Discovery · back to all discoveries
A mass shell admits the same product form.
The correspondence is exact inside a bounded special-relativistic parameterization. The physics is inherited and credited. Mapping the factors to seeing and doing is our interpretation, not a result of relativity.
The claim, exactly
For a massive timelike on-shell particle, define separate relativistic factors r+ = (E + pc)/(mc²) and r− = (E − pc)/(mc²). Then the inherited mass-shell equation gives r+r− = 1. Calling those factors seeing and doing is a separate wager.
The seam, undressed
One factorization, stated exactly.
Start with the special-relativistic mass shell for a massive particle and factor it:
Here w is rapidity along the selected boost direction. The relativistic factors are reciprocal because the mass-shell relation makes their product one. For this comparison only, set xSR=eʷ, so r+=xSR and r−=1/xSR; this label is not the sphere radius x by default. This is inherited special-relativistic algebra [A]. It excludes the massless case, for which division by m is unavailable. No ethical or person-level conclusion follows from the factorization.
The dictionary
Word for word, the mapping.
Why the algebra matches
A declared hyperbolic parameterization.
The match occurs because both descriptions use positive reciprocal factors. Special relativity supplies them through rapidity and the factored mass shell. The sphere chart separately uses raw radii x and 1/x, then defines normalized projection factors VMO=2x/(1+x) and SKA=2/(1+x), whose selected product is EB=VMO×SKA=4x/(1+x)². The raw-radius identity is not Exergy. Chart balance B=2/(x+1/x)=sin θ and Exergy are distinct functions whose maxima coincide under the selected definitions. Reuse of a mathematical form is not evidence that coherence, capability, persons, or institutions are relativistic particles.
The algebra is inherited and exact in scope. The seeing/doing interpretation is our wager.
Why this matters to you
A calibration, not a proof.
Finding a known algebraic form does not validate the framework or make its variable meanings non-arbitrary. It gives one precise comparison to audit. The open question is whether independently defined person-level variables predict anything beyond the chosen normalization; that remains unestablished and belongs in the trial record.
One more discipline: the relativistic identity r+r−=1 is not the sphere's raw-radius identity x(1/x)=1; neither is the selected Exergy profile EB=VMO×SKA or the finite-node score Pnode=min(VMO₄,SKA₄). The chart balance shares a functional form with the proper-time rate under a separate mapping. Keep every domain and quantity distinct.
Live animation: rapidity rising, the clock-rate falling — real hyperbolic functions of one slider, drawn on a family of tori: horn at rest, spindle in motion, a sphere as the limit approached and never occupied. The relativity is computed first and painted onto the geometry; the torus rendering is an emblem of the algebra, not its derivation (its mouth is a cusp, not the light cone — 151_HORN_TORUS_SR_FORMAL_AUDIT_2026_07_20.md; the strong claim is in our graveyard, dated).
The passage. What the slider walks — horn, then spindle, then the sphere-limit it never reaches — is a story of its own, told at its tier: The Crossing · [I] emblem riding credited geometry.
Held against adversarial audit — see the trial record.
[A] mass-shell factorization inherited from special relativity · [I]/[C] seeing/doing mapping — interpretive and unvalidated