Compute the neutron sector keystone: the two region gate in OpenMC at k < 1

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 AGENTS.md
 .DS_Store
 __pycache__/
+neutron/data/
+neutron/work/

README.md

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 **Phase B. Settle the keystone.** Now two questions, one per sector.
 - **B1, photon sector.** Measure (σ_trig, β, C_in) for the best NEEC or IGE candidate (precision EBIT or Penning trap NEEC; resonant IGE of a mid energy isomer) and locate it on the keystone figure. The required cross sections per facility are tabulated in [gates/experiment_menu.md](gates/experiment_menu.md); the ongoing nondestructive Penning trap test of the ⁹³ᵐMo claim largely decides this sector.
-- **B2, neutron sector.** Existence is already answered by library data; what remains is architecture. Reproduce the two region NAND of coupled subcritical assemblies in OpenMC with ENDF data, all inputs public, everything at k<1: no laboratory required, and the coupling matrices, absorber programming, and clock rates it yields are inherited by the photon sector when its gate arrives.
+- **B2, neutron sector.** *Status: done, in [neutron/](neutron/).* Existence was already answered by library data; the architecture is now computed. The two region subcritical gate runs in OpenMC with the official ENDF/B-VIII.0 library, calibrated first against three handbook criticals, and yields the coupling matrix (0.9140 against a transport k of 0.9132), the absorber's transfer curve, the drain per driver neutron (1.49 with the gate open), the level restoring emission spectrum, the clock (Λ = 106 µs, 1/Λ = 9.4 kHz) and the temperature coefficient (-25 pcm/K). All of it is inherited by the photon sector, including the two findings that were not sought: an absorber programs a weight rather than a threshold, and the sector has no signal controlled inversion worth the name.
 - *Kill criterion B (the big one, now confined to B1):* if no compact state can be shown to satisfy the leak condition, Γ>1, *and* level restoration at any achievable flux, then nuclear computing at benchtop scale is permanently confined to the stochastic tier plus memory, a real but bounded result, and the amplifier tier remains a reactor scale curiosity. The theory's *high* ambition lives or dies here.
 
 **Phase C. Tier 2 coherence.** Demonstrate a two qubit operation on Mössbauer or nuclear spin DOF, or a usable Bell measurement on annihilation γ pairs after Compton analysis.
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 | [gates/](gates/) | the criterion evaluated against NUBASE2020 and ENSDF for all 1870 known isomers; curated candidates with measured multiplicities; the experiment menu of required cross sections per facility |
 | [transport/](transport/) | Monte Carlo demonstrations of every routine gate (exact coincidence multiplier, absorption complement, measured Green's matrix, saturable sigmoid, Bernstein universality), the degree checker, and the compiler: the adjoint assembler that turns a target weight matrix into a material layout, three ways |
 | [simulator/](simulator/) | the digital twin: a Tier 1 machine run decay by decay, validated against exact enumeration, with decays per sample and energy per sample measured |
+| [neutron/](neutron/) | the keystone at reactor scale, computed: the two region subcritical gate in OpenMC with the official ENDF/B-VIII.0 library, calibrated against three ICSBEP handbook criticals; gain, the fission matrix and its eigenvalue check, transfer curves under two absorbers, the truth tables, the superposition check, level restoration spectra, the clock and gain bandwidth product, and the temperature coefficient that prices the veto |
 | [transistor/](transistor/) | the reference transistor: one pinout at three scales, the valve, the datasheet with characteristic curves and maximum ratings; with four accompanying notes: [SEALED.md](transistor/SEALED.md) (the machine), [EMBODIMENT.md](transistor/EMBODIMENT.md) (the build, at assembly grade), [VALVE.md](transistor/VALVE.md) (the logic), [COMPONENTS.md](transistor/COMPONENTS.md) (the full inventory of a working computer, missing pieces simulated), [POWER.md](transistor/POWER.md) (the metabolism), and [SCALING.md](transistor/SCALING.md) (the trajectory) |
 
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@@ -366,9 +367,29 @@ Where this all goes is the subject of [transistor/SCALING.md](transistor/SCALING
 
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+## The keystone, computed: the neutron gate at k below one
+
+The claim that the keystone already exists in the neutron sector has, until now, been arithmetic on library constants and a citation to a 1958 paper. It is now a calculation. [neutron/](neutron/) builds the reference transistor's third scale in OpenMC with the official ENDF/B-VIII.0 library: two 72 litre tanks of 4.9 percent enriched uranyl fluoride solution (the composition of a handbook criticality benchmark, so every atom density is published), each at k ≈ 0.901 alone, standing 3 cm of water apart in a water bath, with an absorber that can be lowered between them and a Cf-252 point source in one tank. Before the gate is computed the same code and data reproduce three handbook criticals ([neutron/results.md](neutron/results.md), Section 0), so the instrument is calibrated the way a criticality safety calculation is.
+
+Every number below carries a Monte Carlo uncertainty, and every configuration is strictly subcritical.
+
+- **Gain, and the synapse.** A tank alone has M = 1/(1 - k) = 10.1; the pair, coupled, k = 0.9132. The fission matrix K (fission neutrons born in tank i per fission neutron born in tank j, one generation) has dominant eigenvalue 0.9140 against a transport k of 0.9133. Avery's coupled region kinetics, this theory's synapse with gain on the diagonal, is confirmed across the three absorber states to between -263 and +138 pcm, and that residual is itself the measured price of describing a tank by one number rather than a shape.
+- **A trap the architecture walks into by construction.** A gate is built to couple its regions weakly, and weak coupling drives the dominance ratio of the eigenvalue problem toward one: 0.957 to 0.983 here, so the tilt between the two tanks decays by one or two percent per generation while sampling noise re excites it just as fast. The geometry is mirror symmetric, so the true fission share is exactly half per tank, yet the fission banks stood off by up to 0.070 while the Shannon entropy sat flat to a part in a thousand: entropy watches the shape inside each tank and is nearly blind to the tilt between them. The first version of this calculation shipped a fission matrix that was wrong in a way that looked entirely reasonable, and only the control run caught it. The fix is a mirror rather than more batches, running the half geometry with a reflective plane so the tilt mode cannot exist, and the correction is visible in the physics: a tank's self multiplication went from a non monotonic 0.8924, 0.9030, 0.8977 to a flat 0.8946, 0.8964, 0.8943, which is what an absorber placed *between* two tanks must do to what each tank does to its own neutrons, namely almost nothing. Any photon sector network of many weakly coupled cells inherits this in a sharper form.
+- **The transistor.** One driver neutron entering tank A becomes **1.49 fission neutrons born in tank B**, across the gap, with the gate open: the output exceeds the input, so the gate has gain. A 1 mm cadmium sheet cuts the coupling by 1.7× and the drain to 0.92; a 2 cm boron carbide blade by 2.6× and to 0.67. The contrast is 2.2, not a MOSFET's million, and the neutron budget says why: what crosses 3 cm of water is still largely fast, and a thin absorber is a filter, not a barrier.
+- **Linearity, and the honest AND.** Two drivers together deliver the sum of what each delivers alone to 0.6 percent, which is counting noise: transport is linear while its cross sections are frozen, exactly as the Green's function reading requires. The coincidence AND is therefore a perceptron AND, a threshold on a weighted sum, with the threshold still at the boundary.
+- **Level restoration.** Driven through the water from A, by Cf-252 at its own axis, or by a 14.1 MeV line, tank B emits the same fission spectrum to a total variation distance of 0.002. Inputs spanning thermal to 14 MeV, one output: the self restoring class of the theory supplement, measured. The photon sector cannot buy this at any price, because an isomer cascade emits its own lines and not its trigger's.
+- **The clock.** Generation time Λ = 106 µs, prompt switching time Λ/(1 - k) = 1.22 ms, gain bandwidth product 1/Λ = **9.4 kHz**: the op amp law of the transistor note, with laboratory grade constants in it.
+- **The veto, priced.** Warming the pair by 56 K costs -25 pcm/K (-13 Doppler and scattering law, -13 expansion of the liquid), a 14 percent modulation of the gain; halving the gain would need about 343 K, which water at atmospheric pressure cannot do.
+
+![The neutron gate, computed](figures/fig15_neutron_gate.svg)
+
+The last result is the one that changes the theory rather than confirming it. Every coupling in the neutron sector is excitatory: a neutron only ever adds fissions, and an absorber that burns up disinhibits. Heat is the only channel by which a signal lowers a neighbour's gain, and it is worth a fraction of that gain, slowly. So the sentence in Section 4 of the supplement that an absorber "programs thresholds and vetoes" is now two corrections: an absorber programs a *weight*, the threshold staying at the boundary; and signal controlled inversion, which NAND and signed Ising weights and Turing completeness all require, is the neutron sector's real weak point as an embodiment. The keystone search of Open Problem 1 inherits it as a second requirement: the compact state must supply an inhibition as well as a gain, or what it builds is a linear amplifier network read by comparators, universal for continuous functions and not for logic.
+
+---
+
 ## Open problems and how to contribute
 
-1. **Keystone:** propose or measure a compact state meeting the leak condition with β>1 *in a level restoring or convertible class* (theory, Section 2). *This is problem number one.* A concrete, computable subproblem: search ENSDF for heterogeneous pairs of isomers whose cascade lines and gateway lines are mutually resonant (gate A's output pumps gate B and vice versa); the search space is already tabulated in `/gates`.
+1. **Keystone:** propose or measure a compact state meeting the leak condition with β>1 *in a level restoring or convertible class* (theory, Section 2). *This is problem number one.* A concrete, computable subproblem: search ENSDF for heterogeneous pairs of isomers whose cascade lines and gateway lines are mutually resonant (gate A's output pumps gate B and vice versa); the search space is already tabulated in `/gates`. The computed neutron gate ([neutron/](neutron/)) adds a second requirement that the photon sector had not been asked for: the state must also supply a *signal controlled inhibition*, because the only one the neutron sector has is heat and it is worth a fraction of the gain. A gate with gain but no inversion is a linear amplifier with a comparator bolted to its boundary, and the machine it builds is the Bernstein machine of Section 6 with better numbers, not a logic family.
 2. Tighten the throughput law for fan out reuse: when is a count "spent"?
 3. A learning rule: aperture update Δα ∝ coincidence(pre,post) is Hebbian or STDP using the *same* coincidence primitive as the AND gate (Appendix B.4); formalize and simulate.
 4. Transport level 𝒢 for a real geometry that implements a target weight matrix (signed weights via complementary channels).