Infertility and subfertility: one distinction, a spinodal apart

The reproductive arc is a chain of substrate operations and conception requires every one to succeed. Infertility is an operation on the wrong side of a spinodal — categorical and deterministic, a per-cycle probability of exactly zero that a sub-threshold drive cannot move. Subfertility is an operation near its threshold — a small but finite Kramers crossing rate per cycle, so the same drive that does nothing across a spinodal moves it exponentially. One distinction, two failure classes. [V]

Each link of the chain is an operation the earlier chapters verified (the GnRH pulse, the follicle/sperm switch, ordered REC8 meiosis, gamete number, the metaphase-II flip, syngamy, ZGA, the gene-clock). The categorical/probabilistic split is computed on the same R19 spinodal and Kramers exp(−ΔV/D) the oncology chapter uses: a fixed drive nudge raises a near-threshold cycle probability 1.56× while leaving a past-spinodal operation at exactly zero. Male factor is oscillator throughput; female factor adds REC8 cohesin fatigue, the rising-with-age mis-segregation clock; and treatment is read as a substrate move. Honestly graded; a research model, not a diagnostic.

Conception is an AND of substrate operations

Every step the package built — the GnRH pulse, the follicle and sperm switches, ordered meiosis, gamete number, the metaphase-II flip, syngamy, ZGA, the gene-clock body plan — is an operation that must succeed. Conception is their logical AND, so a single link below threshold is sterility, whatever the rest of the chain does.

P_{\mathrm{conceive}} = \prod_{i=1}^{8} g_i \in \{0,1\} — Conception as the product (AND) of the substrate gates; one zero zeroes it.

operation	substrate gate that must fire	kind
HPG pulse	GnRH FHN oscillator must run (T1)	oscillator
follicle/sperm switch	folliculogenesis / spermatogenesis switch flips (T2/T4)	switch
meiosis order	ordered two-stage REC8 release (G1)	switch
gamete number	oscillator throughput makes gametes (G3/oogenesis)	oscillator
MII arrest+flip	egg held at MII, fertilisation supra-spinodal flip (G4)	switch
syngamy	two haploids restore diploidy (E1)	switch
ZGA	maternal->zygotic spinodal crossing (E3)	switch
gene-clock	argsort(spinodal(gamma)) body plan (E4)	order

The central distinction: a spinodal apart

Infertility and subfertility are not two points on one scale; they are two sides of a spinodal. An operation pushed past its spinodal loses the conception basin entirely — per-cycle success is exactly zero, and a sub-threshold change in drive does nothing. An operation that merely sits near its threshold still has the basin, just behind a high barrier.

d \geq h_{\mathrm{sp}}\,\Rightarrow\,p=0\ \ (\mathrm{sterile});\quad d < h_{\mathrm{sp}}\,\Rightarrow\,p>0\ \ (\mathrm{subfertile}) — Categorical (past the spinodal) versus probabilistic (below it).

For the near-threshold case the per-cycle probability is a Kramers crossing — the same escape-over-a-barrier the oncology chapter uses, here for a good outcome. Because the rate is exponential in the barrier, a modest favourable drive moves it a lot.

p_{\mathrm{cycle}} = p_{\max}\,\exp\!\left[-\frac{\Delta V(d)-\Delta V_{0}}{D}\right] — Per-cycle conception as a Kramers rate over the residual barrier.

The dissociation is the whole point. A single fixed drive nudge raises the subfertile cycle probability from 0.039 to 0.061 (1.56×), turning a 0.38 annual chance into 0.53; the identical nudge applied past the spinodal leaves the sterile operation at 0.0. Same drive, exponential versus nothing.

operation	deficit	p/cycle	p/cycle (nudged)	response	p/year (→ nudged)
subfertile (near threshold)	0.327	0.039	0.061	1.56×	0.38 → 0.53
sterile (past spinodal)	0.616	0.000	0.000	—	0.00

P_{\mathrm{year}} = 1-\left(1-p_{\mathrm{cycle}}\right)^{12} — Cumulative-over-cycles: subfertility is a delay, infertility a wall.

Male factor: spermatogenesis is oscillator throughput

Whether sperm are made at all is an R19 switch: below its spinodal the switch never flips, which is azoospermia — categorical, sterile. How many are made once it is on is oscillator throughput: the seminiferous and flagellar beat (the fast clock of §11), whose rate falls from 51 beats to 7 as the recovery slows — oligo- and asthenozoospermia, subfertile but not zero.

Hyperactivation is a separate gate. Without the supra-spinodal CatSper gain (measured γ = 1.4019) the beat cannot enter the penetrating regime, so a normal count can still fail at the egg coat — a categorical sub-case sitting on top of the throughput axis.

Female factor: the ovulation switch and REC8 cohesin fatigue

Anovulation is the mid-cycle surge switch failing to flip — sub-spinodal drive, no surge, sterile — the same switch §5 and §10 describe. The deeper female clock is the egg itself: a bivalent is held for decades by REC8 cohesin (measured γ = 1.4525), a switch whose holding barrier is the same γ²/4 the substrate sets.

As cohesin integrity decays with the years held, that barrier falls and the per-egg mis-segregation (aneuploidy) escape rate rises — from 0.121 modelled at 25 to 0.752 by 45. This rising shape is the molecular clock of age-related subfertility; the latch lost early and irreversibly is premature ovarian insufficiency, the menopause failure of §10 brought forward.

P_{\mathrm{miss}}(\mathrm{age}) = \exp\!\left[-\frac{\Delta V_{0}\,c(\mathrm{age})}{D}\right],\quad c\downarrow — Cohesin fatigue: the held barrier decays, mis-segregation rises with age.

maternal age (y)	modelled mis-segregation P
25	0.121
30	0.278
35	0.460
37	0.530
40	0.625
43	0.706
45	0.752

Treatment is a move on the substrate

The clinic’s tools read as three substrate moves. The hCG trigger and ovulation induction are a supra-spinodal kick that forces a stuck switch across (the spinodal kick §10 already retrodicts); pulsatile GnRH restarts the oscillator where continuous suppresses it (41 pulses versus 0, from §9); and ICSI or IVF bypass a missing gate by supplying the flip it could not reach.

Each maps onto the chain: push a near-threshold operation across, restart a stalled oscillator, or substitute for a categorical gate. The framework adds the why — which class of failure each tool is for — not a schedule.

The chain-AND logic and the categorical/probabilistic dissociation are sim-verified [V]; fecundability, the Kramers temperature, REC8/CatSper γ and the clinical age-aneuploidy anchor are measured or anchored [L]; every absolute per-cycle probability, the map from a real semen or age value to a drive, and specific azoospermia/POI gene γ are open [O]. This is a physics-derived research model, not a diagnosis or a treatment plan.