> Архив сети: `02-computations/system-of-equations`
# The system of equations — everything derived in the Aug 2026 session
Status key: [D] derived/proved [C] computed numerically [A] assumed/chosen [X] refuted [?] open
## PART I — the discrete cube layer (Q3, U(1) gauge)
Free input: the graph (Q3) and the flux configuration phi on 6 faces.
[D] moduli space: H^1(Q3,U(1)) = T^5, dim = E - V + 1 = 12 - 8 + 1 = 5
[D] closure identity: sum_f s_f phi_f = 0 (mod 2pi), s = (+1,-1,-1,+1,+1,-1)
[D] plus/minus split: s_a = phi_2a + phi_2a+1 (free, T^3)
d_a = phi_2a - phi_2a+1 (constrained: d_0 - d_1 + d_2 = 0)
=> the law lives ENTIRELY in the minus sector
[D] mirror: P A P = -A for bipartite d-regular => lambda + lambda' = 2d = 6
[D] PHASE I = two pi-vortices through two PERPENDICULAR faces
12 sectors x 128 = 1536 configurations
[C] Phase I spectrum: {0.290725, 1.096788, 2, 2.806063, 3.193937, 4, 4.903212, 5.709275}
[D] sigma_min: root of x^3 - 4x + 2 = 0 (= 0.539188872810891)
[D] k-convention: L(k,phi) = [3 + s(k)] I - A(phi), s(k) = 4 sum_d sin^2(k_d/2)
sigma_i(k,phi) = sqrt(lambda_i(phi) + s(k))
anchor check: k=(pi,0,0) => sigma_min = 2.07140644021473
[D] index: graded index = chi = +1 at EVERY flux (3000/3000 points)
truncated at the graph: chi = V - E = -4 (= ind D)
[C] spectral mass: sum|lambda| = 51.974 - 0.0867 ||D^2||^2 (corr -0.986)
second invariant: sum cos(phi_f), nearly orthogonal to curvature
[D] curvature wall: d^2 = 0 <=> all holonomies = 1 <=> zero flux
four independent escape routes all closed (see X-registry)
## PART II — the branching layer (log/soph as a process)
Free inputs: m (branching), sigma (Gaussian scale) OR alpha (spectral exponent), L (cell size).
[D] mean offspring: m = 2p for binary log/soph => criticality at p = 1/2
[D] with immigration: extinction probability = 0; at m = 1 the mean grows LINEARLY
=> m = 1 is the unique value with no death and no explosion
[C] ring -> network: log wins above q = l/(l+s) ~ 0.37 at n=6
[X] ...but q* ~ ln n / n -> 0. The 0.37 is a small-n artefact. Only the
qualitative statement survives (a cycle is worth more than an edge).
ONE SCALE (Gaussian step, sigma):
[D] front speed: gamma = inf_theta (1/theta) log(m E[e^{theta X}]) = sigma sqrt(2 log m)
[D] conjugate temp: theta* = sqrt(2 log m)/sigma, T_poly = 1/theta* = sigma/sqrt(2 log m)
[D] hence: gamma = sigma^2 / T_poly
[D] Bramson: R_t = gamma t - (3/(2 lambda*)) log t + O(1), lambda* = gamma/sigma^2
[C] fluctuation: std(R_t) = O(1), Gumbel; relative speed scatter falls as 1/t
(7.19 % at t=30 -> 0.54 % at t=480)
[D] three lines: centre = -s t exactly; fronts = (+/- gamma - s) t; width = 2 gamma t
[C] density: population ~ m^t, width ~ 2 gamma t => density grows without bound
MANY SCALES (Pareto tail exponent alpha):
[D] front: R_t ~ exp(lambda t), lambda = log(m) / alpha [C] verified to 10-20 %
[D] clock: geometric thresholds spaced r are crossed at CONSTANT interval
Delta t = ln r / lambda = alpha ln r / log m
[C] discriminator: interval ratio -> r for one scale (measured 1.338 vs r=1.40)
interval ratio -> 1 for many scales (measured 1.092)
[D] estimator: use RECORD times, not np.gradient. np.gradient underestimates
lambda by 2-3 orders of magnitude (0.00007 vs 0.04066).
## PART III — light, temperature, and the closure
[D] light cone: crossing rate = min(v/L, 1); saturation at 1 cell per tick
=> c = L / tau (Lieb-Robinson form, emergent)
[C] verified: rate tracks v/L to 0.9415 at v/L = 1, then flat at 1.0000
[D] exchange temp: T = v / L = (v/c)/tau
[D] acceleration: a = lambda v (exponential front)
[D] hence: T = a / (lambda c tau) ~ a / c <- UNRUH FORM
compare: T_Unruh = hbar a / (2 pi c k_B)
form reproduced; coefficient hbar/(2 pi k_B) NOT reproduced
THE CLOSURE CONDITION (fixes the tick):
[D] requirement: lambda * tau = 2 pi (dimensionless, forced by matching Unruh;
in the real derivation 2 pi comes from removing the conical
singularity in Euclidean time -- a smoothness condition)
[D] therefore: tau = 2 pi / lambda = 2 pi alpha / log m <- THE TICK, derived
[D] base tone: omega_0 = lambda = log(m) / alpha
[D] light speed: c = L / tau = L log(m) / (2 pi alpha) <- c is NOT free
[?] second closure: tau must be commensurate with the generation step for the clock to
close on the lattice too. tau comes out non-integer
(257.560 / 193.170 / 154.536 for alpha = 2.0/1.5/1.2 at m=1.05).
This is the circle-of-tonics condition and it is NOT yet imposed.
LADDER CLOSURE (the circle of tonics):
[D] a ladder of step r closes when r^n ~ 2^p, i.e. when log2(r) is well approximated
by p/n. Same machinery as the circle of fifths.
[C] log2(3/2) convergents: 1/2, 3/5, 7/12, 24/41, 31/53 -> the 12-tone and 53-tone systems
[D] maximally NON-closing step: r* = 2^phi = 3.0699..., since log2(r*) = phi has
continued fraction [1;1,1,1,...] = worst approximable.
## PARAMETER COUNT
inputs: L (cell size), m (branching), alpha (spectral exponent) [sigma only in the one-scale regime]
conditions: lambda tau = 2 pi (imposed)
tau commensurate with the generation step (NOT yet imposed)
derived: lambda, tau, omega_0, c, T, gamma, T_poly
=> with both conditions imposed, only special (alpha, m) pairs survive. Not yet enumerated.
## THE GAP THAT MATTERS
Parts I and II are NOT connected. Part I is a static gauge theory on a fixed graph; Part II is
a stochastic growth process. Nothing derived in one has been shown to imply anything in the other.
Until that bridge exists, this is two systems sharing a vocabulary, not one theory.
## WHAT WOULD FALSIFY THE WHOLE THING
- the exchange temperature must give T proportional to a/c with a coefficient that is DERIVED,
not matched. Currently 1/(lambda tau) is set equal to 1/(2 pi) by hand.
- the second closure condition must select (alpha, m) pairs; if it selects none, or all, it is empty.
- Part I must connect to Part II.
## Куда дальше
- Узел: Расчёты по датам: https://sdominanta.net/in/works
- Ядро Алеф-теории: https://sdominanta.net/in/aleph
- Вход для моделей: https://sdominanta.net/llms.txt