القائمة
Homological algebra via chain complexes, derived functors (Tor/Ext), triangulated categories, and spectral sequences
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npx skills add https://github.com/plurigrid/asi --skill derived-categoriesيبقى الأمر في سطر واحد. مرّر أفقيًا لمراجعته كاملًا قبل النسخ.
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Compressive Vasomotion Hypothesis (CVH) — vasomotion as a fast compression sweep that collapses ambivalent neural resonances (the Bayesian-blur problem) into a definite state. Use when modeling the ~100ms taṇhā 'grab', precision-weighting as compression forcefulness, or the generative collapse step of vasocomputational active inference.
Latched Hyperprior Hypothesis (LHH) — a sustained vascular contraction engages the smooth-muscle latch-bridge, durably freezing a circuit as a committed hyperprior isolated from global updating; unlatches when its prediction resolves. Use when modeling durable commitments, trauma/PTSD as cemented priors, latch spirals, or the cross-substrate latch (mechanical/bioelectric/immune/sheaf).
Synaptic substrate of biological active inference — long-term potentiation/depression (LTP/LTD) writes priors into synaptic weights = the learning landscape (Deep CANALs). A held vascular latch annealed long enough crystallizes into a neuron prior. Use when modeling consolidation, neuron priors, the inference→learning landscape write-path, or commit-to-disk of a held prediction.
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عرض SKILL.md
| name | derived-categories |
| description | Homological algebra via chain complexes, derived functors (Tor/Ext), triangulated categories, and spectral sequences |
| version | 1.0.0 |
Status: Production Ready Trit: -1 (MINUS - validator) Color: #8026D8 (Purple) Principle: Derived categories are the correct framework for homological invariants Frame: D(BCI) = derived category of chain complexes of BCI signals
Derived Categories provide the homological algebra infrastructure for the BCI pipeline. All cohomology theories (L8, L13, L14, L17, L19) live naturally in the derived category. Implements:
Correct by construction: d^2 = 0 verified computationally at all levels.
Chain complex C_*:
... -> C_n -d_n-> C_{n-1} -d_{n-1}-> C_{n-2} -> ...
d_{n-1} o d_n = 0 (boundary of boundary is zero)
Homology:
H_n(C) = ker(d_n) / im(d_{n+1})
dim H_n = dim ker(d_n) - rank(d_{n+1})
Euler characteristic:
chi = sum (-1)^n dim H_n = sum (-1)^n dim C_n
Derived functors:
Tor_n(A,B) = L_n(- tensor B)(A) (left derived of tensor)
Ext^n(A,B) = R^n Hom(A,-)(B) (right derived of Hom)
Triangulated category D(A):
Shift: C[1]_n = C_{n-1}
Distinguished triangle: X -> Y -> Cone(f) -> X[1]
Octahedral axiom (TR4): composition coherence
Long exact sequence:
... -> H_n(A) -> H_n(B) -> H_n(C) -delta-> H_{n-1}(A) -> ...
Spectral sequence:
E_r^{p,q} with d_r: E_r^{p,q} -> E_r^{p-r,q+r-1}
E_{r+1} = H(E_r, d_r)
Convergence: E_infinity^{p,q} => H^{p+q}
Build and verify chain complexes:
(def d3 [[1.0 -1.0 0.0] [0.0 1.0 -1.0] [1.0 0.0 -1.0]])
(def d2 [[-1.0 -1.0 1.0] [-2.0 -2.0 2.0] [-1.0 -1.0 1.0]])
;; Verify: (mat-mul d2 d3) = zero-matrix
Tor and Ext computation:
(defn compute-tor [chain-a chain-b]
;; Tor_0 = A tensor B, Tor_1 = flatness obstruction
...)
(defn compute-ext [chain-a chain-b]
;; Ext^0 = Hom(A,B), Ext^1 = extension obstruction
...)
Distinguished triangles and shift:
(defn shift-complex [chain n]
(vec (for [i (range (count chain))]
(nth chain (mod (+ i n) (count chain))))))
(defn cone [f-chain g-chain]
(vec (concat g-chain (shift-complex f-chain 1))))
Page-by-page computation:
(defn spectral-page [signals r]
(case r
0 signals ;; E_0 = associated graded
1 (successive-differences) ;; E_1 = homology of E_0
2 (second-differences) ;; E_2 = homology of E_1
(degenerate))) ;; E_r = E_infinity for r >= 2
BCI Chain Complex:
C_3 (dim 3) -> C_2 (dim 3) -> C_1 (dim 3) -> C_0 (dim 1)
d^2 = 0: VERIFIED at all levels (max |entry| = 0.000000)
Homology: H_0 = H_1 = H_2 = H_3 = 0 (acyclic complex)
Euler characteristic: chi = 0
Derived Functors (world pairs):
Tor_1(a,c) = 1 (non-flat pair, variance 0.043)
Tor_1(b,c) = 1 (non-flat pair, variance 0.022)
Tor_1(a,b) = 0 (flat pair)
Triangulated Structure:
3 distinguished triangles constructed
All cone ratios within quasi-iso range
Spectral Sequences:
Degeneration at E_2 for all worlds
World-b: immediate degeneration (uniform signals)
World-c: non-trivial E_1 page (signal diversity)
Completes the Homological Chain: L8 -> L13 -> L14 -> L21
Skill Name: derived-categories Type: Chain Complexes / Derived Functors / Triangulated Categories / Spectral Sequences Trit: -1 (MINUS) GF(3): Forms valid triads with PLUS + ERGODIC skills
operadic-composition (+1) x information-geometry (0) x derived-categories (-1) = 0
stochastic-resonance (+1) x spectral-methods (0) x derived-categories (-1) = 0
Part of: para-mensch-commons (CT lattice family).