""" Build ZP-M Illustrated Companion Version 1.2 | May 2026 v1.2: fix HTML entities in String() drawing primitives (rendered literally); add validate_drawing() to both diagram functions; increase triangle_diagram dh 3.2→3.5 in to clear top-circle geometry overflow; remove duplicate internal caption from triangle_diagram (overlapped "2-adic convergence" edge label at same y). v1.1: vocab fix: null state → ⊥. v1.0: Initial release. Covers snapEmbed type bridge, triangle diagram, diagonalization unification, and R-M.1 on DA-1 Path 2 boundary. """ import os from zp_utils import * from reportlab.graphics.shapes import Drawing, Line, String, Rect, Circle, Polygon from reportlab.graphics import renderPDF VERSION = '1.2' def triangle_diagram(): """The Kleene-Ordinal-2adic triangle.""" dw = TW dh = 3.5 * inch # 3.5 * 72 = 252 pts; top circle top = 222, dh-10 = 242 ✓ d = Drawing(dw, dh) cx = dw / 2 # Three vertices of the triangle top_x, top_y = cx, 200 # ε₀ / c₁ (ordinal snap) left_x, left_y = cx - 160, 30 # c₀ = ⊥ (Kleene quine) right_x, right_y = cx + 160, 30 # 0 ∈ ℤ₂ (2-adic limit) # Triangle edges d.add(Line(left_x, left_y, top_x, top_y, strokeColor=INDIGO, strokeWidth=1.5)) d.add(Line(right_x, right_y, top_x, top_y, strokeColor=INDIGO, strokeWidth=1.5)) d.add(Line(left_x, left_y, right_x, right_y, strokeColor=INDIGO, strokeWidth=1.5)) # Vertex circles r = 22 for (vx, vy) in [(top_x, top_y), (left_x, left_y), (right_x, right_y)]: d.add(Circle(vx, vy, r, fillColor=INDIGO_LITE, strokeColor=INDIGO, strokeWidth=1.5)) # Vertex labels (inside circles) — Unicode directly; String() does not parse entities d.add(String(top_x - 14, top_y - 6, 'ε₀ / c₁', fontSize=10, fontName='DV-B', fillColor=INDIGO)) d.add(String(left_x - 18, left_y - 6, 'c₀ = ⊥', fontSize=10, fontName='DV-B', fillColor=INDIGO)) d.add(String(right_x - 18, right_y - 6, '0 ∈ ℤ₂', fontSize=10, fontName='DV-B', fillColor=INDIGO)) # Edge labels d.add(String(left_x - 80, (top_y + left_y) / 2, 'ordinal snap', fontSize=8, fontName='DV-I', fillColor=GREY_TEXT)) d.add(String(right_x + 8, (top_y + right_y) / 2, 'snapEmbed', fontSize=8, fontName='DV-I', fillColor=GREY_TEXT)) # Bottom edge label — y=16 clears the caption zone; internal caption removed # (ccaption() in build() already provides it externally) d.add(String(cx - 55, 16, '2-adic convergence', fontSize=8, fontName='DV-I', fillColor=GREY_TEXT)) return validate_drawing(d, dh, 'triangle_diagram') def diag_pattern_diagram(): """Side-by-side: Kleene diagonal vs ordinal diagonal.""" dw = TW dh = 2.0 * inch # 144 pts; two boxes side by side d = Drawing(dw, dh) mid = dw / 2 box_w = mid - 20 box_h = 100 box_y = (dh - box_h) / 2 # ~22 pts — stays above 5pt minimum # Left box: Kleene d.add(Rect(10, box_y, box_w, box_h, fillColor=SLATE_LITE, strokeColor=SLATE, strokeWidth=1)) d.add(String(10 + box_w / 2 - 55, box_y + box_h - 22, 'Kleene (computability)', fontSize=9, fontName='DV-B', fillColor=SLATE)) d.add(String(10 + 8, box_y + box_h - 44, 'Operation: eval c = selfApply c', fontSize=8, fontName='DV', fillColor=GREY_TEXT)) d.add(String(10 + 8, box_y + box_h - 60, 'Fixed point: ∃ c, IsComputationalQuine c', fontSize=8, fontName='DV', fillColor=GREY_TEXT)) d.add(String(10 + 8, box_y + box_h - 76, 'Domain: codes + Gödel numbers', fontSize=8, fontName='DV', fillColor=GREY_TEXT)) d.add(String(10 + 8, box_y + box_h - 92, 'Forced by: second recursion theorem', fontSize=8, fontName='DV', fillColor=GREY_TEXT)) # Right box: ordinal d.add(Rect(mid + 10, box_y, box_w, box_h, fillColor=INDIGO_LITE, strokeColor=INDIGO, strokeWidth=1)) d.add(String(mid + 10 + box_w / 2 - 50, box_y + box_h - 22, 'Ordinal (set theory)', fontSize=9, fontName='DV-B', fillColor=INDIGO)) d.add(String(mid + 18, box_y + box_h - 44, 'Operation: α ↦ ω^α', fontSize=8, fontName='DV', fillColor=GREY_TEXT)) d.add(String(mid + 18, box_y + box_h - 60, 'Fixed point: ε₀ = ω^ε₀ (minimal)', fontSize=8, fontName='DV', fillColor=GREY_TEXT)) d.add(String(mid + 18, box_y + box_h - 76, 'Domain: ordinals + ω-tower iteration', fontSize=8, fontName='DV', fillColor=GREY_TEXT)) d.add(String(mid + 18, box_y + box_h - 92, 'Forced by: least fixed-point theorem', fontSize=8, fontName='DV', fillColor=GREY_TEXT)) # Caption at bottom d.add(String(14, 10, 'Same schema, different domains — no shared machinery between the two.', fontSize=7.5, fontName='DV-I', fillColor=GREY_TEXT)) return validate_drawing(d, dh, 'diag_pattern_diagram') def build(): out_path = os.path.join(PROJECT_ROOT, 'ZP-M_Illustrated_Companion.pdf') print(f'[build_zpm_companion] Output: {out_path}') def footer_cb(canvas, doc): canvas.saveState() canvas.setFont('DV-I', 8) canvas.setFillColor(colors.grey) canvas.drawCentredString( LETTER[0] / 2, 0.6 * inch, 'Zero Paradox ZP-M | Illustrated Companion | May 2026') canvas.restoreState() doc = SimpleDocTemplate( out_path, pagesize=LETTER, leftMargin=LM, rightMargin=RM, topMargin=TM, bottomMargin=BM, title='ZP-M Illustrated Companion', author='Zero Paradox Project', onFirstPage=footer_cb, onLaterPages=footer_cb) E = [] print('[build_zpm_companion] Building header...') hdr_ts = TableStyle([('BACKGROUND',(0,0),(-1,-1),INDIGO), ('TOPPADDING',(0,0),(-1,-1),8),('BOTTOMPADDING',(0,0),(-1,-1),8), ('LEFTPADDING',(0,0),(-1,-1),10)]) hdr = Table([[Paragraph('ZP-M Illustrated Companion', ParagraphStyle('hdr',fontName='DV-B',fontSize=11,textColor=WHITE))]], colWidths=[TW]) hdr.setStyle(hdr_ts) E += [hdr, sp(6), Paragraph('THE ZERO PARADOX', CS['title']), Paragraph('ZP-M: Kleene–Ordinal Bridge', CS['title']), Paragraph('ZP Companion | Version ' + VERSION + ' | May 2026', CS['meta']), Paragraph( 'This companion explains the ideas in plain language. It is not the formal ' 'document — every claim here restates a result already proved in ' 'ZP-M: Kleene-Ordinal Bridge. Consult that document for the authoritative ' 'mathematics.', CS['disc']), sp(4), hr(), sp(4), ] # ── §1: What ZP-M Does ───────────────────────────────────────────────────── print('[build_zpm_companion] Building §1...') E += [Paragraph('1. What This Layer Does', CS['h1']), sp(4)] E.append(cbody( 'ZP-M is a consolidation layer. Its job is to connect two results that were ' 'proved separately in ZP-K and ZP-L and show that they are looking at the same ' 'structure from different angles.')) E.append(cbody( 'ZP-K proved that the initial state c₀ (⊥) is a ' 'Kleene fixed point — a program that is its own program, with no external ' 'executor required. ZP-L proved that the ordinal ε₀ is the exact snap ' 'threshold: the tower ω, ω^ω, ω^ω^ω, … approaches ε₀ from below, and ' 'any monotone map that sends tower stages to c₀ must send ε₀ to c₁. ' 'ZP-L also showed that the tower encodings in ℤ₂ converge to 0.')) E.append(cbody( 'ZP-M builds the bridge connecting these: a formal map snapEmbed that sends ' 'c₁ (the snap state) to 0 in ℤ₂, and proves that all three objects ' '— the ordinal ε₀, the snap state c₁, and the 2-adic zero — are ' 'formally co-witnessed in a single theorem (zpm_triangle). It also closes ' 'a gap in ZP-L\'s minimality theorem by showing that its free hypothesis ' 'follows from simpler conditions.')) E.append(sp(6)) # ── §2: The Type Bridge ──────────────────────────────────────────────────── print('[build_zpm_companion] Building §2...') E += [hr(), Paragraph('2. The Type Bridge (snapEmbed)', CS['h1']), sp(4)] E.append(cbody( 'The two layers — ZP-E\'s machine state space {c₀, c₁} and ZP-B\'s ' '2-adic integers ℤ₂ — are mathematically unrelated types. One is a two-element ' 'set; the other is an infinite algebraic structure. snapEmbed is the map that ' 'connects them structurally.')) E.append(label_box('snapEmbed', [ 'c₀ (pre-snap) ↦ 1 ∈ ℤ₂ (a 2-adic unit, valuation 0)', 'c₁ (snap state) ↦ 0 ∈ ℤ₂ (the 2-adic zero, infinite valuation)', ])) E.append(sp(6)) E.append(cbody( 'The map is injective (c₀ and c₁ land in distinct places) and ' 'preserves the algebraic structure: joining c₁ with anything gives c₁, ' 'and multiplying 0 by anything gives 0. The same absorbing-element pattern ' 'holds in both types, under different operations.')) E.append(cbody( 'The key geometric fact: 0 in ℤ₂ is divisible by every power of 2. ' 'That is what "infinite 2-adic valuation" means. This is the same mathematical ' 'signature as ⊥ = {⊥} — a structure that contains itself at every level of ' 'depth, with no bottom that is not also the top.')) E.append(sp(6)) # ── §3: The Triangle ─────────────────────────────────────────────────────── print('[build_zpm_companion] Building §3...') E += [hr(), Paragraph('3. The Triangle', CS['h1']), sp(4)] E.append(cbody( 'Three objects that appear in three different layers of the framework are ' 'co-witnessed in a single Lean theorem: zpm_triangle.')) E.append(triangle_diagram()) E.append(ccaption('The Kleene–Ordinal–2-adic triangle: three edges, one formal proof.')) E.append(sp(8)) E.append(cbody( 'The left-side edge is the ordinal snap: below ε₀, the tower stages map ' 'to c₀; at ε₀, the canonical map flips to c₁. ' 'The right-side edge is the type bridge: snapEmbed sends c₁ to 0. ' 'The bottom edge is the 2-adic convergence proved in ZP-L: the tower encodings ' 'in ℤ₂ converge to 0 as the ordinal stages approach ε₀.')) E.append(cbody( 'All three edges are theorems, and zpm_triangle assembles them into a single ' 'formal statement. This is the first place in the framework where the ordinal, ' 'computational, and 2-adic threads appear in the same proof.')) E.append(sp(6)) # ── §4: Closing the Gap in ZP-L ─────────────────────────────────────────── print('[build_zpm_companion] Building §4...') E += [hr(), Paragraph('4. Closing a Gap in ZP-L', CS['h1']), sp(4)] E.append(cbody( 'ZP-L\'s central minimality theorem (snap_exactly_at_epsilon_zero) carried a ' 'free hypothesis called hfp. It asserted that for any map from ordinals to ' 'machine phases, every fixed point of ω-exponentiation must be sent to c₁. ' 'This was not proved from the ordinal structure — it was an extra assumption.')) E.append(cbody( 'ZP-M §II closes this gap. The argument is short: if a map is monotone and ' 'sends ε₀ to c₁, then for any other fixed point α ≥ ε₀, ' 'monotonicity forces the map to agree with c₁ at α (because c₁ absorbs ' 'every join). The minimality theorem now holds under weaker hypotheses: just ' 'monotonicity and "snap at ε₀" together imply the full result, without hfp ' 'as a separate commitment.')) E.append(callout( 'hfp is not an additional assumption — it is a consequence of the ordinal ' 'structure plus the alignment condition φ(ε₀) = c₁. The gap in ZP-L is closed.', bg=GREEN_LITE, border=GREEN )) E.append(sp(6)) # ── §5: The Diagonalization Pattern ─────────────────────────────────────── print('[build_zpm_companion] Building §5...') E += [hr(), Paragraph('5. One Pattern, Two Domains', CS['h1']), sp(4)] E.append(cbody( 'The deepest result in ZP-M is not a new theorem but a recognition: Kleene\'s ' 'second recursion theorem (computability) and the construction of ε₀ ' '(ordinal theory) are the same argument running in different mathematical worlds.')) E.append(diag_pattern_diagram()) E.append(ccaption('The diagonalization schema in two domains: same structure, no shared machinery.')) E.append(sp(8)) E.append(cbody( 'In each case: you define an operation that refers to itself, and the framework ' 'forces a fixed point to exist. In computability theory, the operation refers to ' 'a program\'s own Gödel number (its address in the code space). In ordinal theory, ' 'the operation is ω-exponentiation, and the fixed point is the first ordinal that ' 'the tower cannot exceed. The mechanisms are entirely different. The schema is identical.')) E.append(cbody( 'The theorem both_fixed_points_exist states this directly: both fixed points ' 'exist, and their proofs live in the same Lean context. This is not a coincidence ' 'that requires explanation — it is the recognition that diagonalization is a ' 'structural fact that transcends any particular domain.')) E.append(sp(6)) # ── §6: R-M.1 and the Boundary ──────────────────────────────────────────── print('[build_zpm_companion] Building §6...') E += [hr(), Paragraph('6. Remark R-M.1: Where the Frame Ends', CS['h1']), sp(4)] E.append(cbody( 'ZP-M establishes the diagonalization frame for two of the three threads in ' 'the DA-1 argument. Path 1 (AFA structural) and Path 3 (Kleene computational) ' 'are both diagonalization instances — they were unified in ZP-K. Now both share ' 'the frame with the ordinal result.')) E.append(cbody( 'Path 2 (informational) remains outside this frame. L-INF (ZP-C) says that ' '⊥ has unbounded information content — no finite description can ' 'capture it. This is structurally analogous to the diagonalization pattern, ' 'but the analogy has not been made formal.')) E.append(cbody( 'The reason, visible from ZP-M, is a framework separation: L-INF is a ' 'measure-theoretic result (about probability distributions and Shannon entropy), ' 'while Kleene\'s theorem is a computability-theoretic result (about codes and ' 'recursive functions). The two use no shared mathematical machinery. The concept ' 'that bridges them — Kolmogorov complexity, which is simultaneously an ' 'information-theoretic and computability-theoretic notion — is not yet formalized ' 'in Mathlib.')) E.append(callout( 'DA-1 Path 2 is not a missing proof step — it is a genuine framework boundary. ' 'ZP-M makes the boundary precise: on one side, diagonalization over codes and ' 'ordinals (in Lean scope). On the other side, the AIT bridge between ' 'incompressibility and self-execution (outside current Lean scope).', bg=AMBER_LITE, border=AMBER )) E.append(sp(8)) # ── Key Result Box ───────────────────────────────────────────────────────── print('[build_zpm_companion] Building key result box...') E.append(hr()) E.append(result_box( 'Key Results — ZP-M v1.0', [ 'snapEmbed: c₀ ↦ 1, c₁ ↦ 0 in ℤ₂ — injective, join-to-multiply morphism. ✓', 'hfp_from_epsilon_zero: hfp derived from monotonicity + φ(ε₀) = c₁ alone. ✓', 'zpm_triangle: all three edges co-proved — ordinal snap, 2-adic convergence, ' 'type bridge. ✓', 'both_fixed_points_exist: Kleene and ordinal fixed points co-witnessed ' 'in one formal context. ✓', 'R-M.1: DA-1 Path 2 boundary made precise — AIT bridge is outside current ' 'Lean scope; gap is framework separation, not missing proof.', 'All 9 theorems proved sorry-free. ' 'Axiom footprint: [propext, Classical.choice, Quot.sound].', ] )) E.append(sp(8)) E += [ hr(), Paragraph( 'ZP-M Illustrated Companion | May 2026 | ' 'Companion to ZP-M: Kleene-Ordinal Bridge | ' 'Formal verification: ZPM.lean (Lean 4 + Mathlib) | ' 'Zero Paradox Project', CS['meta']), ] print('[build_zpm_companion] Building document...') doc.build(E) print(f'[build_zpm_companion] Done: {out_path}') if __name__ == '__main__': build()