Count retained records first, then specify their response. The same 137 records can have different capacities and different normalized interaction strengths.
Calculate the record and its response
N(10, 3) = 137 retained records. Normalized capacity C* = 137; effective stiffness σ_eff = 1. The conditional model coupling is g = 1/137 ≈ 0.007299270. This assumes the displayed response and normalization. It is not a measured value of α.
- Record count N
- 137distinguishable records
- Normalized capacity C*
- 137C × s_ref; dimensionless
- Model coupling g
- 0.007299270dimensionless
Recurrence: 0 → 1 → 13 → 137.
Effective stiffness σ_eff = 1; primitive normalization σ_eff/(AV) = 1; coupling reciprocal 1/g = 137.
| Terminal depth k | Branches 2ᵏ−1 | Continuation factor qᵈ⁻ᵏ | Records nₖ | Stiffness σₖ | Capacity nₖ/σₖ |
|---|---|---|---|---|---|
| 1 | 1 | 100 | 100 | 1 | 100 |
| 2 | 3 | 10 | 30 | 1 | 30 |
| 3 | 7 | 1 | 7 | 1 | 7 |
Free retention, ordinary trace coupling, positive diagonal sector stiffness and a supplied continuum matching are assumptions. No laboratory constants are fitted by this calculator.
1. Count the retained records
At depth k, retain every nonempty exterior branch of a k-dimensional innovation space. Its rank is bₖ = 2ᵏ−1. Each earlier record can continue through q distinguishable modes; terminal branches stay distinct in a direct sum. This gives the exact free-grammar recurrence:
N(q,d)=Σₖ₌₁ᵈ qᵈ⁻ᵏ(2ᵏ−1).
The selected q=10 and d=3 give 100+30+7=137. Changing q or depth explores a different supplied grammar; it does not demonstrate that all such grammars describe physical receivers. The original ten-mode choice has a separate symmetric-receiver argument. Algebraic depth three also does not replace the six nested inner roles or their completion interface.
These are distinguishable basis records. The count alone supplies neither a probability law nor a physical duration. A nonuniform sampler can use the same record space.
2. Let stiffness change capacity
For a positive response operator K and ordinary trace source f=I_N, minimizing the quadratic energy at fixed current gives C=⟨f,K⁻¹f⟩ and E_min=j²/(2C). Uniform effective stiffness s gives C=N/s. In the diagonal sector family used here, the more general result is C=Σₖnₖ/sₖ.
The factor h changes only the oldest terminal sector. At q=10,d=3 this is the 100-record sector. Keeping the other stiffnesses fixed while setting h=2 gives normalized capacity 100/2+30+7=87. The record count remains 137. The effective scalar stiffness is then σ_eff=137/87, so C*=N/σ_eff still holds.
In a general coupled receiver, hidden directions first relax. Their Schur complement s=u−⟨b,D⁻¹b⟩ determines N/s. This workbench uses a positive diagonal family to make the dependence visible without asserting that every hidden sector is dynamically independent.
3. State the units and physical matching
The continuum convention is g=s_cont/(4πacN). Both s_cont and ac have units of energy times length. Here the sector stiffnesses are assumed already matched to that convention. This supplied finite-to-continuum identification is not derived by the calculator.
Choose fixed reference scales a_ref (action) and c_ref (speed), and set s_ref=4πa_ref c_ref. The numerical inputs are σₖ=sₖ/s_ref, A=a/a_ref and V=c/c_ref. Then:
g=σ_eff/(AVN)=1/(AVC*).
Thus the displayed capacity is dimensionless C*, while dimensional capacity is C*/s_ref. A value σ_eff/(AV)=1 is a physical normalization assumption. It cannot be forced by changing units, since all quantities in that ratio are dimensionless.
At the baseline, C*=137 and g=1/137. Doubling every stiffness gives C*=68.5 and g=2/137. Doubling only the oldest sector gives C*=87 and g=1/87. Changing A or V changes the coupling while leaving both record count and this receiver capacity unchanged. These contrasts isolate the required action–speed dictionary.
What the calculator establishes
The recurrence and positive-response arithmetic are exact under the displayed assumptions. The model does not derive the stiffness, select the physical current, construct a three-dimensional continuum, or identify that current with electromagnetism. In particular, 137 alone does not predict measured α. A source theory must establish these interfaces together and test an additional observable without re-fitting the target.
Follow the derivation and its limits
- The structural integer 137
Exact recurrence, discrimination minimum, sampling and phase distinctions.
- Ten normalized symmetric receiver modes
The separately supplied reason for choosing q=10.
- From retained response to interaction strength
Capacity, hidden-sector relaxation and the physical normalization.
- Trace response and the Schur complement
Working-draft argument and source references.
- The conditional ideal value 1/137
Why the ideal model value is distinct from measured electromagnetism.