""" Build ZP-D: State Layer (Hilbert Space) (v1.11) v1.11: K-11/K-18 vocab fixes — "topological isolation" -> "clopen separation" throughout DP-1; "first atomic state" -> "minimum nonzero state" in D1. v1.10: Version number removed from Open Items Register section header. v1.9: Adversary-review pass — Import I-B stale reference "Topological isolation of 0" corrected to "Clopen gap at 0" to match ZP-B v1.7+ terminology throughout. v1.8: T5 strengthened — T5-b (t5_strict_orthogonal) added in Lean and PDF. T5-b is the non-trivial content of T5: distinct consecutive states map to orthogonal T-images via DP-1. T5 norm result disclosed as tautology of construction (all norms = 1). v1.7: T5 DP-1 dependency made explicit — status line, Open Items Register, and Validation Status table now consistently state "given DP-1" for T5, matching T4's treatment. v1.6: R3 added — topological type of T stated explicitly: locally constant, continuous from (Q2, 2-adic topology) to H; connected-space concern addressed. v1.5: D1 updated — n = 2 foundational minimum; higher n derived, not foundational. v1.4: T5 proof corrected — ball-boundary argument replaces D2(v) citation. """ import os from zp_utils import * VERSION = '1.11' def build(): out_path = os.path.join(PROJECT_ROOT, 'ZP-D_State_Layer.pdf') doc = make_doc(out_path, 'ZP-D: State Layer (Hilbert Space)', 'ZP-D', 'Version ' + VERSION) E = [] E += [Paragraph('THE ZERO PARADOX', S['title']), Paragraph('ZP-D: State Layer (Hilbert Space)', S['subtitle']), Paragraph('Version ' + VERSION + ' | May 2026', S['subtitle']), sp(10), body('This document operates within functional analysis. It imports from ZP-A and ZP-B and constructs the Hilbert space state layer on top of them. No information theory from ZP-C is imported. Cross-framework synthesis is deferred to ZP-E.'), body('Illustrated Companion: A paired ZP-D 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. Imported Structure', S['h1'])) E.append(Paragraph('1.1 From ZP-A: Algebraic Structure of States', S['h2'])) E.append(label_box('Import I-A — From ZP-A: Lattice Algebra', [ '(L, ∨, ⊥): join-semilattice with bottom. Axioms A1–A4.', '≤ partial order: x ≤ y :⟺ x ∨ y = y (D1, T1).', '⊥ is the global minimum: ⊥ ≤ x for all x ∈ L (T2).', 'State transitions are joins: f(x) = x ∨ α (D2).', 'State sequences are monotone: Sn ≤ Sn+1 (T3).', 'CC-1: S0 = ⊥ — Conditional Claim; modelling commitment.', ])) E.append(sp(4)) E.append(Paragraph('1.2 From ZP-B: Topological Domain', S['h2'])) E.append(label_box('Import I-B — From ZP-B: p-Adic Topology', [ 'AX-B1: Binary Existence Axiom.', 'MP-1: Minimality Principle.', 'T0: p = 2 derived from AX-B1 and MP-1.', 'Q2 with 2-adic metric d (D1, D2).', 'T1: Ultrametric (strong triangle inequality).', 'T2: Every ball is clopen.', 'T3: Clopen gap at 0.', 'T5: Q2 is totally disconnected.', 'C3: Snap is topologically irreversible (corollary of T5).', 'ε0: Minimum viable deviation, universe-contingent parameter (D5).', ])) E.append(Paragraph('II. The Hilbert Space State Layer', S['h1'])) E.append(label_box('Definition D1 — State Layer H', [ 'H = ℂn is a complex Hilbert space with orthonormal basis {e0, e1, …}.', 'The foundational minimum is n = 2, corresponding to the two ontological states of the framework: ' '⊥ (null state, mapped to e0) and ε0 (minimum nonzero state, mapped to e1). ' 'These two orthogonal vectors express the binary existence/non-existence distinction that is the ' 'framework\'s central object. All further states are derived from this pair as joins in (L, ∨, ⊥) ' '— they require no additional foundational dimension.', 'The framework\'s core claims (T4: snap produces orthogonal shift; T5: monotone norms) are ' 'established at n = 2. Extensions to higher n are consistent and natural: for level-k ' 'approximations using the clopen ball partition of Q2 at depth k, n = 2k. ' 'No foundational claim of the framework requires n > 2.', ])) E.append(sp(4)) E.append(label_box('Remark R1 — Decoupling of Topological and State Layers', [ 'Q2 and H are categorically distinct structures. Q2 is a topological field; H is a Hilbert space over ℂ. They share no operations.', 'The transition operator T: Q2 → H is the only bridge. It is constructed explicitly in T2.', 'No operation in H is assumed to inherit a topological property of Q2 without proof. Every cross-layer claim must go through T.', ])) E.append(Paragraph('III. The Transition Operator T: Q₂ → H', S['h1'])) E.append(Paragraph('3.1 The Design Commitment — Orthogonality', S['h2'])) E.append(label_box('Design Principle DP-1 — Orthogonality as the Representation of Clopen Separation [reclassified from T1 in v1.1]', [ 'Clopen separation in Q2 (T3: 0 is clopen-separated from all nonzero elements; clopen balls are mutually separated) is represented in H by orthogonality: elements that are clopen-separated in Q2 map to orthogonal vectors in H.', 'Motivation: Orthogonality in H is the natural algebraic analogue of topological separation. ⟨ei, ej⟩ = 0 for i ≠ j; two clopen balls are maximally distinct in the topological sense.', 'Status: DESIGN PRINCIPLE — DP-1 is chosen, not derived. It is the natural and consistent choice, stated explicitly. T4 and T5 below depend on DP-1 as a premise.', ])) E.append(sp(4)) E.append(Paragraph('3.2 The Construction Target', S['h2'])) E.append(label_box('Definition D2 — Transition Operator T (Requirements)', [ 'T: Q2 → H must satisfy:', '(i) T(0) = e0 (null state maps to the designated base vector)', '(ii) T(ε0) = e1 (minimum deviation maps to the first non-base vector)', '(iii) T is injective on the clopen ball partition', '(iv) If x and y are in disjoint clopen balls, then ⟨T(x), T(y)⟩ = 0 (DP-1)', '(v) ‖T(x)‖ ≥ ‖T(0)‖ for all x (norm-increasing: additive ontology preserved)', ])) E.append(sp(4)) E.append(Paragraph('3.3 Existence of T', S['h2'])) E.append(label_box('Theorem T2 — Existence of T (Basis Assignment)', [ 'There exists a function T: Q2 → H satisfying all five requirements of D2.', 'Proof: Construct T by basis assignment. The clopen ball partition of Q2 at level k consists of 2k disjoint clopen balls. Assign each ball to a distinct basis vector of H. T(x) = ei where i is the index of the ball containing x.', '(i) 0 maps to e0 by assignment. ✓ (ii) ε0 maps to e1 by assignment. ✓ (iii) Distinct balls → distinct basis vectors. ✓ (iv) Disjoint balls → orthogonal basis vectors. ✓ (v) All basis vectors have norm 1 ≥ ‖e0‖ = 1. ✓', ])) E.append(sp(4)) E.append(label_box('Remark R3 — Topological Type of T', [ 'T is locally constant: it is constant on each clopen ball of Q2. This follows directly from ' 'DP-1 — clopen separation maps to orthogonality, so T does not interpolate between basis ' 'vectors across ball boundaries.', 'Continuity: T is continuous from (Q2, 2-adic topology) to H. The image of T is a ' 'discrete set of basis vectors {e0, e1, …}. Each preimage T−1(ei) ' 'is a clopen ball of Q2, which is open in the 2-adic topology. Therefore preimages of ' 'open sets in H are open in Q2, and T is continuous. ✓', 'Note on connected spaces: Q2 is totally disconnected; H = ℂn with the norm ' 'topology is path-connected. A continuous map from a totally disconnected space to a ' 'connected space need not be constant — it must only have a totally disconnected image. ' 'T\'s image is a discrete (hence totally disconnected) set of basis vectors, ' 'which is consistent with both the total disconnectedness of Q2 and the ' 'path-connectedness of H as a whole.', ])) E.append(sp(4)) E.append(Paragraph('3.4 Uniqueness of T', S['h2'])) E.append(label_box('Proposition T3 — Uniqueness of T up to Unitary Equivalence', [ 'Any two operators T, T\': Q2 → H satisfying D2 are related by a unitary transformation U: H → H such that T\' = U ∘ T.', 'Proof: T and T\' both assign e0 to the image of 0, which is the unique additive identity (A4). The ball structure of Q2 is fixed; only the labelling of basis vectors varies. A unitary map U taking T(0) to T\'(0) and preserving orthogonality relations defines the equivalence. ✓', ])) E.append(sp(4)) E.append(label_box('Remark R2 — What T Is Not', [ 'T is not a ring homomorphism. Q2 has field operations; H does not. T does not preserve addition or multiplication from Q2.', 'T is not a topological embedding. The topology of H is the norm topology; the topology of Q2 is the 2-adic ultrametric. T is a structure-preserving assignment: ontological distinctions (clopen separation in Q2) map to algebraic distinctions (orthogonality in H), as specified by DP-1.', ])) E.append(Paragraph('IV. The Binary Snap in H', S['h1'])) E.append(label_box('Theorem T4 — Snap Produces Orthogonal Shift in H', [ 'The Binary Snap 0 → ε0 in Q2 maps to an orthogonal shift in H: T(0) = e0 and T(ε0) = e1, and ⟨e0, e1⟩ = 0.', 'Proof: By D2(i), T(0) = e0. By D2(ii), T(ε0) = e1. Since 0 and ε0 are in disjoint clopen balls of Q2 (T3), D2(iv) and DP-1 give ⟨T(0), T(ε0)⟩ = ⟨e0, e1⟩ = 0. ✓', 'Status: Derived — unconditional theorem given DP-1.', ])) E.append(sp(4)) E.append(label_box('Proposition T5 — Monotone Sequences Map to Non-Decreasing Norms', [ 'Let (Sn) be a monotone state sequence in L (ZP-A T3). Then ‖T(Sn)‖ ≤ ‖T(Sn+1)‖ for all n.', 'T5 norm result: Derived — unconditional given DP-1. ' 'Note: in the basis-assignment construction (D2), all T-images are unit basis vectors — all norms equal 1. ' 'The norm inequality ‖T(Sn)‖ ≤ ‖T(Sn+1)‖ is therefore satisfied trivially (1 ≤ 1). ' 'The Lean proof (ZPD.t5_monotone_norms) verifies this tautology; it does not verify the ball-boundary argument. ' '✓', 'T5-b — Strict Orthogonality (non-trivial content, Lean-verified): For any strictly monotone state sequence — where Sn ≠ Sn+1 — consecutive T-images are orthogonal: ' '⟨T(Sn), T(Sn+1)⟩ = 0. ' 'Proof: Sn ≠ Sn+1 implies they index distinct clopen balls. By DP-1, distinct ball-indices map to orthogonal basis vectors. Therefore T(Sn) ⊥ T(Sn+1). ' 'Lean: ZPD.t5_strict_orthogonal (uses DP-1 via t2_orthogonal; axiom-free). ✓', 'T5-b is the load-bearing result: it captures the genuine content of the ball-boundary structure and is the Hilbert-space expression of ZP-B C3 (topological irreversibility) applied to consecutive states.', ])) E.append(Paragraph('V. Open Items Register', S['h1'])) E.append(data_table( ['Item', 'Status', 'Description'], [['DP-1: Orthogonality commitment', 'Design Principle — explicit', 'Reclassified from Theorem T1. Orthogonality is chosen, not derived. Content unchanged.'], ['T2: Existence of T', 'Closed', 'Basis assignment construction. All five requirements verified.'], ['T3: Uniqueness of T', 'Closed', 'Unique up to unitary equivalence.'], ['T4: Snap → orthogonal shift', 'Closed — unconditional', 'Proven from T2 and ZP-B T3. Depends on DP-1 as premise.'], ['T5: Monotone norms + T5-b: Strict orthogonality', 'Closed — unconditional given DP-1', 'T5 norm result: trivially satisfied (all norms = 1 by construction). ' 'T5-b: distinct consecutive states produce orthogonal T-images — non-trivial, Lean-verified (ZPD.t5_strict_orthogonal).']], [1.6*inch, 1.5*inch, 3.4*inch] )) E.append(Paragraph('VI. Validation Status', S['h1'])) E.append(data_table( ['Component', 'Status / Notes'], [['H = ℂn (D1)', 'Valid — Defined; foundational minimum n = 2 (binary existence/non-existence); ' 'extensions to n = 2k consistent; no core claim requires n > 2'], ['Decoupling of Q2 and H (R1)', 'Valid — Structural; Q2 and H are categorically distinct; T is the bridge'], ['Import I-A from ZP-A', 'Valid — Received; CC-1 reclassification noted'], ['Import I-B from ZP-B', 'Valid — Received; MP-1 included; C3 noted'], ['DP-1: Orthogonality', 'Valid — Design Principle; reclassified from T1; well-motivated and explicit'], ['D2: T requirements', 'Valid — Defined; five requirements stated; all satisfied by T2'], ['T2: Existence of T', 'Valid — Derived; basis assignment; all five requirements verified; ' 'R3 (v1.6) names topological type: locally constant, continuous'], ['T3: Uniqueness of T', 'Valid — Proposition; derived; unique up to unitary equivalence'], ['T4: Snap → orthogonal shift', 'Valid — Theorem; derived; unconditional; depends on DP-1'], ['T5 / T5-b: Monotone norms + strict orthogonality', 'Valid — T5 norm result: tautology of construction (all norms = 1); Lean: ZPD.t5_monotone_norms. ' 'T5-b: non-trivial Lean-verified result — distinct consecutive states produce orthogonal T-images via DP-1; Lean: ZPD.t5_strict_orthogonal (axiom-free).']], [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()