Papers Index and Dependency Graph (EN)

`docs/papers` Paper Index and Dependency Graph

This directory contains a set of interconnected manuscripts (each uses main.tex as the entry point). They share a common mechanism chain: unitary scan → projection/readout (finite resolution) → canonical coding (Ostrowski/Zeckendorf, etc.) → mismatch/discrepancy → regulated-to-continuum (orbit trace / finite part).

Back: docs/index.md | docs/index_cn.md

Chinese version: index.md

Paper Overview

Grouped by role (each item uses main.tex as the entry; if main.pdf exists you can open it directly).

Core Interfaces (Tools / Axioms / Main Physics)

2025_holographic_polar_arithmetic/ (HPA): Holographic Polar Arithmetic: Multiplicative Ontology, Unitary Scanning, and the Geometric Origin of Quantum Uncertainty (HPA)
- PDF: main.pdf
- TeX: main.tex
- Role: tools / mathematics (Paper I) — scan–projection, Sturmian/Fibonacci, Ostrowski/Zeckendorf, orbit regularization
- Depends on: —
2025_foundations_of_physics_submission/ (FoP): Omega Theory: Axiomatic Foundations of Holographic Spacetime and Interactive Evolution (FoP)
- PDF: main.pdf
- TeX: main.tex
- Role: main physics manuscript — global static state, finite information / holographic mapping, QCA/quasicrystal substrate, phenomenology/cosmology templates
- Depends on: HPA (toolchain for O5/O6)
2025_holographic_polar_omega_theory/ (HPΩT): Holographic Polar Omega Theory: An Axiomatic Upgrade and the Continuous–Discrete Bridge (HPΩT)
- PDF: main.pdf
- TeX: main.tex
- Role: axiomatic interface note — minimal derivation chain fixing O1–O6 + R1 as a reusable interface
- Depends on: HPA + FoP

Constructive Spacetime, Resolution Folding, and Particle Interface

2025_holographic_hilbert_universe_hpa_omega/ (HHU): The Holographic Hilbert Universe: Constructive Spacetime, Computational-Lapse Gravity, and a Conditional Riemann Critical-Line Rigidity Theorem in HPA–Ω (HHU)
- PDF: main.pdf
- TeX: main.tex
- Role: constructive spacetime — Hilbert folding addressing, routing-overhead lapse gravity, Abel finite parts, (under HTF) conditional RH rigidity
- Depends on: HPA + HPΩT + CAP + RGS
2025_resolution_folding_phi_pi_e_hpa_omega/ (RF): Resolution Folding under φ–π–e Triple-Operator Constraints: A Zeckendorf–Hilbert–Abel Framework for the 64→21 Projection and Recursive Uplift
- PDF: main.pdf
- TeX: main.tex
- Role: pure math / symbolic dynamics–topology–complex analysis interface — Zeckendorf filtering (21), loop-closure split (18+3), Abel–ζ pole barrier, Hilbert address family; includes reproducible scripts
- Depends on: HHU (Hilbert folding) + HPA (Zeckendorf shift)
2025_z128_standard_model_stable_sector_hpa_omega/ (Z128/SM): A Standard Model of Stable Sectors for $Z_{128}$ Scanning Readout: $φ$ – $π$ – $e$ Resolution Folding, Hilbert-Chiral Addressing, and Antimatter Duality
- PDF: main.pdf
- TeX: main.tex
- Role: particle interface — reproducible 21→SM field labeling, mass spectrum (incl. W/Z/H), 20-round bounded-complexity rigidity search and audit summaries
- Depends on: RF + HHU + CAP + PCG

Variational Unification and Dynamical Closure

2025_computational_action_principle_hpa_omega/ (CAP): Computational Action Principle: Least-Discrepancy Dynamics and Field Unification in HPA–Ω (CAP)
- PDF: main.pdf
- TeX: main.tex
- Role: variational unification — least-discrepancy principle → Ω action (Fisher + routing overhead) → GR + information stress; gauge/matter as phase-error compensation and topological defects
- Depends on: HPA + FoP + HPΩT
2025_computational_action_principle_ii_dynamics_hpa_omega/ (CAP-II): Computational Action Principle II: Dynamical Einstein Gravity and Quantum Interfaces from Routing Overhead in HPA–Ω (CAP-II)
- PDF: main.pdf
- TeX: main.tex
- Role: dynamical closure — ADM closure from auditable lapse (routing overhead) and interfaces for quantum/scattering calibration
- Depends on: CAP + HHU + HPA + HPΩT (+ FoP)

Arithmetic Geometry, “Climbing”, and Periods/Motives

2025_ramanujan_holographic_scanning_principle_hpa_omega/ (RHSP): Ramanujan Holographic Scanning Principle: Modular Curves, Hecke Dynamics, and an Arithmetic Constitution for HPA–Ω (RHSP)
- PDF: main.pdf
- TeX: main.tex
- Role: arithmetic-geometry interface — modular curves/Hecke as the “mother geometry” and prime skeleton for scan–projection; includes reproducible experiments
- Depends on: HPA + HPΩT
2025_stairway_to_infinity_holographic_renormalization_flow/ (STI): The Stairway to Infinity: A Holographic Renormalization Flow from Noncommutative Scanning to the Langlands Program (STI)
- PDF: main.pdf
- TeX: main.tex
- Role: renormalization “climbing” — modular geodesic flow / Gauss suspension; finite- $N$ error bounds via star discrepancy and Ostrowski digits; scale-exchange templates (S-inversion, Morita/Fourier)
- Depends on: RHSP + HPA
2025_motive_at_infinity_holographic_scanning_principle/ (MAI): The Motive at Infinity: Functorialization of the Holographic Scanning Principle, Period Realizations, and a Selection Principle (MAI)
- PDF: main.pdf
- TeX: main.tex
- Role: periods/motives — functorialize scan protocols to Kontsevich–Zagier period data with auditable error budgets; includes reproducible examples
- Depends on: STI + HPA (+ PCG rigidity example)
2025_protocol_stable_period_data_computational_teleology/ (PSPD): Protocol-Stable Period Data in Computational Teleology: From Holographic Scanning Protocols to Variational Dynamics of Minimal Computational Discrepancy and Constant Selection (PSPD)
- PDF: main.pdf
- TeX: main.tex
- Role: teleological variational dynamics — Wish data, discrepancy×variation certificates, damped inertial gradient flow on protocol space; constant rigidity/uniqueness-gap toy model
- Depends on: MAI + CAP (+ PCG)

RH / Trace-Formula Topic

2025_riemann_ground_state_hpa_omega/ (RGS): Riemann Ground State: Holographic Trace Formulas, Abel Finite Parts, and a Protocol Derivation of the Riemann Hypothesis in HPA–Omega (RGS)
- PDF: main.pdf
- TeX: main.tex
- Role: RH / trace — protocolize explicit formulas as Abel-traces; under Ω+HTF derive RH as a protocol consequence; includes reproducible experiments
- Depends on: HPA + HPΩT (+ RHSP)

Applications and Topics (Black Holes, Thermodynamics, Constants, Architecture, Bio/Chem)

2025_holographic_polar_dynamics/ (HPD): Holographic Polar Dynamics: Topological Inversion of the Schwarzschild Singularity and the Phase Origin of Gravity (HPD)
- PDF: main.pdf
- TeX: main.tex
- Role: application (Paper II) — apply HPA discrepancy/quantum gap to black-hole/singularity and information-paradox templates (Phase Pressure)
- Depends on: HPA (+ CAP as a unifying lens)
2025_holographic_phase_thermodynamics_hpa_omega/ (HPT): Holographic Phase Thermodynamics: Arithmetic Statistical Mechanics, Computational Lapse, and the Geometric Origin of Intelligence in HPA–Ω (HPT)
- PDF: main.pdf
- TeX: main.tex
- Role: topic (thermodynamics/intelligence) — arithmetic statistical mechanics, phase friction (discrepancy entropy), computational lapse and entropy-flow rescaling, intelligence as active error-correction phase transition
- Depends on: HPA + FoP + HPΩT + CAP (+ HPD Phase Pressure template)
2025_physical_constants_geometry_hpa_omega/ (PCG): The Geometry of Physical Constants in HPA–Ω: From the Fine-Structure Constant to Particle Spectra and Black-Hole/Cosmological Invariants (PCG)
- PDF: main.pdf
- TeX: main.tex
- Role: topic (constants/geometry) — constants relations and numerical verification under HPA–Ω
- Depends on: HPA + FoP
2025_computational_teleology_hpa_omega/ (CT): Computational Teleology in Holographic Polar Arithmetic: Scan Complexity, Readout Resolution, and an Undecidable Quantum-Cellular Universe (CT)
- PDF: main.pdf
- TeX: main.tex
- Role: topic (architecture/decidability) — complexity–geometry–observation, routing overhead and computational lapse, and computability boundaries
- Depends on: HPA + FoP
chemistry/2025_geometric_origin_chemical_bond_hpa_omega/ (Chem): Geometric Origin of Chemical Bonding in HPA–Ω: Deriving Molecular Stability from the Three-Channel Geometric Impedance α and the Internal Phase Volume μ (Chem)
- PDF: main.pdf
- TeX: main.tex
- Role: topic (chemistry/bonding) — atomic units and BO separation from geometric invariants ( $a lp ha$ , $m u$ ); includes scripts
- Depends on: PCG + HPΩT + HHU + HPT + CAP
biology/2025_biological_computational_teleology_hpa_omega/ (BioCT): Biological Computational Teleology in HPA–Ω: Life as Predictive Active Error Correction Against Phase Friction (BioCT)
- PDF: main.pdf
- TeX: main.tex
- Role: topic (biology/teleology) — life as predictive active error correction; phase friction, geometric Landauer, survival inequalities and control laws
- Depends on: HPT + CT + PSPD
biology/2025_arithmetic_origin_genetic_code_hpa_omega/ (GenCode): Zeckendorf–Hilbert Folding Characterization of the Genetic Code: Start–Stop Topological Symmetry and Z-Spectrum Analysis Centered on 6-Bit Resolution Folding
- PDF: main.pdf
- TeX: main.tex
- Role: topic (genetic code decompilation) — codons as 6-bit microstates; Fold $_{6}$ as 64→21 compression; exhaustive search over 2-bit encodings under boundary constraints; Z-spectrum tooling
- Depends on: RF + BioCT

Note: if main.pdf is not present in a directory, compile main.tex locally to generate it (instructions below).

Dependency Graph (Papers)

flowchart TB
  HPA["HPA<br/>Holographic Polar Arithmetic<br/>Tools / Mathematics (Paper I)"]
  FoP["FoP<br/>Omega Theory<br/>Main physics manuscript"]
  HPOT["HPΩT<br/>Omega Theory<br/>Axiomatic interface note"]
  HHU["HHU<br/>The Holographic Hilbert Universe<br/>Constructive spacetime / computational-lapse gravity"]
  RHSP["RHSP<br/>Ramanujan Holographic Scanning Principle<br/>modular curves / Hecke interface"]
  RGS["RGS<br/>Riemann Ground State<br/>trace formula / Abel finite part / RH"]
  STI["STI<br/>Stairway to Infinity<br/>holographic renormalization flow"]
  MAI["MAI<br/>The Motive at Infinity<br/>periods / motives / selection"]
  PSPD["PSPD<br/>Protocol-Stable Period Data<br/>teleological variational dynamics"]
  CAP["CAP<br/>Computational Action Principle<br/>least-discrepancy variational unification"]
  CAPII["CAP-II<br/>Dynamical closure"]
  HPT["HPT<br/>Holographic Phase Thermodynamics<br/>thermodynamics / intelligence"]
  HPD["HPD<br/>Holographic Polar Dynamics<br/>black hole / singularity application (Paper II)"]
  PCG["PCG<br/>Physical Constants Geometry<br/>constants / geometry"]
  CT["CT<br/>Computational Teleology<br/>architecture / computability boundary"]

  HPA -->|"scan–projection, coding, orbit regularization"| FoP
  HPA -->|"constructive details and proofs"| HPOT
  FoP -->|"framework narrative and physical development"| HPOT
  HPA -->|"Hilbert folding addressing, lapse and Abel–trace construction"| HHU
  HPOT -->|"O1–O6 + R1 audit interface"| HHU
  CAP -->|"computational lapse / phase-potential template"| HHU
  RGS -->|"Abel–trace and HTF→RH conditional rigidity mechanism"| HHU
  HPA -->|"modular embedding, Hecke prime skeleton"| RHSP
  HPOT -->|"O1–O6 + R1 audit interface"| RHSP
  HPA -->|"orbit trace / finite-part regularization"| RGS
  HPOT -->|"Abel finite-part convention path"| RGS
  RHSP -->|"primes as periodic orbits / explicit-formula motivation"| RGS
  RHSP -->|"modular geodesic flow / Gauss suspension, slice–coefficients–sampling"| STI
  HPA -->|"scan algebra and discrepancy control"| STI
  STI -->|"scan = period realization, functorialization"| MAI
  HPA -->|"scan–readout protocols and error-budget templates"| MAI
  PCG -->|"low-complexity constant rigidity example"| MAI
  MAI -->|"Wish data + auditable error budgets"| PSPD
  CAP -->|"least-discrepancy variational closure"| PSPD
  PCG -->|"bounded-complexity rigidity toy model"| PSPD
  HPA -->|"least-discrepancy / readout mismatch"| CAP
  FoP -->|"Ω action, lapse, Fisher"| CAP
  HPOT -->|"O1–O6 + R1 interface"| CAP
  CAP -->|"dynamical closure (ADM, interfaces)"| CAPII
  HHU -->|"from auditable lapse to dynamics"| CAPII
  HPA -->|"ASM, phase friction (discrepancy entropy)"| HPT
  FoP -->|"O1–O6 readout interface and thermodynamic semantics"| HPT
  HPOT -->|"O1–O6 readout interface and R1 regularization convention"| HPT
  CAP -->|"computational lapse and entropy-flow rescaling"| HPT
  HPD -->|"Phase Pressure / phase-potential template"| HPT
  CAP -->|"variational closure for Phase Pressure"| HPD
  HPA -->|"discrepancy/quantum gap → Phase Pressure"| HPD
  HPA -->|"Weyl pair, readout and coding tools"| CT
  FoP -->|"implementation-layer routing overhead and phenomenology"| CT
  HPA -->|"arithmetic/coding and regularization tools"| PCG
  FoP -->|"constants/geometry derivation interface"| PCG

  classDef tool fill:#4CAF50,stroke:#2E7D32,color:#FFFFFF;
  classDef core fill:#2196F3,stroke:#1565C0,color:#FFFFFF;
  classDef note fill:#FF9800,stroke:#EF6C00,color:#FFFFFF;
  classDef app fill:#9C27B0,stroke:#6A1B9A,color:#FFFFFF;
  classDef unify fill:#009688,stroke:#00695C,color:#FFFFFF;
  classDef topic fill:#607D8B,stroke:#455A64,color:#FFFFFF;

  class HPA tool;
  class FoP core;
  class HPOT note;
  class HHU note;
  class RHSP note;
  class RGS note;
  class STI note;
  class MAI note;
  class HPT topic;
  class HPD app;
  class CAP unify;
  class CAPII unify;
  class PCG topic;
  class CT topic;
  class PSPD topic;

Paper Briefs (Interfaces with Other Papers)

`2025_holographic_polar_arithmetic/` (HPA, Paper I)

Core question: treat “rotation/phase/multiplication” as the ontological layer; explain why linear additivity and continuous composition cannot both be preserved under discrete readout; turn non-closure residuals into computable objects.
Key components:
- multiplicative skeleton: polar-style embedding from a multiplicative monoid.
- unitary scan: Koopman-type unitary scan + window projection readout.
- canonical coding: irrational rotation → Sturmian; golden branch → Fibonacci; Ostrowski → Zeckendorf.
- orbit calculus / finite part: orbit trace and Abel finite part as fixed conventions for controlled discrete→continuous limits.
Role in the series:
- provides concrete models and reusable tools/proofs for FoP O5/O6.
- provides discrepancy/quantum-gap objects and quantitative language for HPD.
Entry points: main.pdf, main.tex, references.bib

`2025_foundations_of_physics_submission/` (FoP, Omega Theory main physics manuscript)

Core question: under “no external time”, describe the universe as a static global state; impose finite-information / holographic constraints; construct a controlled microscopic framework that generates effective spacetime and dynamics.
Main layers:
- axioms (O1–O6): static global state, finite information, causal locality, holographic map, scan–projection readout, Weyl pairs.
- models (M1–M3, etc.): QCA/quasicrystal substrates, internal algebraic structures, golden-spectrum updates, tensor-network holographic coding.
- phenomenology: information-geometric action (Omega Action), constants/dispersion/noise templates, cosmological consequences.
Interfaces:
- uses HPA toolchain for O5/O6, Sturmian/Zeckendorf ticks, and orbit regularization.
- HPΩT extracts the common axiomatic interface between FoP and HPA.
Entry points: main.pdf, main.tex, references.bib

`2025_holographic_polar_omega_theory/` (HPΩT, axiomatic interface note)

Positioning: a minimal, citable axiomatic interface for the continuous–discrete bridge in Omega Theory.
What it does:
- keeps O1–O4 and upgrades O5–O6 (scan–projection + Weyl pair) and Convention R1 (orbit trace / finite part).
- provides a shortest derivation chain: scan–projection → canonical coding (golden-branch Zeckendorf) → incompatibility/uncertainty → readout-induced probability (instrument/POVM) → regularization convention.
Interfaces: fixes the interface between HPA (constructions/proofs) and FoP (full physical narrative) for later applications.
Entry points: main.pdf, main.tex, references.bib

`2025_holographic_hilbert_universe_hpa_omega/` (HHU, The Holographic Hilbert Universe)

Positioning: an auditable construction chain for “how a 1D scan protocol yields multi-dimensional spatial locality”, unified with computational-lapse gravity templates and Abel–trace (under HTF) conditional RH rigidity in one manuscript.
What it does:
- Hilbert folding axiom (H1): at each resolution level $n$ , map tick-time to a holographic-screen lattice via discrete Hilbert addressing; locality becomes an address/compile effect.
- computational-lapse gravity: define lapse $N = κ_{0} / ka pp a$ from routing overhead $ka pp a$ ; give redshift templates and Poisson closures verifying near-field $1/ r$ phase potential.
- critical-line rigidity (conditional): under HTF, use holomorphy inside the Abel unit disk to rule out interior poles corresponding to off-line zeros; derive RH as a protocol consequence.
- reproducible scripts: locality checks, star-discrepancy bounds, Abel pole-barrier toy models, Poisson solving, Wigner–Smith $ka pp a (E)$ interfaces.
Interfaces: follows HPΩT O5/O6+R1 and completes “spacetime construction” via explicit Hilbert folding; aligns with CAP/HPD lapse/phase-potential templates; structurally matches RGS’s HTF→RH conditional rigidity mechanism.
Entry points: main.pdf, main.tex, references.bib, scripts/

`2025_resolution_folding_phi_pi_e_hpa_omega/` (RF, Resolution Folding)

Positioning: an auditable Zeckendorf–Hilbert–Abel framework for the $64 t o 21$ projection consistent with the $v a r p hi$ – $p i$ – $e$ triple-operator constraints; includes reproducible experiments.
What it does:
- Zeckendorf filtering (21): compress discrete readout to a stable 21-sector core via canonical coding.
- closure split (18+3): provide a loop-closure split as the core input for later field-layer labeling.
- Abel–ζ pole barrier: present controlled regularization and pole-barrier mechanisms in an Abel viewpoint.
- Hilbert address family: align with HHU Hilbert-folding addressing.
Interfaces: provides the provable core input for Z128’s 21→SM labeling and audit scripts.
Entry points: main.pdf, main.tex, references.bib, scripts/

`2025_z128_standard_model_stable_sector_hpa_omega/` (Z128, Stable Sectors → Standard Model)

Positioning: on top of the provable $64 t o 21$ and $18 o pl u s 3$ core, provide reproducible 21→SM field labeling and mass-spectrum closure, together with bounded-complexity audits demonstrating uniqueness gaps and failure baselines.
What it does:
- field-layer labeling: auditable labeling rules for chirality/antimatter/family structures over 21 stable sectors.
- mass spectrum and mixing closure: fits including W/Z/H, and CKM/PMNS interface closures (with scripts and audit summaries).
- rigidity audit: fixed-budget multi-round searches with counterfactual baselines and pass/fail statistics.
Interfaces: RF+HHU provide folding/addressing inputs; CAP/PCG provide variational and constant-geometry templates.
Entry points: main.pdf, main.tex, references.bib, scripts/

`2025_ramanujan_holographic_scanning_principle_hpa_omega/` (RHSP, Ramanujan Holographic Scanning Principle)

Positioning: place scan–projection (O5/O6) into the arithmetic-geometry “mother space” of modular curves $X (1)$ and the Hecke algebra, yielding an auditable “arithmetic–holographic–computational constitution”.
What it does:
- mother geometry: fundamental domain of $ma t h r m PS L_{2} (ma t hbb Z)$ , cusps, and $0 s im in f t y$ cusp identification as a scale-exchange template.
- discrete interface: cusp $q$ -expansions as continuous→discrete coefficient interfaces; $t a u (n)$ from $De lt a$ as a computable example.
- prime skeleton: Hecke operators are defined for all $n$ but generated by primes; prime-power recursions and stable spectrum interpretations.
- canonical coding: continued fractions → Ostrowski; golden branch → Zeckendorf; “integer time → bit readout” interfaces.
- reproducible checks: pure-standard-library Python verifying Fibonacci words, discrepancy, $t a u (n)$ recursions, and $j$ -invariant S-invariance.
Interfaces: reuses HPA scan/coding toolchain and aligns with HPΩT’s O1–O6+R1 interface for use in FoP/CAP and beyond.
Entry points: main.pdf, main.tex, references.bib

`2025_riemann_ground_state_hpa_omega/` (RGS, Riemann Ground State)

Positioning: with HPA–Ω scan–readout and Abel finite parts as closed-layer conventions, read ζ’s explicit formula as a “holographic trace formula”; under Ω+HTF provide a protocol derivation chain for RH.
What it does:
- closed-layer derivation: combine Omega’s “Abel-trace defined for $0 < r < 1$ + existence of finite parts along a canonical path” with HTF’s “zero-index modes enter the spectral side”, yielding an Abel radius barrier excluding $R e (r h o) n e q 1/2$ .
- reproducible mechanism checks: golden-branch 1D star discrepancy remains logarithmically stable; shifting toy zero modes’ real parts triggers Abel thresholds and energy blow-up.
Interfaces: directly reuses HPA orbit calculus/finite parts and HPΩT’s axiomatic interface; compatible with RHSP’s “primes = periodic orbits” view, but the closed-layer argument only needs HTF as a structural bridge.
Entry points: main.pdf, main.tex, references.bib, scripts/

`2025_stairway_to_infinity_holographic_renormalization_flow/` (STI, The Stairway to Infinity)

Positioning: advance RHSP’s “static constitution” into the “climbing process” itself: formalize cross-scale lifting via modular geodesic flow (continuous scale) and its Gauss suspension / continued-fraction digits (discrete scale).
What it does:
- renormalization-flow object: connect modular geodesic flow and the Gauss map via a series theorem; roof functions provide computable scale-time.
- slice–coefficients–sampling: projection formulas recovering $q$ -coefficients from slices at height $y > 0$ ; replace integrals by scan-orbit sampling; derive finite- $N$ bounds via star discrepancy and Ostrowski digits.
- scale-exchange templates: place S-inversion (endpoint identification, deep/shallow swap) alongside noncommutative-torus Morita equivalence / Fourier exchange as unified templates.
- Langlands upgrade tasks: articulate the functorialization goal from scan-protocol categories to automorphic-representation categories.
Interfaces: inherits modular/Hecke prime skeleton from RHSP and reuses HPA discrepancy tools as the bottom layer for sampling error control.
Entry points: main.pdf, main.tex, references.bib

`2025_motive_at_infinity_holographic_scanning_principle/` (MAI, The Motive at Infinity)

Positioning: without breaking Layer 0/1 audit discipline, anchor “scan–readout protocols” as computable realizations of Kontsevich–Zagier periods, and functorialize protocols to period-data objects on a controllable subcategory; state selection as a finite-resource stability/complexity optimization principle (programmatic, not a closure premise).
What it does:
- controllable protocol subcategory: define Scan_alg (Kronecker scan + rational-kernel readout + explicit regularization/truncation rules), encoding finite resolution and truncation budgets into protocol objects.
- functorial upgrade: construct a holographic scanning functor HSP: Scan_alg -> PerDatum mapping protocols to period-data objects (field + rational kernel + regularization rule).
- main closed theorem: under irrational/rational independence, Birkhoff averages converge to the target period integral; under resonance, collapse to Haar averages on subtori as the standard alternative.
- auditable error budgets: decompose finite-resource errors into “sampling discrepancy” + “regularization/truncation”; provide truncation bounds templates for $ze t a (2), ze t a (3)$ with error propagation across protocol composition.
- rigidity signal example: bounded-complexity integer-relation search for $a lp h a^{- 1}$ showing $(4, 1, 1)$ as the uniquely significant optimum under a given budget (consistent with PCG).
Interfaces: continues STI’s functorialization program and aligns with HPA’s audit language; forms a mutual-check interface with PCG on “low-complexity rigidity signals”.
Entry points: main.pdf, main.tex, references.bib, scripts/

`2025_protocol_stable_period_data_computational_teleology/` (PSPD, Protocol-Stable Period Data)

Positioning: close MAI’s “Wish = protocol-stable period data” and CAP’s “least-discrepancy variational skeleton” into an auditable dynamics model on protocol parameter space: certified discrepancy potential + explicit complexity costs + damped inertial gradient flow.
What it does:
- axiomatic objectification of Wish: fix Wish as protocol-stable period data, treated as a reproducible target object rather than interpretive language.
- finite-resource error certificates: sampling-error certificate via discrepancy × Hardy–Krause variation, combined with regularization/truncation error decomposition into a total auditable budget.
- variational dynamics on protocol space: treat certified discrepancy potential and complexity costs as computational potential; derive damped inertial gradient flow and show Lyapunov monotonicity.
- reproducible verification: pure Python protocols for $l o g 2, p i, ze t a (2), ze t a (3)$ ; low-complexity search for $a lp h a^{- 1} (0)$ with a visible uniqueness gap.
Interfaces: inherits MAI’s audit discipline and advances the selection principle into explicit dynamical closure; mutually checks with PCG via constant rigidity examples.
Entry points: main.pdf, main.tex, references.bib, scripts/

`2025_computational_action_principle_hpa_omega/` (CAP, Computational Action Principle)

Core question: elevate “laws = error-correction / steady-state mechanisms” into a closed variational chain: least discrepancy drives a unified expression from readout mismatch to field equations.
Key claims:
- least-discrepancy principle: treat accumulated discrepancy/mismatch as a cost that must be suppressed to sustainable levels; promote it to a continuous source (phase potential / stress source).
- Ω action skeleton: Fisher information (information geometry) + implementation complexity (routing overhead / computational lapse) as minimal coupling; discrepancy penalty enters as potential/source terms.
- unique GR closure: under local covariance and second-order closure conditions, the macroscopic metric equation uniquely converges to Einstein equations (incl. $L amb d a$ ), with discrepancy/complexity entering effective stress and potential.
- gauge/matter interpretation: gauge connections as local phase-error compensation; matter as topologically locked phase defects (requiring sustained budgets).
Interfaces: takes HPA scan/coding/discrepancy tools as inputs and uses FoP/HPΩT axioms and the Ω-action dictionary as the backbone; offers a variational closure perspective for HPD’s Phase Pressure template.
Entry points: main.pdf, main.tex, references.bib

`2025_computational_action_principle_ii_dynamics_hpa_omega/` (CAP-II, Computational Action Principle II)

Positioning: from auditable routing-overhead / computational-lapse quantities, provide ADM-type dynamical closure and consistent calibration paths for quantum/scattering interfaces.
Interfaces: continues CAP’s variational skeleton, reuses HHU’s lapse construction, and maintains compatibility with HPA/HPΩT readout interfaces.
Entry points: main.pdf, main.tex, references.bib

`2025_holographic_phase_thermodynamics_hpa_omega/` (HPT, Holographic Phase Thermodynamics)

Positioning: take HPA–Ω readout mismatch and computational-lapse inputs to build arithmetic statistical mechanics and a “phase-friction” thermodynamics, and interpret intelligence as an active error-correction phase transition.
Interfaces: reuses HPD’s Phase Pressure viewpoint with variational closure via CAP; connects to CT/BioCT on teleological active-correction applications.
Entry points: main.pdf, main.tex, references.bib

`2025_holographic_polar_dynamics/` (HPD, Omega Dynamics / Paper II)

Core question: reinterpret GR endpoint pathologies (e.g., Schwarzschild essential singularity) as “readout coordinate endpoints”, and propose a macroscopic gravity template driven by scan–projection mismatch.
Key claims:
- Phase Pressure: coarse-grain accumulated discrepancy/quantum gap into a phase potential $P hi$ source, reproducing Newtonian potential in the weak-field limit and closing to the Schwarzschild exterior geometry under standard assumptions.
- inversion continuation: use the isotropic-radius inversion symmetry to elevate the Einstein–Rosen throat into an “extension rule”, replacing endpoint termination with an inversion channel.
- information-paradox template: treat evaporation as coarse-grained readout; thermalization-edge statistics and correlation-carrying information coexist.
Interfaces: uses HPA as Paper I and borrows its arithmetic/symbolic readout language; can be read as an application module compatible with FoP (same scan–projection axis) though it mainly cites HPA.
Entry points: main.pdf, main.tex, references.bib

`2025_physical_constants_geometry_hpa_omega/` (PCG, Physical Constants Geometry)

Positioning: a topical manuscript using HPA–Ω geometric/information structure for physical-constant relations and numerical checks.
Entry points: main.pdf, main.tex, references.bib

`chemistry/2025_geometric_origin_chemical_bond_hpa_omega/` (Chem, chemistry/bonding topic)

Positioning: take constant-geometry invariants ( $a lp ha, m u$ ) from PCG as closed-layer inputs; map to atomic units and BO separation; propose falsifiable spectroscopy interfaces; includes scripts.
Entry points: main.pdf, main.tex, references.bib, scripts/

`biology/2025_biological_computational_teleology_hpa_omega/` (BioCT, biology/teleology topic)

Positioning: with HPT’s phase friction and CT’s architectural language, treat “life” as a predictive active error-correction phase and provide auditable templates for survival inequalities and control laws.
Entry points: main.pdf, main.tex, references.bib

`biology/2025_arithmetic_origin_genetic_code_hpa_omega/` (Genetic-code decompilation topic)

Positioning: treat codons as 6-bit microstates and implement Fold $_{6}$ as 64→21 folding; exhaustively decompile 2-bit encodings under start/stop boundary constraints, yielding a unique optimum and transcript-level Z-spectrum tooling.
Entry points: main.pdf, main.tex, references.bib

`2025_computational_teleology_hpa_omega/` (CT, Computational Teleology)

Positioning: system architecture layer for complexity–geometry–observation: routing overhead and computational lapse, time/space complementarity, and decidability boundaries (open conditions at the computability level).
Entry points: main.pdf, main.tex, references.bib

Building `main.pdf` Locally

To make the main.pdf links available, compile main.tex in each paper directory.

Recommended (with latexmk):

latexmk -pdf -interaction=nonstopmode -halt-on-error main.tex

Generic (without latexmk):

pdflatex -interaction=nonstopmode -halt-on-error main.tex
bibtex main
pdflatex -interaction=nonstopmode -halt-on-error main.tex
pdflatex -interaction=nonstopmode -halt-on-error main.tex

The Omega Framework