A104Draft-based articleExpanded article4 min read

A common dimensional dictionary

A shared dictionary converts source outputs into calibrated actions, lengths, times, masses, and curvature coefficients. Dimensional checks expose missing scales and distinguish coefficient matching from an independent derivation.

In this article

Why a dimensional dictionary is necessary
Action, phase, and the free scale
Length, time, and propagation
Mass from a matched gap
Gravity and curvature coefficients
Dimensionless combinations reveal the physical target

Key ideasSources & reading

01Why a dimensional dictionary is necessary

A finite generative model may naturally produce integers, phases, eigenvalue ratios, and dimensionless stiffnesses. Laboratory quantities are expressed through calibrated lengths, durations, masses, and actions. A dictionary connects the two. Without it, a numerical match can conceal a unit error or an arbitrary scale choice.

SMM/DTT aims to make the dictionary common across sectors: the same temporal generation should underlie matter, propagation, and geometry. Dimensional consistency is the first check of that aim. It is necessary, but it is not enough to select an absolute value.

02Action, phase, and the free scale

If additive action S maps continuously and nontrivially into a phase character, then χ(S)=exp(iS/a) for some positive action scale a after choosing orientation. The character law fixes the exponential form because continuous additive phase lifts are linear.

Every positive a is permitted by those assumptions. Identifying a with physical ℏ requires a quantum matching and a calibration to standards. A recurrence time already calculated with ℏ cannot then be used as an independent input to derive the same ℏ.

χ(S₁+S₂)=χ(S₁)χ(S₂)

χ(S)=exp(iS/a); [a]=energy×time

Two identical energy landscapes are vertically shifted. Their normalized probability distributions are identical even though their absolute baselines differ. — **Figure 1. Normalized probabilities forget additive offsets.** Adding an energy constant c₀ multiplies every Boltzmann weight by the same factor and leaves normalized probabilities unchanged. Such response alone cannot fix an absolute vacuum-energy offset. View full size ↗

03Length, time, and propagation

Let ℓ* and τ* convert dimensionless spatial and temporal readouts into physical units. If υ is a dimensionless sharp propagation coefficient, c=υℓ*/τ*. This has units length per time.

The formula does not prove that a lattice bound, group velocity, and relativistic signal cone are identical. Those are different measurements and may differ in a candidate model. A universal speed law requires the selected matter and signal sectors to share the appropriate operational coefficient.

c=υℓ*/τ*

04Mass from a matched gap

For a positive physical transfer problem, define ε_s=−log(t_s/t₀) using a common reference eigenvalue. If the relevant physical time and relativistic rest-energy interpretation have been established, the mass dictionary is m_s=aε_s/(c²τ*). The numerator divided by τ* is an energy; division by c² gives mass.

A spatial radius R in m=na/(Rc²) would give the wrong units. With a spatial radius and a supplied linear dispersion, the corresponding mass form is na/(Rc). With a temporal radius, the c² denominator can be appropriate. The symbol’s physical meaning must therefore be stated before manipulating the formula.

ε_s=−log(t_s/t₀); m_s=aε_s/(c²τ*)

A curve in a two-parameter plane represents models sharing the same calibrated observable. A second contour crosses it at a point and removes that illustrative ambiguity. — **Figure 2. Calibration can leave hidden parameter directions.** A fixed measurement may select a curve of models rather than one point. A genuine prediction must be constant on that calibration fiber, or require additional independently measured diagnostics. View full size ↗

05Gravity and curvature coefficients

Suppose a dimensionless geometric action contains κ times curvature and a constant term −r. Matching it to the stated Einstein-action convention gives G=c³ℓ*²/(16πaκ) and Λ=r/(2κℓ*²). The dimensions are respectively length cubed per mass-time squared and inverse length squared.

The matching does not calculate κ or r. Nor does it establish the correct sign of all dynamical modes, universal matter coupling, or a Lorentzian limit. A geometric inequality coefficient also needs an independently fixed energy normalization before it can be connected to Newton’s constant.

G=c³ℓ*²/(16πaκ)

Λ=r/(2κℓ*²)

06Dimensionless combinations reveal the physical target

Combining these dictionaries removes the arbitrary conversion scales from ratios such as μ, Gm_e²/(ac), and Λ[a/(m_ec)]². Those are appropriate targets for a source theory because changing the unit labels cannot alter them. A prediction of their values still requires the source to determine the remaining dimensionless coefficients.

The practical discipline is simple: define every symbol’s units and operational role, check each equation, distinguish calibration from selection, and use one dictionary consistently. This makes SMM/DTT’s common-source ambition testable. It also prevents a dimensional identity from being advertised as an independent derivation of an action scale, an electron mass, or a gravitational coupling.

KEEP THE CONNECTION

Key ideas

Dimensional consistency checks the dictionary but does not select its scales.
Action-to-phase conversion leaves a positive action scale free.
Dimensionless cross-sector ratios are the meaningful targets after unit freedom cancels.

Scope of the argument. Conditional unit and coefficient matching; no absolute action, length, time, or gravitational scale is selected.

Sources & further reading

dtt_constants_reconstruction.pdfWorking draft, pp. 5–6, 11–14

Source editions and catalogue entries

Working manuscript

Physical Constants from Temporal Completion: Dependency Structure, Identifiability, and Cross-Sector Tests in SMM/DTT

Working manuscript. Article-level references identify the consulted sections and the scope of the interpretation. No public URL was supplied for this exact manuscript version.

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