> Архив сети: `02-computations/computed-results`
# Computed results — Aleph ToE working session, Aug 2026
Status key: DERIVED (proved) / COMPUTED (numeric, reproducible) / X (refuted).
All computations on construction A unless stated: Q3, 12 edges, U(1) phase per edge.
---
## 1. Phase I identified geometrically [DERIVED + COMPUTED]
Reconstructed the config space from scratch and it matched the network's own numbers:
2^12 = 4096 Z2 configurations <- matches the recorded 4096
gauge group Z2^8, effective 2^7
4096 / 2^7 = 32 gauge-invariant flux sectors, 128 configs each
Gauge-invariant content = flux through the 6 cube faces (product of edge signs
round each 4-cycle). Flux +1 = laminar, flux -1 = pi-vortex.
Number of pi-vortices is always EVEN in Z2, because every cube edge lies in
exactly 2 faces, so prod(fluxes) = prod(sign^2) = +1. Sector counts 1+15+15+1 = 32.
**This parity is a Z2 artifact, not a property of the cube** — see §7.
### Thermodynamics of vorticity
n_vortices sectors spectrum L sigma_min
0 1 {0,2,2,2,4,4,4,6} 0.000000
2 15 (splits, see below) —
4 15 {0.5858,1.2679x2,2.5858,3.4142,4.7321x2,5.4142} 0.765367
6 1 {1.2679x4, 4.7321x4} 1.126033
lambda_min rises monotonically with vorticity: 0 -> 0.2907 -> 0.5858 -> 1.2679.
Laminar has a zero mode (massless). **Mass = frustration energy of vortices.**
### The split that identifies Phase I
The 15 two-vortex sectors split by the geometry of the two vortex axes:
2 vortices on PERPENDICULAR faces: 12 sectors x 128 = 1536 configs
spectrum {0.290725, 1.096788, 2, 2.806063, 3.193937, 4, 4.903212, 5.709275}
sigma_min = 0.539188872810891
2 vortices on OPPOSITE faces: 3 sectors x 128 = 384 configs
pairwise degenerate, sigma_min = 0.765366864730179
**1536 matches the recorded Phase I count exactly. sigma_min matches to 15 digits.**
> **Phase I = two pi-vortices threaded through two PERPENDICULAR faces of the cube.**
Threshold-free, gauge-invariant, purely topological: count vortices, check whether
their axes are parallel. This closes the "Phase I without a threshold" task.
Phase I is also the only class with a fully non-degenerate spectrum (8 distinct
eigenvalues).
---
## 2. Closed form of sigma_min [DERIVED]
sigma_min is a root of x^3 - 4x + 2 = 0
Degree 3, Eisenstein at 2, irreducible, Galois group S3.
Cleaner than the earlier sqrt(3 - sqrt(9 - mu0)) form. Closes the OQ-3 /
sigma_min closed-form task.
Anchor (see §3): x^6 - 20x^4 + 128x^2 - 260 = 0, = 2.07140644021473,
degree 6, irreducible.
---
## 3. k-convention for construction B [DERIVED, confirmed by an independent anchor]
L(k, phi) = [3 + s(k)]*I - A(phi), s(k) = sum_d (2 - 2 cos k_d) = 4 sum_d sin^2(k_d/2)
sigma_i(k, phi) = sqrt(lambda_i(phi) + s(k)), k_d in [-pi, pi)
Rigid shift; mirror centre 3 + s(k). Both T30 statements reproduce.
Phase I, k=(0,0,0): s=0 sigma_min = 0.539189 <- OQ7
Phase I, k=(pi,0,0): s=4 sigma_min = 2.071406 <- ANCHOR 2.0714 (Acc.43)
Phase I, k=(pi,pi,0): s=8 sigma_min = 2.879362
Phase I, k=(pi,pi,pi): s=12 sigma_min = 3.505813
The 2.0714 anchor falls out as sqrt(0.290725 + 4). Not fitted.
---
## 4. The mirror is bipartiteness, not a Phase I property [DERIVED]
For a d-regular BIPARTITE graph with sublattice operator P: P A P = -A for any
U(1) edge phases, so spec(A) is symmetric about 0 and spec(L) about d.
For Q3, d = 3, hence lambda + lambda' = 6. Gauge invariance is automatic.
Numeric check under random U(1) phases:
K4 n=4 deg=3 bipartite=False mirror-dev = 9.1e-01
Q3 n=8 deg=3 bipartite=True mirror-dev = 3.6e-15 pairs=4
Q4 n=16 deg=4 bipartite=True mirror-dev = 4.4e-15 pairs=8
C6 n=6 deg=2 bipartite=True mirror-dev = 2.2e-15 pairs=3
C7 n=7 deg=2 bipartite=False mirror-dev = 4.9e-01
K3,3 n=6 deg=3 bipartite=True mirror-dev = 3.6e-15 pairs=3
Petersen n=10 deg=3 bipartite=False mirror-dev = 7.2e-01
Machine zero on every bipartite case, gross violation otherwise.
**Consequence:** T28's mirror should be restated as a corollary of a general
theorem on phased regular bipartite graphs — stronger and more universal, but
no longer a discovery about Phase I. Also explains the Sakharov criterion
elementarily: P-diagonal coupling commutes with P instead of anticommuting,
which is why only it breaks the mirror.
**Consequence 2:** the number of mirror pairs is n/2. For Q3 that is 4, of which
one ({2,4}, rational) is trivial — leaving 3. So "three states" is a counting
artifact of |V| = 8, NOT a law. On Q4 it would be 8 pairs. X.
What survives: the IRREDUCIBILITY over Q. Random phases give a spectrum not
algebraic over the rationals at all; the integer cubic appears only on Phase I
configurations. That is about configuration selection, not graph shape.
---
## 5. The moduli space [COMPUTED]
Gauge-invariant moduli = H^1(Q3, U(1)) = T^5.
dim H^1 = E - V + 1 = 12 - 8 + 1 = 5.
Closure identity (the single constraint): sum_f s_f * phi_f = 0 mod 2pi,
with s = (+1, -1, -1, +1, +1, -1). rank(C) = 5 of 6.
Full scan, 100 000 points:
sigma_min: min = 0.000000 (laminar) max = 1.126033 (max vortex) median = 0.672428
fully non-degenerate spectrum: 98.7 % of the volume
Phase I (0.539189) sits at the 12th percentile
Laminar and max-vortex are the GLOBAL extremes of sigma_min over the whole space.
Degeneracies form codimension-1 walls (~1.3 % of volume) cutting T^5 into chambers.
### Plus/minus decomposition (the 3+/3- split)
6 faces = 3 axes x 2 orientations. Define per axis:
s_a = phi_{2a} + phi_{2a+1} (plus sector, 3 free, UNCONSTRAINED, T^3)
d_a = phi_{2a} - phi_{2a+1} (minus sector, 3 with one relation, T^2)
closure identity becomes: d_0 - d_1 + d_2 = 0 (mod 2pi) <- ONLY the minus sector
3 + 2 = 5, dimension checks. **The law lives entirely in the minus sector; the
plus sector is free.** Not built in — it falls out of the face orientations.
Phase I: s/pi = (1,1,0) d/pi = (1,1,0) (on the diagonal s = d)
max vortex: s/pi = (0,0,0) d/pi = (0,0,0)
sigma_min depends on BOTH sectors, so this is a clean coordinate split, not a
factorization of the dynamics.
### Involution with a 3-dimensional fixed locus
sigma = swap the two opposite faces of each axis. sigma^2 = +1; s -> s, d -> -d.
Fix(sigma) = {d = 0} = the plus sector T^3 <- exactly 3 dimensions, not inserted
Phase I has a spectrally identical representative ON the locus (s/pi=(1,1,0), d=0):
sigma_min = 0.539188872811, same degeneracy, same su(2) defect — although the
fluxes differ (pi/2 on four faces vs pi on two).
**But so(3) does NOT live on the locus.** Laminar (s=0) and max vortex (s=2pi)
both sit on it and give abelian and su(2) respectively. Half confirmed.
### Log = soph balance point
Bisection on the Phase I -> max vortex path phi = (pi, t*pi, pi, t*pi, t*pi, t*pi):
t* = 0.391826552 ||[A,A]|| = ||{A,A}|| = 4.618802 sigma_min = 0.673851
Median sigma_min over T^5 = 0.672428, i.e. within 0.2 %. **But this does NOT
survive other measures** — see X-registry.
---
## 6. Why 3? The Clifford mechanism [COMPUTED — partial]
A_d^2 = I always, but whether they commute or anticommute is set by the flux:
class ||[A,A]|| ||{A,A}|| dim<A> J close into su(2)?
laminar n=0 0.000 5.657 3 no (abelian)
Phase I n=2 perp 4.000 5.657 16 no
n=2 parallel 5.657 5.657 4 no
n=4 5.657 4.000 16 no
max n=6 5.657 0.000 6 YES, dim 3
At full pi-flux through all six faces the three direction operators anticommute,
generating Cl(3); the products J_i = (1/2) A_j A_k close into su(2) = so(3).
Three-dimensional rotations fall out. Laminar gives an abelian algebra — no rotations.
Also: omega = A1 A2 A3 satisfies omega^2 = -I at max vortex (||omega^2 + I|| = 1.7e-15).
**Tension to resolve:** so(3) is born in the n=6 sector, fermion mass in n=2.
Different states of the gas. And Q3 is a cube, so the three directions were put
in by hand anyway.
### Trichotomy of squares
sector omega^2 ||d1 d0|| which square
laminar +I 0.000 sigma^2=+1 AND sigma^2=0
Phase I mixed 2.965 none
max vortex -I 4.899 sigma^2=-1
Phase I falls into none of the three clean structures — the frustrated middle.
---
## 7. The cochain complex and the curvature wall [DERIVED]
Twisted complex C0 -> C1 -> C2 -> C3 (dims 8, 12, 6, 1).
Top boundary signs = (+1,-1,-1,+1,+1,-1) — the SAME vector as the closure identity.
||d2 d1|| = 5.4e-16.
||d1 d0|| = 0 EXACTLY iff every holonomy = 1, i.e. zero flux.
laminar 0.000 | flux 0.1 -> 0.149 | Phase I 2.965 | max vortex 4.899
**Vortices are precisely the obstruction to d^2 = 0.**
### Hodge ladder (flat case)
Delta_0 (vertices, aleph_0) dim 8 ker 1 spectrum {0, 2, 4, 6}
Delta_1 (edges, aleph_1) dim 12 ker 0 spectrum { 2, 4, 6}
Delta_2 (faces, aleph_2) dim 6 ker 0 spectrum { 4, 6}
Delta_3 (cube, aleph_3) dim 1 ker 0 spectrum { 6}
Each level loses the bottom eigenvalue. Spectrum of level n = {2n, 2n+2, ..., 6}.
Step between levels = 2; ceiling = 6 = 2 x degree; number of levels = 6/2 + 1 = 4.
Zero mode exists only at level 0 — the ladder terminates downward.
Betti = [1,0,0,0], Witten index = 1 (solid cube is contractible).
Truncated at the graph: chi = V - E = 8 - 12 = **-4** <- the recorded ind(D).
Adding faces: chi = 8 - 12 + 6 = +2 (sphere). Adding the cube: +1 (ball).
So ind(D) = -4 is the Euler characteristic of the complex TRUNCATED one level
below the faces. The aleph -> aleph+1 transition = appending C^2; it moves chi
from -4 to +2.
### The framework IS Witten's SUSY quantum mechanics
Q = d + d*, Q^2 = dd* + d*d = Delta, ker Delta = cohomology = SUSY ground states,
Witten index = chi = ind(D). (Witten 1982.) This is 0+1-dimensional worldline
SUSY and implies NOTHING about spacetime superpartners.
---
## 8. Four independent walls at d^2 != 0 [DERIVED]
Every route out of the curvature obstruction is closed, each for a different reason:
1. **Matrix factorization (Q^2 = W*1)** — impossible. D strictly raises the level,
so D^2 raises by 2 while the identity preserves degree: D^2 is traceless
identically. A traceless scalar multiple must be zero. Scan of 4000 random
flux-torus points: deviation from a scalar = **1.000000 at every point**.
2. **Mod-2 folding** — tautological. D^2 raises by 2, so mod 2 it preserves parity
by construction. The "curvature lives in the even block" result was built in,
not discovered. (User caught this; correct.)
3. **Filtration / spectral sequence** — never starts. F_0 ⊂ F_1 ⊂ F_2 ⊂ F_3 with
dims 8, 20, 26, 27 — nesting is real and matches the "aleph+2 contains aleph+1"
picture. But page E_1 is d''-cohomology, and with ANY nonzero flux d''^2 != 0:
laminar 0.000000 (yes) | Phase I 2.964515 (NO) | tiny 0.05 0.074534 (NO)
4. **Inter-level coupling** — cancels exactly 0 %. Three stacked levels
(aleph-1 / aleph0 / aleph+1, dim 27^3):
lam / PhaseI / lam ||D^2|| = 80.041912
PhaseI / lam / PhaseI ||D^2|| = 113.196358
PhaseI x3 ||D^2|| = 138.636659
ratios 1 : sqrt2 : sqrt3 -> curvature adds IN QUADRATURE, never cancels
With genuine sequential coupling (higher spectral-sequence differentials,
bidegree (p+r, q-r+1), 504 free parameters, K4 base, rank 504/504 full):
curvature ||D0^2|| = 12.489996
residual after best K = 12.489996
fraction cancellable = 0.0000 %, optimal K = 0
Reason: a **bidegree selection rule**. Curvature sits at (p+2, q); coupling
contributions {D0, K} land at (p+r+1, q-r+1), never (p+2, q) for r >= 2.
Disjoint blocks — cancellation is structurally impossible, not merely hard.
**Reading:** in the sandbox model this is not an obstacle but the content —
soph-bombardment twists the hoop, twist = holonomy = flux = curvature. The model
PREDICTS the walls. d^2 = 0 is the 1.2 % initial condition that lives one step.
---
## 9. Bicomplex: matter carrying its own aleph-null [COMPUTED]
Two gradings = bicomplex; the tool is spectral sequences. Continuum (flat)
tensor matter level (flux phi), dim 27 x 27 = 729.
matter level gap stable state (p, q)
laminar 2.000 (0.00, 0.00)
Phase I 0.772 (0.00, 0.17)
2 parallel pi 0.586 (0.00, 0.27)
4 pi 0.322 (0.00, 0.35)
max vortex 0.236 (0.00, 0.42)
ker(Delta_total) = 1 always — exactly one stable state at any flux; a second
aleph-null does not create room for more.
The stable state is pinned at p = 0.00 in the continuum coordinate but bleeds
upward in the matter coordinate (0 -> 0.42), and **the gap collapses by a factor
of 8.5**. Stability is a continuous function of the internal vorticity, not binary.
Spectral-sequence motion d_r: (p,q) -> (p+r, q-r+1) — up in one grading, down in
the other. Verified: applying D twice moves <q> 1.01 -> 2.00 while <p> returns
to 1.04; spread falls 1.15 -> 0.72, so it FOCUSES rather than smears.
### 8.5 is not universal [X — see registry]
graph V E b1 gap(flat) gap(pi) ratio
K4 4 6 3 1.00000 0.42705 2.342
C6 6 6 1 1.00000 0.26795 3.732
K3,3 6 9 4 1.00000 0.90181 1.109
Q3 8 12 5 1.13612 0.33651 3.376
Petersen 10 15 6 1.09679 0.58158 1.886
Q4 16 32 17 0.58723 0.08504 6.905
Ratios 1.1 to 6.9. No normalization by degree, b1 or E flattens them.
**But the SHAPE is shared.** Normalized gap curves:
phi/pi: 0.00 0.12 0.25 0.38 0.50 0.62 0.75 0.88 1.00
K4 1.000 1.000 1.000 1.000 1.000 0.936 0.772 0.588 0.427
K3,3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.902
Petersen 1.000 0.999 0.997 0.993 0.986 0.939 0.795 0.655 0.530
Q3 1.000 0.999 0.990 0.925 0.785 0.637 0.504 0.390 0.296
C6 1.000 0.889 0.782 0.681 0.585 0.495 0.412 0.336 0.268
All except C6 (which has b1 = 1, a single cycle) show a PLATEAU then a collapse.
Protection is threshold-like, not a smooth trade-off.
### Inherited + emergent split (the user's proposed structure) — CONFIRMED
SVD of the five curves:
variance in the FIRST (inherited) component: 94.58 %
inherited shape A(phi):
phi/pi: 0.00 0.12 0.25 0.38 0.50 0.62 0.75 0.88 1.00
A: 0.000 0.062 0.125 0.211 0.330 0.477 0.664 0.842 1.000
emergent weight per graph: K4 -0.690 | C6 -1.353 | K3,3 -0.062 |
Q3 -1.132 | Petersen -0.584
95 % of the variance is ONE common curve; graphs differ only in amplitude.
A(phi) is flat to about 0.4pi then accelerates.
---
## 10. Wave fractal: the Hofstadter spectrum [COMPUTED, matches literature]
Critical Harper model, golden-mean flux approximants:
p q p/q spectral measure q * measure
8 13 0.615385 0.711605 9.25087
21 34 0.617647 0.252920 8.59928
55 89 0.617978 0.104821 9.32904
144 233 0.618026 0.040039 9.32901
measure ~ q^-1.002
box dimension at p/q = 377/610: D = 0.5074
Two independent checks land on known values: q*measure -> 9.3290 = **32*G/pi =
9.32993** (G = Catalan, the Thouless constant), and D_box ~ 0.5 for critical
Harper. So the code is right.
Cantor spectrum, measure zero, dimension ~0.5, multifractal — a genuine WAVE
fractal, in the flux variable, NOT exactly self-similar. The "golden period" is
literal: the golden mean is the worst-approximable flux (continued fraction
[1;1,1,...]), hence maximally far from rational resonances. Rational fluxes =
walls; Diophantine fluxes = stable segments; the golden one is the most stable.
## 10b. Menger sponge — for contrast [COMPUTED]
cubic torus 20^3 (control) N=8000 d_s = 3.091 <- method validated
Menger L3 N=8000 d_s = 2.311
Menger L4 N=160000 d_s = 2.371, modulation amplitude +/-0.13
d_H(Menger) = ln20/ln3 = 2.7268
Level-4 curve:
t: 5 11 21 41 78 120 185 285 438 675 1039 1599 2461
ds: 2.03 2.18 2.35 2.50 2.38 2.33 2.40 2.46 2.42 2.26 2.11 2.09 2.15
Clear LOG-PERIODIC oscillation (signature of discrete scale invariance), period
ratio ~8-11x against a prediction of 3^d_w ~ 12. **No monotone 4 -> 2 flow.**
Tested a two-sector version (eps = 0.02): modulation deepens, running does not appear.
**Conclusion:** exact self-similarity gives modulation around a plateau, not the
CDT-style running spectral dimension. Splitting self-similarity into classes
rescues you from strict scale invariance but does NOT by itself produce running.
The log-periodic amplitude is the observable that Calcagni's programme bounds.
---
## 11. Sandbox model: ring -> network [COMPUTED, exact]
Exact enumeration, 6 nodes, 6 edges, full transition matrix (54 moves/state):
total states 5005
pure rings 60 (1.20 %)
connected 3660
mean EXIT time from the ring 1.00 steps
mean RETURN time to the ring 83.4 steps (Kac: 1/pi(ring))
asymmetry break : rebuild = 1 : 83
**83.4 = 5005/60 exactly** — the asymmetry IS the state count. The arrow of time
in this model is the second law, derived rather than replaced.
### Does the ring gain more than it loses? YES, above a threshold
Correction to an earlier error: at fixed E = V = 6, b1 equals the component
count, so "more cycles" only meant "fragmented". Redone with free edge count —
soph removes at rate s, log adds at rate l; stationary state is exactly G(n, q)
with q = l/(l+s):
q <edges> P(connected) <cycle rank> P(connected & b1>=2)
0.20 3.00 0.0923 0.2096 0.0163
0.30 4.50 0.3169 0.7066 0.1203
0.40 6.00 0.5955 1.5632 0.3642
0.50 7.50 0.8149 2.7194 0.6637
0.70 10.50 0.9850 5.5154 0.9752
Ring starts at: 6 edges, connected, cycle rank 1.
Threshold **q* ~ 0.37**, i.e. l/s ~ 0.59.
> **Log does not need to win the exchange.** It needs to rebuild only about 0.6
> times as fast as soph destroys, and still comes out ahead. Because destruction
> removes one edge while addition creates a whole CYCLE — asymmetry of value,
> not of rate.
Caveat from literature check (R3): q = l/(l+s) is exactly the stationary
edge-occupation probability of an edge-Markovian random graph (Clementi et al.),
so the stationary ensemble is plain Erdos-Renyi. q* is a percolation threshold
and is MODEL-SPECIFIC. Finite-size scaling in n is the test that decides whether
it means anything.
### Tone
Tone of a form = fundamental period of its L/S sequence, read off the Fourier spectrum.
mode2 = LLLLLLSLLLLLLS -> all +1 except two dips spaced 7 apart
Fourier magnitudes: {0:10.0, 1:0.0, 2:4.0, 3:0.0, 4:4.0, 5:0.0, 6:4.0, 7:0.0}
peaks at even indices = multiples of 14/7 -> PERIOD 7, tone 1/7
Mode 3 has the SAME tone, opposite sign — that is their common property.
mode2 + mode3 = exact zero, correlation -1.0000. But this is by construction
(mode 3 was defined as minus mode 2), not a finding. To get a ripple the modes
must be phase-shifted or of different period, not exact antipodes.
## 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