively excluded. Widening the affinity spread sharpens the focus — the effective number falls from 11.8 at equal affinity to 6.2 at the widest spread — and a larger repertoire concentrates further still relative to its size, the shared pool sustaining only a few committers. At equal affinity the response stays evenly shared, with no spurious winner. The absolute depth is [O]; the concentration and its scaling with spread and repertoire size are measured.

Affinity maturation matures the repertoire round over round

A germinal centre is the repertoire contest run as a Darwinian loop. Each round the higher-affinity clones commit first and deplete the shared pool — excluding the slow clones exactly as immunodominance does — and the committed parents are re-seeded with symmetric somatic hypermutation before the next round. Nothing about “improvement” is imposed: mutation is an unbiased random walk and selection is only the competition outcome. Yet the measured mean affinity of the responding set rises round over round, climbing from 0.857 to 1.088 over 7 rounds (gain 0.231).

The rise is causally dissected, not assumed. It needs mutation: at zero hypermutation the gain stalls (0.0262), and it accelerates as the mutation step grows (gain 0.066 → 0.211 → 0.403). It needs competition: an effectively unlimited pool removes the selection pressure and the affinity drifts neutrally (gain -0.0028 ≈ 0). And the pressure itself has an honest optimum — too tight a pool starves even the winners, so the gain peaks at an intermediate pool (0.214 at the optimum vs 0.036 in the weak regime) rather than rising forever. Direction, mutation-dependence, competition-dependence and the inverted-U are measured; the absolute maturation rate stays [O].

Imprinting biases recall toward experienced clones

Re-exposure to a drifted antigen is a two-clone contest of a different kind: an experienced memory clone with a recall head-start (it starts nearer the ridge) against a fresh naïve clone that is fully OFF but better matched to the new antigen. The memory clone's affinity degrades with antigenic distance, so beyond a crossover distance (here d≈0.375) the naïve clone is strictly the better fit. The bias is then measured: at an intermediate distance the experienced clone is still preferentially recalled even though the naïve clone is better matched — P(memory wins)=0.767 at the first distance where the naïve clone already holds the affinity edge. This is original antigenic sin emerging from competition, not a fitted rule.

The bias is bounded and causally controlled. It decays monotonically with antigenic distance — P(memory) falls 0.767 → 0.493 → 0.183 across the better-matched zone — until the naïve clone takes over (P(naïve)=0.753 at the farthest distance). And it is experience-driven, not affinity-driven: removing the recall head-start abolishes it entirely, so in the same zone the better-matched naïve clone always wins (P(memory) collapses to 0.197, 0.037, 0.000). The existence of the imprint, its decay with distance and the head-start control are measured; the absolute imprinting magnitude stays [O].

Central tolerance deletes self-reactive clones

Negative selection is clonal selection read with the opposite sign. The same rising-affinity ramp is run on a self-reactive clone in the thymic context, but now a crossing to ON marks the clone for deletion, not activation. The measured deletion drive is again each organ's spinodal (deletion-drive ÷ spinodal = 0.998, 0.997, 0.997, 0.997 for marrow, spleen, thymus, lymphoid) and orders by γ exactly as activation does — the very same saddle-node, now culling rather than committing. A sub-threshold self-drive is ignored (P(delete)=0.006 on marrow) and a supra-threshold one is reliably deleted (P=1.000).

The consequence is a measured tolerance ceiling on what leaves the thymus. Educating a uniform spread of self-affinities through the deletion channel caps the exported self-affinity at the spinodal (max exported = 0.980× spinodal) and empties the autoreactive tail (exported-autoreactive fraction 0.000 with the channel on), while 54.9% of the self-reactive repertoire is deleted. Switching the channel off lets that tail straight through — exported-autoreactive jumps to 0.450 and the ceiling rises to 1.50× — so central tolerance is required, not incidental. Only the absolute deletion rate and the width of the escaped near-threshold sliver are [O].

Prime-boost scheduling has an interior optimum

A vaccination campaign is the maturation loop asked a temporal question: at a fixed campaign length, how does the matured affinity depend on the spacing of the boosts? The only added physics is antigen pharmacokinetics — each boost tops up a slowly-clearing depot, and every round runs the measured maturation competition at the current depot level. The answer is measured, an inverted-U in the boost interval with an interior optimum: boosting every round over-supplies the depot (mean pool 4.23) into the weak large-pool regime of maturation, so the gain is starved to 0.154; the optimum sits at an intermediate interval (gain 0.439 at interval 4); and spacing the boosts too far under-uses the fixed campaign (gain 0.292).

The interior optimum is the temporal face of the maturation inverted-U, and it is causally controlled. Clamping the depot's carrying capacity down to the productive pool removes the over-supply penalty — now even the most frequent schedule only ever holds a productive pool, so more boosts is strictly better and the optimum collapses to the most-frequent boundary (gain 0.493 at interval 1). The existence of an interior optimum and the over-supply mechanism are measured; the absolute optimal interval (depot clearance, dose, campaign length, noise scale D) stays [O].

Peripheral tolerance re-contains the escaped clone

Central deletion leaves an escaped near-threshold sliver, and (next volume) an inflammatory insult can break it. A second, peripheral layer of tolerance — regulatory suppression — guards exactly that sliver. The volume models it as a suppressor field that subtracts from the escaped self-clone's drive, and measures the suppression at which the clone is re-contained. The threshold that comes out is the saddle-node complement: σ_crit = (residual self-drive + insult) − spinodal, organ by organ (σ_crit = 0.250, 0.254 for marrow and lymphoid, both matching the 0.250× overshoot). The regulatory field cancels exactly the amount by which the escaped clone's total drive overshoots the switch — suppression and drive are one currency read with opposite sign.

The two tolerance layers add. Sweeping the escapee's residual self-drive (its negative-selection depth), the required suppression rises one-for-one with it (measured slope 0.931 ≈ 1): a deeper escapee, closer to having broken through central deletion, needs proportionally more peripheral suppression to stay quiescent. And suppression is required — with the field off the escaped clone driven by a supra-threshold insult breaks (P=0.980), and engaging sufficient suppression re-contains it (P=0.096). Only the absolute suppression strength (the map from σ to a regulatory-cell count) and the escaped-sliver width are [O].

Chronic antigen exhausts the response, reversibly

Affinity maturation is the productive arc of a response; exhaustion is its dynamical mirror. The volume drives a committed clone under persistent antigen and couples in an accumulating feedback — an inhibitory tone that builds while the cell is ON and antigen is present, and relaxes only when antigen is withdrawn. The response is measured to rise to a peak (1.000) and then collapse to a hyporesponsive floor (0.000, drop 1.000) as the feedback accumulates: chronic stimulation extinguishes the very response it drives. The collapse needs the feedback — with the coupling off the same chronic drive holds the response up indefinitely (floor 1.000).

The silencing is reversible, and gated by the switch's own bifurcation. Because the committed cell sits in a bistable ON basin, the feedback must push it past its negative saddle-node to force it OFF — so the onset is measured at a coupling (1.685) near the closed-form reverse-saddle-node estimate (1.875). And it is a functional silencing, not a deletion: withdrawing antigen lets the feedback decay, so a re-challenge recovers the response (1.000) while a continuously-stimulated clone stays extinguished (0.000). Only the absolute exhaustion rate (accumulation time, coupling, noise scale D) is [O].