"""
Zero Paradox — ZP-G: Category Theory PDF Builder
Version 1.11 | May 2026
v1.11: Version line style fixed (bodyI → subtitle); local make_doc override removed.
v1.10: Hash sync — script was modified without full workflow; rebuilt to bring
hash into alignment with register.md.
v1.9: Version number removed from Open Items Register section header.
v1.8: "Internal Working Document" marker removed from footer and endnote — not a working
document; removing public-facing scaffolding label.
v1.7: Remark R-AX added after AX-G2 — addresses non-triviality of the axioms; explains that
the initial object is not an abstract placeholder but is grounded in ZP-H via four concrete
domain functors; distinguishes categorical generalisation from trivial initial-object renaming.
v1.6: D7' well-definedness note corrected — invariance scoped to what ZP-G claims
(finite/zero/undefined), not asserted as a general AIT property.
v1.5: T6-b/T6-c Lean scope disclosure strengthened — Lean proofs verify only
Nat.zero_le _ (non-negativity by type); K-theoretic content not Lean-verified.
v1.4: Lean scope note added after T6-c.
v1.3: Remark R2 added (Categorical Expression of Self-Containment).
v1.2: Theorem/Proposition/Lemma hierarchy applied throughout.
v1.1: D7 replaced by D7' (native Kolmogorov surprisal); BA-G1 demoted to Remark.
v1.0: Initial release.
"""
import os
from zp_utils import *
VERSION = '1.11'
# ZP-G uses a slightly different amber shade; override zp_utils default
AMBER = colors.HexColor('#B07800') # ZP-OVERRIDE: ZP-G import_box label text uses darker amber
# ZP-G import box uses amber-colored label text (not white)
S['labelAmber'] = ParagraphStyle('labelAmber', fontName='DV-B', fontSize=9, leading=13,
textColor=AMBER)
def _box(title, title_style, hdr_bg, status, status_bg, rows):
"""ZP-G box with a status row between header and content rows."""
data = [
[Paragraph(fix(title), S[title_style])],
[Paragraph(fix(status), S['cellI'])],
]
for r in rows:
data.append([Paragraph(fix(r), S['cell'])])
ts = TableStyle([
('BACKGROUND', (0,0), (-1,0), hdr_bg),
('BACKGROUND', (0,1), (-1,1), status_bg),
('BACKGROUND', (0,2), (-1,-1), GREY_LITE),
('BOX', (0,0), (-1,-1), 0.5, hdr_bg),
('LINEBELOW', (0,0), (-1,0), 0.5, hdr_bg),
('LINEBELOW', (0,1), (-1,-2), 0.5, colors.HexColor('#CCCCCC')),
('TOPPADDING', (0,0), (-1,-1), 5),
('BOTTOMPADDING', (0,0), (-1,-1), 5),
('LEFTPADDING', (0,0), (-1,-1), 8),
('RIGHTPADDING', (0,0), (-1,-1), 8),
('VALIGN', (0,0), (-1,-1), 'TOP'),
])
t = Table(data, colWidths=[TW], repeatRows=1)
t.setStyle(ts)
return t
def def_box(title, status, rows): return _box(title, 'label', BLUE, status, BLUE_LITE, rows)
def result_box(title, status, rows): return _box(title, 'label', GREEN, status, GREEN_LITE, rows)
def axiom_box(title, status, rows): return _box(title, 'label', ORANGE, status, ORANGE_LITE, rows)
def remark_box(title, status, rows): return _box(title, 'label', SLATE, status, SLATE_LITE, rows)
def import_box(title, status, rows):
"""ZP-G import box — amber background, amber-text header (not white)."""
data = [
[Paragraph(fix(title), S['labelAmber'])],
[Paragraph(fix(status), S['cellI'])],
]
for r in rows:
data.append([Paragraph(fix(r), S['cell'])])
ts = TableStyle([
('BACKGROUND', (0,0), (-1,-1), AMBER_LITE),
('BOX', (0,0), (-1,-1), 0.5, AMBER),
('LINEBELOW', (0,0), (-1,0), 0.5, AMBER),
('LINEBELOW', (0,1), (-1,-2), 0.5, colors.HexColor('#CCCCCC')),
('TOPPADDING', (0,0), (-1,-1), 5),
('BOTTOMPADDING', (0,0), (-1,-1), 5),
('LEFTPADDING', (0,0), (-1,-1), 8),
('RIGHTPADDING', (0,0), (-1,-1), 8),
('VALIGN', (0,0), (-1,-1), 'TOP'),
])
t = Table(data, colWidths=[TW], repeatRows=1)
t.setStyle(ts)
return t
def build():
out_path = os.path.join(PROJECT_ROOT, 'ZP-G_Category_Theory.pdf')
print(f'[build_zpg] Output: {out_path}')
doc = make_doc(out_path, 'ZP-G: Category Theory', 'ZP-G: Category Theory', 'Version ' + VERSION, date_str='May 2026')
E = []
print('[build_zpg] Building title block...')
# ── TITLE BLOCK ───────────────────────────────────────────────────────────
E += [
sp(12),
Paragraph('THE ZERO PARADOX', S['title']),
Paragraph('ZP-G: Category Theory', S['subtitle']),
Paragraph('Version ' + VERSION + ' | May 2026', S['subtitle']),
sp(8),
hr(),
sp(4),
]
# ── INTRO PARAGRAPHS ──────────────────────────────────────────────────────
E += [
body('This document is self-contained within category theory, with one explicit import from '
'ZP-C: the conditional Kolmogorov complexity K(x|n) and the coding theorem connecting '
'it to Shannon entropy. That import is named and labelled; it replaces Bridge '
'Axiom BA-G1 (Leinster categorical entropy characterization) with a native derivation. All '
'other content from ZP-A, ZP-B, ZP-D, and ZP-E remains excluded from this document. '
'Cross-framework connections are deferred to ZP-H.'),
body('Honest labelling is the governing discipline. Every claim is marked as Axiom, Definition, '
'Derived, Import, Design Commitment, or Remark. Nothing slides between categories.'),
body('Illustrated Companion: A paired ZP-G Illustrated Companion provides concrete examples '
'and visual intuitions for the results here. Examples are kept separate from the formal '
'layers to distinguish illustrative material from proofs.', 'bodyI'),
sp(4),
hr(),
]
print('[build_zpg] Building Section I: Categorical Primitives...')
# ── I. CATEGORICAL PRIMITIVES ─────────────────────────────────────────────
E.append(Paragraph('I. Categorical Primitives', S['h1']))
E.append(body('Definitions D1 through D6 and results T1 through T5 are reproduced here for completeness and self-reference.'))
E.append(Paragraph('1.1 The Definition of a Category', S['h2']))
E.append(def_box(
'Definition D1 — Category',
'Status: Definition — foundational',
[
'A category C consists of:',
'(i) Objects: A collection ob(C), written A, B, X, 0, ...',
'(ii) Morphisms: For each ordered pair (A, B), a collection hom(A, B) of morphisms, written f: A → B.',
'(iii) Composition: ∘: hom(B,C) × hom(A,B) → hom(A,C), written g ∘ f.',
'(iv) Identity: For each A, a morphism idA: A → A.',
'Associativity: h ∘ (g ∘ f) = (h ∘ g) ∘ f.',
'Unit laws: idB ∘ f = f = f ∘ idA.',
]
))
E.append(sp(6))
E.append(def_box(
'Definition D2 — Morphism Uniqueness Notation',
'Status: Definition',
[
'A morphism f: A → B is unique if for any g, h: A → B, g = h. Written: ∃! f: A → B.',
]
))
E.append(sp(8))
E.append(Paragraph('1.2 Initial and Terminal Objects', S['h2']))
E.append(def_box(
'Definition D3 — Initial Object',
'Status: Definition — load-bearing',
[
'An object 0 ∈ ob(C) is initial if for every X ∈ ob(C), there exists a unique morphism '
'ιX: 0 → X.',
]
))
E.append(sp(6))
E.append(result_box(
'Proposition T1 — Uniqueness of the Initial Object',
'Status: Derived — standard category theory [unchanged from v1.0]',
[
'If 0 and 0\' are both initial in C, ∃! isomorphism 0 ≅ 0\'. '
'The initial object is unique up to unique isomorphism.',
'Proof: ∃! f: 0 → 0\' and ∃! g: 0\' → 0. Initiality forces g ∘ f = id0 and '
'f ∘ g = id0\'. Therefore f is a unique isomorphism. ✓',
]
))
E.append(sp(6))
E.append(def_box(
'Definition D4 — Terminal Object',
'Status: Definition — defined for exclusion',
[
'An object 1 ∈ ob(C) is terminal if for every X ∈ ob(C), ∃! morphism '
'τX: X → 1. The Zero Paradox framework is not a zero object category (0 ≇ 1).',
]
))
print('[build_zpg] Building Section II: Foundational Axioms...')
# ── II. THE FOUNDATIONAL AXIOMS OF ZP-G ──────────────────────────────────
E.append(Paragraph('II. The Foundational Axioms of ZP-G', S['h1']))
E.append(axiom_box(
'Axiom AX-G1 — Asymmetry Axiom',
'Status: Axiom — foundational structural commitment [unchanged from v1.0]',
[
'The category C possesses an initial object 0 and no terminal object.',
'Formally: ∃ 0 ∈ ob(C) satisfying D3. ¬∃ 1 ∈ ob(C) satisfying D4.',
'Correspondence: In ZP-A: join-semilattice without ⊤ and without ∧. '
'In ZP-B: Q2 has no element to which all paths converge. '
'The present axiom is the categorical generalization.',
]
))
E.append(sp(6))
E.append(axiom_box(
'Axiom AX-G2 — Source Asymmetry',
'Status: Axiom — foundational [unchanged from v1.0]',
[
'For any non-initial object X ≠ 0: hom(X, 0) = ∅.',
'Motivation: The categorical expression of irreversibility. Morphisms '
'ιX: 0 → X exist for all X. Their reversal does not exist. '
'AX-G2 is consistent with AX-G1 but not derivable from it.',
]
))
E.append(remark_box(
'Remark R-AX — On the Non-Triviality of AX-G1 and AX-G2',
'Status: Remark [new in v1.7]',
[
'AX-G1 and AX-G2 are satisfied by many categories — they are structural conditions, not exotic ones. '
'What distinguishes ZP-G from a trivial application of initial-object asymmetry is that the initial '
'object here is not an abstract placeholder: it is ⊥, the algebraically minimal element of '
'ZP-A\'s lattice. This identification is not asserted in ZP-G; it is demonstrated in ZP-H via '
'four concrete domain functors (FA, FB, FC, FD), '
'each of which maps the natural number depth hierarchy into its domain category and preserves '
'the initial object. The axioms are not postulated in isolation; they are shown to hold in each '
'of the four domain categories that constitute the framework\'s subject matter. The categorical '
'layer generalises a phenomenon that is independently grounded in four distinct mathematical domains.',
]
))
print('[build_zpg] Building Section III: Universal Constituent and Unreachability...')
# ── III. UNIVERSAL CONSTITUENT AND UNREACHABILITY ─────────────────────────
E.append(Paragraph('III. Universal Constituent and Unreachability', S['h1']))
E.append(result_box(
'Lemma T2 — Universal Constituent',
'Status: Derived — from D3 and AX-G1 [unchanged from v1.0]',
[
'For every X ∈ ob(C), ∃! ιX: 0 → X. '
'The initial object 0 is the universal categorical source.',
'Proof: Immediate from D3 and AX-G1. ✓',
]
))
E.append(sp(6))
E.append(result_box(
'Lemma T3 — Unreachability of 0',
'Status: Derived — from AX-G2 [unchanged from v1.0]',
[
'For any X ≠ 0: hom(X, 0) = ∅. The initial object 0 is unreachable from any non-initial object.',
'Proof: Direct from AX-G2. ✓',
]
))
E.append(sp(6))
E.append(remark_box(
'Remark R1 — Structural Inversion — The Categorical Zero Paradox',
'Status: Remark [unchanged from v1.0]',
[
'T2 and T3 together constitute the categorical Zero Paradox. 0 reaches every object (T2); '
'no non-initial object reaches 0 (T3). This is not a logical contradiction. It is a structural '
'inversion: the unique universal source is the unique object with no incoming non-trivial morphisms.',
]
))
E.append(sp(6))
E.append(remark_box(
'Remark R2 — Categorical Expression of Self-Containment',
'Status: Remark — connecting note to ZP-A CC-2 [new in v1.3]',
[
'R1 frames the structural inversion of 0. This remark connects that structure to ZP-A CC-2: '
'⊥ = {⊥}. The null state is its own extension — a Quine atom. A self-containing object has '
'no external interpreter by structure; it IS its own interpretation.',
'In categorical terms, this corresponds to two conditions together: (1) AX-G2: '
'hom(X, 0) = ∅ for all X ≠ 0 — no morphism can reach inside 0 from outside; and '
'(2) T2: ∃! ιX: 0 → X for all X — 0 is structurally present in every object. '
'Together these are the categorical image of undifferentiated self-containment: unreachable '
'from without, yet constitutive of everything.',
'0 "points in all directions" (T2) because it is the undifferentiated ground from which all '
'differentiation proceeds — not because it selects a direction. The uniqueness of each '
'ιX is not a choice among alternatives; it is the absence of internal '
'structure that would allow differentiation among morphisms.',
'This remark bridges ZP-G to ZP-A CC-2. The formal correspondence between the categorical '
'initial object structure and the set-theoretic Quine atom ⊥ = {⊥} is made explicit in ZP-H.',
]
))
print('[build_zpg] Building Section IV: Monotone Structure...')
# ── IV. MONOTONE STRUCTURE AND THE ADDITIVE ONTOLOGY ─────────────────────
E.append(Paragraph('IV. Monotone Structure and the Additive Ontology', S['h1']))
E.append(def_box(
'Definition D5 — Morphism Chain',
'Status: Definition [unchanged from v1.0]',
[
'A morphism chain of length n from 0 is a sequence:',
'0 = X0 → X1 → ... → Xn',
'where each Xk → Xk+1 is a morphism in C.',
]
))
E.append(sp(6))
E.append(result_box(
'Proposition T4 — Chains are Forward-Only',
'Status: Derived — from AX-G2 [unchanged from v1.0]',
[
'No morphism chain from 0 can return to 0 through non-initial objects.',
'Proof: A return morphism Xn → 0 for Xn ≠ 0 would contradict AX-G2. ✓',
]
))
E.append(sp(6))
E.append(def_box(
'Definition D6 — Functor',
'Status: Definition — standard category theory [unchanged from v1.0]',
[
'A functor F: C → D consists of an object map F: ob(C) → ob(D) and a morphism map '
'F: homC(A,B) → homD(F(A), F(B)), preserving composition and identity.',
]
))
E.append(sp(6))
E.append(result_box(
'Proposition T5 — Functors Preserve Initial Objects',
'Status: Derived — [OQ-G2 closed in ZP-H T-H1; unchanged from v1.0]',
[
'For each instantiation functor F ∈ {FA, FB, FC, FD}, '
'F(0) is an initial object in the codomain. '
'Verified by direct universal property check in ZP-H T-H1. ✓',
]
))
print('[build_zpg] Building Section V: Kolmogorov Import...')
# ── V. THE KOLMOGOROV IMPORT FROM ZP-C ───────────────────────────────────
E.append(Paragraph('V. The Kolmogorov Import from ZP-C', S['h1']))
E.append(body('This section contains the single import from outside category theory that ZP-G '
'requires. It is named, scoped, and labelled explicitly. It replaces BA-G1 (the Leinster '
'Bridge Axiom) as the informational foundation of D7\'.'))
E.append(import_box(
'Import I-KC — Conditional Kolmogorov Complexity from ZP-C',
'Status: Import — from ZP-C D1 and standard algorithmic information theory',
[
'What is imported: The conditional Kolmogorov complexity K(x|y), defined as the length of the '
'shortest program p such that U(p, y) = x, where U is a fixed universal Turing machine:',
'K(x|y) = min { |p| : U(p, y) = x }',
'Key property — The Coding Theorem: For any computable probability measure P, Kolmogorov '
'complexity and Shannon entropy are related up to an additive constant c (depending only on U, '
'not on x or y):',
'K(x|y) ≈ −log2 P(x|y) + O(c)',
'The coding theorem is a standard result of algorithmic information theory (Li and Vitanyi, An '
'Introduction to Kolmogorov Complexity and Its Applications). It is not derived within ZP-G. '
'It is imported as a named result.',
'Scope of import: I-KC imports K(x|y) and the coding theorem only. No other content from ZP-C is '
'imported into ZP-G. The Kolmogorov complexity machinery is already present in ZP-C D1 '
'(incompressibility threshold P0), so I-KC introduces no new external dependency into the overall '
'Zero Paradox framework — it only introduces a dependency within ZP-G specifically.',
'Computability note: K(x|y) is not computable in general (it is approximable from above by standard '
'results). This is not a defect for the present purposes: the framework requires that K(x|y) be '
'well-defined, not that it be computable. The ontological claims of ZP-G do not depend on computability.',
'Status: This is an import, not a bridge axiom. A bridge axiom is a claim that cannot be derived '
'from either side and must be assumed. K(x|y) is a defined mathematical object with a complete '
'internal theory. I-KC is a decision to use that object within ZP-G, not an assumption about it.',
]
))
print('[build_zpg] Building Section VI: Categorical Information Theory...')
# ── VI. CATEGORICAL INFORMATION THEORY [REBUILT IN v1.1] ─────────────────
E.append(Paragraph('VI. Categorical Information Theory [Rebuilt in v1.1]', S['h1']))
E.append(Paragraph('6.1 Native Categorical Surprisal — Definition D7\'', S['h2']))
E.append(body('Version 1.0 defined categorical surprisal via the Shannon entropy functor H imported '
'through BA-G1. Version 1.1 replaces this with conditional Kolmogorov complexity, '
'imported via I-KC. The definition is native to the morphism structure of C.'))
E.append(def_box(
'Definition D7\' — Native Categorical Surprisal',
'Status: Definition — from D5, I-KC [replaces D7 from v1.0]',
[
'Let f: A → B be a morphism in C. Represent A and B as binary strings xA and xB '
'via any injective encoding consistent with the morphism structure of C. The categorical surprisal of f is:',
'I(f) = K(xB | xA)',
'the conditional Kolmogorov complexity of the target given the source.',
'Interpretation: I(f) measures the minimum description length of B given knowledge of A. It is the '
'irreducible informational content added by the transition f: A → B, independent of any probability distribution.',
'Well-definedness: K(xB|xA) depends on the encoding of objects as strings. Different encodings '
'yield values differing by at most an additive constant c (the coding theorem constant of I-KC). '
'All claims in ZP-G are stated in terms of whether I(f) is finite, zero, or undefined — qualitative '
'properties that are invariant under any additive constant. This invariance is a consequence of what '
'ZP-G chooses to claim, not a general property of K-complexity (additive constants can matter '
'for precise K-complexity comparisons in AIT).',
'Relationship to v1.0 D7: By the coding theorem (I-KC), K(xB|xA) ≈ −log2 P(xB|xA) + O(c) '
'for any computable measure P. Therefore D7\' and D7 are equivalent up to O(c). The choice of D7\' '
'over D7 is not a change in what is being measured; it is a change in how the measure is defined — '
'natively versus via import.',
]
))
E.append(sp(8))
E.append(Paragraph('6.2 Properties of the Native Surprisal', S['h2']))
E.append(result_box(
'Lemma T6-a — Surprisal of the Identity Morphism is Zero',
'Status: Derived — from D7\' and I-KC',
[
'Claim: I(idA) = K(xA|xA) = 0 up to the additive constant c.',
'Proof: The shortest program producing xA given xA is the empty program (output the input). '
'Therefore K(xA|xA) = 0 up to c. The identity morphism adds no informational content. ✓',
]
))
E.append(sp(6))
E.append(result_box(
'Lemma T6-b — Surprisal is Non-Negative for Forward Morphisms',
'Status: Derived — PDF-level (from D5, D7\', AX-G2); NOT Lean-verified — see Lean scope remark below',
[
'Claim: For any morphism f: A → B in a forward morphism chain from 0, I(f) ≥ 0 up to c, '
'with strict inequality when A ≠ B.',
'Proof: K(xB|xA) ≥ 0 by definition (program length is non-negative). Strict inequality holds '
'when xB cannot be computed from xA by the empty program — i.e., when A and B are distinct objects '
'encoding distinct states. In a forward morphism chain (D5), each step adds content by the additive '
'ontology (AX-G2). ✓',
]
))
E.append(sp(6))
E.append(result_box(
'Proposition T6-c — Surprisal Accumulates Along Chains',
'Status: Derived — PDF-level (from D5, T6-b, subadditivity of K); NOT Lean-verified — see Lean scope remark below',
[
'Claim: For a morphism chain 0 = X0 → X1 → ... → Xn, the total surprisal '
'∑ I(Xk → Xk+1) ≥ 0, with monotone accumulation as n increases.',
'Proof: By subadditivity of Kolmogorov complexity: K(xn|x0) ≤ '
'∑ K(xk+1|xk) + O(n·c). Each term is ≥ 0 by T6-b. '
'Adding distinct objects strictly increases the total. ✓',
]
))
E.append(sp(6))
E.append(remark_box(
'Remark — Lean Scope of T6-b and T6-c',
'Status: Scope note [strengthened v1.5] — T6-b and T6-c are NOT Lean-verified for their mathematical claims',
[
'T6-b and T6-c are not Lean-verified. The ZPG.lean proofs for these results compile '
'without error, but they verify nothing about Kolmogorov complexity. The ZPSurprisal typeclass '
'defines surp : hom → ℕ (surprisal as a natural number) and the proofs reduce to '
'Nat.zero_le _, which states that any natural number is ≥ 0. This is trivially true by '
'type for any ℕ-valued function, regardless of its mathematical content. '
'A compiling Lean proof here does not mean the K-theoretic claims have been verified.',
'What the Lean proofs do NOT establish: (1) T6-b strict inequality — '
'K(xB|xA) > 0 when A ≇ B (distinct objects cannot have zero description length). '
'(2) T6-c subadditivity — K(xn|x0) ≤ '
'∑ K(xk+1|xk) + O(n·c) (total surprisal along a chain). '
'These are standard and correct results from algorithmic information theory, but they require '
'a K-formalization that does not exist in Lean 4 / Mathlib.',
'What IS Lean-verified: T6-a (identity morphism has zero surprisal — from surp_id), '
'T6 Part II (inward surprisal is undefined — from AX-G2 and the absence of morphisms to 0), '
'and T7 insofar as it depends on Parts II, III, V, VI. T6 Part I (outward accumulation) '
'and T7 Part IV rely on T6-b and T6-c and are therefore also PDF-level only.',
'Readers citing "the Lean-verified ZP-G framework" should note this boundary. '
'The PDF-level arguments for T6-b and T6-c are mathematically valid — these are standard '
'K-theoretic results (Li and Vitanyi). The gap is Lean scope, not mathematical correctness.',
]
))
print('[build_zpg] Building Section VII: Informational Singularity...')
# ── VII. THE INFORMATIONAL SINGULARITY OF 0 ─────────────────────────────
E.append(Paragraph('VII. The Informational Singularity of 0', S['h1']))
E.append(body('This is the central theorem of the information-theoretic section. It is proved from D7\' and AX-G2 alone, with I-KC as the only external dependency.'))
E.append(result_box(
'Theorem T6 — Informational Singularity of 0',
'Status: Derived — from AX-G2, D7\', I-KC [rebuilt from v1.0, OQ-G1 closed]',
[
'Setup: Let 0 be the initial object of C (AX-G1). Let I(f) = K(xB|xA) be the categorical '
'surprisal (D7\'). Let I-KC provide the Kolmogorov framework.',
'Part I — Outward surprisal accumulates (from T6-b, T6-c): For any morphism chain '
'0 = X0 → ... → Xn, ∑ I(Xk → Xk+1) ≥ 0, '
'with strict accumulation as n increases. ✓',
'Part II — Inward surprisal is undefined (from AX-G2): For any X ≠ 0, hom(X, 0) = ∅ (AX-G2). '
'Therefore D7\' cannot be applied to any morphism f: X → 0 from outside 0 — no such morphism '
'exists. I(X → 0) is undefined not because K diverges to infinity, but because there is no morphism '
'to apply D7\' to. The undefined-domain condition is strictly stronger than divergence. ✓',
'Part III — The singularity: 0 is the unique object in C for which outward surprisal is defined and '
'accumulates (Part I) while inward surprisal is undefined by absence of morphisms (Part II). This '
'is the informational singularity: the initial object is informationally accessible in the outward '
'direction and categorically inaccessible in the inward direction. The singularity is structural, not '
'numerical. It does not require K to diverge — it requires only AX-G2 and D7\'. ✓',
'Comparison with ZP-C (to be reconciled in ZP-H T-H2): ZP-C establishes that the discrete surprisal '
'DF diverges (numerically, to ∞) along infinite sequences approaching 0 in Q2. T6 Part II '
'establishes that I(X → 0) is undefined (domain-absent) for any X ≠ 0 in C. These are compatible: '
'undefined is stronger than infinite. ZP-H T-H2 proves they describe the same obstruction under '
'the functor FC.',
'Status: DERIVED. Depends on AX-G1, AX-G2, D3, D5, D7\', I-KC, T6-a, T6-b, T6-c. OQ-G1 is closed. '
'BA-G1 is no longer a premise of T6. ✓',
]
))
E.append(sp(8))
E.append(Paragraph('7.1 Compatibility with Shannon Entropy — BA-G1 Demoted to Remark', S['h2']))
E.append(remark_box(
'Remark R-BA — Compatibility of D7\' with the Shannon Entropy Functor',
'Status: Remark — BA-G1 demoted from Bridge Axiom [v1.0] to Compatibility Remark [v1.1]',
[
'Version 1.0 introduced BA-G1 as a bridge axiom: it imported Leinster\'s categorical '
'characterization of Shannon entropy (naturality, maximality, chain rule) to define the surprisal '
'functor. BA-G1 was the only bridge axiom in ZP-G v1.0 and was the source of OQ-G1.',
'In v1.1, BA-G1 is no longer a premise of any theorem. It is retained here as a compatibility remark: '
'the coding theorem (I-KC) guarantees that D7\' and the Shannon functor of BA-G1 are equivalent '
'up to an additive constant c. Specifically:',
'K(xB|xA) ≈ H(F(B)) − H(F(A)) + O(c)',
'for any computable probability measure P consistent with the morphism structure of C. This '
'means all quantitative results that v1.0 derived from BA-G1 remain valid under D7\' — they differ '
'only by the additive constant c, which does not affect any structural (finite/zero/undefined) claim.',
'BA-G1 is not false. It is not retired. It is now a derived compatibility result rather than an assumed '
'premise. Any reader who finds the Shannon characterization more intuitive than Kolmogorov '
'complexity may use BA-G1 as an equivalent formulation, knowing that I-KC and the coding '
'theorem connect them.',
]
))
print('[build_zpg] Building Section VIII: Categorical Zero Paradox...')
# ── VIII. THE CATEGORICAL ZERO PARADOX — FORMAL STATEMENT ─────────────────
E.append(Paragraph('VIII. The Categorical Zero Paradox — Formal Statement', S['h1']))
E.append(body('Theorem T7 is the closing theorem of ZP-G. Part IV (informational singularity) rests on T6, which does not depend on BA-G1.'))
E.append(result_box(
'Theorem T7 — The Categorical Zero Paradox',
'Status: Derived — Closing Theorem [Part IV strengthened in v1.1]',
[
'Setup: Let C satisfy AX-G1 and AX-G2. Let I be the categorical surprisal from D7\'. '
'Let I-KC provide the Kolmogorov framework.',
'Part I — Universal Constituent (T2): ∀X ∈ ob(C), ∃! ιX: 0 → X. '
'The initial object 0 is the universal categorical source.',
'Part II — Unreachability (T3): ∀X ≠ 0, hom(X, 0) = ∅. No non-initial object reaches 0.',
'Part III — Forward Irreversibility (T4): No morphism chain from 0 can return to 0 through '
'non-initial objects.',
'Part IV — Informational Singularity (T6, rebuilt): I(X → 0) is undefined for all X ≠ 0 '
'(no such morphism exists, AX-G2). Outward surprisal from 0 accumulates along any morphism '
'chain (T6-b, T6-c). 0 is an informational singularity: undefined inward, accumulating outward. '
'This part no longer depends on BA-G1.',
'Part V — The Structural Inversion: Parts I and II together constitute the paradox. 0 is the unique '
'universal source of all objects, and simultaneously the unique object unreachable from outside. '
'The foundation is the one object the morphism machinery cannot return to.',
'Part VI — Resolution: The paradox is not a logical contradiction. It is a structural inversion. The '
'correct tools for characterizing 0 are the universal property (D3) and D7\' applied to outward '
'morphisms from 0. Under these tools, 0 is fully characterized. The paradox is the precise boundary '
'between what can reach 0 and what cannot.',
'Status: DERIVED — Closing Theorem. Depends on D3, D5, D7\', AX-G1, AX-G2, I-KC, T2, T3, T4, '
'T6. BA-G1 is not a dependency. ✓',
]
))
print('[build_zpg] Building Section IX: Open Items Register...')
# ── IX. OPEN ITEMS REGISTER FOR ZP-G v1.2 ────────────────────────────────
E.append(Paragraph('IX. Open Items Register', S['h1']))
oq_rows = [
['OQ-G1',
'Closed — D7\', T6',
'Native categorical derivation of surprisal without importing Shannon entropy. Closed by replacing D7 '
'with D7\' (conditional Kolmogorov complexity K(B|A)). BA-G1 demoted from Bridge Axiom to Compatibility '
'Remark R-BA. The single remaining external dependency is I-KC (Kolmogorov framework from ZP-C), '
'which is an import, not a bridge axiom.'],
['OQ-G2',
'Closed — ZP-H T-H1',
'Left adjoint verification for instantiation functors. Resolved in ZP-H v1.0 by direct universal '
'property verification for each functor.'],
['OQ-G3',
'Closed — ZP-H C-H1 through C-H4',
'Explicit construction of four instantiation functors. Resolved in ZP-H v1.0.'],
['OQ-G4',
'Closed — ZP-H T-H2',
'Reconciliation of categorical and ZP-C singularity characterizations. Resolved in ZP-H v1.0. '
'Undefined domain (ZP-G) and infinite accumulation (ZP-C) shown to be the same obstruction '
'under the functor FC.'],
['I-KC',
'Import — named dependency',
'Conditional Kolmogorov complexity K(x|y) and the coding theorem, imported from ZP-C D1 and '
'standard algorithmic information theory. This is an import, not a bridge axiom: K(x|y) is a fully '
'defined mathematical object. ZP-G is no longer purely categorical; this dependency is explicitly stated.'],
['AX-G1',
'Axiom — intentional',
'Asymmetry: initial object 0, no terminal object. Foundational structural commitment. Not a gap.'],
['AX-G2',
'Axiom — intentional',
'Source asymmetry: hom(X, 0) = ∅ for X ≠ 0. Foundational irreversibility commitment. Not a gap.'],
['R-BA',
'Remark — BA-G1 demoted',
'Leinster Shannon entropy characterization is now a compatibility remark, not a bridge axiom premise. '
'Derivable from D7\' and I-KC via the coding theorem. Not a gap.'],
]
E.append(data_table(
['Item', 'Status', 'Description'],
oq_rows,
[TW*0.10, TW*0.18, TW*0.72]
))
print('[build_zpg] Building Section X: Validation Status...')
# ── X. VALIDATION STATUS ──────────────────────────────────────────────────
E.append(Paragraph('X. Validation Status', S['h1']))
val_rows = [
['D1: Category',
'Valid — Definition. Standard; foundational. Unchanged.'],
['D2: Uniqueness notation',
'Valid — Definition. Unchanged.'],
['D3: Initial object',
'Valid — Definition. Load-bearing for T2, T7. Unchanged.'],
['D4: Terminal object',
'Valid — Definition. Defined for exclusion. AX-G1 prohibits it. Unchanged.'],
['D5: Morphism chain',
'Valid — Definition. Native to C. Unchanged.'],
['D6: Functor',
'Valid — Definition. Standard. Unchanged.'],
['D7\': Native categorical surprisal',
'Valid — Definition [new in v1.1]. K(xB|xA) via I-KC. Replaces D7. '
'Well-defined up to additive constant c. Structurally invariant.'],
['I-KC: Kolmogorov import',
'Import — named [new in v1.1]. K(x|y) and coding theorem from ZP-C. '
'Not a bridge axiom. Introduces explicit ZP-C dependency into ZP-G.'],
['AX-G1: Asymmetry Axiom',
'Axiom — intentional. Unchanged.'],
['AX-G2: Source Asymmetry',
'Axiom — intentional. Unchanged.'],
['R-BA: BA-G1 compatibility remark',
'Remark — [BA-G1 demoted from Bridge Axiom in v1.0]. Shannon entropy functor '
'compatible with D7\' up to O(c) by coding theorem. No longer a premise of any theorem.'],
['Proposition T1: Uniqueness of initial object',
'Valid — Derived. Relabelled Proposition in v1.2 (subsidiary uniqueness result). Unchanged. ✓'],
['Lemma T2: Universal constituent',
'Valid — Derived. Relabelled Lemma in v1.2 (stepping-stone result). Unchanged. ✓'],
['Lemma T3: Unreachability of 0',
'Valid — Derived. Relabelled Lemma in v1.2 (stepping-stone result). Unchanged. ✓'],
['R1: Structural inversion',
'Valid — Remark. Unchanged.'],
['R2: Categorical expression of self-containment',
'Valid — Remark [new in v1.3]. Connects T2 + AX-G2 to ZP-A CC-2 (⊥ = {⊥}). '
'No new derivation; explanatory bridge note.'],
['Proposition T4: Chains are forward-only',
'Valid — Derived. Relabelled Proposition in v1.2. Unchanged. ✓'],
['Proposition T5: Functors preserve initial objects',
'Valid — Conditional on ZP-H T-H1 (closed). Relabelled Proposition in v1.2. Unchanged. ✓'],
['Lemma T6-a: Identity surprisal is zero',
'Valid — Derived [new in v1.1]. Relabelled Lemma in v1.2. K(xA|xA) = 0 up to c. ✓'],
['Lemma T6-b: Non-negative outward surprisal',
'Valid (PDF-level) — Derived from D5, D7\', AX-G2. K ≥ 0; strict inequality for distinct objects. '
'NOT Lean-verified: Lean proof reduces to Nat.zero_le _ (trivially true for any ℕ-valued function; '
'verifies nothing about K).'],
['Proposition T6-c: Surprisal accumulates along chains',
'Valid (PDF-level) — Derived from D5, T6-b, subadditivity of K. '
'NOT Lean-verified: Lean proof reduces to Nat.zero_le _. '
'Subadditivity of K is a standard AIT result but requires K-formalization absent from Mathlib.'],
['Theorem T6: Informational singularity',
'Valid — Derived [rebuilt in v1.1]. Does not depend on BA-G1. '
'Part II: undefined domain (AX-G2) — fully Lean-verified. '
'Parts I, III: accumulation via T6-b, T6-c — PDF-level only (T6-b/T6-c not Lean-verified).'],
['Theorem T7: Categorical Zero Paradox',
'Valid — Derived [Part IV strengthened in v1.1]. All six parts derived. BA-G1 not a dependency. ✓'],
['OQ-G1: Native surprisal derivation',
'Closed — D7\', T6. No bridge axiom remains as a theorem premise.'],
]
E.append(data_table(
['Component', 'Status / Notes'],
val_rows,
[TW*0.38, TW*0.62]
))
E += [
sp(12),
Paragraph(
'Zero Paradox ZP-G: Category Theory | Version ' + VERSION + ' | May 2026 |'
'Supersedes v1.4 | T6-b and T6-c: PDF-level only; Lean proofs verify non-negativity by type only (Nat.zero_le _), '
'not K-theoretic content | T6 Part II: Lean-verified',
S['endnote']),
]
print(f'[build_zpg] Calling doc.build() with {len(E)} elements...')
doc.build(E)
print(f'[build_zpg] Written: {out_path}')
if __name__ == '__main__':
build()