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Applied MODEL Mathematics & foundations v5

Dimensional-Quantity Type System

Definitions for L, T, M dimensions and dimensioned positive quantities

Definitions for L, T, M dimensions and dimensioned positive quantities. Dimensional signature: [Length, Time, Mass] exponents. Used to track physical dimensions through calculations.

Equations

[ J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y) ]

Recognition Composition Law.

Derivation chain (Lean anchors)

Each row links to the corresponding Lean 4 declaration in the Recognition Science canon. A resolved anchor has a green check; an unresolved anchor flags a registry/canon mismatch.

  1. 1 Dimension structure structure checked
    IndisputableMonolith.Constants.Dimensions.Dimension Open theorem →
  2. 2 Dimensioned quantity structure checked
    IndisputableMonolith.Constants.Dimensions.DimensionedQuantity Open theorem →
  3. 3 Positive dimensioned structure checked
    IndisputableMonolith.Constants.Dimensions.PositiveDimensionedQuantity Open theorem →

Narrative

1. Setting

Dimensional-Quantity Type System is anchored in Constants.Dimensions. The page is not a loose explainer: it is a public map from the Recognition Science forcing chain into one Lean-checked declaration bundle. The primary anchor determines what is proved, and the surrounding declarations show how the result is used.

2. Equations

(E1)

$$ J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y) $$

Recognition Composition Law.

3. Prediction or structural target

  • Structural target: Constants.Dimensions must keep resolving in the Lean canon, and all downstream pages that cite this anchor must continue to type-check.

This page is currently a structural derivation. Where the claim has direct empirical content, the prediction table gives the measurable target; otherwise the claim is a formal bridge inside the Lean canon.

4. Formal anchor

The primary anchor is Constants.Dimensions..Dimension.

    Used to track physical dimensions through calculations. -/
structure Dimension where
  L : ℤ  -- Length exponent
  T : ℤ  -- Time exponent
  M : ℤ  -- Mass exponent
  deriving DecidableEq

/-! ## Fundamental Dimension Constants -/

/-- Dimensionless quantity: [L⁰T⁰M⁰] -/

5. What is inside the Lean module

Key definitions:

  • Dimension
  • dim_one
  • dim_L
  • dim_T
  • dim_M
  • dim_c
  • dim_hbar
  • dim_G

6. Derivation chain

7. Falsifier

A Lean-checkable counterexample to the named theorem or to the upstream functional equation refutes this derivation.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

9. Reading note

The minimal way to audit this page is to open the first Lean anchor and then walk the supporting declarations listed above. If the primary theorem is a module-level anchor, the key theorems section names the internal declarations that carry the mathematical load. This keeps the public derivation readable without severing it from the proof object.

10. Audit path

To audit dimensional-quantity-system, start with the primary Lean anchor Constants.Dimensions.Dimension. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.

11. Why this belongs in the derivations corpus

The corpus is organized around load-bearing consequences, not around file names. This entry is included because Constants.Dimensions contributes a reusable theorem or definitional bridge that other pages can cite. Keeping the page public gives readers a stable URL, a JSON record, and a direct path into the Lean theorem page. If the entry becomes redundant with a stronger derivation later, the current slug should be retired rather than silently rewritten; the replacement should absorb its anchors and preserve the audit history.

Falsifier

A Lean-checkable counterexample to the named theorem or to the upstream functional equation refutes this derivation.

References

  1. lean Recognition Science Lean library (IndisputableMonolith)
    https://github.com/jonwashburn/shape-of-logic
    Public Lean 4 canon used by Pith theorem pages.
  2. paper Uniqueness of the Canonical Reciprocal Cost
    Washburn, J.; Zlatanovic, B.
    Axioms (MDPI) (2026)
    Peer-reviewed paper anchoring the J-cost uniqueness theorem.
  3. lean Recognition Science Lean library (IndisputableMonolith)
    https://github.com/jonwashburn/shape-of-logic
    Public Lean 4 canon used by Pith theorem pages.
  4. paper Uniqueness of the Canonical Reciprocal Cost
    Washburn, J.; Zlatanovic, B.
    Axioms (MDPI) (2026)
    Peer-reviewed paper anchoring the J-cost uniqueness theorem.
  5. lean Recognition Science Lean library (IndisputableMonolith)
    https://github.com/jonwashburn/shape-of-logic
    Public Lean 4 canon used by Pith theorem pages.
  6. paper Uniqueness of the Canonical Reciprocal Cost
    Washburn, J.; Zlatanovic, B.
    Axioms (MDPI) (2026)
    Peer-reviewed paper anchoring the J-cost uniqueness theorem.
  7. lean Recognition Science Lean library (IndisputableMonolith)
    https://github.com/jonwashburn/shape-of-logic
    Public Lean 4 canon used by Pith theorem pages.
  8. paper Uniqueness of the Canonical Reciprocal Cost
    Washburn, J.; Zlatanovic, B.
    Axioms (MDPI) (2026)
    Peer-reviewed paper anchoring the J-cost uniqueness theorem.
  9. lean Recognition Science Lean library (IndisputableMonolith)
    https://github.com/jonwashburn/shape-of-logic
    The full forcing chain and the supporting machinery for this derivation are checked in Lean 4.
  10. paper Uniqueness of the Canonical Reciprocal Cost
    Washburn, J., Zlatanovic, B.
    Axioms (MDPI) (2026)
    Peer-reviewed source for the Law of Logic cost theorem uniqueness theorem that anchors RS at T5.

How to cite this derivation

  • Stable URL: https://pith.science/derivations/dimensional-quantity-system
  • Version: 5
  • Published: 2026-05-14
  • Updated: 2026-05-14
  • JSON: https://pith.science/derivations/dimensional-quantity-system.json
  • YAML source: pith/derivations/registry/bulk/dimensional-quantity-system.yaml

@misc{pith-dimensional-quantity-system, title = "Dimensional-Quantity Type System", author = "Recognition Physics Institute", year = "2026", url = "https://pith.science/derivations/dimensional-quantity-system", note = "Pith Derivations, version 5" }