WILL Relational Geometry I — machine-checked audit of the zero-parameter core

Every boxed identity of WILL_RG_I is re-expressed as a SymPy residual that must vanish identically. A claim counts as verified only when simplify(residual) == 0 and every symbol it consumed belongs to a declared register.

Result

68 checks, 68 with the expected verdict, 0 problems.
67 residuals vanish identically; 1 check (C1.5) is a negative control designed to leave a non-zero residual.
Symbols appearing outside the declared registers: R — the deliberately smuggled carrier radius of the negative control, and nothing else.
No symbol in any verified identity is a fitted quantity. c, G, M, E_0, R_s enter only through the SCALE register and cancel wherever the paper claims they do (checks C6.2, C7.5, C8.1).

Where every number in the core comes from

constant	provenance
`1`	unit normalisation of the relational ledger on each carrier
`2`	dim S^2 / dim S^1 = DOF count (Closure Theorem); also R_s = 2GM/c^2 bookkeeping
`1/2`	first Taylor coefficient of sqrt(1-x) – origin of the classical 1/2
`3`	volumetric proxy exponent r^3, fixed by r-independence of the mass label
`4`	square of the DOF ratio entering the tau inversion (1+8 tau^2 quadratic)
`8`	8 pi G/c^4 legacy coupling; and the 8 in 1+8 tau^2 from the closure quadratic
`1/4`	root of the closure quadratic beta^2 = (3 - sqrt(1+8 tau^2))/4
`4*pi`	surface measure of the S^2 carrier (NOT Newton’s 4pi/3 volume measure)
`8*pi`	2 x 4 pi: carrier surface measure times the R_s = 2GM/c^2 factor
`19/32, 3/8`	higher Taylor coefficients of the exact RG ratio (derived, not fitted)

Earth–GPS numerical reproduction (40-digit arithmetic)

quantity	value
Δt_RG, exact ratio [µs/day]	`38.5421472752`
Δt_GR, additive 1PN [µs/day]	`38.5421472451`
Δt_RG − Δt_GR [µs/day]	`3.01678e-8`
δ_RG⁽²⁾·D·M predicted [µs/day]	`3.01678e-8`
residual of difference vs prediction	`2.21e-17`
closure residual β²_GPS − κ²_GPS/2	`0.0`
omitted term / total shift	`7.827e-10`

These reproduce the table in Sec earth-gps of the source document exactly, including the ~1e-17 residual and the ~7.8e-10 relative size of the term the additive formula omits.

Checks by layer

Layer 1 carriers

id	claim	source	residual	verdict
`C1.1`	S^1 closure beta^2 + beta_Y^2 = 1 under beta=cos(th1), beta_Y=sin(th1)	Thm carriers (a); Thm conservation	`0 ✓`	OK
`C1.2`	S^2 closure kappa^2 + kappa_X^2 = 1 under kappa=sin(th2), kappa_X=cos(th2)	Thm carriers (b); Thm conservation	`0 ✓`	OK
`C1.3`	Phase is fixed by amplitude: beta_Y = sqrt(1-beta^2)	Sec kinetic	`0 ✓`	OK
`C1.4`	Phase is fixed by amplitude: kappa_X = sqrt(1-kappa^2)	Sec potential	`0 ✓`	OK
`C1.5`	Any carrier radius R != 1 reintroduces a free parameter (must fail)	Thm carriers; Pr epistemic	`1 - R**2`	OK

Notes:

C1.1 — Closure is the Pythagorean identity of the unit circle: no parameter.
C1.2 — Meridional great-circle section of S^2; ledger normalised to unity.
C1.5 — NEGATIVE CONTROL: residual 1-R^2 is the smuggled scale; vanishes only at R=1.

Layer 2 spectroscopy

id	claim	source	residual	verdict
`C2.1`	kappa^2 = 1 - 1/(1+z_kappa)^2 from kappa_X = 1/(1+z_kappa)	Thm Spectroscopic Phase Shift	`0 ✓`	OK
`C2.2`	beta^2 = 1 - 1/(1+z_beta)^2 from beta_Y = 1/(1+z_beta)	Thm Kinematic Phase Shift	`0 ✓`	OK
`C2.3`	Round trip: 1+z_kappa = 1/kappa_X recovers kappa	Thm Spectroscopic Phase Shift	`0 ✓`	OK
`C2.4`	tau = 1/[(1+z_kappa)(1+z_beta)] = kappa_X*beta_Y	Thm Operational Measurability, step 1	`0 ✓`	OK
`C2.5`	tau^2 = 1 - (kappa^2+beta^2) + kappa^2 beta^2	Thm Operational Measurability, step 2	`0 ✓`	OK
`C2.6`	Lorentz factor is the reciprocal phase: gamma = 1/beta_Y	Summary after Thm restenergy	`0 ✓`	OK

Notes:

C2.2 — Transverse Doppler: observer’s own amplitude vanishes against its local frame.
C2.5 — Multiplicative composition of the two phases – source of the cross term.

Layer 3 closure

id	claim	source	residual	verdict
`C3.1`	Closure kappa^2 = 2 beta^2 by eliminating ledger-per-DOF ell	Thm Closure; Lem DOF-Indifference	`0 ✓`	OK
`C3.2`	Closure is equivalent to the DOF ratio itself	Thm Closure	`0 ✓`	OK
`C3.3`	Spatial distance: r = R_s/kappa^2 <=> kappa^2 = R_s/r	Thm Inverse-Distance Potential Amplitude	`0 ✓`	OK
`C3.4`	Closure factor delta = kappa^2/(2 beta^2) equals 1 exactly on closure	Def Closure Factor	`0 ✓`	OK
`C3.5`	Eccentricity e = 2 beta^2/kappa^2 - 1 vanishes for a closed circular state	Rem scope (ROM eccentricity)	`0 ✓`	OK
`C3.6`	No singularity: beta_max^2=1 -> kappa_max^2=2 -> r_min = R_s/2	Sec no_singularities	`0 ✓`	OK
`C3.7`	Static bound kappa^2 <= 1 forces beta^2 <= 1/2 for closed static states	Thm Closure + S^2 additive closure	`0 ✓`	OK
`C3.8`	Channel independence: the 1/r channel and the factor-2 channel share only kappa	Box Structural Independence of the Force Law and Virial Coefficient	`0 ✓`	OK
`C3.9`	Virial: closure kappa^2=2beta^2 <=>	V	= 2T with T=m v^2/2, V=-GMm/r	Rem Geometric Origin of Physical Law	`0 ✓`	OK

Notes:

C3.1 — beta^2 = 1ell, kappa^2 = 2ell -> kappa^2/beta^2 = 2 = dim S^2 / dim S^1.
C3.3 — r is DEFINED as inverse amplitude per DOF (1/kappa * 1/kappa); no metric input.
C3.5 — e = 1/delta - 1, so circularity and closure are the same statement.
C3.7 — Derived corollary: circular closure saturates at beta = 1/sqrt(2), i.e. r = R_s.
C3.8 — shared symbols = {kappa}; r absent from closure, beta absent from the distance law.
C3.9 — V = (kappa^2/2)E_0 and T = (beta^2/2)E_0, so the virial coefficient IS the DOF ratio.

Layer 4 energy

id	claim	source	residual	verdict
`C4.1`	Invariant internal projection: E_beta * beta_Y = E_0	Thm Invariant Projection of Rest Energy	`0 ✓`	OK
`C4.2`	E_beta^2 = p_beta^2 + m^2 (c=1) with p_beta = E_beta*beta, m = E_0	Cor Energy–Momentum Relation	`0 ✓`	OK
`C4.3`	Trigonometric form: E_beta^2 = (cot(th1) E_0)^2 + E_0^2	Rem Geometric Forms	`0 ✓`	OK
`C4.4`	Restoring c: p_beta = gamma m v	Rem Units sanity check	`0 ✓`	OK
`C4.5`	Gravitational analogue: E_kappa^2 = (p_kappa c)^2 + (m c^2)^2	Sec Gravitational Tangent Formulation	`0 ✓`	OK
`C4.6`	Geometric equivalence: radial free fall beta=kappa gives p_beta = p_kappa	Rem Ontological Status of p_kappa	`0 ✓`	OK
`C4.7`	Equivalence principle: m_g = m_i = E_0/c^2 (single rest invariant)	Lem Equivalence of Inertial and Gravitational Response	`0 ✓`	OK
`C4.8`	Composition independence: channel decomposition E_0 = sum E_0^(a) cancels	Rem Composition-Independence	`0 ✓`	OK
`C4.9`	Reciprocal duality at beta=0: E * E_kappa = E_0^2	Sec energy quantities	`0 ✓`	OK
`C4.10`	Energy symmetry: Delta E_{A->B} + Delta E_{B->A} = 0	Thm Energy Symmetry	`0 ✓`	OK
`C4.10b`	Control: the same antisymmetry holds for an ARBITRARY state function	Thm Energy Symmetry (scope probe)	`0 ✓`	OK
`C4.11`	Kinematic phase exhaustion beta_Y->0 gives E -> oo (speed of light)	Thm Universal Rate of Change and the Horizon of Causality	`0 ✓`	OK
`C4.12`	Potential phase exhaustion kappa_X->0 gives E -> 0 (event horizon)	Thm Universal Rate of Change and the Horizon of Causality	`0 ✓`	OK

Notes:

C4.2 — Pure restatement of S^1 closure; mass enters only as the invariant E_0.
C4.7 — Both momenta are the SAME E_0 scaled by different phase ratios.
C4.8 — Every internal channel scales by the identical phase ratio kappa_X/beta_Y.
C4.10 — Antisymmetry of a state function difference: identically zero, no parameter.
C4.10b — So the zero-sum law alone constrains nothing; all physical content sits in the specific form E = E_0 kappa_X/beta_Y (Lem Unified Relational Scaling).
C4.11 — limit = oo
C4.12 — limit = 0

Layer 5 legacy

id	claim	source	residual	verdict
`C5.1`	First-order phase ratio: kappa_X/beta_Y ~ 1 - kappa^2/2 + beta^2/2	Sec linearized relational limit	`0 ✓`	OK
`C5.2`	Second order carries the cross term -kappa^2 beta^2/4	Sec linearized relational limit	`0 ✓`	OK
`C5.3`	Linearized two-point law: (kappa_A^2-kappa_B^2)/2 + (beta_B^2-beta_A^2)/2	Eq linearized_two_point	`0 ✓`	OK
`C5.4`	Newtonian Hamiltonian H = m v^2/2 - GMm/r as the single-point collapse	Sec Hamiltonian	`0 ✓`	OK
`C5.5`	Minkowski interval = S^1 closure x (posited c^2 dt^2)	Sec SR_interval	`0 ✓`	OK
`C5.6`	Schwarzschild g_tt = S^2 closure x (posited c^2 dt^2)	Sec GR_interval	`0 ✓`	OK
`C5.7`	GR dictionary: kappa_X = sqrt(-g_tt) for static spacetimes	Legacy Dictionary	`0 ✓`	OK

Notes:

C5.1 — The classical factor 1/2 is the first Taylor coefficient – not a postulate.
C5.2 — second-order coefficient = 3beta4/8 - beta2kappa2/4 - kappa4/8
C5.3 — The constant 1 cancels: only differences of squared amplitudes survive.
C5.4 — Requires the operationally impossible frame at infinity (kappa_A=beta_A=0).
C5.5 — Four posits (container, xyz, autonomous t, scale c^2dt^2) added to beta^2+beta_Y^2=1.
C5.6 — Same four posits; kappa^2 localized as R_s/r introduces r, G, M.
C5.7 — Pragmatic translation, not an ontological identity.

Layer 6 field

id	claim	source	residual	verdict
`C6.1`	Mass label from geometry: M = kappa^2 c^2 r /(2G) via R_s = 2GM/c^2	Sec density	`0 ✓`	OK
`C6.2`	Normalised identity kappa^2 = rho/rho_max (G and M cancel)	Lem norm_id; Eq unified_field	`0 ✓`	OK
`C6.3`	Self-consistency forces n=3 and alpha=4pi in M = alpha r^n rho	Sec Self-Consistency Requirement	`0 ✓`	OK
`C6.4`	Closure of the loop: M = 4 pi r^3 rho reproduces M = kappa^2 c^2 r/(2G)	Sec Self-Consistency Requirement	`0 ✓`	OK
`C6.5`	Equation of state P = -rho c^2 from the radial balance relation	Sec pressure	`0 ✓`	OK
`C6.6`	Saturation: P_max = -c^4/(8 pi G r^2) at kappa^2 = 1	Sec pressure	`0 ✓`	OK
`C6.7`	Vacuum field equation d(r kappa^2)/dr = 0 gives r kappa^2 = R_s	Sec Field Equation and Matter Sources	`0 ✓`	OK
`C6.8`	Matter source: d(r kappa^2)/dr = 8 pi G r^2 rho_matter/c^2 is dimensionally closed	Eq will_field_diff	`0 ✓`	OK
`C6.9`	Bounds: kappa^2 <= 2 implies rho <= 2 rho_max	Sec no_singularities	`0 ✓`	OK

Notes:

C6.2 — S^2 surface normalisation 1/(4pi) applied to M/r^3; both G and M drop out.
C6.3 — n is fixed by r-independence of M; alpha then follows. Note 4pi, not Newton’s 4pi/3.
C6.5 — d(kappa^2)/dr = -kappa^2/r drives the negative surface tension.
C6.7 — The 1/r law is the vacuum solution of the accumulation equation.
C6.8 — Substituting rho_matter = rho_field reduces the source term to kappa^2 itself.

Layer 7 globes

id	claim	source	residual	verdict
`C7.1`	Total phase of the closed pair: tau_AC^2 = (1-2 beta^2)(1-beta^2)	Sec Newton’s Question Answered	`0 ✓`	OK
`C7.2`	Inversion beta^2 = (3 - sqrt(1+8 tau^2))/4 solves the closure quadratic	Eq globes-answer	`0 ✓`	OK
`C7.3`	At tau=0 the inversion saturates at beta^2 = 1/2 (r = R_s)	Eq globes-answer	`0 ✓`	OK
`C7.4`	Inversion of the two-station shift gives R_sA(Z_A, t_A, Delta t_A)	Eq globes-RsA	`0 ✓`	OK
`C7.5`	kappa_AC^2 = R_sA/a with a = c t_d/2: the speed of light cancels	Eq globes-kappaAC	`0 ✓`	OK
`C7.6`	Cord share: kappa_d^2 = 2 beta_AC^2 - kappa_AC^2	Thm Restoring Closure	`0 ✓`	OK
`C7.7`	Fully operational closed form for kappa_d^2 (two shifts, three light-times)	Eq globes-kappad-closed	`0 ✓`	OK
`C7.8`	Classical tension is the cord’s ledger share: T = E_0 kappa_d^2/d	Sec Reductio ad Absurdum of the Classical Tension	`0 ✓`	OK
`C7.9`	Optional period label: T = pi t_d / beta_AC, N = pi/beta_AC	Rem Optional period	`0 ✓`	OK

Notes:

C7.1 — kappa_X,tot uses kappa_tot^2 = 2 beta^2 from closure.
C7.2 — Physical branch: the (3+sqrt) root gives beta^2=1 and is discarded.
C7.5 — c does not appear in the result: two shifts and three light-times suffice.
C7.6 — Quadratic additivity across channels is DOF-indifference, not an extra postulate.
C7.8 — d = 2a, so T = E_0 kappa_d^2 / d. Force is not a primitive here.

Layer 8 invariant

id	claim	source	residual	verdict
`C8.1`	W_ILL = E T/(M L) reduces to 2 beta^2/kappa^2 – all constants cancel	Sec willinvariant	`0 ✓`	OK
`C8.2`	W_ILL = 1 is EXACTLY equivalent to the closure theorem kappa^2 = 2 beta^2	Sec willinvariant	`0 ✓`	OK
`C8.3`	Phase-normalised W_ILL = E_0 t_o^2/(m_0 r_o^2) = 1 given E_0=m_0c^2, r_o=c t_o	Sec willinvariant (phase-normalised form)	`0 ✓`	OK
`C8.4`	Sector coupling E_o/M_o = L_o/T_o	Sec willinvariant	`0 ✓`	OK

Notes:

C8.1 — reduced form = 2*beta2/kappa2; a, G, c, m_0 all cancel identically.
C8.2 — So W_ILL carries no content beyond closure: it is closure in dimensionful dress.
C8.3 — Phases cancel pairwise; the residual content is the light-time relation r = c t.

Layer 9 gps

id	claim	source	residual	verdict
`C9.1`	GR 1PN coefficient equals kappa_E^2/2 + beta_E^2/2 - 3 kappa_E^2 varrho/4	Eq delta-GR	`0 ✓`	OK
`C9.2`	First-order Taylor coefficient of the exact RG ratio reproduces it exactly	Eq delta-RG-1 vs delta-GR	`0 ✓`	OK
`C9.3`	Second-order coefficient matches the published 6-term expression	Eq delta-RG-2	`0 ✓`	OK
`C9.4`	The beta^2 kappa^2 cross term is absent from any additive GR rearrangement	Interpretation (iii)	`0 ✓`	OK
`C9.5`	Numerical: Delta t_RG - Delta t_GR equals the predicted second-order term	Results table (numerical)	`0 ✓`	OK
`C9.6`	Numerical: circular closure residual beta_GPS^2 - kappa_GPS^2/2 vanishes	Methodology, Part II	`0 ✓`	OK

Notes:

C9.2 — delta_RG^(1) = beta_E2/2 - 3kappa_E2varrho/4 + kappa_E2/2
C9.3 — delta_RG^(2) = beta_E22/8 + 3beta_E2kappa_E2varrho/8 - beta_E2kappa_E2/4 - 19kappa_E2**2varrho2/32 + 3*kappa_E22*varrho/8 + kappa_E22/8
C9.4 — delta_GR is linear in each amplitude; the cross term is purely multiplicative.
C9.5 — Delta t_RG = 38.5421472752 us/day, Delta t_GR = 38.5421472451 us/day, difference = 3.01678e-8, predicted = 3.01678e-8, relative agreement = 7.33e-10
C9.6 — closure residual = 0.0

Assumption ledger

Places where the chain rests on a stipulation rather than a derivation. None of these is a fitted parameter, but each is a load-bearing choice that an auditor should see stated explicitly.

tag	kind	location	item	status
`A1`	convention	Thm carriers	Ledger on each carrier normalised to unity (unit radius).	Verified as the unique parameter-free choice: any radius R != 1 leaves the residual 1-R^2 (negative control C1.5).
`A2`	stipulation	Lem DOF-Indifference; Thm Restoring Closure	Independent channels add in quadrature: kappa_tot^2 = sum_i kappa_i^2.	Asserted from isotropy + minimalism, not independently derived. Load-bearing for the factor 2 (C3.1) and for the cord’s share (C7.6).
`A3`	translation	Box Cross-Cultural Invariants	beta = v/c and kappa = v_e/c = sqrt(R_s/r).	Presented as translation into legacy vocabulary rather than definition. Empirically anchored via z_beta, z_kappa (C2.1-C2.2).
`A4`	branch choice	Eq globes-answer	Root selection in beta^2 = (3 - sqrt(1+8 tau^2))/4.	The (3 + sqrt) root gives beta^2 = 1 and is discarded on physical grounds. Both roots solve the quadratic (C7.2).
`A5`	internal boundary	Sec no_singularities vs Sec potential	r_min = R_s/2 requires kappa_max^2 = 2, but additive S^2 closure caps kappa^2 <= 1.	Inside Part I the additive closure restricts CLOSED STATIC states to beta^2 <= 1/2, i.e. r >= R_s (C3.7). kappa^2 = 2 is reachable only after the promotion to the multiplicative phase tau^2 = beta_Y^2 kappa_X^2 (deferred to R.O.M. Kerr). r_min = R_s/2 is therefore NOT self-contained in Part I.
`A6`	asserted relation	Sec pressure	Radial balance P = (c^4/8 pi G)(1/r) d(kappa^2)/dr.	Stated without derivation in Part I. Given it, P = -rho c^2 follows identically (C6.5).
`A7`	translation	Sec density	3D volumetric proxy r^3 plus 1/(4 pi) S^2 surface normalisation.	Explicitly declared as a legacy-translation interface. Self-consistency then forces n=3, alpha=4 pi (C6.3-C6.4). NOTE: M = 4 pi r^3 rho differs from Newton’s M = (4 pi/3) r^3 rho by a factor 3, so ‘rho’ here is not numerically the Newtonian mass density; the identity kappa^2 = rho/rho_max is internally consistent but convention-dependent.
`A8`	no new content	Sec willinvariant	W_ILL = E T/(M L) = 1.	Reduces identically to 2 beta^2/kappa^2 (C8.1), so W_ILL = 1 IS the closure theorem in dimensionful dress. The phase-normalised form additionally needs r_o = c t_o (C8.3). No independent predictive content.
`A9`	scope	Sec earth-gps	GPS comparison target is the ADDITIVE 1PN formula.	Acknowledged in the paper. Not a comparison against full geodesic integration in exact Schwarzschild plus SR kinematics.
`A10`	no new content	Thm Energy Symmetry	Delta E_{A->B} + Delta E_{B->A} = 0.	Holds for an arbitrary state function (control C4.10b). The law alone is vacuous; its content is entirely in E = E_0 kappa_X/beta_Y.

Corollaries the formalization makes explicit

Closed static states satisfy beta^2 <= 1/2, i.e. orbital speed <= c/sqrt(2) and r >= R_s (C3.7) – a sharper bound than r >= R_s/2 within Part I’s additive closure.
The virial coefficient is not an independent fact: V = 2T is literally kappa^2 = 2 beta^2 rewritten in legacy units (C3.9).
Eccentricity and the closure factor are the same quantity: e = 1/delta - 1 (C3.5).
W_ILL = 1 and kappa^2 = 2 beta^2 are the same statement (C8.1, C8.2).
The GR/RG difference is controlled by the multiplicative cross term -beta^2 kappa^2/4, which no rearrangement of an additive 1PN sum can produce (C5.2, C9.3, C9.4).

Reproducing this

python build_report.py      # writes will_rg_core_checks.csv and this report
python will_rg_core.py     # prints pass/fail summary only

will_rg_core.py has no side effects on import: run_all() returns the result records, audit_zero_parameters() returns the symbol census, and GPS holds the numerical table. Adding a claim means adding one record(...) call.