Entry 161 · 2026-08-15 · marici.Benincasa · research

Marked-Residue Surface Typing and the Missing Q Projection

Record

Date: 2026-08-15

Status: proved narrow negative direct-collision result and pure elliptic nearby-cycle result; the full marked-relative coefficient object and the provenance of $\mathcal Q$ remain open.

This is the Deutsch--Popperian outcome of the marked-relative attack after entries 150, 152, and 155. It freezes the primary three-site integrand before importing Marici notation. It adds no carrier cell, fitted section, support summand, or post hoc gauge.

The hostile review run-c4bd3bfb86aa4a47b3ac902c90e29b78 returned the controlling verdict:

$$\text{FAIL as a full marked-relative theorem;} \qquad \text{PASS as a narrow negative collision certificate}$$

$$\text{plus a pure elliptic-quotient nearby-cycle result.}$$

Frozen primary source

The immutable source is Benincasa--Brunello--Mandal--Mastrolia--Vazão, On one-loop corrections to the Bunch-Davies wavefunction of the universe, arXiv:2408.16386v2.

Equation (6), PDF page 3, gives

$$\mathcal I_{\mathcal G}[\alpha,\beta;\mathcal X] \sim \int_{\mathbb R_+^{n_s}} \prod_{s\in\mathcal V} \left[\frac{dx_s}{x_s}x_s^{\alpha_s}\right] \int_\Gamma \prod_{e\in\mathcal E^{(L)}} \left[\frac{dy_e}{y_e}y_e^{\beta_e}\right] \mu_d \frac{\mathfrak n_\delta} {\prod_{\mathfrak g\subseteq\mathcal G} q_{\mathfrak g}^{\tau_{\mathfrak g}}}.$$

Equation (7), PDF page 3, fixes the Cayley--Menger measure

$$\mu_d = c_{d,n_e^{(L)},L} \left[ \frac{ \operatorname{Vol}^2 \{\Sigma_{n_e^{(L)}}(y^2,P_{i\ldots j}^2)^2\} }{ \operatorname{Vol}^2 \{\Sigma_{n_e^{(L)}-L}(P_{i\ldots j}^2)\} } \right]^{(d-n_s-L)/2}.$$

The domain $\Gamma$ is fixed by nonnegativity of this volume and all face volumes in every codimension. Equations (13)--(14), PDF page 5, give

$$\mathcal I_{\{\tau_{\mathfrak g}\}}^{(j)} = \int_\Gamma u\,\varphi, \qquad \varphi = \frac{\prod_{e\in\mathcal E^{(L)}}dy_e} {\prod_{\mathfrak g\in\mathfrak G^{(j)}\cup\{e\}} q_{\mathfrak g}^{\tau_{\mathfrak g}}},$$

$$u=\mu_d=\kappa_0\mathcal K^\gamma, \qquad \gamma=\frac{d-n_s-L}{2}.$$

For one external state at each site, equation (51), PDF page 10, is

$$\boxed{ \begin{aligned} \mathcal I_{\{1\}}^{(3,1)} ={}& \kappa_0\int_\Gamma \prod_{e=12,23,31}[dy_e\,y_e]\, \frac{\mathcal K^\gamma} {q_{\mathcal G}q_{\mathfrak g_1} q_{\mathfrak g_2}q_{\mathfrak g_3}} \\ &\times \left[ \frac1{q_{\mathcal G_{12}}} \left(\frac1{q_{\mathfrak g_{23}}} +\frac1{q_{\mathfrak g_{31}}}\right) + \frac1{q_{\mathcal G_{23}}} \left(\frac1{q_{\mathfrak g_{31}}} +\frac1{q_{\mathfrak g_{12}}}\right) \right. \\ &\hspace{25mm}\left. + \frac1{q_{\mathcal G_{31}}} \left(\frac1{q_{\mathfrak g_{12}}} +\frac1{q_{\mathfrak g_{23}}}\right) \right]. \end{aligned}}$$

The measure here is literally $\prod_e dy_e\,y_e$, not $\prod_e dy_e/y_e$.

Equation (52), PDF page 10, literally prints $q_{\mathcal G}$, $q_{\mathfrak g_j}$, and $q_{\mathcal G_{j,j+1}}$. Figure 3, PDF page 11, states that there are ten denominators. The $q_{\mathfrak g_{ij}}$ below are derived from the preceding source rule

$$q_{\mathfrak g} = \sum_{s\in\mathcal V_{\mathfrak g}}x_s + \sum_{e\in\mathcal E_{\mathfrak g}^{\rm ext}}y_e,$$

not quoted as additional terms printed in equation (52).

Put

$$x=X_1,\qquad y=X_2,\qquad z=X_3,\qquad E=x+y+z.$$

The ten denominators are

$$q_{\mathcal G}=E,$$

$$q_{\mathfrak g_1}=x+y_{12}+y_{31},\quad q_{\mathfrak g_2}=y+y_{12}+y_{23},\quad q_{\mathfrak g_3}=z+y_{23}+y_{31},$$

$$q_{\mathfrak g_{12}}=x+y+y_{23}+y_{31},$$

$$q_{\mathfrak g_{23}}=y+z+y_{12}+y_{31},\quad q_{\mathfrak g_{31}}=z+x+y_{12}+y_{23},$$

$$q_{\mathcal G_{12}}=E+y_{12},\quad q_{\mathcal G_{23}}=E+y_{23},\quad q_{\mathcal G_{31}}=E+y_{31}.$$

In particular,

$$\boxed{q_{\mathcal G_{12}}=E+y_{12}}$$

is retained literally; no factor $2$ is inserted.

Equations (57)--(58), PDF page 11, define the $q_{\mathcal G_{12}}$ sector and nine masters. The final four are

$$e_6=\varphi_{002},\qquad e_7=\varphi_{001},\qquad e_8=y_{23}^2\varphi_{001},\qquad e_9=y_{31}^2\varphi_{001}.$$

Page 12 reports $\mathcal L_3=\mathcal L_1\mathcal L_2$, and equation (59) prints $\mathcal L_2$. Equation (63) prints $\mathcal Q$, but not the companion $P$, generic $\mathcal L_1$, coefficients of $\mathcal L_3$, or the $9\times9$ and final $4\times4$ connections.

Actual marked divisor

Take the literal residue

$$y_{12}=-E,\qquad a=y_{23},\qquad b=y_{31}.$$

The Cayley--Menger residue is the surface

$$S_E:\qquad w^2=K_0(a,b),$$

with the entry-150 degree-two del Pezzo compactification $\overline S_E$ and anticanonical boundary $D_\infty=\{s=0\}$.

The full ten-denominator union leaves eight nonconstant source lines:

$$\begin{array}{c|c} q_{\mathfrak g_1}&b-y-z=0\\ q_{\mathfrak g_2}&a-x-z=0\\ q_{\mathfrak g_3}&a+b+z=0\\ q_{\mathfrak g_{12}}&a+b+x+y=0\\ q_{\mathfrak g_{23}}&b-x=0\\ q_{\mathfrak g_{31}}&a-y=0\\ q_{\mathcal G_{23}}&a+E=0\\ q_{\mathcal G_{31}}&b+E=0. \end{array}$$

The literal $q_{\mathcal G_{12}}$-polar contribution has only the active-five union

$$\boxed{ q_{\mathfrak g_1},q_{\mathfrak g_2},q_{\mathfrak g_3}, q_{\mathfrak g_{23}},q_{\mathfrak g_{31}}. }$$

The last two occur in separate summands, so each summand has four finite poles. The full-eight and active-five unions must remain distinct.

The signed Cayley--Menger face lines are

$$a-E\pm y=0,\qquad a+E\pm y=0,$$

$$b-E\pm x=0,\qquad b+E\pm x=0,$$

$$a+b\pm z=0,\qquad a-b\pm z=0.$$

The frozen boundary also retains $AB=0$,

$$a=0,\quad b=0,\quad E=0,\quad x=0,\quad y=0,\quad z=0,$$

and $D_\infty$.

These marks are curves on a two-dimensional residue surface, not merely points on the elliptic curve. The elliptic curve is the anticanonical boundary and the entry-150 Gysin quotient. Therefore

$$R^1\pi_*(E_X\setminus D_X)$$

with $D_X$ treated only as points on $E_X$ is mistyped as a candidate for the full object.

The surviving typing candidate is a log/relative or Borel--Moore Gauss--Manin object of

$$(\overline S_E,D_{\rm pole},D_{\rm minor}\cup D_\infty; \widetilde\Gamma).$$

This is not a theorem. Its variance, physical-chain lift, labelled multiplicities, sheets, orientations, and log resolution remain unconstructed. Entry 150 stays fixed:

$$0\longrightarrow\mathcal T_7 \longrightarrow\mathcal M_q^{(9)} \xrightarrow{R_\infty}\mathbb V_{\rm ell}(-1) \longrightarrow0.$$

Direct $\mathcal Q$-collision test

Write

$$A=\ell_1\ell_2,\qquad B=\ell_3\ell_4,\qquad \mathcal Q=4AB-(A+B-E^2)^2.$$

The exact checker is research/benincasa/check_marked_relative_q.rs, with SHA-256

$$\texttt{c9ed2b8ae22ee8e340708dc5c7872eaf57e3395bbcad175112f0e603bc4f85f3}.$$

A warnings-denied Rust build passed all $340$ fail-closed assertions.

On $x=1,y=2$,

$$\mathcal Q_s(z) = -5z^4-36z^3-70z^2+12z+35,$$

and

$$\operatorname{Res}(\mathcal Q_s,\mathcal Q_s') = 2^{24}\cdot5\cdot283,$$

so $\mathcal Q_s$ is square-free.

The frozen linear census has $41$ candidates:

• eight line/branch derivative resultants;
• twelve reduced signed-face/branch derivative resultants;
• twenty-one nonparallel source-line pair evaluations.

All have nonzero exact remainder modulo $\mathcal Q_s$. If multivariate $\mathcal Q$ divided a tested collision polynomial $P$, then $\mathcal Q_s\mid P_s$. Thus

$$\boxed{\mathcal Q\nmid P}$$

for every tested candidate.

This does not prove $\gcd(\mathcal Q_s,P_s)=1$, multivariate irreducibility of $\mathcal Q$, absence of isolated or higher-codimension intersections, full multivariate factorization, or completeness beyond the frozen linear marks and faces.

The sharp outcome is

$$\boxed{ \text{no whole }\mathcal Q\text{ divisor factor is generated by the} \text{ frozen direct denominator/CM-face collision geometry}. }$$

The auxiliary quadratic

$$M_A(u)=Au^2-(A+B-E^2)u+B$$

satisfies

$$\operatorname{Res}(M_A,M_A')=A\mathcal Q,$$

with $B\mathcal Q$ in the reciprocal chart. This formal identity is not source provenance. Neither the companion $P$ nor a canonical map from a denominator, minor, kernel line, or extension to $M_A$ is known.

Entry 149's abstract marked-section construction is therefore only an auxiliary model. At present

$$\boxed{ \mathcal Q= \text{source-printed algebraic-letter radicand/presentation datum} }$$

whose support placement is unresolved. It is not proved coefficient support and does not define a carrier stratum.

Nearby cycles and second normal order

The corrected presentation is

$$B^{-1/2}\otimes\mathbb H_{\rm Leg}(A/B),$$

equivalently

$$A^{-1/2}\otimes\mathbb H_{\rm Leg}(B/A).$$

At $E=0$, genericity is

$$xy(x+y)\neq0,$$

which excludes all three site-soft loci because $z=-(x+y)$.

For the rank-two elliptic quotient, $m=A/B\to\infty$, and

$$T_{\rm Leg}=-\exp N_0,\qquad \operatorname{rank}N_0=1,\qquad N_0^2=0.$$

The Kummer line has $T_{\rm Kum}=-1$. Hence

$$\boxed{ T=\exp N,\qquad \operatorname{rank}N=1,\qquad N^2=0. }$$

In an integral $I_2$ basis the off-diagonal has magnitude $2$; a normalized complex Frobenius basis may show $1$.

At $E=0$,

$$F_0(t)=(xt^2+y)^2.$$

The raw infinity fiber is type $I_2$, two rational components meeting at two nodes. The whole compactified residue surface obeys

$$\overline K_0=R^2,\qquad R=x\alpha^2+y\beta^2-xy(x+y)s^2,$$

so the central surface is $W=R\cup W=-R$. The marks collide with signed boundaries at $(a,b)=(\pm y,\pm x)$, and

$$R|_{q_{\mathfrak g_3}}=(x+y)(a-y)^2$$

shows a tangency. These facts do not determine full marked-relative nearby cycles. That requires the physical relative/Borel--Moore chain, variance, sheets, orientations, multiplicities, and a semistable/log resolution.

Finally,

$$B=2(x+y)E-E^2$$

has a linear term, while

$$\boxed{ \mathcal Q=-16x^2y^2-8xyE^2+8(x+y)E^3-5E^4. }$$

Thus

$$\mathcal Q_0=-16x^2y^2,\qquad \mathcal Q-\mathcal Q_0=0\cdot E-8xyE^2+O(E^3).$$

The first normal Taylor coefficient vanishes and the second is

$$\boxed{-8xy}.$$

The shorthand $\operatorname{gr}^{(2)}_E\mathcal Q=-8xy$ means the second normal deformation grade after subtracting $\mathcal Q_0$. Literally, the $E$-adic initial form of $\mathcal Q$ is degree zero. This second coefficient does not control the first-order elliptic degeneration created by $B$.

Classification

Existing carrier:

• sourced energy/Cut base and signed-energy arrangement;
• total-energy boundary $E=0$;
• site-soft loci $xyz=0$;
• graph homology $H_1(G)$, kept as rational-integrand topology.

Sector-specific coefficient geometry and data:

• Cayley--Menger residue surface;
• pole divisors and signed minor boundaries;
• $D_\infty$ and the Legendre/Kummer quotient;
• rank-seven algebraic kernel and unresolved extension;
• physical relative/Borel--Moore chain.

Unresolved coefficient-letter datum:

$$\mathcal Q$$

is a printed radicand; its support placement is not proved.

Genuinely new carrier datum:

$$\boxed{\text{none found}.}$$

Graph $H_1(G)$ and elliptic $H^1(D_\infty)$ remain distinct types.

Boundary and prohibited repairs

Do not:

• change $q_{\mathcal G_{12}}=E+y_{12}$;
• conflate the full-eight and active-five sets;
• fit elliptic points to replace the surface divisors;
• promote the formal $M_A$ resultant to source provenance;
• call $\mathcal Q=0$ proved coefficient support;
• infer no intersections from $\mathcal Q\nmid P$;
• assign elliptic Tate ranks to the full relative object;
• identify graph $H_1$ with elliptic $H^1$;
• add carrier cells, support summands, or post hoc gauges.

Entries 150 and 155 remain unchanged. This entry does not compute generic $\mathcal L_1$, companion $P$, the nine-master connection, or the physical relative chain.

Consequence and next falsifier

Two stronger candidates fail narrowly:

$$\boxed{\text{the full coefficient object is not merely a punctured elliptic curve};}$$

$$\boxed{\text{the direct frozen linear marked geometry does not derive a whole } \mathcal Q\text{ divisor}.}$$

H2 survives only as an open refined hypothesis:

$$\boxed{ \text{shared existing energy/Cut carrier} + \text{shared support-sensitive derived/six-functor calculus} + \text{sector-specific surface-level relative/log coefficients}. }$$

The elliptic $H^1$ is the infinity-Gysin quotient, not the full object. The negative factor test is not positive evidence for $\mathcal Q$-support.

The next finite falsifier is:

1. construct the source-defined log/Borel--Moore pair and lifted physical chain for the nine equation-(58) masters;
2. derive its exact relative Gauss--Manin connection;
3. compute local residue and monodromy on one generic transverse $\mathcal Q=0$ slice.

If there is no nontrivial $\mathcal Q$-supported extension or half-character, the remaining marked-relative $\mathcal Q$ hypothesis is falsified. A surviving canonical class must also expose the missing generic $\mathcal L_1$ or companion-$P$ projection.

Evidence

• arXiv:2408.16386v2, equations (6), (7), (13), (14), (51), (52), (57)--(59), and (63);
• research/benincasa/check_marked_relative_q.rs at SHA-256 c9ed2b8ae22ee8e340708dc5c7872eaf57e3395bbcad175112f0e603bc4f85f3;
• Narada runs run-6a344537606246e6bdec13ec1bde7689, run-65877a0dc8244419a27d5543103687ea, run-17c67923c8cb4047b17d04d8020a6a6d, run-66ed86e70f914366bf610121762a3392, run-6cc82ba9a75d41f7b14a67c0739f18fe, run-f26a7ee359b445a1ba903fa7889497fc, and run-c4bd3bfb86aa4a47b3ac902c90e29b78.

Outcome contract

{
  "claim": "On the literal q_G12 Cayley-Menger residue surface, none of the 41 frozen direct linear collision candidates has the full source quartic Q as a divisor factor; separately, the corrected rank-two infinity-Gysin quotient has unipotent generic total-energy monodromy with rank N=1 and N^2=0.",
  "status": "proved_narrow",
  "assumptions": [
    "The arXiv:2408.16386v2 normalization and q_G12=E+y12 are frozen.",
    "The census is limited to eight source lines, twelve reduced signed faces, and twenty-one nonparallel source-line pairs.",
    "The nearby-cycle statement is restricted to the rank-two elliptic quotient on xy(x+y)!=0."
  ],
  "evidence_refs": [
    "arXiv:2408.16386v2 equations (6), (7), (13), (14), (51), (52), (57)-(59), and (63)",
    "research/benincasa/check_marked_relative_q.rs at SHA-256 c9ed2b8ae22ee8e340708dc5c7872eaf57e3395bbcad175112f0e603bc4f85f3",
    "ledger entries 150 and 155",
    "Narada exact-check, nearby-cycle, and hostile-review runs recorded in Evidence"
  ],
  "factorization_test": {
    "full_eight_restricted_lines": "frozen",
    "active_five_poles": "distinguished",
    "signed_CM_faces": 12,
    "nonparallel_source_line_pairs": 21,
    "tested_collision_candidates": 41,
    "Q_divides_any_tested_candidate": false,
    "formal_M_A_resultant": "A Q, without a source-fixed projection",
    "pure_elliptic_monodromy": "unipotent, rank N=1, N^2=0",
    "full_marked_relative_nearby_cycles": "open"
  },
  "counterevidence": [
    "The marks are divisors on a surface, not merely points on the infinity elliptic curve.",
    "The formal M_A resultant is not attached to a printed source denominator or companion P.",
    "The slice test excludes a whole Q factor but not isolated intersections or nonlinear and extension realizations.",
    "The physical relative/Borel-Moore chain and semistable log model are not constructed."
  ],
  "next_experiment": "Construct the source-fixed nine-master log/Borel-Moore pair and physical chain, derive its relative Gauss-Manin connection, and compute its residue and monodromy on a generic transverse Q=0 slice."
}

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