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Shadow Theory

Source / readout Follow-on research · Version 3.1

Sealed or Leaky

The source tetralemma, Bell-certified hidden information, and finite-resource witnesses.

Information can remain present in the whole while being inaccessible to an observer inside it. This work turns that distinction into quantitative theorems: how much an initial readout leaves unresolved, when reversible dynamics keep that information sealed, and what finite physical resources can reveal.

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Jeremy Rodgers · Independent Researcher · Mathematical revision: 21 September 2026
doi:10.5281/zenodo.23077474 · Web edition prepared 1 October 2026

A quantitative result

Bell violation puts a floor under missing information.

Hmin⁡(Z∣T)≥−log⁡2 ⁣(32−S4),H(Z∣T)≥S−22H_{\min}(Z\mid T)\geq-\log_2\!\left(\frac32-\frac{S}{4}\right),\qquad H(Z\mid T)\geq\frac{S-2}{2}

For CHSH values 2 ≤ S ≤ 4, these bounds are sharp in the conditional no-signalling class. Here T is the nominated initial classical readout and Z the generated outcome record. The source tetralemma connects unresolved outcomes to source incompleteness under definite source outcomes, measurement independence, and no-signalling conditional on T. The complete statements, constructions and proofs are below.

Search every technical section, including proofs and references.

Complete technical reading edition

Every step of the argument.

Read all 17 sections in sequence or enter at the question you need. The edition retains the publication’s numbered statements, full proofs, calculations, tables and bibliography. The companion article connects the results to the wider framework.

  1. PUBLICATIONOverview and publication identity
  2. Section 1Question, main result, and scope
    Explore this section1 Question, main result, and scope Main implication. What the tetralemma does not say. Published-data anchoring. Attribution and contribution.1.1 Conventions and levels of conclusion
  3. Section 2The source tetralemma
    Explore this section2 The source tetralemma2.1 Setting and premises2.2 Guessing and hidden information2.3 The four explicit escape models and their exact logical status
  4. Section 3Experimental anchoring
    Explore this section3 Experimental anchoring3.1 A finite-sample likelihood statement, not a central-value entropy claim3.2 Published CHSH values: illustrations only3.3 Bierhorst-admissible information and the actual protocol word Alphabet and symbol translation. Causal padding is part of the record map. Source and numerical quantities. The exact imports used.3.4 Extracted randomness translated into source-response entropy3.5 A stronger finite-sample consequence without an extractor or source–setting independence3.6 Who certifies what, and what remains a gap Explicit unresolved empirical scope.
  5. Section 4The escape routes and their boundaries
    Explore this section4 The escape routes and their boundaries Branching. Fundamental chance. Measurement dependence. Accessible nonlocality.
  6. Section 5Operational equivalence and its restricted alternatives
    Explore this section5 Operational equivalence and its restricted alternatives Temporal and intervention consistency.
  7. Section 6Deterministic completions require unresolved information
    Explore this section6 Deterministic completions require unresolved information
  8. Section 7Conservation of complete information and operational sealing
    Explore this section7 Conservation of complete information and operational sealing A finite reversible counterexample. Persistent forward sealing under reversible dynamics. What retained records prevent.
  9. Section 8What a quantum preparation description necessarily omits
    Explore this section8 What a quantum preparation description necessarily omits8.1 A sharp finite-table separation and a robust witness
  10. Section 9Operational sealing and quantitative failures of POVM form
    Explore this section9 Operational sealing and quantitative failures of POVM form
  11. Section 10The measurement constitutions and the scope of their sealing
    Explore this section10 The measurement constitutions and the scope of their sealing10.1 Pointwise and uniform sealing in the Bell limit10.2 Source distinctions that are not automatically records10.3 What is and is not transferred between constitutions
  12. Section 11Finite pilot resources: exact inventories and retained records
    Explore this section11 Finite pilot resources: exact inventories and retained records11.1 Microscopic input and exchangeability11.2 A coherent writer preserving a drained count11.3 An operational same-density-matrix witness11.4 A retained distant-setting witness in the same model11.5 Joint finite resources and empirical scope11.6 Why randomizing only the initial residue does not restore Born endpoints11.7 A proved undrained asymptotic and its access boundary
  13. Section 12Setting dependence, controlled nonequilibrium, and access
    Explore this section12 Setting dependence, controlled nonequilibrium, and access12.1 A sharp quantitative version of the transition-set argument12.2 A finite massive realization: replacing an exact-record premise12.3 Delayed action meters and the locality of their access12.4 Statistical design for a conditional finite-pilot test
  14. Section 13Assumptions and result status
    Explore this section13 Assumptions and result status
  15. Section 14What the results establish about source reality
    Explore this section14 What the results establish about source reality Remaining gaps, not strengthened claims. Other no-go results and the operational ceiling.
  16. PUBLICATIONResearch transparency and reproducibility
    Explore this section Research transparency and reproducibility AI assistance. Funding and collaboration. Computational materials.
  17. PUBLICATIONReferences

From the foundation to information access

The seven-paper foundation establishes when readout loss matters, what completion requires, and what internal experiments can identify. Sealed or Leaky adds explicit information bounds and physical access conditions.

The framework →Paper 6 →Paper 7 →The full model →Reading manifest