Exposure-Gate Beam-Mask Demonstration

Reading Position

Read this after the main RSG v1.4 paper and the probability note, or alongside Section 5 of the v1.4 paper. The main paper defines the locked analogue protocol; the probability note explains the count-normalisation layer; this companion note makes the gate logic concrete with reproducible numerical demonstrations.

The page is especially useful for readers who ask, "What does the exposure gate actually do?" It answers by replacing abstract histories with visible masks, finite supports, tabled exposures, gate transmissions, and control comparisons.

Claim Status

The note is deliberately modest. It does not derive a new optical law, a new probability law, or a new matter law. It demonstrates a protocol pattern: prepare candidate amounts, compute a declared exposure, multiply by the gate, normalise, then compare against a control or held-out readout.

prepared amount
  -> declared exposure W
  -> gate transmission G = exp(-eta W)
  -> transmitted amount
  -> represented output fraction

The failure conditions are also clear. The demonstration fails as a protocol example if masks, incident support, support valuation, or normalisation are changed after the reveal. It fails as an RSG-specific positive-control example if the exposure-gated rule does not improve on the path-only control in the declared regime.

Beam-Mask Exposure

The first demonstration fixes an incident beam image and several lossy masks before comparison. The exposure weight of each mask is the fraction of incident support inside that mask:

W_m = sum_{(x,y) in M_m} I_in(x,y) / sum_{(x,y)} I_in(x,y)

Ordinary path-only attenuation treats every mask as equally exposed:

T_path = exp(-eta)

The exposure-gated model instead uses:

T_m^gate = exp(-eta W_m)

In the rendered example, eta = 1.05. The five masks have exposures approximately 0.086, 0.234, 0.300, 0.198, and 0.065. Because those values are fixed from the incident support and mask geometry, the output comparison is not tuned after the synthetic reveal.

Beam-Mask Results

The path-only control predicts the same transmission for every mask: approximately 0.350. The exposure-gated predictions vary with the mask support and match the synthetic held-out values closely.

path-only total absolute error: 2.428
gate total absolute error:      0.030

path-only RMSE: 0.491
gate RMSE:      0.006

The correct reading is not "RSG proves optics." The useful reading is that a support-weighted exposure law is operationally distinguishable from ordinary path-only attenuation when the support geometry is fixed in advance.

Surtea Support Version

The second demonstration replaces the beam image with a valued finite Surtea-style support universe. A support X has an interior, closure, and boundary, and a positive valuation nu_D on cells. Boundary valuation supplies the exposure:

bd_D(X) = cl_D(X) \ int_D(X)

W_X = nu_D(bd_D X) / nu_D(cl_D X)

G_X = exp(-eta W_X)

Compact supports have lower boundary exposure because more of their closure is proper interior. Hollow or fragmented supports expose more of their valued closure to the boundary channel and therefore receive lower gate transmission.

This does not claim that boundary exposure is mass, gravity, or matter. It says that Surtea boundary valuation can be used as a declared exposure variable before a survival gate is applied.

Support-Gate Results

The rendered support examples compare compact, hollow, fragmented, and paired supports. The gate transmissions decrease as the boundary share increases:

compact:    W_X approx 0.562, G_X approx 0.586
hollow:     W_X approx 0.760, G_X approx 0.486
fragmented: W_X approx 0.940, G_X approx 0.410
paired:     W_X approx 0.806, G_X approx 0.465

For paired disjoint supports, the script also computes the interaction measure:

mu_hat(X,Y) = nu_D(cl_D X intersection cl_D Y), where X intersection Y is empty

In the rendered case this value is approximately 1.382. The interaction measure is carried alongside the boundary exposure; it is not itself the survival gate.

Catalogue Of Gate Examples

The note gathers the small numerical examples used across the probability note and the v1.4 paper. The value of the catalogue is consistency: each example follows the same operational count map.

C_i,0 = C_0 q_i
C_i,N = C_0 q_i G_i
p_i   = C_i,N / sum_j C_j,N
      = q_i G_i / sum_j q_j G_j

Relation To Probability

The gate is not a second probability law. It is the multiplier applied before normalisation. Probability appears only after the transmitted represented amounts are divided by their total.

C_i,0 = C_0 q_i
C_i,N = C_0 q_i G_i
p_i = C_i,N / sum_j C_j,N

This is why the companion note belongs between the probability page and the analogue-test sections: it shows the same algebra with visible support data, finite masks, and concrete controls.

Reproducibility

The note cites an executable Colab notebook for the beam-mask and Surtea support demonstrations. The code is described as trig-free and is used for reproducibility of the displayed figures and metrics, not as laboratory validation.

The practical rule for future work is simple: the incident support, masks, Surtea valuation, exposure law, loss coefficient, output rule, and failure metric must all be locked before the result is inspected.

Fit With The Site

This page is the best bridge between the live visualisation and the formal PDF stack. It explains why coloured support, boundary, and light/mask language can be useful on the site without turning the visualisation into an empirical claim. The visual scenes are explanatory; validation still belongs to locked comparison protocols.