Incoprea Research · 30 October 2026 · Mathematical/model test
Finding: Competing reaction pathways determine yield in a simple excited-state model, and stronger absorption alone changes the amount of product rather than the yield per absorbed photon. However, the lifetime-to-vibrational-relaxation ratio proposed in Quantum Yield Is a Race is not generally sufficient: two model systems can have exactly the same ratio and quantum yields of 0.1 and 0.8.
This is an exact kinetic derivation and synthetic stochastic check. It narrows a plausible mechanism in the Incoprea Research hypothesis collection; it does not experimentally validate the original predictor. Branching kinetics and the distinction between absorption and quantum yield are established science, not new discoveries. This note has no external peer review.
IUPAC's photochemistry glossary defines quantum yield by events per photon absorbed. Product count and quantum yield therefore answer different questions. Under fixed conditions, absorbing ten times as many photons can make ten times as much product without changing the fraction of absorbed photons that produces the chosen event.
Our declared model starts with one reactive excited state. It has three independent first-order exits: desired product at rate k_p, a vibrational-relaxation sink at rate k_v, and all other losses at rate k_o. The synthetic rate constants are expressed in inverse nanoseconds. The corresponding total excited-state lifetime is tau = 1/(k_p+k_v+k_o), and the putative vibrational time is tau_v = 1/k_v.
Calling vibrational relaxation a sink is an explicit simplifying assumption. In real molecules, vibrational cooling may leave a state that still reacts, and sequential pathways can require a multistate model. This test is intentionally more favorable to the proposed single-ratio mechanism than many real systems would be.
The excited population is E(t) = exp[−(k_p+k_v+k_o)t]. Integrating the product flux k_p E(t) from zero to infinity gives
Phi_product = k_p/(k_p+k_v+k_o) = k_p × tau.
This is the branching fraction for the desired pathway. The lifetime ratio tau/tau_v = k_v/(k_p+k_v+k_o) instead specifies the fraction entering the vibrational sink. Without knowing the remaining channels, it cannot say how the rest is divided between product and other losses.
Only in the special two-exit case k_o = 0 does the proposed ratio determine yield: Phi_product = 1 − tau/tau_v. Even there, the direction needs care. A larger total-lifetime/vibrational-time ratio means a larger loss fraction in this model, not automatically a higher product yield. A lifetime that already includes competing decay rates is not an independent measure of how long the molecule “has to react.”
The plan fixed four rate triples before outcome calculation: (k_p,k_v,k_o) = (1,9,0), (9,1,0), (1,1,8) and (8,1,1). For each, 200,000 independent excitations were simulated with separate exponential clocks for each available exit. The earliest clock decides the outcome; this is independent of directly sampling a precomputed branching probability.
Exact product yields are 0.1, 0.9, 0.1 and 0.8. Simulated yields are 0.099720, 0.900025, 0.100345 and 0.798810. All four meet the prespecified absolute tolerance of 0.005. The 800,000 events are synthetic trials, not laboratory photons.
The final two systems are the crucial control. Both have total rate 10/ns, lifetime 0.1 ns, vibrational time 1 ns and ratio 0.1. One allocates most exits to another loss channel and yields 0.1; the other allocates most to product and yields 0.8. This exact counterexample establishes that a universal mapping from that ratio to product yield is impossible within the declared three-channel family.
Holding the rates fixed at yield 0.1, 1,000, 10,000 and 100,000 absorbed photons give expected product counts of 100, 1,000 and 10,000. Dividing each by its absorbed-photon count returns the same yield. This supports the narrow distinction in the original hypothesis, while avoiding the misleading inference that absorption cannot matter to practical productivity.
It also does not imply that changing a molecule's absorption spectrum can never change yield. Different excitation energies, electronic states, quenching, heating, concentration effects or saturation can change the rates themselves. Chain reactions may produce more than one product event per absorbed photon, outside this one-excitation/one-exit model. Absorbed and incident photons must not be interchanged.
The productive follow-up is a specified reaction scheme with measured rate constants and independently measured product yield. The current result gives that test a sharper target: estimate the desired branching rate and every important competitor. A lifetime ratio becomes predictive only after the missing-channel assumptions are justified.