Constitutions for Calculations
The Assumption-List Project, with the Two-Body Problem as Entry One
Version 1.0 — public draft, open for amendment
J. Rogers, SE OhioAbstract
Physics is not wrong. It is unreadable about its own assumptions. Every standard calculation performs three operations it does not publish: a cut (which objects exist), a gauge (which conventions are in force), and a law (which ratios hold among the cut quantities). The measured value is a product — data × cut × gauge — and the textbook presents only the product. The factors are recoverable but never written, and the cost is reification: convention is billed to nature as "fundamental constants," and arbitrary object-formation is billed to reality as "bodies." We propose that every standard calculation be accompanied by a typed, versioned, publicly amendable assumption list — a constitution — and we supply Entry One in full: the two-body orbital problem. Its constitution shows that the textbook never assumed the universe away. The equivalence principle, measured to parts in 10¹⁵, converts the total gravitational field of the universe into the chosen reference frame; what survives of "the rest of everything" is a tidal tensor, a cosmological constant term, and gravitational radiation — all measured, all bounded, all absent from the textbook's presentation. The two-body problem is the licensed art of forgetting everything else, and the license, the price, and the expiry date are entries on a list the textbook has never shown. We invite amendments to this list and propose the project's extension to the canon: one constitution per standard calculation.
1. Physics is not wrong; it is read-only
Newton's Principia opens its dynamical definitions with a warning aimed at precisely one error. After introducing absolute and relative time, space, place, and motion, he distinguishes the two registers by separate names — "mathematical" and "common" — lest the ambiguity of words produce confusion, and states the discipline plainly:
"In philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them."
Newton thus possessed the distinction between the quantity and its sensible representation — between what we will call data and gauge — and built a firewall around it in the founding document of the field. The subsequent history consists of two violations in sequence. First, the textbook tradition reified Newton's absolutes, promoting the axes to substances. Then, corrected by positivism, it reified the measurements: the sensible measure was declared the only beable, the clock became the definition of time, and "shut up and calculate" — coined as satire by Mermin in 1989, adopted as policy thereafter — became the official stance toward everything above the product.
The operationalist tradition, the most careful statement of this stance, smuggles in two premises it never declares. First, the cut precedes the operations: to specify "the operations that measure temperature" presupposes temperature as a kind of quantity before any operation runs; Hempel (1954) showed the symptom — under operationalism, concepts fragment into disjunctive lists of procedures, and the framework cannot say why a mercury column, a gas law, and a blackbody spectrum measure "the same" temperature. Second, the gauge rides inside the procedure: every operation embeds a calibration and a standard, and operationalism counts this arbitrary scaling as part of the "objective operation." A constant reified by this laundering looks discovered rather than chosen.
The cost of leaving assumptions unlisted is measurable in the history. A listed assumption can be amended; an unlisted one must be overthrown. Euclid listed his postulates; that is why the fifth could be debated within the list for two millennia and finally edited — and the edit turned out to be empirical physics. Newton listed; that is why Mach could file a critique against specific entries and Einstein could execute it. Copenhagen never listed: there is no document, no entry addresses, no version number. When Bell wished to engage its assumptions, he was reduced to reverse-engineering the list from fifty years of practice — extracting local causality, outcome realism, and setting independence from prose that never stated them — at the cost of a career spent on nights and weekends (Bell 1964). The extraction worked; the theorem, the experiments, and the 2022 Nobel are the receipt. But the method — archaeology before critique — is what an unversioned document forces.
This paper proposes the minimal fix. Not new physics — no prediction in this document is new, and every calculation cited runs identically with or without the lists. The proposal is about provenance: that each standard calculation be accompanied by a typed, versioned list of the assumptions that constitute it, maintained as a public document open to amendment. A framework whose assumptions are listed is not a different framework. It is the same framework, made writable.
2. What a calculation actually contains
2.1 The factorization
Every measured value factorizes:
measured value = data × gauge
where data is what survives every legitimate change of convention and gauge is the convention-dependent remainder. The statement is ordinary quantity calculus, known since Maxwell: a quantity is a number in a unit, and conversion is unit algebra. What is not ordinary is the discipline of asking, for each factor of a published result, which one it is.
The same factorization has a second, deeper input. Before any gauge is chosen, the calculation must decide what objects exist — that this bounded collection is "a body," that this excitation is "a particle," that this degree of freedom is "the system." This act is not measurement and not convention; it is the formation of the quantities that measurement will quantify. We call it the cut. The full factorization is:
measured value = data × cut × gauge
and the standard presentation of every textbook result is this product with all three factors collapsed into one noun-laden object. The two "sneaks" of §1 are exactly the collapse performed twice: the cut billed into the procedure (what you measure presupposes what exists), and the gauge billed into the number (what units you used rides inside the "measured" value).
2.2 The five entry types
An assumption list is a set of typed entries:
The deletion test sorts any contested entry: edit it and check the predictions. Invariant predictions mark a gauge entry; moved predictions mark a cut entry; a residue is what no edit touches. This is dimensional analysis promoted from trick to sorting procedure.
2.3 The compiler
Every worked calculation performs a three-step operation that is almost never written:
- Cancel input units — divide the measured inputs by their unit standards (or, most cleanly, by the corresponding Planck-scale quantities), leaving pure ratios.
- Compute the ratio — the physics proper: a relation among dimensionless quantities.
- Decorate the output — multiply by the output unit standards to return a human-readable value.
Newton wrote step 2 only: the Principia states gravitation as F ∝ m₁m₂/r², with no constant, because the law is a proportionality among ratios. The constant appears when the chart hardens: Cavendish's experiment (1798) measured Earth's density, and the symbol G enters the literature only in the late nineteenth century. The cancellation step was formalized last — the 2019 SI redefinition is precisely a treaty-level performance of step 1, with the seven defining constants serving as the cancellation standards (BIPM 2019). The historical order — law, then constant, then cancellation — is the ontological order: the morphism first, the Jacobian when the chart hardens, the gauge-discipline last.
A worked check, using Planck-scale quantities as the cancellation standards (Appendix B for conventions): Planck force is F_P = c⁴/G and l_P/m_P = G/c², whence
G = F_P · (l_P/m_P)²
The "fundamental constant" G is exactly the Jacobian converting an input decoration of kg²/m² into an output decoration of newtons. It does not disappear in a better theory; it disappears in a better-typed ledger, where it belongs to the gauge column.
2.4 What the list does not do
Dimensional structure fixes the scaling of a law and never its dimensionless residue. The compiler reproduces the forms E = mc², E = hf, λ = h/p, r_s = 2GM/c² exactly; it does not produce Wien's 4.965114, Hawking's 16Ï€², the Stefan–Boltzmann Ï€⁴/60, the anomalous magnetic moment series, or the digits of α. Those are residues — dynamical content, the part nature actually writes — and this paper makes no claim to derive them. A companion manuscript develops the categorical formulation (a fibration of measured quantities over conceptual types) and the observation that the celebrated pairwise laws are, under Planck normalization, a single equivalence chain; the present paper needs none of that apparatus. The lists are prior to it. What the list prices, the dynamics determines.
Conventions: throughout, "Planck system" means the h-based (non-reduced) system; the ħ-based alternative differs by factors of √2Ï€ in the mass, length, and time scales. Both are gauge choices (Appendix B); the framework's habit of carrying both without a criterion is one of the small pathologies the typed ledger cures.
3. The document: format, versioning, governance
3.1 Entry schema
Each entry carries: an address (C1, L1, G2, R3 …), a statement, a type, the evidence (measurement or theorem) supporting it, an error theory (what fails, and how fast, if the entry is false), and the consequence of deletion (what the calculation loses). The addresses are what make debate possible: one can file an amendment against C2; one cannot file an amendment against a vibe.
3.2 The edit taxonomy
- Gauge-edit: free, conventional, accepted on demonstration of invariance (the deletion test).
- Cut-edit: empirically constrained; must arrive with an error theory — a statement of what the new cut gets wrong that the old one didn't, and vice versa.
- License-challenge: requires a measurement, not an argument. The licenses are the strongest entries on any list; that is their job.
- Residue-addition: requires a measured effect absent from the current list.
- Data-amendment: requires a better law. This is rare and decisive when it happens.
3.3 Precedent and limits
Metrology already constitutionalized itself. The 2019 SI is a versioned document: seven defining constants with exact values, an amendment procedure (CGPM resolution), scheduled review. The kilogram went from hardcoded — a platinum-iridium cylinder, an assumption with no address, whose drift was undebatable for a century — to declared: an entry, h, with a changelog. The practice exists; the proposal is only to extend it from units to calculations.
Two limits, stated upfront. First, no list is complete: the list is written in a background language whose own assumptions are not on the page. The list is therefore not the complete set but the editable set — the assumptions one is willing to change, given addresses. Second, this document's own typology is entry zero: the five-type scheme above is itself a cut, and is open to amendment by the same procedure.
3.4 Why this is a group project
Where assumptions are listed, editing them is editorial work: normal science with a paper trail. Where they are unlisted, every edit looks like insurrection, and insurrections are punished — the sociological record of foundations (Bell's exile, Everett's departure, the fifty-year delay between theorem and experiment) is the cost of running a read-only foundation. The list converts the deepest arguments of the field from warfare into diff review. That conversion cannot be performed by any single author, because the value of the list is exactly its public amendability: the changelog of the lists, across contributors, is the provenance record of physics itself.
4. Entry One: the two-body problem
The two-body problem is the canonical "solved" problem of mechanics, and its constitution is fully assemblable. Here it is.
4.1 The constitution
4.2 Commentary
(a) What the textbook shows. The ellipse, with F = Gm₁m₂/r² stated as unconditioned fact. Zero entries declared. The residues it later mentions ("perturbations," the 43″) are R-entries filed as complications. The license that makes the whole pretense possible — the best-measured identity in physics — appears nowhere, because the pretense it licenses is never stated as a pretense.
(b) How the universe is in the problem. The popular slogan "you are in orbit with every mass in the universe" is C2 restored, and it decomposes under the list into three live components and one dead one:
- The uniform sum of every external pull — gauge. In free fall the pair falls with the total field of everything: Sun, galaxy, supercluster, all of it. This is real, and it enters the description as G2 — as the frame you are permitted to choose. Falling-with-everything is what a geodesic is.
- The differential — data. The tidal tensor (R1) is the part of "every mass" that no frame choice removes: the universe's grip on the pair's relative motion, negligible at 1 AU, dominant at the Oort cloud.
- The curvature — data. Λ (R2) requires no permission and no field; it survives every gauge.
- The naive reading — dead. "Everything pulls directly on the ellipse" is exactly false at first order, by L1.
(c) The validity boundary is observed nightly. Because tide scales as r/R³ while binding scales as 1/r², they cross at the Jacobi radius. Inside it, C2 holds and the pretense is safe; outside it, the universe reaches in and the two-body problem becomes a three-body problem with the galaxy as third body. The comet flux from the Oort cloud is the boundary of the two-body cut, measured.
(d) The pretense is stable because it is pretense. Two bodies is the unique N with a closed-form solution; Poincaré showed that N = 3 destroys integrability. (The solar system's ~5 Myr Lyapunov time is an N-body property, not a two-body one.) And the "exactly solved" problem was doubly fictional even as stated: general relativity has no closed-form two-body solution for comparable masses — the exact solutions are test-particle limits. The two-body problem is solvable to the degree that it is fictional, and the fiction is priced.
(e) The half-conventional gauge. G3 carries an asterisk because unit choice is free in a way the rotation standard is not: the local inertial frame is dragged by the mass distribution (frame-dragging, measured). Mach's demand — that rotation be defined relative to the cosmic matter distribution rather than absolute space — was partially executed by general relativity. The empty-universe-with-inertia solutions that remain are the unexecuted remainder. A typed ledger is where such half-executions get recorded instead of forgotten.
(f) The summary. The two-body problem never assumed the universe away. The equivalence principle converted the universe into the frame; the remainder — tide, curvature, radiation — is the universe's signature on the pretense. The problem of two bodies is the licensed art of forgetting everything else, and the license, the price, and the expiry are all entries on a list the textbook has never once shown.
5. The pipeline and the standing invitation
5.1 Next entries
The canon is full of constitutions waiting to be written, most of them one page:
- Harmonic oscillator — cut: linearization (sin θ ≈ θ); residue: the amplitude-dependent period the cut hides.
- Ideal gas — cut: non-interacting; residue: the virial coefficients, the Joule–Thomson effect.
- Hydrogen atom — cut: proton fixed at infinite mass; residues: reduced-mass corrections, fine structure, the Lamb shift (the residue of cutting away the field).
- Born–Oppenheimer molecule — cut: nuclei fixed; known failure locus: conical intersections, where the cut provably bleeds.
- Thermodynamic limit — cut: N → ∞; failure regime: long-range forces, finite-size effects.
- S-matrix scattering — cut: asymptotic free states; recorded defect: Haag's theorem (1955), the proved impossibility of that cut for interacting fields, currently filed as a technicality.
- The lattice and the loop — every numerical scheme is a cut with a known discretization error; these constitutions are already half-written by the practitioners and need only typing.
And the grand challenge: the constitution of quantum measurement — the one standard procedure whose license entry has never been written. The Heisenberg cut is a cut entry with no address, no error theory, and no stated location; the interpretation debate is a decades-long dispute over the C and L entries of that list. Typed, the debate becomes diff review: Copenhagen proposes no list; Everett proposes deleting the cut entry; Bohm and GRW propose specific cut-and-license pairs. The list does not adjudicate. It makes the proposals addressable — which is the precondition the debate has been missing.
5.2 Invitation
Amendments are invited to Entry One and to the typology (entry zero). A proposal consists of:
- the entry's address (or a proposed new address),
- its type,
- the statement,
- the evidence,
- the error theory,
- the consequence of deletion.
Acceptance criteria are the deletion test and the edit taxonomy: gauge-edits on demonstrated invariance, cut-edits with error theories, license-challenges with measurements, residues with data. Two standing rules: no new physics, and no replaced calculations — the list documents, it does not substitute. The document is versioned and forkable; the changelog is the provenance record.
5.3 Objections, briefly
"The assumptions are obvious; why write them?" They were obvious enough that fifty years of physicists did not extract them, until Bell did it by hand and paid for it. Euclid's fifth postulate sat in plain sight for two thousand years.
"This is philosophy, not physics." It is provenance — the same discipline effective field theory already applies to Lagrangians, where every neglected term is named, priced, and suppressed by a stated power. This paper applies EFT's own bookkeeping to the frame instead of the action. If it is philosophy, it is philosophy that physics already performs daily, unnamed.
"It will overload the student." The constitution is an appendix, not a preface. The calculation runs identically; the list costs one page and buys the ability to say what kind of thing went wrong when the student's answer disagrees with experiment.
"Unmeasurable distinctions are empty." The list enforces the honest version of this principle: a distinction may be deleted only when a gauge has been exhibited making it redundant — a proof, not a shrug. Where no gauge is exhibited, the entry stands and the unmeasurability is recorded as a property of current instruments, not of the world.
6. Closing
An unstated assumption is not a modest assumption. It is an unaddressable one, and "how things are" is what an unaddressable assumption looks like from the inside. The alternative to a listed foundation is not a foundationless physics — the foundations exist and are load-bearing, as Entry One demonstrates — but a read-only one: two centuries of running a constitution that was never printed.
The lists will not change a single prediction. They will change what kind of thing a disagreement is. With addresses, a disagreement is an amendment; without them, it is an insurrection. The two-body problem — the first "solved" problem, the cleanest fiction in the canon — turned out to have a constitution with a measured license, a priced error, a nightly-observed validity boundary, and a cosmological constant already fitted into the operational pipeline without anyone naming the transaction. If the cleanest case was this rich, the rest of the canon is richer.
Physics will keep computing. The lists are how it starts keeping receipts.
Appendix A: Entry One as a standalone document
The constitution of §4.1, maintained at version 1.0. Changelog: initial release. Forks welcome; merge criteria per §5.2.
Appendix B: Conventions and numerical checks
Planck system (h-based, non-reduced): m_P = √(hc/G), l_P = √(Gh/c³), t_P = √(Gh/c⁵), E_P = √(hc⁵/G), F_P = c⁴/G. Planck temperature T_P = E_P/k_B.
Named Jacobian coordinates (the conversion factors the standard framework uses without naming):
- Hz_kg = h/c² = 7.3725×10⁻⁵¹ kg·s (mass per unit frequency)
- K_Hz = k_B/h = 2.0837×10¹⁰ Hz/K (frequency per unit temperature)
Checks:
- G = F_P·(l_P/m_P)²: (c⁴/G)·(G/c²)² = G. ✓
- Residue factors cited in §2.4: Wien (wavelength form) carries 4.965114; Hawking carries 16Ï€² (h-form); Stefan–Boltzmann carries Ï€⁴/60 (h-form). These are dynamical content; the compiler does not produce them, by design and by theorem.
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