The Debt-1 calc notes (#5–#8)
The Debt-1 calc notes are four working notes, artifacts #5 to #8, on the most severe of the lab notebook’s open problems: what equations govern the geometry of momentum space. The first, written overnight, showed that the original postulates left the program with no content beyond general relativity and one model of self-interacting dark matter, repaired them, and gathered the geometry into a single modular field; on the way it found the mechanism of the velocity gate. The second showed that the field carrying momentum-space geometry must be a second graviton, and a massive one. The third tested whether that field avoids the Boulware–Deser ghost, the instability that generic massive-graviton theories carry, and found that it does when the field’s potential is written in frame variables, sets of local reference axes from which a metric is assembled; written in metric variables it fails, and the note argues that the frame formulation is the natural one. The fourth used Born reciprocity to constrain the remaining coefficients, leaving two free parameters, and to cancel the sector’s vacuum energy. Together they closed Debt 1 in structure; their results are now stated in checkpoint #11 §5–§8, and the battery checks their explicit algebra.
Status: Record, final for its scope — the notes tag their own results ("[D — rigorous]", “[D — structural]”, “[D + T]”); the battery T1–T7 in verify/battery.py checks the explicit algebra [V], which #11 §2 calls Tier 1 — HANDOFF.md §3 marks all four “final”, and #8 “final — closes Debt 1 (structural)”; the root CLAUDE.md files them with the audit trail, “superseded as reference but kept”. The files are unchanged since their import on 2026-10-19 (commit 8d588f6; moved to notes/ in 008c3ee).
What the notes are#
Debt 1 was the first item of the lab notebook’s open problems: “The field equation for H beyond minisuperspace — Born’s 88-year debt” (#1 §11), H being the generalized metric on phase space. It is the question Born’s question leaves: if position and momentum are symmetric, what is the dynamics of the geometry of momentum space? See the three debts.
The work came in four chunks, each a working note: Chunk A (notes/debt1a-geometric-core.md, #5), A-ii (notes/debt1a-ii-X-classification.md, #6), A-iii (notes/debt1a-iii-tuning-test.md, #7) and A-iv (notes/debt1a-iv-beta-coefficients.md, #8). The notes are undated; they follow the Debt-3 notes in HANDOFF.md §3’s creation order, and the notebook’s addenda v0.3 to v0.6 summarize them, one each. #5 calls itself an “Overnight working note”, and its addendum is headed “overnight”.
Each note ends with a list of what it settled and what it left open, and the next note takes up the first open item. The chain therefore reads as one argument: the axioms are repaired (#5), the new field is classified (#6), its potential is tested (#7), and the potential’s coefficients are fixed (#8).
Their standing#
The handoff marks all four “final” and describes each: #5 “compatibility theorem proof; P2→P2′ inconsistency + repair; Siegel/modular field identification; gate mechanism = X-exchange (pays Debt 2’s mechanism)”; #6 “shift symmetry; X = second graviton; massless/ungauged excluded; massive spin-2 via Higgsed dual diffs”; #7 “{J,I[Z]} blocks computed; metric tier fails FP twice; frame tier (HR wedge class) passes; constrained-bimetric picture”, FP being Fierz–Pauli and HR Hassan–Rosen; #8 “palindrome βₙ = β₄₋ₙ; V(1) = 2T ⇒ Λ_wedge = 0; parameters 5→2; spectrum/parities” (HANDOFF.md §3).
The wiki treats the calc notes as “final for their own scope” (wiki/CLAUDE.md). Their results are restated, with later tags and corrections, in #11 §5 (the compatibility theorem), §6 (the modular field), §7 (the massive momentum graviton) and §8 (the gate’s mechanism), and the wiki cites #11 first (the checkpoint).
#5, debt1a: the geometric core#
“Debt 1, Chunk A — the geometric core, linearized groundwork”, subtitled “(plus an unplanned payment on Debt 2)”. Status: “two rigorous results, one axiom correction forced, one mechanism identified, one decisive computation defined but not completed.”
- §A, the compatibility theorem [D — rigorous]. With ω and η both in standard form, the positive H satisfying (η⁻¹H)² = +1 and (ω⁻¹H)² = −1 “are exactly” H = diag(g, g⁻¹), with a short block proof and its corollaries: det H = 1, no Kalb–Ramond analog, and “The momentum metric is slaved: g̃ = g⁻¹ pointwise” (the compatibility theorem; its corollaries).
- §B, the inconsistency and the forced repair [D — structural]. Under the original P2, with ω and η rigid, the compatible polarizations are “discrete — the 2^d coordinate-type splittings, GL(d)-rigid”: no Lagrangian Grassmannian, no Maslov circle, “no τ field”, and no geometric origin for the gate. “Call this Branch R (rigid): BBT reduces to GR + one bespoke SIDM model”, BBT being the Broken-Born theory, GR general relativity and SIDM self-interacting dark matter. The repair, P2′: “Freeze ω only — that is genuinely ℏ — and let η be emergent: the order parameter itself.” The “Program-nontriviality theorem”: “BBT has content beyond GR + SIDM iff η is emergent.” See Original P2 (superseded), P2′ and the nontriviality theorem. The word “discrete” is queried in
review/wiki-build-flags.mditem 1: for d ≥ 2 such polarizations form continuous families, though the conclusion of no Maslov circle survives. - §C, the modular field [D — structural; math facts standard]. Positive H compatible with ω alone form the Siegel domain, coordinatized by Z = X + iY: “Im Z = Y = g”, “Re Z = X, a new symmetric tensor: the momentum graviton”, reciprocity “as Z → −Z⁻¹”, the cosmological constraint “the |Z| = 1 self-dual circle”, and Λ(d) as the Shilov boundary carrying τ (the modular field Z; the Siegel domain; reciprocity as modular inversion). “Born’s debt, restated exactly: the field equation for momentum-space geometry is the Re Z sector of S_H.” The identification of a·ã = 1 with |Z| = 1 is queried in
review/wiki-build-flags.mditem 23. - §D, the quadratic spectrum [partial]. The field content of the broken vacuum, with “Λ(d) σ-modes (10: τ plus nine, of which inverse-Higgs constraints likely eliminate most [O])” (inverse-Higgs constraints), and the decisive check posed as a three-way classification of X: healthy, auxiliary or ghost (Open: field content of the broken vacuum).
- §E, Debt 2’s mechanism [D — structural]. Exchange of X between the dual stresses of two χ quanta “yields a four-point amplitude that depends on the incoming relative momenta of the pair”: “gate = momentum-graviton exchange”. The gate’s shape stays open (the gate mechanism; dual stress; Open: the gate shape).
- §F, the record after the night. Under the note’s heading “Paid”: the theorem, the repair, the reframing of Born’s debt, Debt 2’s mechanism, unimodularity and the modular unification. “Now precisely open”: the X classification ("do this next"), the gate shape, the origin of τ’s potential under P2′, the “inverse-Higgs accounting of the nine heavy moduli”, and connection uniqueness for the (ω, H)-only structure (the Born connection).
#6, debt1a-ii: the classification of X#
“Debt 1, Chunk A-ii: The classification of X”. Status: “X is forced to be a massive spin-2 field, with exactly one known consistent structure available, and the massless option excluded by theorem. One tuning question remains, now precisely posed.”
- §1, shift symmetry [D — rigorous]. The rigid Sp(2d, ℝ) contains Z → Z + B, so constant shifts of X are exact symmetries: X is “axion-like”, and “‘auxiliary X’ (the soft outcome 2) is off the table” (shift symmetry).
- §2, gauge structure and the no-go [D + T]. Local dual diffeomorphisms act as δX_μν = ∂₍μλ_ν₎, “a second copy of linearized diffeomorphisms”; hence “the symplectic partner of the graviton is another graviton.” Without the gauge invariance no ghost-free kinetic term exists (the signature obstruction); with it, X is a second massless graviton, excluded by the Boulanger–Damour–Gualtieri–Henneaux theorem. “Therefore X must be massive.” See dual diffeomorphisms, the symplectic partner of the graviton and the multi-graviton no-go theorem. The phrase “η antisymmetric-pairing vs ω” is queried in
review/wiki-build-flags.mditem 15, since η is the symmetric structure. - §3, the mass mechanism [D — structural]. The polarization choice “spontaneously breaks the local dual diffeomorphisms”: “the four would-be Goldstones π_μ of the broken λ-gauge are eaten”, and X becomes a massive spin-2 with “mass M_X = the polarization-breaking scale” (the gravitational Higgs mechanism). Massive spin-2 “has exactly one known consistent structure”, the Fierz–Pauli and Hassan–Rosen class, so the geometric sector “is healthy iff the potential generated by the η-selection lands in the Hassan–Rosen class” (why X must be massive). A signature caveat is “Flagged, not glossed”: the chain is “the Euclidean-rigorous skeleton” (Open: Lorentzian dressing).
- §4, consequences, “with one honest downgrade”. The gate comes from the χ quanta’s extent on the momentum fiber, and “Wavepacket overlap of Gaussian profiles gives exactly e^{−(p₁−p₂)²/2μ²} — the postulated gate, with μ = the χ quantum’s own p-width”; but generic vertices give power-law tails, so “row 1’s ’exponential cliff’ weakens to ‘sharp knee at v*, with the tail shape (Gaussian vs rational) a diagnostic of the vertex completion.’” (the gate mechanism; row 1). The relation of this exponent to the notebook’s e^{−k²/μ²} is queried in
review/wiki-build-flags.mditem 17. The note also logs “A new door, deliberately left closed today”: X-condensate dark matter (the X-condensate candidate). - §5, the record. The single remaining computation: derive the X-potential from the η-selection sector “and test it against the dRGT coefficient family”, dRGT being de Rham–Gabadadze–Tolley massive gravity. “Pass ⇒ the geometric core stands and ’theory’ becomes the defensible word.”
#7, debt1a-iii: the tuning test#
“Debt 1, Chunk A-iii: The tuning test”. Status: “computed. Outcome: conditional pass — the condition identified, both branches computed or classified, and the failure mode exhibited explicitly.”
- §1, the object. The X mass must come from “a misalignment potential penalizing incompatibility between the polarization J and the modular field Z”, measured by the anticommutator {J, I[Z]} (the misalignment potential).
- §2, the metric tier [D — rigorous]. Block multiplication gives {J₀, I[Z]} = [[S, A], [A, S]] with S = Y + XY⁻¹X − Y⁻¹ and A = [X, Y⁻¹]. Any positive quadratic invariant then gives “m²Tr(h²) — a graviton mass without the −(Tr h)² companion”, which violates Fierz–Pauli at linear order; and the nonlinear structure Tr(g²) + Tr(g⁻²) lies outside the Hassan–Rosen family. “Every metric-tier (second-order, H-coupled) misalignment invariant tested lands off the tuning” (the metric tier fails twice; the Fierz–Pauli mass term). Three statements of this section are queried in
review/wiki-build-flags.md: item 3, the zero set of the potential; item 4, a missing −h² in the expansion of S; and item 14, which square-root eigenvalues p₄ refers to. Items 4 and 14 leave the conclusion unchanged; item 3 asks whether the larger zero set affects the claims about the vacuum structure. - §3, the frame tier [D — structural, resting on T]. In vielbein variables the ghost-free potentials are wedge polynomials, and the symplectic vielbein E with H = EᵀE lives “on the same geometric tier” as J. Wedge couplings of (E, E₀[J]) are “precisely the Hinterbichler–Rosen class: ghost-free by construction”. “Verdict of the tuning test: the theory fails in second-order (metric) variables and passes in first-order (frame) variables” (the frame tier; vielbein formulation and wedge potentials; the verdict: a conditional pass).
- §4, constrained bimetric gravity [D — structural]. “Hassan–Rosen bimetric gravity with the second frame constrained to a compact moduli space Λ(d) — a predictive truncation of bimetric theory” (constrained bimetric gravity).
- §5, the record. Remaining: the βₙ coefficients, “the last computation”, and the checks that τ is not eaten, the Lorentzian dressing, and the couplings of ordinary matter against graviton-mass bounds. “Program-level note: three structural tests in a row (multi-graviton no-go, signature obstruction, tuning test) could each have killed the geometric sector outright”.
#8, debt1a-iv: the β coefficients#
“Debt 1, Chunk A-iv: The β coefficients — closing the debt”. Status: “computed. Reciprocity fixes the potential’s structure, cancels its vacuum energy by theorem, and cuts the free parameters from five to two. Debt 1 is structurally paid.”
- §1, the palindrome [D]. Reciprocity acts on the Hinterbichler–Rosen sector as the exchange of the two frames, which forces “the palindromic constraint” βₙ = β₄₋ₙ, the “interchange-symmetric branch” of bimetric theory: “its ℤ₂ is Born reciprocity.” (the palindrome).
- §2, the vacuum-energy theorem [D — rigorous, four lines]. With eₙ(1) = C(4, n) and the tadpole T, the palindrome gives V(1) = 2T; a flat aligned vacuum needs V(1) = T, so “T = 0 and V(1) = 0 exactly.” The massive-gravity sector “contributes nothing to the cosmological constant”. The note flags its novelty “to verify” (the vacuum-energy identity).
review/wiki-build-flags.mditem 6 notes that §2’s wording, which counts the flatness condition as one condition, is the accurate one, against “by symmetry rather than tuning” elsewhere. - §3, spectrum and parameters [D + T]. “Parameter count: five βₙ → palindrome (−2) → T = 0 (−1) → two: the Fierz–Pauli mass combination m_FP² ∝ β₁ + 2β₂ + β₃ = 2(β₁+β₂), i.e. M_X, and one orthogonal shape parameter”. The massless graviton is reciprocity-even and X reciprocity-odd (on the parities under #11 §6’s Z → −Z⁻¹, see
review/wiki-build-flags.mditem 39); safety for ordinary matter “is by range, not decoupling”; graviton-mass bounds from gravitational waves “apply to the even eigenstate, which is exactly massless”; and τ is not eaten [D — leading order]. See spectrum, parameters and reciprocity parity, safety by range and τ is not eaten. Its “five-function freedom” is queried inreview/wiki-build-flags.mditem 8, since the βₙ are constants; item 5 records that #9 and #11 later wrote “M_X ∝ 2(β₁+β₂)” where this note has m_FP². - §4, the answer to Born. “Momentum-space geometry is carried by X = Re Z, the reciprocity-odd partner of the metric. It is a massive spin-2 field”, with “two free parameters, vacuum energy exactly zero by the palindrome theorem” (Born’s answer).
- §5, the record. “Debt 1: structurally paid. Open residue, none of it existential”: the wedge reduction from S_H, Lorentzian dressing, τ–Stückelberg mixing, the dual-stress enhancement, and the literature check on Λ = 0 (the partially-massless comparison). “Debt 2: mechanism paid (X-exchange, μ = fiber width); shape completion open”. “Debt 3: closed (negative)”.
What the machine checks cover#
#11 §2 calls the explicit algebra Tier 1, “verified this session by an independent numerical/exact-integer battery”, and all seven tests passed [V]. The battery is HANDOFF.md’s Appendix A as verify/battery.py; each test checks a statement of the notes (the verification battery):
| Test | What it checks | Note | Current output |
|---|---|---|---|
| T1 | the compatibility theorem in both directions, and the proof’s block identities | #5 §A | PASS |
| T2 | the Siegel parametrization: I[Z]² = −1, H_Z symmetric, positive and of unit determinant, at one random point | #5 §C; #7 §2 | det=1.000000000000 |
| T3 | the anticommutator blocks [[S, A], [A, S]] | #7 §2 | PASS |
| T4 | the linearized potential: Tr(S²)/ε² against 4Tr(h²) and the Fierz–Pauli form | #7 §2 item 2 | 66.5770 vs 4Tr(h^2)=66.5770 vs FP=63.7221 |
| T5 | the palindrome identity V(1) = 2T in exact integers | #8 §2 | V1=[2, 8, 6] T=[1, 4, 3] |
| T6 | the binomial linearization coefficients C(3, n−1) | #8 §2 | [0.0, 1.0, 3.0, 3.0, 1.0] |
| T7 | p₄ outside the linear span of the eₙ | #7 §2 item 3 | rms=3.2853 |
The outputs are those of the current run of scripts/check.sh. The checks confirm the algebra, not the physics it is attached to (verification checks arithmetic, not the model). Steps stated in words are not covered: the derivation of the palindrome from reciprocity, the parameter count, the frame-tier verdict and the exclusion of Tr(g²) + Tr(g⁻²), which rests on the derivation since T7 tests p₄ alone. The wiki tags those [D].
Where the results went#
The notebook’s addenda v0.3 to v0.6 summarized each note as it was written. The program v1.0 (#9 §3–§6) consolidated them under “(Debt 1: paid)” and “(Debt 2: mechanism paid)”. #11 states them in §5–§8, under the headings “The kinematic core and the compatibility theorem”, “The modular field”, “The massive momentum graviton (Debt 1)” and “Conjugate matter and the gate (Debt 2)”. #8’s open residue became #11 §14 items 1, 3, 6, 5 and 2 [O] (open problems and the queue).
What has changed since#
- The compatibility theorem’s tag. “[D — rigorous]” in #5; #11 since v2.1 tags it [T][V], the statement being Freidel, Leigh and Minic’s and the block proof and machine check the program’s (broken Born duality).
- The novelty of the branch and of Λ = 0. #8 said the interchange-symmetric branch “now has a reason to exist” and that the Λ = 0 property, if new, “is a standalone publishable observation”. #11 §7.4 found the partially-massless point inside the branch and downgraded the claim to “requires detailed comparison with the PM and exchange-symmetric literature”, PM being partially massless (Open: the partially-massless comparison; partially massless gravity).
- “Exactly one known” structure. #6 §3 keeps the qualifier; #11 §7.1 writes “The unique consistent structure” (
review/wiki-build-flags.mditem 7). - The safety argument. #5 §E and #6 §4 combined X’s short range with a weak coupling through ordinary matter’s dual stress (“suppressed” in #5, “unenhanced” in #6); #7 §4 required the massless combination to dominate the couplings of ordinary matter; #8 §3 rests on range alone, with couplings of gravitational strength: “safety is by range, not decoupling”. #8’s form is #11’s (safety by range).
- The graviton-mass bound. #7 used it as a check on a mixing angle; #8 applies it to the exactly massless even eigenstate, #11’s row 6 (spectrum and parity).
- Counts and wording. #7 counts “three structural tests in a row”, #11 §15 “four consecutive consistency tests”, and #11 governs. What X eats is “the four would-be Goldstones π_μ” in #6 but “the vector π_μ and the longitudinal scalar of the λ-gradient sector” in #8; #11 §7.1 follows #6, and no document reconciles the two. #6 §5 makes a pass the point where “’theory’ becomes the defensible word”; #11 §15 says the word “still waits on items 1–3” (conflicts between sources).
Sources#
papers/broken-born-checkpoint-v2.md§5–§8 — the current statements of the notes’ results and their tags; §2 — Tier 1 and the battery; §7.4 — the novelty downgrade; §14 — the open items the residue became; §15 — “four consecutive consistency tests”notes/debt1a-geometric-core.md— §A–§F: the compatibility theorem, P2 → P2′, the modular field, the quadratic spectrum, the gate’s mechanismnotes/debt1a-ii-X-classification.md— §1–§5: shift symmetry, the no-go, the mass mechanism, the gate overlap and the X-condensate doornotes/debt1a-iii-tuning-test.md— §1–§5: the metric tier, the frame tier, constrained bimetric gravitynotes/debt1a-iv-beta-coefficients.md— §1–§5: the palindrome, the vacuum-energy theorem, the parameter count and spectrum, the answer to Born, the residuenotes/broken-born-theory.md‘Addendum (v0.3, overnight)’ to ‘Addendum (v0.6)’ — the notebook’s summariesnotes/broken-born-program-v1.md§3–§6 — the v1.0 consolidationHANDOFF.md§3 rows 5–8 — the notes’ standing and descriptionsCLAUDE.md‘Layout’ andwiki/CLAUDE.md‘The wiki is derivative’ — their place among the documentsverify/battery.py— tests T1–T7 and their current outputreview/wiki-build-flags.mditems 1, 3, 4, 5, 6, 7, 8, 14, 15, 17, 23 and 39 — suspected errors in statements of the notes or their later restatements