"""
Build ZP-C: Information Theory (v1.17)
v1.17: Version references removed from Remark R5 label_box and T-BUF li() call (Gemini catch — build gate does not cover li()/label_box()).
v1.16: Vocabulary fixes — "null state" → "⊥"; "non-null state" → "nonzero state"; "first atomic state" → "minimum nonzero state (ε₀)" throughout body prose. Palette rebuild.
v1.15: K-19/K-21 vocab fixes — "First Atomic State" -> "Minimum Nonzero State" in T1; "Informational Extremity" -> "Unbounded Surprisal" in L-INF section header, label, and status lines.
v1.14: Version number removed from Open Items Register section header.
v1.13: Adversary-review pass — "forcing result" terminology replaced with "structural
constraint" (avoids incorrect borrowing of set-theoretic forcing vocabulary).
v1.12: Encoding commitment and information-theoretic failure mode added to preamble — makes
the conditionality of the 1-bit claim (T1b) visible before the result appears, and states the
unbounded-surprisal failure mode at ⊥ (L-INF) upfront as the information-theoretic instance of
the foundational claim.
v1.11: L-INF semantic content corrected — "compressed limit of all possible binary programs"
replaced with "limit point of the binary ball hierarchy under the 2-adic metric"; stays within
what the proof actually establishes (ball-hierarchy depth, not Kolmogorov programs).
v1.10: R-BRIDGE "not a coincidence" clause given cross-reference to ZP-E R-AFA, where the
structural argument for why K and I(n) converge at the same threshold is made explicitly.
v1.9: Remark R-TQ added after TQ-IH — external expert confirmation that TQ-IH holds and is
domain-independent: holds in any setting with a non-null first step, independent of any
Turing-specific or Kolmogorov machinery.
v1.8: Remark R-BRIDGE added after L-INF — explicit statement of the relationship between
Kolmogorov complexity K and 2-adic surprisal I(n): distinct measures that converge at P₀;
used as independent routes to the same conclusion, not as a unified measure.
"""
import os
from zp_utils import *
VERSION = '1.17'
def build():
out_path = os.path.join(PROJECT_ROOT, 'ZP-C_Information_Theory.pdf')
doc = make_doc(out_path, 'ZP-C: Information Theory', 'ZP-C', 'Version ' + VERSION)
E = []
E += [Paragraph('THE ZERO PARADOX', S['title']),
Paragraph('ZP-C: Information Theory', S['subtitle']),
Paragraph('Version ' + VERSION + ' | May 2026', S['subtitle']),
sp(10),
body('This document is self-contained within information theory and discrete analysis on Q2. The topological structure of Q2 — specifically total disconnectedness (ZP-B T5), the clopen ball hierarchy, and the binary existence axiom (AX-B1) — is imported from ZP-B as a dependency. Every claim is marked as Derived, Axiomatic, Defined, or Candidate.'),
body('Encoding commitment. The two ontological states of AX-B1 are represented here as point-mass (Dirac) distributions over {0, 1} — the minimum-sufficient probabilistic encoding (RP-1, Section II). Under this encoding, the information-theoretic separation between ⊥ and the minimum nonzero state (ε0) is exactly 1 bit (T1b). The 1-bit result is conditional on this encoding; RP-1 declares and justifies the commitment. Separately, at ⊥, the primary descriptive tool of information theory — surprisal — is unbounded above (L-INF, Section III): no finite external description can contain ⊥. These two results — the 1-bit cost of the transition and the infinite descriptive cost of the origin — are the information-theoretic expression of the same structural constraint on ⊥.'),
body('Illustrated Companion: A paired ZP-C 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. Kolmogorov Complexity and the Incompressibility Threshold', S['h1']))
E.append(body('Let x be a binary string of length n. The conditional Kolmogorov complexity is:'))
E.append(body('K(x|n) = min { |p| : U(p, n) = x }'))
E.append(body('where U is a fixed universal Turing machine and |p| is the length of program p.'))
E.append(sp(4))
E.append(label_box('Definition D1 — Incompressibility Threshold', [
'P0 = limn→∞ K(x|n) / n',
'P0 is the algorithmic entropy rate of x. When P0 reaches its maximum, x has exhausted its capacity for self-description.',
]))
E.append(sp(4))
E.append(label_box('Remark R1 — Scope of P₀', [
'What P0 establishes (Derived): the point at which x becomes incompressible.',
'What P0 does not establish (prior versions: Axiomatic): that incompressibility causes the Binary Snap. Version 1.4 updates this: the generative claim is now a Candidate Theorem (T-BUF), not a bare axiom. See Section V.',
]))
E.append(Paragraph('II. Informational Work: State Representations and JSD', S['h1']))
E.append(Paragraph('2.1 The Representation Principle', S['h2']))
E.append(label_box('Principle RP-1 — Minimum Sufficient Probabilistic Representation', [
'The minimum sufficient probabilistic representation of a binary ontological state is a point-mass (Dirac) distribution over {0,1}: all probability mass is assigned to the value the state occupies, and none to the other value.',
'Justification: A point-mass distribution is the unique distribution that is (a) faithful — assigns non-zero probability only to the ontologically actual value, and (b) non-redundant — carries no uncertainty about which state is occupied. Any distribution with mass at both values represents partial existence and partial non-existence simultaneously. AX-B1 excludes this.',
'Status: PRINCIPLE — representational commitment parallel to MP-1 in ZP-B.',
]))
E.append(sp(4))
E.append(label_box('Theorem T1 — State Representations are Uniquely Derived', [
'Given AX-B1 and RP-1, the distributions are unique:',
'P = (1, 0) (Null State: all mass at 0 — non-existence)',
'Q = (0, 1) (Minimum Nonzero State: all mass at 1 — existence)',
'Proof: AX-B1 establishes two ontological states. RP-1 requires point-mass representation of each. The Null State occupies value 0: P = (1,0). The minimum nonzero state occupies value 1: Q = (0,1). No distribution (p, 1−p) with 0 < p < 1 is consistent with AX-B1. P and Q are unique. Status: DERIVED from AX-B1 and RP-1. ✓',
]))
E.append(sp(4))
E.append(label_box('Corollary T1b — JSD = 1 bit', [
'For P=(1,0) and Q=(0,1) with M=(1/2, 1/2):',
'DKL(P‖M) = 1 bit, DKL(Q‖M) = 1 bit, JSD(P‖Q) = 1 bit',
'E = JSD(P‖Q) = 1 bit [information-theoretic, dimensionless]',
'Status: Derived from AX-B1 and RP-1 via T1. E is not physical energy in joules. Dimensional bridge belongs to ZP-E as BA-1.',
]))
E.append(sp(4))
E.append(label_box('Definition D3 — Dirac Measure δ₀', [
'∫Ω f dδ0 = f(0)',
'δ0 places unit mass at 0. Compatible with the discrete topology of {0,1} and with the totally disconnected topology of Q2 (ZP-B T5). δ0 governs behaviour exactly at x = 0.',
]))
E.append(sp(4))
E.append(label_box('Remark R2 — Hamming Cross-Validation', [
'dH(P, Q) = 1 [Hamming, on {0,1}]',
'The agreement between dH = 1 and E = 1 bit is a consistency check, not a proof that Hamming distance and JSD are equivalent in general. Both are computed by independent methods on the same AX-B1-derived distributions.',
]))
E.append(Paragraph('III. The Discrete Surprisal Field on Q₂', S['h1']))
E.append(label_box('Remark R3 — Why the Smooth Embedding Remains Retired', [
'ZP-B T5 establishes Q2 is totally disconnected. ZP-B T2 establishes every ball is clopen. Importing smooth calculus operators onto a totally disconnected space imports a smoothness assumption that contradicts ZP-B. The smooth embedding, MO-1, and P1 from v1.1 remain retired.',
]))
E.append(sp(4))
E.append(label_box('Definition D4 — Discrete Surprisal Function I', [
'For x ∈ Q2 with x ≠ 0 and P(x) > 0:',
'I(x) = −log2 P(x)',
'As v2(x) → +∞ (x approaches 0 in the 2-adic metric), I(x) → +∞: states 2-adically close to 0 are informationally extreme.',
'The probability measure P on Q2 \\ {0} is the branching measure induced by the binary ball hierarchy of Q2 (RP-2 below): at each level k, the two sub-balls of B(0, 2−k) each receive half the probability mass of their parent ball.',
]))
E.append(sp(4))
E.append(label_box('Principle RP-2 — Branching Measure on Q₂ \\ {0}', [
'The probability measure P on Q2 \\ {0} is the canonical branching measure induced by the binary ball hierarchy: at each level k, each sub-ball receives half the probability mass of its parent. This is a representational commitment — a specific measure choice not uniquely forced by AX-B1 alone.',
'Justification: The branching measure assigns equal weight to each binary branch, consistent with AX-B1\'s symmetric treatment of existence and non-existence. Alternative measures (e.g. non-uniform branching weights) would depart from this symmetry without additional motivation.',
'Parallel: RP-1 fixes the point-mass representation of ontological states. RP-2 fixes the probability measure on the 2-adic domain. Both are representational commitments sitting alongside, not derived from, AX-B1.',
'Status: PRINCIPLE — representational commitment. Required by T2 and L-INF.',
]))
E.append(sp(4))
E.append(label_box('Definition D5 — Discrete Surprisal Difference Operator DF', [
'For x, y ∈ Q2 \\ {0}:',
'DF(x, y) = I(y) − I(x) = log2[P(x) / P(y)]',
'DF is antisymmetric: DF(x,y) = −DF(y,x). No smoothness is assumed; DF is defined entirely pointwise.',
]))
E.append(sp(4))
E.append(label_box('Definition D6 — Discrete Circulation (Extended)', [
'FINITE CASE: A discrete loop γn = (x0, x1, …, xn, x0) in Q2 \\ {0} returning to its start. Circulation: C(DF, γn) = ∑i=0n DF(xi, xi+1) where xn+1 = x0.',
'Note: By telescoping, C(DF, γn) = I(x0) − I(x0) = 0 for all finite loops. DF is conservative on all finite loops.',
'INFINITE CASE: An infinite sequence σ = (x1, x2, …) in Q2 \\ {0} where v2(xi) = i for all i ≥ 1. Partial circulation: Cn(DF, σ) = I(xn+1) − I(x1).',
'Circulation of σ: C(DF, σ) = limn→∞ Cn(DF, σ) if the limit exists or diverges.',
'An infinite sequence σ exhibits non-conservative behaviour if C(DF, σ) diverges.',
]))
E.append(sp(4))
E.append(label_box('Theorem T2 — DF Exhibits Non-Conservative Behaviour on Infinite Sequences Approaching 0', [
'DF is conservative on all finite loops (C = 0 by telescoping). On infinite sequences through the ball hierarchy approaching 0, the circulation diverges: C(DF, σ) → +∞.',
'Proof: Let σ = (x1, x2, …) with xi = 2i. Then P(xi) = 2−i, so I(xi) = i. Partial circulation Cn = I(xn+1) − I(x1) = n. As n → ∞: C(DF, σ) = +∞.',
'Divergence arises from the unbounded depth of the 2-adic ball hierarchy (ZP-B). Finite conservation and infinite divergence are distinct regimes; no contradiction arises.',
'Status: DERIVED from ZP-B ball hierarchy structure and branching measure on Q2. OQ-C1 closed. ✓',
]))
E.append(Paragraph('III-B. Unbounded Surprisal of the Bottom Element', S['h1']))
E.append(label_box('Lemma L-INF — Unbounded Surprisal of ⊥', [
'The surprisal I(n) = n at ball-hierarchy depth n is unbounded above: for any finite bound M, there exist depths n with I(n) > M.',
'⊥ = c0 corresponds to the limit point 0 ∈ Q2 — the limit of the binary ball hierarchy at infinite depth. The binary branching measure assigns equal probability mass at each branch level (D4), and the surprisal diverges without bound as depth increases (T2). At this limit, no finite bound M contains the informational content of ⊥.',
'Proof: Let M ∈ ℝ. By the Archimedean property, ∃ n ∈ ℕ with n > M. Then I(n) = n > M. Since M was arbitrary, surprisal is unbounded above. ✓',
'Formal content: surprisal is not bounded above by any real M.',
'Semantic content: ⊥ is informationally extreme — it is the limit point of the binary ball hierarchy under the 2-adic metric, the accumulation point approached by sequences of increasing depth. No finite bound M contains the surprisal at that limit; therefore no finite external interpreter can hold ⊥ as a static description. This is the mathematical premise for DA-1 (ZP-E § I-DA1). ZP-A CC-2 (⊥ = {⊥}) provides a structural second grounding for the same conclusion: a self-containing object has no external interpreter by structure (ZP-A R3). The informational argument from the ball hierarchy and the structural argument from self-containment are independent derivations converging on the same fact.',
'Note: the connection from unbounded surprisal (L-INF) to forced execution is a named design principle (DA-1 in ZP-E), not a mathematical consequence of L-INF alone. L-INF supplies the formal premise; DA-1 supplies the ontological bridge.',
'Status: DERIVED from D4 and T2. Structural corroboration: ZP-A CC-2 (⊥ = {⊥}) and R3. Lean: ZPC.l_inf (purity check: no non-Mathlib axioms).',
]))
E.append(sp(4))
E.append(label_box('Remark R-BRIDGE — On the Relationship Between K and 2-Adic Surprisal', [
'ZP-C uses two complexity measures: Kolmogorov complexity K(x|n)/|x| (Section I) and '
'2-adic surprisal I(n) = n (Section III). These measure related but distinct structures. '
'Their relationship warrants explicit statement.',
'What each measures. K(x|n)/|x| = 1 is a property of a specific binary string x: '
'no program shorter than x generates it. It is a pointwise statement about a particular '
'configuration at the incompressibility threshold. I(n) = n is a property of position in '
'the ball hierarchy: the surprisal of any string at 2-adic depth n is n, by the branching '
'measure RP-2. It is a structural property of depth, not of any particular string.',
'Where they coincide. At P0, both conditions hold simultaneously for '
'the same configuration: K(c1|n)/|c1| = 1 (no shorter external '
'description exists) and I(n) = n → ∞ (surprisal grows without bound as depth '
'increases). This is not a coincidence of definition. The 2-adic limit point 0 ∈ '
'Q2 is the unique accumulation point of the binary ball hierarchy; algorithmic '
'incompressibility is the K-complexity characterisation of the same extremality. Both '
'locate the same threshold from independent directions. The structural argument for why '
'this convergence is not accidental — that these two independently defined measures are '
'forced to meet at the same limit by the framework\'s architecture — is developed in '
'ZP-E Remark R-AFA.',
'Where they diverge. Away from P0, the measures are not interchangeable. '
'K(2n) = O(log n): the n-fold power of 2 is compactly described by a '
'short program, yet v2(2n) = n grows without bound. A string '
'can be 2-adically deep without being Kolmogorov-incompressible, and vice versa. The '
'two measures agree at the limit but diverge throughout the interior.',
'How ZP-C uses them. ZP-C uses K and I as independent convergent routes to the '
'same conclusion — that ⊥ at P0 admits no finite external interpreter — '
'not as a single unified measure. L-INF is the 2-adic surprisal argument: I(n) unbounded '
'→ no finite interpreter can hold ⊥. DA-1 Path 3 (ZP-E) is the K-incompressibility '
'argument: K = 1 eliminates the static-description state via D7\'s dichotomy. Their '
'independence is preserved deliberately. The convergence of two structurally distinct '
'complexity measures on the same threshold is part of the argument for DA-1, not a '
'circularity within it. Axiomatising an equivalence between K and I would merge these '
'two routes into one and eliminate that independence.',
]))
E.append(sp(4))
E.append(Paragraph('IV. Execution as State: The Hardware Lemma', S['h1']))
E.append(label_box('Definition D7 — Machine Configuration', [
'A machine configuration c is a complete description of a Turing machine at a given moment: the current state of the control unit, the position of the read/write head, and the contents of all tape cells.',
'c0: the initial configuration — the machine exists but has not yet begun execution (no instruction has been fetched).',
'c1: the first running configuration — the machine has fetched and begun executing its first instruction.',
'Note: D7 does not import physics. It is a structural distinction between two discrete machine states within the standard Turing model.',
]))
E.append(sp(4))
E.append(label_box('Lemma L-RUN — Execution is a Non-Null State Change', [
'The transition from c0 (initial: not yet running) to c1 (running: first instruction fetched) is a discrete, irreducible state change in the machine configuration.',
'Proof:',
'Step 1 — c0 and c1 are distinct configurations by D7: the control unit is in a different state (idle vs. executing). c0 ≠ c1.',
'Step 2 — By AX-B1, a state either exists or it does not. The machine at c1 occupies a configuration distinct from c0. The transition c0 → c1 is a binary state change in the sense of AX-B1.',
'Step 3 — The transition is irreducible: there is no intermediate configuration between c0 and c1. The first instruction fetch is the minimal unit of execution.',
'Step 4 — c0 is modeled as corresponding to ⊥ in the semilattice (CC-2 — see below). c1 is strictly above ⊥: the machine configuration has gained content (an active execution context) not present in c0.',
'Conclusion: c1 ≠ ⊥. The act of execution is itself a nonzero state, regardless of what the output tape contains. ✓',
'Status: DERIVED from AX-B1 and D7. No Coding Theorem required. No output-tape contents required.',
]))
E.append(sp(4))
E.append(label_box('Conditional Claim CC-2 — c₀ = ⊥ (Parallel to CC-1 in ZP-A)', [
'We model the initial machine configuration c0 as corresponding to ⊥ in the semilattice (L, ∨, ⊥). This is not derived from D7 or from A1–A4 — it is a modeling choice.',
'Motivation: c0 is the machine state prior to any execution: no instruction has been fetched, no content has accumulated. ⊥ is the algebraic element below all others — the state of no accumulated content. The identification is motivated by structural correspondence, not derivation.',
'Parallel: CC-1 in ZP-A commits to S0 = ⊥ for state sequences. CC-2 makes the same commitment for machine configurations. Both are modeling choices, not derivations from the core axioms.',
'Effect: Step 4 of L-RUN depends on CC-2. The conclusion c1 ≠ ⊥ follows from L-RUN\'s proof; the identification of c0 with ⊥ is the additional modeling commitment supplied by CC-2.',
'Status: CONDITIONAL CLAIM — modeling commitment; not derived from D7 or A1–A4.',
]))
E.append(sp(4))
E.append(label_box('Remark R4 — Configuration and Output Are Independent', [
'D-TQ-2 (Output Definition): A computation\'s output is defined by the contents of its tape/register at termination.',
'L-RUN establishes that configuration state and output state are independent. A program may terminate with null output while still having passed through non-null configuration states during execution.',
'The question "can a program output ⊥ without any non-null intermediate state?" is answered by looking at the execution trace, not the output tape. This distinction is load-bearing for T-BUF and TQ-IH.',
]))
E.append(Paragraph('V. The Test Question and the Buffer Overflow Theorem', S['h1']))
E.append(label_box('Test Question TQ-IH — Can a Program Output ⊥ Without a Non-Null Intermediate State?', [
'Question: Does there exist a program p such that U(p) = ⊥ and the execution trace τ(p) contains no configuration ci with ci ≠ ⊥?',
'Answer: No.',
'Proof:',
'Step 1 — Any program that executes must pass through c1 (by definition of execution, D7).',
'Step 2 — By L-RUN, c1 ≠ ⊥.',
'Step 3 — Therefore the execution trace τ(p) = (c0, c1, …) contains c1, which is a non-null configuration state, regardless of what appears on the output tape at termination.',
'Step 4 — The output of p being ⊥ does not imply that all configurations in τ(p) are ⊥. Output state and configuration state are distinct (R4).',
'Conclusion: No program can produce ⊥ as output without passing through a non-null intermediate configuration state. TQ-IH answered negatively. ✓',
'Status: DERIVED from L-RUN, D7, and R4. No Kolmogorov machinery required. AX-1 derivability pathway now open.',
]))
E.append(sp(4))
E.append(label_box('Remark R-TQ — External Confirmation of TQ-IH (April 2026)', [
'TQ-IH was independently reviewed by an external PhD mathematician.',
'Confirmation: "As long as you have c₁ ≠ c₀, this implies cᵢ ≠ c₀ for some 1 ≤ i < n, at least for n ≥ 2. This has nothing to do with computation or anything at all, it\'s a triviality."',
'This confirms two things. First, TQ-IH holds. Second, and more significantly, the result is domain-independent: it does not depend on any Turing-specific machinery, any Kolmogorov complexity argument, or any property of the semilattice. It holds in any setting where a non-null first step c₁ ≠ c₀ is given. The derivation from L-RUN alone is complete and unassailable.',
'Status: EXTERNALLY CONFIRMED — April 2026.',
]))
E.append(sp(4))
E.append(label_box('Candidate Theorem T-BUF — Incompressibility Forces Non-Null Execution State', [
'Statement: At the incompressibility threshold P0, the Binary Snap ⊥ → ε0 is a structural consequence of execution, not an external trigger.',
'Step 1 — P0 identifies the configuration x at which K(x|n)/n = 1: the configuration string is incompressible. (D1)',
'Step 2 — An incompressible configuration at P0 is informationally extreme (L-INF): its surprisal is unbounded — no finite external program bounds its informational content. A configuration with unbounded informational content has no finite external interpreter and cannot be a static description. Therefore the configuration at P0 is a live machine state. The design principle connecting informational extremity to forced execution is DA-1 (ZP-E § I-DA1, citing L-INF).',
'Step 3 — Any execution passes through c1 (L-RUN). c1 ≠ ⊥ (L-RUN conclusion).',
'Step 4 — In (L, ∨, ⊥), this non-null configuration state is c1 = ⊥ ∨ ε0. By ZP-A D2, this is the Binary Snap.',
'Conclusion: At P0, execution is structurally guaranteed. Execution guarantees a non-null configuration state. That state is ε0 in the semilattice. The derivation pathway is open: P0 + L-RUN + TQ-IH + ZP-A D2, pending cross-framework integration via DA-1 in ZP-E. ✓',
'Status: CANDIDATE THEOREM — structurally complete within ZP-C. The step from unbounded surprisal (L-INF) to forced execution (DA-1) is a design principle, not a mathematical consequence. Full derivation owned by ZP-E.',
]))
E.append(sp(4))
E.append(label_box('Remark R5 — Updated Status of AX-1', [
'AX-1 (Binary Snap Causality) was previously labeled Axiomatic in ZP-C.',
'AX-1 is a Candidate Theorem. The derivation pathway: P0 (D1) identifies the threshold. L-RUN establishes that execution at the threshold constitutes a nonzero state change. TQ-IH establishes that no program avoids this. ZP-A D2 establishes that a nonzero state change from ⊥ is the Binary Snap.',
'Remaining work: DA-1 (Definitional Alignment) must formally tie instantiation of P0 to an execution event. This is owned by ZP-E.',
'Status label: CANDIDATE THEOREM — gap identified and named (DA-1). Closed in ZP-E DA-1 insert.',
]))
E.append(Paragraph('VI. Open Items Register', S['h1']))
E.append(data_table(
['Item', 'Status', 'Description'],
[['S1: Distribution stipulation', 'Closed — T1', 'T1 derives P and Q from AX-B1 and RP-1.'],
['OQ-C1: Non-conservation', 'Closed — T2 rebuilt', 'Telescoping critique resolved; infinite divergence proven within extended D6.'],
['Smooth embedding, MO-1, P1', 'Retired', 'Remain retired from v1.2. Inconsistent with ZP-B.'],
['RP-1: Representation principle', 'Principle — explicit', 'Bridge between AX-B1 and probabilistic tools.'],
['RP-2: Branching measure on Q₂ \\ {0}', 'Principle — explicit', 'Canonical branching measure; representational commitment; required by T2 and L-INF.'],
['CC-2: c₀ = ⊥', 'Conditional Claim', 'Modeling commitment — c₀ identified with ⊥ in semilattice. Parallel to CC-1 in ZP-A. Required by L-RUN Step 4.'],
['D7: Machine configuration', 'Defined', 'Foundation for L-RUN and TQ-IH.'],
['L-RUN: Hardware Lemma', 'Derived — Lemma', 'Execution is a nonzero state change. Derived from AX-B1 and D7.'],
['TQ-IH: Test Question', 'Closed — Confirmed', 'No program can output ⊥ without a non-null intermediate configuration state. Proven by L-RUN. Externally confirmed April 2026 (R-TQ): domain-independent, requires no Turing-specific or Kolmogorov machinery.'],
['T-BUF: Buffer Overflow Theorem', 'Candidate Theorem', 'Incompressibility forces non-null execution state. DA-1 bridge in ZP-E closes this fully.'],
['AX-1: Binary Snap Causality', 'Candidate Theorem', 'Derivation pathway formalized. Closed as T-SNAP in ZP-E DA-1 insert.']],
[1.6*inch, 1.5*inch, 3.4*inch]
))
E.append(Paragraph('VII. Validation Status', S['h1']))
E.append(data_table(
['Component', 'Status / Notes'],
[['K(x|n) and P0 (D1)', 'Valid — standard algorithmic IT'],
['R1: Scope of P0', 'Valid — updated: AX-1 now Candidate Theorem, not bare axiom'],
['RP-1: Representation Principle', 'Valid — Principle; explicit bridge; resolves reviewer gap in T1'],
['T1: Distributions from AX-B1 + RP-1', 'Valid — Derived; reviewer gap closed'],
['D2: JSD definition', 'Valid — standard'],
['T1b: E = JSD = 1 bit', 'Valid — Derived from AX-B1 + RP-1'],
['D3: Dirac measure δ0', 'Valid — standard; compatible with discrete Q2 topology'],
['R3: Smooth embedding retired', 'Valid — Retired; inconsistent with ZP-B'],
['D4: Discrete surprisal I(x)', 'Valid — pointwise on Q2 \\ {0}; branching measure stated'],
['RP-2: Branching measure', 'Valid — Principle; explicit representational commitment; canonical binary branching measure'],
['CC-2: c0 = ⊥', 'Valid — Conditional Claim; modeling commitment; parallel to CC-1 in ZP-A; load-bearing for L-RUN Step 4'],
['D5: Difference operator DF', 'Valid — antisymmetric; no smoothness assumed'],
['D6: Circulation (extended)', 'Valid — finite and infinite cases both defined; finite conservation acknowledged'],
['T2: Non-conservation rebuilt', 'Valid — Derived; telescoping critique addressed; divergence at σ→0 established'],
['R2: Hamming cross-validation', 'Valid — Consistency check'],
['D7: Machine configuration', 'Valid — Defined; standard Turing model; no physics imported'],
['L-RUN: Hardware Lemma', 'Valid — Derived from AX-B1 and D7'],
['R4: Configuration vs. output independence', 'Valid — load-bearing distinction for TQ-IH and T-BUF'],
['TQ-IH: Test question answered', 'Valid — Derived by L-RUN; no Kolmogorov machinery required'],
['T-BUF: Candidate Theorem', 'Candidate — structurally complete in ZP-C; DA-1 bridge in ZP-E closes fully'],
['R5: AX-1 status updated', 'Valid — AX-1 is Candidate Theorem; prior Axiomatic status corrected'],
['R-BRIDGE: K vs. 2-adic surprisal', 'Valid — Remark; states relationship explicitly: distinct measures converging at P0; independence of L-INF and K paths preserved; cross-reference to ZP-E R-AFA for structural convergence argument']],
[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()