"""
Build ZP-B: p-Adic Topology (v1.9)
v1.9: K-17 vocab fix — "First Atomic State" -> "minimum nonzero state (ε₀)" in AX-B1 box.
v1.8: Adversary-review pass — R1 remark (Section VI) "universal ontology of state emergence"
replaced with precise scope statement: results hold for any AX-B1 + MP-1 instantiation.
v1.7: "Topological isolation of 0" renamed to "Clopen gap at 0" throughout — "isolated" has a
specific topological meaning ({p} is open) that does not hold for {0} in Q2. The correct claim
is clopen-ball separation: v2(0) = +inf places 0 in a distinct clopen class from every nonzero
element. Section header IV, label_box, export table, and validation table all updated.
v1.6: Remark added after AX-B1 clarifying that 0 and 1 are both present as mathematical objects;
"existence" in AX-B1 is a property the states represent, not a statement about set membership.
v1.5: T3 ZP interpretation bullets labelled explicitly — "ZP Interpretation:" prefix added to
distinguish pure topology (the theorem statement) from ZP-specific framing (the Binary Snap).
"""
import os
from zp_utils import *
VERSION = '1.9'
def build():
out_path = os.path.join(PROJECT_ROOT, 'ZP-B_pAdic_Topology.pdf')
doc = make_doc(out_path, 'ZP-B: p-Adic Topology', 'ZP-B', 'Version ' + VERSION)
E = []
E += [Paragraph('THE ZERO PARADOX', S['title']),
Paragraph('ZP-B: p-Adic Topology', S['subtitle']),
Paragraph('Version ' + VERSION + ' | May 2026', S['subtitle']),
sp(10),
body('This document is self-contained within p-adic analysis and topology. No abstract algebra from ZP-A, no probability, and no Hilbert space is imported. Cross-framework connections are deferred to ZP-D and ZP-E.'),
body('Illustrated Companion: A paired ZP-B 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.'),
sp()]
E.append(Paragraph('I. The Foundational Distinction', S['h1']))
E.append(Paragraph('1.1 The Binary Existence Axiom', S['h2']))
E.append(label_box('Axiom AX-B1 — Binary Existence', [
'The foundational distinction of the Zero Paradox framework is binary: a state either exists or it does not. There is no third option at this level.',
'0 — non-existence (the Null State, corresponding to ⊥ in ZP-A)',
'1 — existence (the minimum nonzero state, ε0)',
'Status: AXIOM. This is the only non-topological commitment in ZP-B. It precedes p-adic analysis and is the premise from which the field selection is derived.',
'Scope: AX-B1 asserts the structure of the ontological distinction, not its physical realisation. It is invariant across all instantiations.',
]))
E.append(body(
'Note on set membership: both 0 and 1 are fully present as mathematical objects — '
'neither is absent from the formal structure. The set is not a collection of things '
'that happen to exist; it is a model of an ontological situation. '
'0 names the null condition: the state in which nothing has been instantiated. '
'1 names the first non-null condition: the state in which something has. '
'Existence in AX-B1 is a property the states represent, not a statement about set membership.'))
E.append(body(
'Lean encoding: AX-B1 is encoded as OntologicalStates — a free inductive type '
'with two named constructors: .null (non-existence) and .exist (existence). '
'This avoids tying the null state to any numeric convention such as ℕ\'s 0. '
'Distinctness (null ≠ exist) is verified by decide via deriving DecidableEq — '
'no classical axioms required. Lean identifiers do not appear in the companion documents, '
'which are Lean-free by design.'))
E.append(sp(4))
E.append(Paragraph('1.2 The Minimality Principle', S['h2']))
E.append(label_box('Principle MP-1 — Minimality of Representation', [
'The representational base of the framework must be the minimum base capable of encoding the ontological distinction of AX-B1 without redundancy and without information loss.',
'Redundancy: a base p > minimum introduces representational states with no ontological counterpart, violating parsimony.',
'Information loss: a base p < minimum cannot distinguish all ontologically distinct states, violating faithfulness.',
'Status: PRINCIPLE — methodological commitment. Given AX-B1 (two ontological states) and MP-1 (minimum sufficient base), the representational base is uniquely determined as 2.',
]))
E.append(sp(4))
E.append(Paragraph('1.3 Derivation of p = 2', S['h2']))
E.append(label_box('Theorem T0 — p = 2 is the Unique Minimum Sufficient Representational Base', [
'Given AX-B1 and MP-1, the p-adic field appropriate for the Zero Paradox framework is Q2.',
'Note on MP-1: MP-1 is a design commitment — the choice to use the minimum sufficient base. '
'Any Qp for prime p ≥ 2 contains elements 0 and 1 capable of representing the AX-B1 '
'distinction; the choice of minimum base is what MP-1 encodes. T0 is a valid derivation '
'from AX-B1 and MP-1, but MP-1 is the load-bearing design choice, not a mathematical necessity. '
'OQ-B1 is closed given MP-1.',
'Proof:',
'Step 1 — AX-B1 establishes exactly two ontological states: non-existence (0) and existence (1).',
'Step 2 — A p-adic field Qp uses coefficients from {0, 1, …, p−1}. The minimum base p capable of representing exactly two distinct values without redundancy is p = 2, with coefficient set {0, 1}. One coefficient per ontological state; no unused coefficients.',
'Step 3 — p = 1 has only one coefficient value {0}. Cannot distinguish existence from non-existence. Fails faithfulness (MP-1).',
'Step 4 — p > 2: coefficient set {0, …, p−1} contains values with no ontological counterpart. Violates no-redundancy condition of MP-1.',
'Step 5 — p = 2 is the unique prime satisfying both conditions simultaneously.',
'Step 6 — The binary branching at every level of Q2\'s ball structure reflects the eventual binary resolution of any representational complexity.',
'Therefore p = 2. Status: DERIVED given MP-1 (design commitment). OQ-B1 closed. ✓',
]))
E.append(Paragraph('II. The 2-Adic Field', S['h1']))
E.append(Paragraph('2.1 The 2-Adic Absolute Value', S['h2']))
E.append(body('Fix p = 2 (derived in T0). Every non-zero rational q ∈ ℚ can be written uniquely as q = 2v · (a/b) where v ∈ ℤ and a, b are integers not divisible by 2. The integer v is the 2-adic valuation v2(q). By convention, v2(0) = +∞.'))
E.append(label_box('Definition D1 — 2-Adic Absolute Value', [
'For q ∈ ℚ: |q|2 = 2−v2(q) for q ≠ 0; |0|2 = 0',
'Elements with high powers of 2 are considered small under |·|2. Elements with no factor of 2 have |·|2 = 1.',
]))
E.append(sp(4))
E.append(Paragraph('2.2 The 2-Adic Field Q2', S['h2']))
E.append(body('Q2 is the completion of ℚ under the metric induced by |·|2. Elements of Q2 are formal power series in 2: x = ∑n=v∞ an · 2n where an ∈ {0,1}. The coefficients an ∈ {0,1} are precisely the binary values of AX-B1.'))
E.append(sp(4))
E.append(Paragraph('2.3 The Minimum Viable Deviation ε0', S['h2']))
E.append(label_box('Definition D5 — Minimum Viable Deviation ε₀', [
'ε0 = 2k for some integer k, where k is the maximum valuation accessible in the instantiation.',
'Structural role (universal): ε0 is always the first element crossed by the Snap. Fixed by the structure of Q2 and AX-B1.',
'Numerical value (contingent): determined by physical constants of the instantiation. Planck-scale quantities in our universe.',
'Status: DEFINED — universe-contingent parameter.',
]))
E.append(Paragraph('III. The Ultrametric', S['h1']))
E.append(label_box('Definition D2 — 2-Adic Metric', [
'For x, y ∈ Q2: d(x, y) = |x − y|2',
]))
E.append(sp(4))
E.append(label_box('Proposition T1 — Strong Triangle Inequality (Ultrametric)', [
'For all x, y, z ∈ Q2: d(x, z) ≤ max( d(x, y), d(y, z) )',
'Proof: Write x − z = (x − y) + (y − z). The ultrametric property of v2 gives v2(a+b) ≥ min(v2(a), v2(b)), from which |a+b|2 ≤ max(|a|2, |b|2). Apply with a = x−y and b = y−z. ✓',
]))
E.append(sp(4))
E.append(label_box('Corollary C1 — All Triangles are Isosceles', [
'If d(x,y) ≠ d(y,z), then d(x,z) = max(d(x,y), d(y,z)).',
'Proof: Suppose d(x,y) < d(y,z). By T1, d(x,z) ≤ d(y,z). Also d(y,z) ≤ max(d(y,x), d(x,z)) = d(x,z) since d(x,y) < d(y,z). Therefore d(x,z) = d(y,z). ✓',
]))
E.append(sp(4))
E.append(Paragraph('3.2 Clopen Ball Structure', S['h2']))
E.append(label_box('Definition D3 — Ball in Q₂', [
'B(a, r) = { x ∈ Q2 : d(x, a) ≤ r } (closed ball)',
'B°(a, r) = { x ∈ Q2 : d(x, a) < r } (open ball)',
]))
E.append(sp(4))
E.append(label_box('Proposition T2 — Every Ball is Clopen', [
'In Q2, every closed ball is also open and every open ball is also closed.',
'Proof (closed ball is open): Let y ∈ B(a, r). For any z ∈ B(y, r), T1 gives d(z, a) ≤ max(d(z,y), d(y,a)) ≤ r. So B(y,r) ⊆ B(a,r). Every point is an interior point. ✓',
'Proof (open ball is closed): Let (xn) → x with all xn ∈ B°(a,r). Ball radii in Q2 are discrete (powers of 2), so d(x,a) < r holds in the limit. Thus x ∈ B°(a,r). ✓',
]))
E.append(sp(4))
E.append(label_box('Corollary C2 — Disjoint Balls Do Not Communicate [v1.2: derived from T2 only]', [
'If B(a, r) and B(b, r) are disjoint (d(a,b) > r), then no continuous path exists from any point in B(a, r) to any point in B(b, r).',
'Proof: By T2, B(a, r) is clopen in Q2. Any continuous f: [0,1] → Q2 with f(0) ∈ B(a,r) and f(1) ∈ B(b,r) would require f to map the connected set [0,1] onto a subset intersecting both B(a,r) and its clopen complement. The preimage of a clopen set under a continuous function is clopen in [0,1]. Since [0,1] is connected, the preimage is either empty or all of [0,1]. It cannot be all of [0,1] (since f(1) ∉ B(a,r)) and cannot be empty (since f(0) ∈ B(a,r)). Contradiction. ✓',
]))
E.append(Paragraph('IV. Clopen Gap at Zero', S['h1']))
E.append(label_box('Theorem T3 — Clopen Gap at 0', [
'For any r = 2−k, the ball B(0,r) = { x ∈ Q2 : v2(x) ≥ k }. Any x outside this ball has d(0,x) ≥ 2−k+1 > r. B(0,r) and its complement are separated by a gap of at least 2−k.',
'ZP Interpretation: this discrete jump across a clopen boundary is the topological expression of the Binary Snap (defined in ZP-E).',
'Relationship to ε0: ε0 = 2k is the smallest non-zero element outside the tightest ball around 0. ZP Interpretation: this is the gap the Binary Snap crosses. ✓',
]))
E.append(Paragraph('V. Topological Structure of Q₂', S['h1']))
E.append(Paragraph('5.1 Total Disconnectedness — proven before C3', S['h2']))
E.append(label_box('Proposition T5 — Q₂ is Totally Disconnected', [
'The only connected subsets of Q2 are singletons.',
'Proof: Let S ⊆ Q2 contain two distinct points a, b with d(a,b) = r > 0. Choose s with 0 < s < r. By T2, B(a,s) is clopen. Then S = [S ∩ B(a,s)] ∪ [S \\ B(a,s)] is a separation of S into two disjoint non-empty clopen sets. Therefore S is not connected. Since S was arbitrary, the only connected subsets are singletons. ✓',
]))
E.append(sp(4))
E.append(Paragraph('5.2 Topological Irreversibility of the Snap', S['h2']))
E.append(label_box('Definition D4 — Topological Irreversibility', [
'A transition from a to b in a topological space X is topologically irreversible if there exists no continuous path γ: [0,1] → X with γ(0) = b and γ(1) = a.',
]))
E.append(sp(4))
E.append(label_box('Corollary C3 — The Snap is Topologically Irreversible [reclassified from T4 in v1.1]', [
'Let x ∈ Q2 with x ≠ 0. There exists no continuous path γ: [0,1] → Q2 with γ(0) = x and γ(1) = 0.',
'Proof: By T5, Q2 is totally disconnected. A continuous path γ with γ(0) = x ≠ 0 and γ(1) = 0 would require γ([0,1]) to be a connected subset of Q2 containing two distinct points. By T5, no such connected subset exists. ✓',
'Derivation chain: T1 → T2 → T5 → C3. Reclassified from Theorem T4: this result is a corollary of T5, not an independent theorem.',
]))
E.append(Paragraph('VI. Universal Structure vs. Contingent Parameters', S['h1']))
E.append(label_box('Remark R1 — Universal Structure vs. Universe-Contingent Parameters', [
'Universal (invariant across all instantiations): AX-B1 (binary distinction — logical, not physical). MP-1 (methodological commitment). T0 (p=2 derived). T1, T2, T3, T5, C1, C2, C3 (all topological results). Structural role of ε0.',
'Universe-contingent (varies across instantiations): Numerical value of ε0 (determined by physical constants). Physical predictions invoking ε0 numerically.',
'The topological and algebraic results of this document hold for any instantiation satisfying AX-B1 and MP-1 — they are not specific to our universe.',
]))
E.append(Paragraph('VII. Boundary Conditions for ZP-D and ZP-E', S['h1']))
E.append(data_table(
['Export', 'Status', 'Receiving Document'],
[['AX-B1', 'Axiom', 'ZP-E: foundational axiom'],
['MP-1', 'Principle', 'ZP-E: bridge between ontological and representational binary'],
['T0: p = 2', 'Derived from AX-B1 + MP-1', 'ZP-D: domain of T is Q2'],
['Q2 with 2-adic metric (D1, D2)', 'Defined', 'ZP-D: topological domain of T'],
['T1: Ultrametric', 'Derived', 'ZP-D: non-Archimedean structure'],
['T2: Clopen balls', 'Derived', 'ZP-D: clopen gap maps to orthogonality in H'],
['T3: Clopen gap at 0', 'Derived', 'ZP-E: grounds ontological claim about the Snap'],
['T5: Total disconnectedness', 'Derived', 'ZP-E: supports C3'],
['C3: Snap topologically irreversible', 'Derived — Corollary of T5', 'ZP-E: cross-framework irreversibility'],
['ε0 (D5)', 'Defined — contingent', 'ZP-E: Snap threshold; value depends on instantiation'],
['R1: Universal vs. contingent', 'Remark', 'ZP-E: framework is instantiation-independent']],
[1.6*inch, 1.8*inch, 3.1*inch]
))
E.append(Paragraph('VIII. Validation Status', S['h1']))
E.append(data_table(
['Component', 'Status / Notes'],
[['AX-B1', 'Axiom — explicit; load-bearing premise'],
['MP-1', 'Principle — explicit bridge; resolves reviewer gap in T0'],
['T0: p = 2', 'Valid — Derived given MP-1 (design commitment). MP-1 encodes the minimality choice; T0 follows. OQ-B1 closed.'],
['D1: 2-adic absolute value', 'Valid — standard definition'],
['D2: 2-adic metric', 'Valid — follows from D1'],
['T1: Strong triangle inequality', 'Valid — Derived'],
['C1: All triangles isosceles', 'Valid — Corollary of T1'],
['T2: Every ball is clopen', 'Valid — Derived from T1'],
['C2: Disjoint balls do not communicate', 'Valid — Derived from T2 only; forward citation to T5 removed'],
['T3: Clopen gap at 0', 'Valid — Derived from D1 and D2'],
['T5: Q2 totally disconnected', 'Valid — Derived from T2; proven before C3'],
['C3: Snap topologically irreversible', 'Valid — Corollary of T5; reclassified from Theorem T4'],
['D5: ε0', 'Valid — Defined; structural role universal; value contingent'],
['R1', 'Valid — Remark; ontological scope clarified']],
[2.5*inch, 4.0*inch]
))
doc.build(E)
print(f'Built: {out_path} ({os.path.getsize(out_path) // 1024} KB)')
if __name__ == '__main__':
build()