PulseCore Dimensional Analysis Framework
The rules on this page are what the calculator follows; the test suite holds the 210 equations that prove it, each with the exact working expected.
Base Dimensions
| Symbol | Name | Description | Domain |
|---|---|---|---|
| ∅ | Dimensionless | Pure numbers, ratios, indices, counts — the empty dimension, the identity of the algebra (the number one). What a dimension leaves behind when it cancels. In BPT a pure number is data: see Numbers and the data dimension below. | Universal |
| ℨ | Universal Unit | Fundamental computational substrate, base unit for all quantities | BPT Core |
| ↁ | Data Dimension | Computational processing operations, underlying property of all phenomena | Data Domain |
| 𝔸 | Action | Computational work, process optimization, energy cost of operations | Action Domain |
| 𝕄 | Mass | Matter dimension (kilograms) | Physics |
| 𝕃 | Length | Spatial dimension (meters) | Physics |
| 𝕋 | Time | Conventional temporal dimension (seconds) | Physics |
| 1ᵇ | Data Bits | Static data content | Data |
| 2ᵇ | 2 Data Bits | Static Physical Bit | Data |
Numbers and the Data Dimension
Standard mathematics first
Dimensional analysis never sees a numeric value: 2 metres and 0 metres are metres. So a coefficient is not a dimension — 2 × 𝔸, 𝔸 + 𝔸 and 𝔸 − 𝔸 are all [𝔸], a typed number is [∅], and a dimension divided by itself is [∅]: 𝔸 ÷ 𝔸 = [∅], ℨ⁰ = [∅]. The innovations of Binary Pulse Theory are the data dimension ↁ, the Zinf level ℨ and bits as kinds; everywhere else the calculator follows standard physics, and the places where the theory departs from it are named as such on this page.
The number is the data
When a dimension cancels — 𝔸 ÷ 𝔸, a ratio, an efficiency, an angle — what is left is a pure number. A pure number is information with nothing material behind it. The number is the data dimension: [∅] is what every dimension leaves behind, and what it leaves behind is data. That meaning lives in the value, not in the dimension: the dimension is [∅], and the calculator says what [∅] means.
Test𝔸 ÷ Test𝔸 = Test∅ ✓ VALID Test𝔸 ÷ Test𝔸 = Testↁ ✗ INVALID — data in meaning, not in dimension.
Open thread: does a cancelled dimension leave 1ᵇ?
Josh's proposal (2026-09-25): after 𝔸 ÷ 𝔸 there is still one bit of data — the fact of the A — which is not material because material takes 2ᵇ; then (𝔸 ÷ 𝔸) × 𝔸 = [𝔸·1ᵇ] would be "A at the data level, the purest form of A", distinct from the A we see.
Why it is not the rule: it makes the dimension depend on the order of the operations. 𝔸 × 𝔸 ÷ 𝔸 ÷ 𝔸 would give [1ᵇ] but (𝔸 ÷ 𝔸) × (𝔸 ÷ 𝔸) would give [1ᵇ²], the same quantity with two dimensions, and division would no longer undo multiplication, so no equation could be rearranged. There is probably something in it — a trace of a cancelled dimension is a real idea — and it is recorded here so it is not lost. The two path-independence tests (std-5, std-6) are what would catch it if it ever came back into the algebra.
Zinf UniSpheral Unit Bridging Rule
Core Principle: Zinf is Compatible with All Dimensions
As the fundamental computational substrate and base unit for all quantities, the Zinf Unit (ℨ) serves as the dimensional foundation of BPT. Any dimension containing Zinf can bridge to any other dimensional combination because everything is ultimately built upon this UniSpheral substrate.
UniSpheral Bridging Examples
[ℨ] = [ↁ·2ᵇ] ✓ VALID
Why: Zinf is the UniSpheral substrate underlying data processing operations
[ℨ²] = [ℨ·ↁ·𝔸·𝕄·𝕃²·𝕋⁻²] ✓ VALID
Why: any power of ℨ alone bridges to any composite; the final line shows the two sides as they reduced, and the calculator notes that ℨ bridged
[ℨ·ↁ] = [𝔸·1ᵇ] ✗ INVALID
Why: ℨ·ↁ is not pure ℨ. A side bridges only when nothing but ℨ is in it — the bridge is the floor and the ceiling, not a factor that can be carried inside a composite
[ℨ] × [𝕄] ÷ [𝕄] = [ℨ·ↁ·𝔸·𝕄·𝕃²·𝕋⁻²] ✓ VALID
Why: a side that reduces to pure ℨ bridges; [ℨ] ÷ [ℨ] = [∅] does not — ∅ is a number, not ℨ
Implementation Rule
- Any side containing only
[ℨ](any non-zero power, nothing else) equals any other side - This reflects the Zinf Unit as the computational substrate underlying all phenomena — the floor and the ceiling
- Not transitive, by design:
[ℨ] = [𝕄]and[ℨ] = [𝕃]do not make[𝕄] = [𝕃]. Every pair of sides is checked, soTestZinf = Test𝕄 = Test𝕃is ✗ INVALID whileTest𝕄 = TestZinf = Test𝕄is ✓ VALID, whatever the order [∅]is not ℨ and does not bridge;ℨ⁰ = [∅]- Bridging applies to whole sides only, never to the terms of a relational chain
①⥂ = (0→1→0) shows [ℨ] = [ↁ·2ᵇ], this is valid because the Zinf Unit represents the fundamental UniSpheral substrate that enables data processing operations. Zinf touches everything without making everything equal: that is what having a floor and a ceiling means, and it is why the bridge is deliberately not transitive. The final line of the calculator shows the two sides as they reduced; the product [ℨ·ↁ·2ᵇ] is not written as a third side, because the two sides of an equation are never multiplied together.
ↁ Data Substrate Coupling Rule
BPT Fundamental Principle: Data is the Computational Substrate of All Physical Phenomena
In Binary Pulse Theory, data (ↁ) is the underlying computational substrate from which all physical dimensions emerge. This foundational principle enables dimensional coupling between physical quantities and data processing operations, recognizing them as different manifestations of the same underlying computational reality.
Physical-Data Equivalences
[𝕋] = [ↁ·2ᵇ] ✓ VALID
Principle: Temporal cycles directly produce computational data - each pulse cycle generates exactly 2 bits
[𝕃] = [ↁ·1ᵇ] ✓ VALID
Principle: Spatial measurements correspond to single-bit computational operations
[𝕄] = [ↁ·3ᵇ] ✓ VALID
Principle: Mass-energy relationships emerge from 3-bit computational processes
[ℨ·ↁ·𝕃] = [ℨ·ↁ²·1ᵇ] ✓ VALID [𝕋⁻¹] = [ↁ⁻¹·2ᵇ⁻¹] ✓ VALID
How the calculator applies it: to judge an equation it puts both sides into data form — every 𝕋, 𝕃, 𝕄 replaced by its coupling with the exponent carried through — and compares. So the coupling holds inside composites and under powers. The final line still shows each side as it reduced; the product [ↁ·𝕋·2ᵇ] is not written, because the sides of an equation are never multiplied together.
[𝕄] = [ↁ·2ᵇ] ✗ INVALID [𝕋] = [ↁ·𝕋] ✗ INVALID — the coupling is an equivalence with a fixed count, not an absorption
Computational Substrate Theory
- Data First: ↁ is not derived from physics - physics emerges from ↁ
- Substrate Coupling: Physical dimensions can couple with data combinations
- Process Equivalence: Temporal/spatial processes ≡ their data outputs
- Dimensional Bridge:
data_substrate_couplingrule enables cross-domain equivalence
Implementation Examples
Pulse Rate → Data Production:
①⥂⧗ = (0→1→0) represents [𝕋] = [ↁ·2ᵇ]
Temporal cycles and binary state transitions are equivalent processes
Spatial Flow → Information:
①⥂⦜ = DataBit represents [𝕃] = [ↁ·1ᵇ]
Spatial measurements and data operations share computational basis
Physics Analogy
- Different manifestations of same underlying reality
- Not unit conversion - fundamental equivalence
- Enables transformations between domains
- Recognized as dimensionally valid by physics
BPT extends this: Time-data, space-data, mass-data equivalences based on computational substrate foundation
[𝕋] = [ↁ·2ᵇ] represent fundamental equivalence - temporal processes and their corresponding data outputs are two aspects of the same underlying computational process. The dimensional analysis system recognizes this through the data form described above.
To unpack: 3ᵇ = 1ᵇ + 2ᵇ, but 𝕄 ≠ 𝕃 + 𝕋
Bits add by count, so the three couplings say mass is data with as many bits as length and time together. Yet Test𝕄 = Test𝕃 + Test𝕋 is an error (Cannot add 𝕃 and 𝕋): unlike dimensions never add, in standard physics or here. Either the bit counts carry a relationship between mass, length and time that the algebra does not yet express, or the counts are labels and the coincidence means nothing. Recorded as a question for the theory (test bpt-19).
Composite Dimensions
| Dimension | Name | Mathematical Form | Physical Meaning |
|---|---|---|---|
| ℨ⁻¹ | Zinf Frequency | 1/ℨ | Fundamental computational frequency |
| ↁ·ℨ⁻¹ | Data Processing Rate | data-ops/zinf-time | Computational operations per Zinf |
| ℨ·ↁ | Zinf-Data Coupling | zinf·data-ops | Substrate-computation interaction |
| ℨ·ↁ·1ᵇ | Single bit with substrate | zinf-data-bit | Single Data Bit with Substrate |
| ℨ·ↁ·2ᵇ | Complete 2-bit cycle | zinf-data-2bit | Binary Pulse Cycle with substrate (complete 2-bit cycle) |
| ℨ·𝔸 | Substrate Action | zinf·action | Computational work at substrate level |
| ↁ·𝔸 | Data Action | data-ops·action | Computational processing work |
| ℨ·ↁ·𝔸 | Complete Computational Action | zinf·data·action | Full computational work description |
| ℨ·ↁ·𝕄 | Mass-Substrate | zinf·data·mass | Matter grounded in computational substrate |
| ℨ·ↁ·𝕃 | Length-Substrate | zinf·data·length | Spatial extension from computational substrate |
| ℨ·ↁ·𝕋 | Time-Substrate | zinf·data·time | Temporal emergence from computational substrate |
| ℨ·ↁ·𝕄·𝕃 | Physical-Substrate | zinf·data·mass·length | Basic physical entities with computational foundation |
| ℨ·ↁ·𝕄·𝕃·𝕋 | Complete Physical | zinf·data·mass·length·time | Full physical description with computational substrate |
| ℨ·ↁ·𝔸·𝕄·𝕃²·𝕋⁻² | Classical Energy | zinf-data-energy | BPT Classical Energy |
| 𝔸·𝕄·𝕃²·𝕋⁻¹ | Classical Action | action·energy·time | Traditional physics action (energy × time) |
| ℨ·ↁ·𝔸·1ᵇ | Information Processing Action | zinf·data·action·bits | Computational work with information content |
| ↁ·1ᵇ | Data-Information (1 bit) | data-ops·bits | Active processing with information content |
| ↁ·2ᵇ | 2bit Computation Cycle (2 bits) | data-ops-2bits | Active Processing 2 bit Cycle |
| ↁ²·1ᵇ | Data-Squared-Info | data-ops²·bits | Recursive data processing with information |
| 𝕄·𝕃²·ℨ⁻² | Energy-Substrate | mass·length²/zinf-time² | Energy expressed in substrate units |
| 𝕄·ↁ | Mass-Data | mass·data-ops | Matter-computation interaction |
| 𝕃·ↁ | Length-Data | length·data-ops | Spatial information processing |
| 𝕋·ↁ | Time-Data | time·data-ops | Temporal computation |
| 𝔸·ↁ·1ᵇ | Action-Information | action·data-ops·bits | Work performed on information processing |
Dimensional Algebra Rules
| Operation | Rule | Example | Result |
|---|---|---|---|
| Same dimensions | Valid | ℨ + ℨ |
ℨ |
| Different dimensions | Invalid | ℨ + 𝕃 |
ERROR |
| With dimensionless | Valid | 𝕄 + (∅·𝕄) |
𝕄 |
| A number plus a dimension | Invalid | 𝔸 + ∅ |
ERROR |
| Subtraction to zero | Valid | 𝔸 − 𝔸 |
𝔸 — zero action is still action |
| Bits add by count (BPT) | Valid | 1ᵇ + 2ᵇ, 2ᵇ − 1ᵇ, 1ᵇ − 1ᵇ |
3ᵇ, 1ᵇ, ∅; 1ᵇ − 2ᵇ is an ERROR |
| Different scales | Invalid | ℨ + 𝕋 |
ERROR |
| Operation | Rule | Example | Result |
|---|---|---|---|
| Dimension multiplication | Combine dimensions | 𝕄 × 𝕃² |
𝕄·𝕃² |
| Exponent addition | Add powers | 𝕋⁻¹ × 𝕋² |
𝕋¹ |
| Dimensionless multiplication | Identity | ∅ × ℨ |
ℨ |
| Same dimension division | Cancel | ℨ ÷ ℨ |
∅ |
| Coefficients are not dimensions | Ignore the number | 2 × 𝔸, ½ × 𝕃 |
𝔸, 𝕃 |
| Bits multiply as factors (BPT) | Add exponents | 1ᵇ × 1ᵇ, 2 × 1ᵇ |
1ᵇ², 1ᵇ — neither is 2ᵇ |
| Path independence | Division undoes multiplication | (𝔸 ÷ 𝔸) × 𝔸, 𝔸 × 𝔸 ÷ 𝔸 |
𝔸 both — the dimension belongs to the quantity, not to the order of the operations |
| Substrate ordering | Always ℨ first | 𝕄·ℨ → ℨ·𝕄 |
Canonical form |
| Action integration | Include in ordering | 𝔸·ℨ·ↁ → ℨ·ↁ·𝔸 |
Canonical form |
Detailed Dimensional Multiplication Rules
When multiplying dimensions, add the exponents for like terms:
Example: [ℨ²·ↁ⁻¹·𝔸³·𝕋⁻¹] × [ↁ] = [ℨ²·𝔸³·𝕋⁻¹]
- ℨ² stays as ℨ² (no like terms)
- ↁ⁻¹ × ↁ¹ = ↁ⁰ = 1 (cancels out completely)
- 𝔸³ stays as 𝔸³ (no like terms)
- 𝕋⁻¹ stays as 𝕋⁻¹ (no like terms)
Result: The ↁ (Data Domain) cancels out because ↁ⁻¹ × ↁ¹ = ↁ⁰ = 1, leaving us with [ℨ²·𝔸³·𝕋⁻¹]
General Rules:
- Exponent Addition: For like dimensions, add exponents (𝕋⁻¹ × 𝕋² = 𝕋¹)
- Dimensional Cancellation: When exponent equals 0, dimension disappears (ↁ⁰ = 1)
- Unchanged Dimensions: Dimensions without like terms remain as-is
- Canonical Ordering: Always reorder result by BPT dimension hierarchy
| Operation | Rule | Example | Result |
|---|---|---|---|
| Integer powers | Multiply exponents | (𝕃²)³ |
𝕃⁶ |
| Fractional powers | Fractional exponents | 𝕄^(1/2) |
√𝕄 — not yet read by the calculator, which takes integer superscripts only (², ⁻¹, ¹⁰) |
| Zero power | Always dimensionless | ℨ⁰ |
∅ |
BPT-Specific Rules
Computational Substrate Hierarchy
- Zinf Unit (ℨ): UniSpheral computational substrate - enables bridging between any dimensional combinations
- ↁ (Data): Physical substrate foundation - underlying computational layer from which all physical phenomena emerge
- 𝔸 (Action): Computational work and optimization layer - energy cost and efficiency of operations
- Physical Dimensions (𝕄, 𝕃, 𝕋): Emergent quantities that can couple with their data substrate origins
- Information (nᵇ): Discrete computational content - couples with data operations
Zinf UniSpheral bridging and ↁ data substrate coupling enable dimensional equivalences that appear incompatible in classical physics but represent fundamental computational relationships in BPT.
Scale Relationships
ℨ = tₚ / (2²⁰³ × f²⁰²) ≈ 1.142 × 10⁻¹⁰⁶ s— the Zinf unit, the pulse of the Zinf universe created by the Genesis Prime Pulse, its own dimension [ℨ]; derived from our Planck values through 202 levels of doubling, each carrying the null-well heritage f ≈ 1.018 (f²⁰² ≈ 36.7) — andℨ∞ = 1/ℨ ≈ 8.759 × 10¹⁰⁵ s⁻¹[ℨ⁻¹]. Zinf units measure every property: each Planck unit has a Zinf value.- Cannot combine different temporal scales:
ℨ + 𝕋 = ERROR - All physical dimensions can be expressed in terms of ℨ base units
- Action represents computational work across all scales
Data-Physical-Action Coupling Rules
| Coupling Type | Dimension | Meaning |
|---|---|---|
| Spatial Data | ℨ·ↁ·𝕃 | Computational spatial processing |
| Temporal Data | ℨ·ↁ·𝕋 | Computational temporal processing |
| Mass Data | ℨ·ↁ·𝕄 | Matter-computation interaction |
| Action Data | ℨ·ↁ·𝔸 | Computational work processing |
| Energy Data | ℨ·ↁ·𝕄·𝕃²·𝕋⁻² | Computational energy processing |
| Complete Action | ℨ·ↁ·𝔸·𝕄·𝕃·𝕋 | Full computational-physical work |
Bits are kinds — the theory's one departure from standard arithmetic
1ᵇ,2ᵇ,3ᵇ… are distinct base dimensions: two bits is a pulse, a different thing from one bit, not twice it. Standard physics says two of a thing is never a new dimension; BPT says the quantity of data becomes a phase change at two bits.- They combine by count under addition and subtraction only:
1ᵇ + 1ᵇ = 2ᵇ,1ᵇ + 2ᵇ = 3ᵇ,2ᵇ − 1ᵇ = 1ᵇ,1ᵇ − 1ᵇ = ∅. - Under multiplication they are ordinary factors:
1ᵇ × 1ᵇ = 1ᵇ², and a coefficient never makes a bit (2 × 1ᵇ = 1ᵇ). - The departure is confined to bare bits; nothing else in the algebra depends on it.
Data Information Processing vs. Action
1ᵇ: Static Data Information Storage
Pure bits without processing operations
ↁ: Active Data Processing
Dynamic computational operations
𝔸: Computational Work
Optimization and energy cost of operations
ↁ·1ᵇ: Processing with Data Information
Operations on information content
𝔸·1ᵇ: Work on Information
Energy cost of information processing
ℨ·ↁ·𝔸·1ᵇ: Complete Computational Work
Full substrate-level information processing
ↁ ≠ 𝔸, 1ᵇ ≠ 𝔸
Accumulation Rules by Operator
[ℨ] → [𝔸] → [1ᵇ] reduces each term and stays three terms — and judges two chains equal when they have the same operators in the same order and each pair of terms is equal; the logical operators are not read at all. The union rule for ⇔, the rightmost rule for → and the dimensionless rule for ∧ ∨ ⊻ below are the theory's proposal, kept here until they are settled; the 39 chain tests expect the term-by-term reading.
Rule: Combine unique dimensions from all components
Input: [ℨ·ↁ·𝔸·𝕄] ⇔ [ℨ] ⇔ [ↁ·1ᵇ]
Unique dimensions: {ℨ, ↁ, 𝔸, 𝕄, 1ᵇ}
Result: [ℨ·ↁ·𝔸·𝕄·1ᵇ] ✓
Rule: Validate dimensional consistency, don't artificially multiply
Input: [ℨ·𝔸] = [ℨ·𝔸] = [ℨ·𝔸]
Result: [ℨ·𝔸] ✓ (consistent)
NOT: [ℨ³·𝔸³] ❌ (artificial exponential growth)
Rule: Always dimensionless
[ℨ·𝔸] ∧ [ↁ] = [∅]
Rule: Take rightmost dimension
[ℨ] → [𝔸] → [1ᵇ] = [1ᵇ]
Validation Examples
Valid Dimensional Equations
ℨ × ℨ⁻¹ = ∅ ✓
substrate × frequency = dimensionless
ℨ·ↁ·𝔸·𝕃² = ℨ·ↁ·𝔸 × 𝕃² ✓
computational action-spatial combination
𝔸·ℨ⁻¹ × ℨ = 𝔸 ✓
action rate × time = action
ℨ·ↁ·𝔸·𝕄 ÷ ℨ = ↁ·𝔸·𝕄 ✓
remove substrate factor
𝔸·𝕄·𝕃²·𝕋⁻¹ ⇔ ℨ·ↁ·𝔸 ✓
classical ⇔ computational action
Invalid Dimensional Equations
ℨ + 𝕃 = ? ✗
cannot add substrate to length
𝔸 + ↁ·1ᵇ = ? ✗
cannot add action to data-info
ℨ + 𝕋 = ? ✗
different temporal scales
1ᵇ = ↁ = 𝔸 ✗
static info ≠ processing ≠ action
𝔸 ÷ 𝔸 = 1ᵇ ✗
a cancelled dimension is a number, [∅], not a bit
𝔸 − 𝔸 = ∅ ✗
zero action is still action
2 × 1ᵇ = 2ᵇ ✗
a pulse is two bits added, not one bit doubled
Implementation Guidelines
Database Storage Requirements
- Canonical ordering: All dimensions must be stored as
[ℨ·ↁ·𝔸·𝕄·𝕃·𝕋·1ᵇ]format - Include Action: Ensure 𝔸 is properly integrated in all composite dimensions
- No unknown dimensions: Remove undefined symbols
- Complete signatures: Ensure all composite symbols include full dimensional representation
Parsing Requirements
- Parse composite dimensions:
ℨ·ↁ·𝔸·𝕄·𝕃² → {ℨ: 1, ↁ: 1, 𝔸: 1, 𝕄: 1, 𝕃: 2} - Handle negative exponents: Recognize superscript notation (⁻¹, ⁻²)
- Process multiplication symbols: Both · and implicit multiplication
- Canonical reordering: Sort all results by BPT dimension order including Action
- Validate base dimensions: Match against complete 8-dimension BPT set
Validation Logic
- Equation parsing: Split on operators to identify components
- Dimensional reduction: Reduce each component to base dimensional form
- Operator-specific rules: Apply appropriate accumulation rules
- Canonical formatting: Present results in standard BPT ordering with Action
- Error reporting: Specify which dimensions don't match and why
Special Handling
- Universal Unit ℨ: Foundational substrate that scales to all other dimensions
- Data coupling: Allow ↁ to combine with any physical dimension
- Action integration: Allow 𝔸 to combine with computational and physical dimensions
- Information content: Treat 1ᵇ as discrete additive units
- Dimensionless operations: ∅ acts as multiplicative identity
- Notation symbols: Handle as structural elements, not dimensional contributors
Framework Summary
Check the calculator against every rule here: run the test suite.
This comprehensive dimensional analysis framework provides rigorous mathematical validation of BPT equations while accommodating the theory's unique computational-physical substrate that extends beyond conventional physics. With ℨ, ↁ, and 𝔸 forming the foundational computational layer, all physical phenomena and optimization principles emerge from this unified substrate architecture, enabling precise dimensional validation across the complete spectrum of BPT mathematical expressions.