-
Notifications
You must be signed in to change notification settings - Fork 11
/
Copy pathindex.html
592 lines (523 loc) · 23.4 KB
/
index.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>AXIO/1 2025</title>
<style>
body {
margin: 0;
height: 200vh; /* Make the page scrollable */
overflow-x: hidden;
font-family: Arial, sans-serif;
color: Black;
background-color: #EFEFFF;
}
/* Fixed Canvas Background */
.canvas-container {
position: fixed;
top: 0;
left: 0;
width: 100vw;
height: 100vh;
z-index: -1; /* Always behind other content */
}
/* Content on top */
.content {
position: relative;
z-index: 1;
padding: 50px;
}
</style>
</head>
<body>
<div class="canvas-container"></div>
<div class="content">
<aside>
<H1>AXIO/1</h1>
<h3>Artificial Experienced Intelligent Ontology v10.0 (2025)</h3>
<div>Groupoїd Infinity fibrational type systems for
mathematical representation, certification (theorem proving),
with extraction to verified interpreter and its runtime.
</div>
<div>
<ul>
<li>Українською: Штучна Досвідчена Інтелектуальна Онтологія</li>
<li>Тибетською: བཟོ་བཀོད་ཀྱི་ཉམས་ཡོད་པའི་རིག་པའི་ངོ་བོ་ལུགས། (bzo bkod kyi nyams yod pa'i rig pa'i ngo bo lugs)</li>
</ul>
</div>
<div>2015—2025 © <a href="https://5ht.co/license/">DHARMA LICENSE</a>
<br><br></div>
</aside>
<main>
<section>
<h2>Table of Contents (དཀར་ཆག dkar chag)</h2>
<ul>
<li>Introduction</li>
<li>Process</li>
<li>Components</li>
<li>Operators</li>
<li>Refinments</li>
<li>Goals</li>
<li>Runtime Languages</li>
<li>Higher Languages</li>
<li>Total Verification of Mathematics</li>
</ul>
<h2>1. Introduction (ངོ་སྤྲོད ngo sprod)</h2>
<p>The <strong>AXIO/1 Framework</strong> is a layered system for <strong>infinite reasoning</strong>, structured into:</p>
<ul>
<li><strong>Runtime Languages</strong>: Execute computations and manage concurrency.</li>
<li><strong>Higher Languages</strong>: Handle theorem proving and formal verification.</li>
</ul>
<p>This framework operates as a <strong>cyclic, iterative system</strong> for formal reasoning,
where an <strong>operator</strong> (human, AI, or hybrid) directs a process that continuously refines itself.</p>
<h2>2. Process (ལས་ཀ las ka)</h2>
<p>AXIO/1 follows a structured flow:</p>
<ol>
<li><strong>Conditions</strong>: Foundational elements (Axioms, Definitions, Types, Propositions, Syntax).</li>
<li><strong>Environment</strong>: The structured setting (Model, Consistency, Completeness, Library).</li>
<li><strong>Thinking</strong>: Reasoning mechanisms (Hypotheses, Computation, Deduction, Conjecture, Inference Rules, General Induction).</li>
<li><strong>Fruit</strong>: Logical results (Proof, Judgment, Theorem).</li>
<li><strong>Insight</strong>: Higher-level understanding (Semantics, Categorical Frameworks, Abstraction).</li>
</ol>
<h2>3. Components (ཆ་ཤས cha shas)</h2>
<h3>Condition (C) རྐྱེན Умова rkyen</h3>
<code>
C = (A, D, T, P, X)
</code>
<ul>
<li><strong>Axioms (A)</strong>: Fundamental truths.</li>
<li><strong>Definitions (D)</strong>: Precise descriptions of entities.</li>
<li><strong>Types (T)</strong>: Categorization of objects.</li>
<li><strong>Syntax (X)</strong>: Structural rules.</li>
</ul>
<h3>Environment (E) ཁོར་ཡུག Середовище khor yug </h3>
<code>
E = (M, C, K, L)
</code>
<ul>
<li><strong>Model (M)</strong>: Formal representation of the system.</li>
<li><strong>Consistency (C)</strong>: No contradictions within the system.</li>
<li><strong>Completeness (K)</strong>: The extent to which all truths can be derived.</li>
<li><strong>Library (L)</strong>: Repository of known results.</li>
</ul>
<h3>Reason (T) རྒྱུ Причина rgyu </h3>
<code>
T = (J, H, C, D, G)
</code>
<ul>
<li><strong>Judgment (J)</strong>: Logical assertions.</li>
<li><strong>Hypotheses (H)</strong>: Presupposition, Assumption, Supposition, Proposition.</li>
<li><strong>Computation (C)</strong>: Lambda Calculus, Pi-Calculus.</li>
<li><strong>Deduction (D)</strong>: Inference Rules, General Induction.</li>
<li><strong>Conjecture/Assertion (G)</strong>: Formulation of potential truths.</li>
</ul>
<h3>Fruit (F) འབྲས་བུ Плід 'bras bu</h3>
<code>
F = (⊢,Θ)
</code>
<ul>
<li><strong>Proof</strong> ⊢ Verified propositions.</li>
<li><strong>Theorem</strong> Θ Established truths.</li>
</ul>
<h3>Insight (I) ལྟ་བའི་ཤེས་པ lta ba'i shes pa</h3>
<code>
I = (S, C, A)
</code>
<ul>
<li><strong>Semantics</strong> Σ: Meaning assignment.</li>
<li><strong>Categorical Frameworks </strong> C: High-level abstractions..</li>
<li><strong>Abstraction</strong> A: Generalization of concepts.</li>
</ul>
<h2>3. Operators (བཀོལ་སྤྱོད་པ bkol spyod pa) </h2>
<p>Three types of operators drive the system:</p>
<ul>
<li><strong>Human</strong>: Chooses propositions, interprets insights, and guides conjectures.</li>
<li><strong>Machine</strong>: Automates computations, checks consistency, and suggests hypotheses.</li>
<li><strong>Hybrid</strong>: Human sets goals, machine executes reasoning steps.</li>
</ul>
<h2>4. Refinements (ལེགས་བཅོས legs bcos) </h2>
<p>Ensuring correctness and progression:</p>
<ul>
<li><strong>Infinite Thinking</strong>: Achieved via iteration <code>Sₙ → ∞</code>.</li>
<li><strong>Finite Steps</strong>: Each step is discrete, <code>Sₙ → Sₙ₊₁</code>.</li>
<li><strong>Operator-Driven</strong>: The direction of reasoning is controlled by <code>O</code>.</li>
</ul>
<p>The cycle repeats indefinitely, refining knowledge.</p>
<code>
S₀ → S₁ → S₂ → ... → Sₙ → Sₙ₊₁ → ...
</code>
<p>Where:</p>
<ul>
<li><code>Sₙ</code> is a finite reasoning step.</li>
<li><code>Sₙ₊₁</code> builds upon <code>Sₙ</code>, ensuring refinement.</li>
<li><strong>Limit process</strong>: <code>lim (n → ∞) Sₙ</code> represents <strong>infinite reasoning</strong>.</li>
</ul>
<h2>5. Design Goals (དམིགས་ཡུལ dmigs yul)</h2>
<ul>
<li><strong>Runtime Languages</strong>: Handle computation and concurrency.</li>
<li><strong>Higher Languages</strong>: Ensure theorem proving and soundness.</li>
<li><strong>Infinite Thinking</strong>: Achieved via refinements cycles.</li>
<li><strong>Operator-Driven</strong>: Collaboration between humans and machines.</li>
</ul>
</section>
<section>
<hr>
</section>
<section>
<p>Verified Lambda Interpreter and Concurrent Parallel Matrix Runtime.
Joe, Bob, and Alice languages share the same Standard ML like BNF grammar.</p>
</section>
<section>
<a name=Joe></a>
<h3><a href="https://github.com/groupoid/joe">Joe</a></h3>
<p><b style="sel">Joe</b> is a certified bytecode stack interpreter and Intel/ARM code compiler.</p>
<P>[1] — MinCaml<br>
[2] — CoqASM<br>
[3] — Verified LISP Interpreter<br>
[4] — Kind<br>
</p>
<figure><pre>
fun a (0, n) = n + 1
| a (m, 0) = a (m - 1, 1)
| a (m, n) = a (m - 1, a (m, n - 1))
</pre></figure>
</section>
<section>
<a name=Bob></a>
<h3><a href="https://github.com/o83/n2o">Bob</a></h3>
<p><b style="sel">Bob</b> is a parallel concurrent non-blocking
zero-copy run-time with CAS cursors [4,5].</p>
<P>[5] — Kernel<br>
[6] — Pony<br>
[7] — Erlang<br></p>
<figure><pre>
fun proc =
let val p0 = pub(0,8)
val s1 = sub(0,p0)
val s2 = sub(0,p0)
in send(p0,11);
send(p0,12);
[ receive(s1);
receive(s2);
receive(s1);
receive(s2)
]
end
</pre></figure>
</section>
<section>
<a name=Alice></a>
<h3><a href="https://tonpa.guru/stream/2023/2023-09-25%20Formal%20Tensor.htm">Alice</a></h3>
<p><b style="sel">Alice</b> is a linear types calculus
with partial fractions [6] for BLAS level 3 programming.</p>
<P>[8] — NumLin<br> </p>
<figure><pre>
fun simpleConvolution (i n: int) (x0: float) (write w: vector float)
: vector float
= begin
if n = i then result.emit(write),
a = [w0,w1,w2] = w.get(0,3),
b = [x0,x1,x2] = [ x0 | write.get(i,2) ],
write.set(i, Dotp(a,b)),
simpleConvolution((i + 1),n,x1,write,w)
end
</pre></figure>
</section>
<section>
<hr>
</section>
<section>
<p>Sound and Consistent Predicative Formal Languages.
Henk, Per, Anders, Dan languages share the same Lean like BNF grammar.</p>
</section>
<section>
<a name=Henk></a>
<h3><a href="https://github.com/groupoid/henk">Henk</a></h3>
<p><b style="sel">Henk</b> is a Pure Type System (PTS-91) in
the style of Coquand/Huet Calculus of Inductive Constructions (CoC-88)
with infinite numbere of universes. Henk also supports AUTOMATH syntax (AUT-68).</p>
<p>
AUT-68 — AUTOMATH<br>
CoC-88 — Calculus of Constructions<br>
PTS-91 — Pure Type System (Π)<br>
</p>
<figure><pre>
def N := Π (A : U), (A → A) → A → A
def zero : N := λ (A : U) (S : A → A) (Z : A), Z
def succ : N -> N := λ (n : N) (A : U) (S : A → A) (Z : A), S (n A S Z)
def plus (m n : N) : N := λ (A : U) (S : A → A) (Z : A), m A S (n A S Z)
def mult (m n : N) : N := λ (A : U) (S : A → A) (Z : A), m A (n A S) Z
def pow (m n : N) : N := λ (A : U) (S : A → A) (Z : A), n (A → A) (m A) S Z
</pre></figure>
</section>
<section>
<a name=Per></a>
<h3><a href="https://github.com/groupoid/per">Per</a></h3>
<p><b style="sel">Per</b> is a ΠΣ (MLTT-72) prover with Calculus of
Inductive Constructions and idenitity types (MLTT-75). The natural
extension of CoC to CIC was done by Frank Pfenning and Christine Paulin (IND-89).</p>
<p>
Mini-TT — OCaml implementation<br>
MLTT-72 — Pi, Sigma<br>
MLTT-75 — Pi, Sigma, Id<br>
MLTT-80 — 0, 1, 2, W, Pi, Sigma, Id<br>
PP-89 — Inductively Defined Types<br>
CIC-2015 — Calculus of Inductive Constructiions<br>
</p>
<figure><pre>
def empty : U := inductive { }
def L¹ (A : U) : U := inductive { nil | cons (head: A) (tail: L¹ A) }
def S¹ : U := inductive { base | loop : Equ S¹ base base }
def quot (A: U) (R : A -> A -> U) : U
:= inductive { quotient (a: A)
| identification (a b: A) (r: R a b)
: Equ (quot A R) (quotient a) (quotient b)
}
</pre></figure>
</section>
<section>
<a name=Anders></a>
<h3><a href="https://github.com/groupoid/anders">Anders</a></h3>
<p><b style="sel">Anders</b> is a Homotopy Type System (HTS-2013)
with Strict Equality and Cubical Agda (CCHM-2016) primitives.</p>
<p>
HTS-2013 — Homotopy Type System<br>
BCH-2014 — Cubical Sets<br>
CCHM-2015 — Cubical Type System<br>
OP-2016 — Topos Axioms<br>
CHM-2017 — Huber Equations<br>
VMA-2017 — Cubical Agda<br>
</p>
<figure><pre>
def idfun (A : U) : A → A := λ (a : A), a
def idfun′ (A : U) : A → A := transp (<i> A) 0
def idfun″ (A : U) : A → A := λ (a : A), hcomp A 0 (λ (i : I), []) a
def isFiberBundle (B: U) (p: B → U) (F: U): U
:= Σ (v: U) (w: surjective v B), (Π (x: v), PathP (<_>U) (p (w.1 x)) F)
def ~~ (X : U) (a x′ : X) : U := Path (ℑ X) (ι X a) (ι X x′)
def 𝔻 (X : U) (a : X) : U := Σ (x′ : X), ~~ X a x′
def unitDisc (X : U) (x : ℑ X) : U := Σ (x′ : X), Path (ℑ X) x (ι X x′)
def starDisc (X : U) (x : X) : 𝔻 X x := (x, idp (ℑ X) (ι X x))
def T∞ (A : U) : U := Σ (a : A), 𝔻 A a
def inf-prox-ap (X Y : U) (f : X → Y) (x x′ : X) (p : ~~ X x x′)
: ~~ Y (f x) (f x′) := <i> ℑ-app X Y f (p @ i)
def d (X Y : U) (f : X → Y) (x : X) (ε : 𝔻 X x) : 𝔻 Y (f x)
:= (f ε.1, inf-prox-ap X Y f x ε.1 ε.2)
def T∞-map (X Y : U) (f : X → Y) (τ : T∞ X) : T∞ Y
:= (f τ.1, d X Y f τ.1 τ.2)
def is-homogeneous (A : U)
:= Σ (e : A) (t : A → equiv A A),
Π (x : A), Path A ((t x).1 e) x
</pre></figure>
</section>
<section>
<a name=Daniel></a>
<h3><a href="https://github.com/groupoid/dan">Dan</a></h3>
<p><b style="sel">Dan</b> is a simplicial CCHM-based verification system with
Simplicial, Simplex, Chain, Monoid, Category, Group primitives built into the type checker core.
Dan is new Rzk/GAP replacement with Kan, Rezk and Segal simplicial
modes for computable ∞-categories.
</p>
<p>
R-HoTT — Rezk Infinity Categories<br>
Hopf-HoTT — Hopf Fibrations, Stable Spherical Homotopy Groups<br>
</p>
<figure><pre>
def path_z2_category : Category
:= П (x y : Simplex),
(f g h : Simplex),
(z2 : Group(П (e a : Simplex), a² = e ⊢ 1 (a | a² = e))),
f ∘ g = h
⊢ 2 (x y | f g h | f ∘ g = h)
def z3 : Group
:= П (e a : Simplex),
a³ = e
⊢ 1 (a | a³ = e)
def Möbius : Simplex
:= П (a b c : Simplex),
(bc ac : Simplex), ab = bc ∘ ac
⊢ 2 (a b c | bc ac ab)
</pre></figure>
</section>
<section>
<a name=Jack></a>
<h3><a href="https://github.com/groupoid/jack">Jack</a></h3>
<p><b style="sel">Jack</b> is a Minimal Framework for Homotopy Groups of Spheres
which encompasses unstable homotopy, stable homotopy (e.g., π₀^S(S⁰) = ℤ),
and chromatic phenomena (e.g., H^*(RP^2), spectral sequences),
inspired by Morava's chromatic vision. It features HopfFibⁿ (n=1,2,3,4),
Susp(A), Truncⁿ(A), ℕ, ℕ∞, Π(x:A).B, Σ(x:A).B, Id_A(u, v), Spec, πₙ^S(A),
S⁰[p], Group, A ∧ B, [A, B], Hⁿ(X; G), G ⊗ H, SS(E, r). It has
Sⁿ, πₙ(Sᵐ), K(G, n), Cohomology Rings, Chromatic Towers as derivables.
</p>
<a name=Urs></a>
<h3><a href="https://github.com/groupoid/urs">Urs</a></h3>
<p><b style="sel">Urs</b> is a Framework for Supergeometry in Cohesive Topos.
It features Graded Universes, Graded Tensor, Group Action, Super Type Theory: Uᵍᵢ| 𝖘 A | 𝔾 → A,
Super Modality: Γ ⊢ A : Uᵢ^g → Γ ⊢ 𝖘 A : Uᵢ^g, Cohesive Type Theory: ∣ ʃ ∣ ♭ ∣ ♯ ∣ ℑ | & | ℜ .</P>
<a name=Laurent></a>
<h3><a href="https://github.com/groupoid/laurent">Laurent</a></h3>
<p><b style="sel">Julius</b> is a type system for Functional Analysis and Calculus.
It features ℝ, C, Nat, Boo, Forall, Exists, Set, Measure, Lebesgue. Seq, Inf, Sup, Lim.
for Real, Complex Functional Analysis, and Calculus</P>
<a name=Ernst></a>
<h3><a href="https://github.com/groupoid/ernst">Ernst</a></h3>
<p><b style="sel">Ernst</b> is a type system for ZFC LEM theories.
It features: 𝑉, Pow(𝐴), 𝑥 ∈ 𝐴, 𝐴 ⊆ 𝐵; LEM: ⊢ 𝑃 ∨ ¬𝑃 for Classical Logic Support.</p>
<a name=Paul></a>
<h3><a href="https://github.com/groupoid/paul">Paul</a></h3>
<p><b style="sel">Paul</b> is a type system for Forced Cardinals.
It features: ⊢ 𝜅 : Card, inaccessible(𝜅), measurable(𝜅), Force(𝑃, 𝐺) : 𝑉 → 𝑉, 𝑝 ⊩ 𝜙,
Generic filter 𝐺 over a poset 𝑃, yielding a new model 𝑉[𝐺], allowing for
adjoin reals and control cardinalities or axioms.</p>
<a name=Fabien></a>
<h3><a href="https://github.com/groupoid/fabien">Fabien</a></h3>
<p><b style="sel">Fabien</b> is a Motivic A^1-Homotopy Theory.
It featues Π,Σ,Path,𝑘:𝑈,0_𝑘,1_𝑘,point_𝑘,𝐴^1:U,point:𝑘→𝐴^1., A^1-contr,
𝐿_{A^1}:U→𝑈, 𝜂_{A^1}, rec_{A^1}, n-Trunc, 𝑁, Suspension,S^{1,1}, Nisnevich Cover. It derives all
structural theorems of A^1-Homotopy Theory—such as A^1-connectivity (X×A^1)≅π_n(A^1),
contractibility of 𝐴^1, and unstable connectivity — while providing a foundation
for stable A^1-homotopy via suspensions and motivic spheres. Explicitly
supporting Nisnevich descent, aligning L_{A^1} with sheaf-like properties,
while keeping k as a placeholder.</p>
<figure><pre>
def k : U
def 0_k : k
def 1_k : k
def A1 : U := inductive { point : k → A1 }
def 0_A1 : A1 := A1.point 0_k
def 1_A1 : A1 := A1.point 1_k
def A^1-contr (a : A1) : Path A1 a 0_A1 := <i> comp A1 [j : I] a (i ∧ j) 0_A1
def L_A1 (X : U) : U := (i : I) → X
def isA1Local (Y : U) : U
:= isEquiv (λ (y : Y) (a : A1). y : Y × A1 → Y)
def eta_A1 (X : U) (x : X) : L_A1 X := <i> x
def rec_A1 (X Y : U) (f : X → Y) (loc : isA1Local Y) : L_A1 X → Y
:= λ z => comp Y [i : I] (f (z i)) (f (z 0))
def S11 : U
:= inductive { base : S11
| loop : A1 → Path S11 base base
| zero : Path S11 (loop 0_A1) (refl base)
}
def NisCover (X : U) : U
:= inductive {
| triv : X → NisCover X
| cover : (U : U) → (f : U → X) → isNisCover f → NisCover X
}
def isSurjective (X Y : U) (f : X → Y) : Type fibrant
:= ∀ (y : Y), ∃ (x : X), Path Y (f x) y
def isNisCover (X U : U) (f : U → X) : U
:= isSurjective f
def rec_NisCover
(X : U) (C : NisCover X → U)
(t : ∀ (x : X), C (NisCover.triv x))
(d : ∀ (U : U) (f : U → X) (h : isNisCover f), C (NisCover.cover U f h))
(c : NisCover X) : C c
:= match c with
| NisCover.triv x => t x
| NisCover.cover U f h => d U f h
def L_A1-NC (X : U) : U
:= Σ (x : (i : I) → X),
∀ (c : NisCover X),
Path X (rec_NisCover (λ x => x) (λ U f h => f (h x).1) c) (x 0)
def isA1Local-NC (X : U) : U
:= ∀ (c : NisCover X),
isEquiv (rec_NisCover (λ x => x) (λ U f h => f) c)
def eta_A1 (X : U) (x : X) : L_A1-NC X
:= ((<i> x), λ c => refl x)
</pre></figure>
</section>
<section>
<H2>Total Verification of Mathematics</H2>
<p><b>Axiomatic Extended Integrated Ordered System for Infinite Structures (AXIOSIS)</b>
is a novel type theory engineered to mechanically verify all existing
theorems across mathematics, from classical analysis to modern set
theory and homotopy. Building on top of advanced frameworks:</p>
<p><ul>
<li><b>Henk Barendregt</b> Type Theory for Pure Dependent Lambda Calculus,</li>
<li><b>Per Martin-Löf</b> Type Theory for Fibrational setting and inductive types,</li>
<li><b>Anders Mörtberg</b> Type Theory for cubical CCHM/CHM/HTS flavor,</li>
<li><b>Dan Kan</b> Simplicial Homotopy Type Theory with Kan, Rezk, Saegal simplicial sets,</li>
<li><b>Jack Morava</b> Type Theory for Chromatic Homotopy Theory and K-Theory,</li>
<li><b>Urs Schreiber</b> Type Theory for Equivariant Supergeometry,</li>
<li><b>Fabien Morel</b> Type Theory for A¹-homotopy theory,</li>
<li><b>Julius Dedekind</b> Type Theory for Reals,</li>
<li><b>Ernst Zermelo</b> Type Theory for ZFC with LEM, and</li>
<li><b>Paul Cohen</b> Type Theory for cardinals system incorporating large cardinals and forcing;</li></ul></p>
<p>this system synthesis unifies synthetic homotopy, stable homotopy spectra, cohesive geometry, real analysis,
and set-theoretic foundations into a single, computationally verifiable formalism. We demonstrate its
power through key theorems:</p>
<p><ul>
<li>Number Theory: Prime Number Theorem,</li>
<li>Fundamental Theorem of Calculus (Analysis),</li>
<li>Analysis: Lebesgue Dominated Convergence Theorem,</li>
<li>Topology: Poincaré Conjecture (3D),</li>
<li>Algebra: Classification of Finite Simple Groups,</li>
<li>Set Theory: Independence of the Continuum Hypothesis (CH),</li>
<li>Category Theory: Adjoint Functor Theorem,</li>
<li>Homotopy Theory: Adams Conjecture (via K-theory),</li>
<li>Cohesive Homotopy Theory: Brouwer Fixed-Point Theorem,</li>
<li>Consistency of ZFC with Large Cardinals,</li>
<li>Fermat’s Last Theorem,</li>
<li>Large Cardinal Theorem: Martin’s Maximum;</li></ul>
<p>showcasing its ability to span algebraic, analytic, topological, and
foundational domains. The verification systems stands as a candidate for a universal mechanized mathematics platform,
rivaling systems like Cubical Type Theory while extending their scope.
The system achieves a landmark synthesis, unifying synthetic and classical mathematics in a mechanically
verifiable framework. Its type formers—spanning simplicial ∞-categories, stable spectra, cohesive modalities,
reals, ZFC, large cardinals, and forcing — cover all known mathematical domains as of 2025.</p>
<a name=Mono></a>
<h3>Monography as Introductory Course</h3>
<ul><li>Published by Axiosis: <a href="https://axiosis.github.io/books/axio/axio.pdf">axiosis.github.io/books/axio/axio.pdf</a></li>
<li>AXIO/AXIOSIS Github Organization: <a href="https://github.com/groupoid/">github.com/groupoid/</a></li></ul>
</section>
<section>
<h3>LaTeX</h3>
<figure><pre>
$ cp *.ttf ~/.local/share/fonts
$ sudo apt install texlive-full
$ sudo fc-cache -f
$ fc-match Geometria
$ make
</pre></figure>
<h3>Sole Copyright</h3>
<p>Namdak Tonpa</p>
<section>
</main>
<script src="https://cdnjs.cloudflare.com/ajax/libs/three.js/r128/three.min.js"></script>
<script>
const scene = new THREE.Scene();
const camera = new THREE.PerspectiveCamera(75, window.innerWidth / window.innerHeight, 0.1, 1000);
const renderer = new THREE.WebGLRenderer({ antialias: true, alpha: true });
renderer.setSize(window.innerWidth, window.innerHeight);
document.querySelector(".canvas-container").appendChild(renderer.domElement);
const starGeometry = new THREE.BufferGeometry();
const starCount = 1500;
const starVertices = [];
for (let i = 0; i < starCount; i++) {
let x = (Math.random() - 0.5) * 1000;
let y = (Math.random() - 0.5) * 1000;
let z = (Math.random() - 0.5) * 1000;
starVertices.push(x, y, z);
}
starGeometry.setAttribute('position', new THREE.Float32BufferAttribute(starVertices, 3));
const starMaterial = new THREE.PointsMaterial({ color: 0xFFFFFF, size: 2 });
const stars = new THREE.Points(starGeometry, starMaterial);
scene.add(stars);
camera.position.z = 300;
function animate() {
requestAnimationFrame(animate);
stars.rotation.y += 0.001;
renderer.render(scene, camera);
}
function onResize() {
renderer.setSize(window.innerWidth, window.innerHeight);
camera.aspect = window.innerWidth / window.innerHeight;
camera.updateProjectionMatrix();
}
window.addEventListener("resize", onResize);
animate();
</script>
</body>
</html>