With one action scale, clock and vacuum reference, mp/me=εp/εe. The separate integer expression 13×137+55=1836 is an exact valuation, but needs a derived baryonic energy law to become a mass prediction.
Why the ratio is a demanding target
The Single Monad proposal seeks one origin for different forms of matter. Its constants programme should therefore obtain the electron and proton from compatible sectors of one dynamics. Giving each sector its own fitted duration could reproduce two masses while leaving their relationship unexplained.
A ratio removes a shared overall mass unit and focuses attention on the relative spectrum. This makes it a particularly useful SMM/DTT target. The six nested inner roles should constrain the carrier, interactions and completion readouts that generate both sectors. Their existence alone does not assign those sectors a specific energy ratio, but a common source law could make that ratio a genuine output.
Derive the exact gap ratio
Use the same normalized positive transfer theory, reference vacuum, action scale a, completion duration τ* and propagation speed c for both sectors. Once their rest-threshold interpretation is established, their mass dictionaries are me=aεe/(c²τ*) and mp=aεp/(c²τ*).
For illustration, gaps δ and 5δ give ratio five for every positive δ and for every common conversion scale. Doubling both gaps preserves the ratio. This elementary cancellation is valuable, but the number five in the example was supplied. A physical prediction must calculate the gaps in independently identified electron and baryon sectors.
Evaluate the structural 1836 proposal
A separate structural proposal begins with X(b)=b²+3b+7. Setting X=137 gives roots b=10 and b=−13. It then declares an inertia valuation combining the magnitude of the negative root with a triangular contribution from the positive root:
The arithmetic is exact. Its physical content resides in the declared energy rule: why should the product term and the triangular term add to a baryonic rest energy measured in electron units? A negative polynomial root is not a positive receiver dimension or an automatic antimatter state. Those interpretations require operators and states that realize the proposed contributions.
What a baryonic derivation must contain
A three-dimensional complex color carrier with a fixed volume form admits an antisymmetric singlet made from three fundamentals. That representation statement supplies a possible invariant tensor; it does not calculate a proton’s binding energy, flavor structure, spin or confinement dynamics. An internal color triplet is also a different object from three real spatial readout directions.
The finite singlet-loop model provides an instructive intermediate example. Its supplied hopping and singlet penalty give an exact ground response and a protected auxiliary gap. To use such a construction in a proton calculation, one would need an interacting physical baryon sector, a justified reference state and controlled limits. The auxiliary rank or gap cannot simply be relabeled as the proton mass.
What remains after successful calibration?
Even a fit to a mass ratio can leave many microscopic choices unresolved. If both gaps scale by the same factor, their ratio is unchanged. Other observables, such as dispersion or transition structure, can distinguish models with the same ratio. A useful calibrated prediction must remain fixed across the admissible family compatible with the calibration, or be reported as a controlled interval.
Adding an unexplained independent correction to every sector removes the common-source test. A justified correction should follow from a specified interaction, boundary condition or spectral change and produce additional consequences. This is the positive role of a failed ansatz: it identifies which proposed connection needs replacement while leaving valid algebraic results available for a better dynamics.
The next useful calculation
A constructive next model should define both charged sectors, their shared update and completion rule before choosing target masses. Calculate the relative rest thresholds, then compare an additional sector-sensitive quantity using the same parameters. Finite approximations should include errors and the regime in which the thresholds are stable.
The mass-ratio workbench separates the spectral route from the structural valuation and its shared-deformation test. The electron-mass guide explains the absolute dictionary that cancels here. Read together, they identify the intended SMM/DTT advance: a source-generated relative spectrum followed by one common physical normalization, with no separate clock chosen merely to restore each target value.
Continue with the articles
- A dependency map for physical constants
A common constants program must connect dimensionless source outputs to response, matter, and geometry through one physical dictionary. A dependency map separates those deductions from supplied scales and empirical calibration.
- Identifiability: can observations determine the model?
Four dimensionless outputs leave two exact parameter directions unresolved in the displayed constants dictionary. The calibration fiber shows which additional diagnostics could distinguish otherwise equivalent models.
- Prediction after calibration
Calibration produces a prediction only when the proposed output is fixed across the remaining admissible models, or constrained to a stated range. This criterion distinguishes cross-sector consequences from reuse of a fitted target.
- What a failed mass-ratio ansatz teaches
A shared deformation of the proposed 137 and 1836 baseline formulas fails to match both benchmark targets. This negative result rejects one precise ansatz and identifies the matter dynamics a stronger mass-ratio model needs.
- 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.
- An exact singlet-loop response
A mobile carrier, stationary dual, and supplied singlet-binding Hamiltonian reduce to one exact resolvent equation. Its unique physical root determines the holonomy-dependent ground response and a conditional completion-cost candidate.
- A protected gap in the finite singlet model
The finite singlet-loop Hamiltonian has a unique ground state and a uniform lower bound on its excitation gap. The derivation identifies the supplied penalty, carrier rank and limits that separate this result from a continuum gauge-theory mass gap.
- The conditional route to S(U(3) × U(2))
Retaining rank-three and rank-two Hermitian carriers together with a product determinant tensor selects S(U(3) × U(2)). The stabilizer calculation leaves carrier selection, chiral matter and empirical Standard-Model matching as separate requirements.