A claim small enough to be wrong
The first machine is not a universal solver for every stationary-action problem. It targets a narrower class that is clean enough to falsify: discretized additive action over a Markov-complete state.
τₖ = τbase + α · aₖ
earliest arrival = argminₚ A[P]
If a local action increment is negative, add the same per-layer baseline delay to all alternatives. Every candidate crosses the same number of layers, so the baseline shifts absolute arrival time without changing the winner.
Known mathematics
Fermat, Bellman optimality and the eikonal equation are mathematically connected. That connection itself is not the novelty.
Known hardware idea
Race logic already encodes values as arrival time and uses temporal primitives for graph and dynamic-programming workloads.
Proposed contribution
Compile local action into programmable photonic delay, preserve cumulative action in time, and prune with a first-arrival photonic state element.
Delay + race + latch
The critical device is not a passive beam combiner. A passive junction preserves late pulses. We need an active first-arrival cell: the first valid pulse fires the output after a fixed latency, records the winning input, and suppresses later arrivals until reset.
predₖ₊₁(j) = argminᵢ [ Tₖ(i) + τₖ(i,j) ]
Timestamp must survive
The cell may add a fixed gate latency g, but must not re-clock the winner to a new global time origin. Physical time is the accumulated action.
State must be complete
If future cost depends on velocity or momentum, position alone is insufficient. A node may need to represent phase-space state such as (q,p).
Why early pruning does not change the optimum
Suppose two different histories reach exactly the same Markov state j. History A arrives first:
Every future continuation from that complete state is identical for the two histories. For any future action F(j):
The late history can never recover. The earliest arrival is therefore a sufficient statistic for every history that collapses into the same state.
Make it fail before building it
The simulator builds random layered action graphs. “Oracle” is exact dynamic programming. “Photonic” propagates physical arrival times, prunes every state to its earliest pulse, and optionally adds edge and gate timing noise.
Experiment A — first-arrival equivalence
Random continuous edge actions; the ideal model almost surely has a unique winner.
Experiment B — noise boundary
Sweep timing error and watch exact-path accuracy fall as physical arrival ordering becomes ambiguous.
Model: every programmed edge delay and every node forwarding latency receives independent Gaussian timing error. Static calibration error and dynamic jitter should eventually be separated experimentally.
Exponential virtual space, polynomial active frontier
If each of d decisions has b possible states, the number of complete histories is b^d. With first-arrival pruning, only one winner per state needs to continue.
Scaling explorer
Compare implicit trajectory count, physical fabric, dynamic-programming work and propagation depth.
Advantage should mean crossover, not rhetoric
The fair comparison is not against exhaustive enumeration. It is against the best optimized electronic algorithm for the same graph, accuracy target and I/O assumptions.
and / or
EDPphotonic < EDPbest-electronic
| Quantity | Photonic architecture | Fair classical baseline | What to measure |
|---|---|---|---|
| Candidate histories | b^d implicit | b^d implicit | Do not use as a speed metric alone |
| Work representation | O(d·b²) spatial delays / gates | O(d·b²) min-plus ops | Area, utilization, memory traffic |
| Critical depth | ≈ O(d) propagation | O(d log b) ideal parallel reduction | Measured end-to-end latency |
| Precision | arrival-time margin / jitter | digital numeric precision | Exact path + regret |
| Energy | laser + gates + programming + detection | compute + SRAM/HBM + control | Joules per solved instance |
From fiber bench to integrated fabric
0 / Fiber digital-physical twin
3 states × 3–8 stages. Fixed fiber delays implement transition cost; switched delay segments implement programmable potentials. Validate action↔arrival calibration and the noise model.
1 / First-arrival cell
Characterize one active node: set/reset behavior, latency distribution, late-pulse extinction, fan-out and predecessor recording.
2 / Integrated lattice
Combine programmable true-time-delay edges with first-arrival cells. Scale width until electronic min-plus reduction becomes the meaningful adversary.
First-arrival cell: what must actually be measured
| Parameter | Why it matters | Initial target |
|---|---|---|
| Forwarding latency g | Adds directly to every stage | <50 ps aspirational; characterize achievable value first |
| Latency jitter σg | Can reorder near-tied candidates | Much smaller than the minimum timing margin |
| Late-pulse extinction | Determines whether the exponential pulse population is actually pruned | >20 dB proof-of-principle target |
| Reset / refractory time | Limits instance throughput, not single-instance latency | Measure separately |
| Winner-port memory | Allows full trajectory reconstruction | One predecessor symbol per state |
The experiments that matter most are adversarial
Near-tie test
Sweep ΔA until αΔA approaches total timing uncertainty. The measured failure curve should match the digital twin.
State-aliasing test
Merge histories that share position but hide different velocity. If pruning fails, show that a phase-space state restores correctness.
Loss adversary
Attenuate the true winner without changing delay. It should still win until detector threshold makes it disappear.
Electronic crossover
Benchmark the same Bellman update on CPU, GPU and ideally FPGA/ASIC-like hardware with symmetric I/O accounting.