For the free depth-three grammar, N=q²+3q+7. A specified positive quadratic receiver gives χ=s²(q²/k₁+3q/k₂+7/k₃). The count and susceptibility agree at unit stiffness and unit source; their identification is an additional choice.
Explore the declared model
Retained count N = 137. Normalized susceptibility χ = 137. Reciprocal model response 1/χ = 0.00729927. Response per record χ/N = 1. These responses are not an independently derived electromagnetic coupling.
| Sector | Count Nᵢ | Stiffness kᵢ | Response s²Nᵢ/kᵢ |
|---|---|---|---|
| 1 | 100 | 1 | 100 |
| 2 | 30 | 1 | 30 |
| 3 | 7 | 1 | 7 |
| Total | 137 | — | 137 |
Exact finite grammar and strictly positive quadratic response in declared dimensionless units. This model does not independently derive a physical electromagnetic coupling, its measured value, or its energy dependence.
What each input changes
The integer q counts distinguishable continuation labels. Its default ten corresponds to the symmetric quadratic receiver developed in the linked research chapter. The sliders k₁, k₂ and k₃ are positive stiffness ratios relative to a chosen common scale. Increasing a stiffness makes that sector harder to displace. The common amplitude s controls how strongly the probe couples to every retained component.
These variables describe a finite mathematical receiver. They are independently adjustable here so their roles can be tested. The workbench does not fit laboratory measurements or insert the observed fine-structure constant. All response numbers use the explicitly normalized units defined below. The table gives the exact sector decomposition behind the chart, which makes unequal or very small contributions easier to inspect on a narrow screen.
Count a free grammar before assigning response
Let R₀ be empty and let the depth-d record retain either a continuation label together with an earlier record, or a newly terminated branch. Use branch counts 2ᵈ−1, giving 1, 3 and 7 at the three depths. The direct-sum alternatives must remain distinguishable and the continuation register must retain all label combinations.
r₁=1, r₂=q+3, r₃=q²+3q+7
The three terminal sectors therefore contain q², 3q and 7 records. With q=10 their sizes are 100, 30 and 7, summing to 137. With q=8 the same stopping rules give 64+24+7=95. Changing q tests a declared information-retention assumption; it does not change an experimental energy scale. These are counts of basis records, not weights or probabilities.
Derive the normalized susceptibility
For sector i let xᵢ be a real vector with Nᵢ components. Choose a dimensionless probe J and a vector 1ᵢ whose components are all one. The declared energy is a strictly convex quadratic form because every kᵢ is positive:
xᵢ*=sJ1ᵢ/kᵢ, min FJ=−½χJ²
χ=s²ΣᵢNᵢ/kᵢ
Differentiating each coordinate sets kᵢxᵢ=sJ1ᵢ, proving the unique minimum. Equivalently, completing the square leaves the stated negative quadratic probe term. The induced readout −∂minF/∂J is χJ, so χ is the linear susceptibility in this convention. Each bar is one positive contribution s²Nᵢ/kᵢ. The response per record χ/N compares the weighted response with the unweighted count.
Calling the energy dimensionless is substantive: the common energy, probe and displacement scales have already been chosen. A dimensionful implementation must restore them consistently. Rescaling an uncalibrated probe changes the numerical susceptibility, which is why the source amplitude belongs among the displayed inputs.
Three comparisons you can reproduce
At the default q=10, k₁=k₂=k₃=1 and s=1, the model gives N=137 and χ=137. Its reciprocal is exactly 1/137 before decimal rounding. Double only k₁: the sectors contribute 50, 30 and 7, so χ=87 while N remains 137. The state count has not changed, but the response has.
Restore unit stiffness and double s. Every contribution grows by four, giving χ=548 and reciprocal 1/548. Alternatively double every stiffness with s=1: χ becomes 68.5. These comparisons establish distinct scaling laws, χ∝s² and χ∝1/k for a common stiffness. They also show why replacing a response derivation by a dimension count leaves a physical normalization undetermined.
Connect the receiver to SMM/DTT
Within SMM/DTT, inner generation can motivate the compositional record construction and outer completion can specify which distinctions remain available. A physical receiver must additionally derive the positive response operator, the coupling of a probe to that receiver, and the common action and clock convention. The six inner roles are nested dependencies; their number alone does not set the three stiffness ratios.
The quantity 1/χ is explicitly labelled a reciprocal model response. Identifying it with α, or another coupling, requires a physical matching law that connects the normalized probe to a specified interaction. A successful source derivation would constrain q, kᵢ and s together instead of selecting the values because their output is desired. The linked chapter develops bounds, multiple probes and the effect of hidden-sector relaxation.
Continue with the articles
- From three-dimensional readout to retained records
A quaternionic readout, ten-mode pair memory, and a selected stopping grammar produce 137 distinguishable records. The construction makes its retention assumptions explicit and leaves physical coupling as a separate problem.
- Why a record count is not yet a coupling
A receiver's static response depends on its source and inverse stiffness, as well as its size. Exact minimization shows why counting 137 records does not by itself determine an electromagnetic coupling.
- Why binding does not fix the coupling
Dilation shows that every positive Coulomb coupling admits binding and the same type of ground-state filter. A source derivation of a particular coupling must therefore fix field, matter, action and clock normalization beyond the existence of an atom.