CSM_RH Paper 88

Polynomial-Threshold Inertia, Extremal-Family Sieve-Level Conservation, and a Threshold-Type-II Gate for the Prime-Excess Tail

Project: CSM_RH
Paper: 88
Version: v0.1
Date: 2026-09-09
Canonical root frontier: F-RH-017-v3
One-sided frontier: F-RH-027+
Entry state: v1.78 / Paper 87 v0.1
Status: POLYNOMIAL-THRESHOLD INERTIA CERTIFIED / EXTREMAL-FAMILY LOCAL SIEVE LEVEL SHOWN NOT TO IMPROVE WITH FAMILY SIZE / INERTIA ENTROPY DOES NOT REMOVE X^ν RESOLUTION CLASS / THRESHOLD-SENSITIVE TYPE-II FRONTIER IDENTIFIED
RH_PROVED: FALSE
RH_DISPROVED: FALSE
GLOBAL_RH_CERTIFICATE: FALSE

Abstract

Paper 87 showed that controlling the prime-excess tail

\mathcal E_\nu^+ = \left\{ x: \Delta_H(x)>HX^{-\nu} \right\}

by global factorial moments requires polynomial order

X^\nu\log X.

It proposed a threshold-native alternative:

use inertia to compress the exceptional set into a maximal separated family and sieve only that family.

The present paper audits this strategy.

Let

H=X^{1-\tau}

and

\boxed{ T=HX^{-\nu}. }

The first result is a polynomial-threshold inertia theorem.

For fixed $H$ and $x,y\asymp X$ ,

\Delta_H(x)=\psi(x+H)-\psi(x)-H.

Since

\Lambda(n)\le\log(4X)

on the relevant range,

\boxed{ |\Delta_H(y)-\Delta_H(x)| \le 2(|y-x|+1)\log(4X). }

Therefore, if

\Delta_H(x)>T,

then

\boxed{ \Delta_H(y)>\frac{T}{2} }

whenever

\boxed{ |y-x| \le L := \frac{T}{8\log(4X)} }

for sufficiently large $X$ .

Thus

\boxed{ L=X^{1-\tau-\nu+o(1)}. }

The same statement holds for the deficiency tail.

This extends the power-scale content of the classical inertia property to the polynomially shrinking threshold relevant to F-RH-017-v3.

Bazzanella and Perelli's published fixed-threshold inertia theorem requires

\delta-\delta' \ge \exp(-\sqrt{\log X}),

which does not include

\delta=X^{-\nu}.

The elementary bounded-jump argument above loses only logarithms and is sufficient for the exponent ledger.

Now select a maximal $L$ -separated family of excess centers

\boxed{ \mathcal C=\{x_1,\ldots,x_J\}. }

A maximal-net covering gives

\boxed{ |\mathcal E_\nu^+| \ll JL }

at fixed-power resolution.

Therefore failure of an exceptional bound

|\mathcal E_\nu^+| \ll X^{1-c}

forces

\boxed{ J\gg X^{\tau+\nu-c-o(1)}. }

A greedy extraction of an $H$ -separated subfamily gives at least

\boxed{ K\gg J\frac{L}{H} = JX^{-\nu-o(1)}. }

Hence

\boxed{ K\gg X^{\tau-c-o(1)}. }

This produces an important regime split.

Strip-dominated exceptional mass

d<\tau,

then F-RH-017-v3 only requires

c>d.

One may test a violation at some

d<c<\tau.

Such a violation forces polynomially many $H$ -separated prime-rich intervals.

Lag-dominated exceptional mass

\tau\le d,

then F-RH-017-v3 requires

c>\tau.

A failure of the desired bound need not produce polynomially many $H$ -separated intervals.

It may be supported in only one or a bounded number of $H$ -scale clusters.

Thus an extremal-family method is structurally better suited to the regime

d<\tau.

Combining this with the Poisson-resolution window

d<\nu<\frac{1-\tau}{2}

gives:

Early-bootstrap family window

There exist parameters satisfying

\boxed{ d<\tau }

and

\boxed{ d<\nu<\frac{1-\tau}{2} }

if and only if

\boxed{ d<\frac13. }

Equivalently,

\boxed{ \kappa<\frac23. }

So a separated-family excess strategy can only live in the clean Poisson large-deviation window as an early-to-middle bootstrap mechanism.

The second main result is the family-size cancellation law for local sieve information.

Associate to the centers the multiset sequence

\boxed{ a_{\mathcal C}(n) = \sum_{j=1}^J \mathbf1_{(x_j,x_j+H]}(n). }

Its total mass is

\boxed{ A_{\mathcal C} = JH+O(J). }

For every modulus $q$ ,

\boxed{ A_{\mathcal C,q} := \sum_{q\mid n} a_{\mathcal C}(n) = \frac{JH}{q} + O(J). }

Therefore, for every fixed divisor weight exponent $B$ ,

\boxed{ \sum_{q\le D} \tau(q)^B \left| A_{\mathcal C,q} - \frac{A_{\mathcal C}}{q} \right| \ll_B JD(\log D)^{O_B(1)}. }

Relative to the total mass,

\boxed{ \frac{\text{sieve remainder to level }D}{A_{\mathcal C}} \ll \frac{D}{H}X^{o(1)}. }

The number of exceptional intervals $J$ cancels completely.

Consequently, collecting a large family does not by itself increase the deterministic local-divisibility level beyond the length of one interval.

At prime-detection scale, classical upper-sieve bounds retain a fixed multiplicative gap.

For a single polynomial interval

H=X^\theta, \qquad \theta=1-\tau,

Brun–Titchmarsh gives

\boxed{ \pi(x+H)-\pi(x) \le \left( \frac{2}{\theta}+o(1) \right) \frac{H}{\log X}. }

Summing over the multiset family gives exactly the same constant:

\boxed{ \sum_j \left( \pi(x_j+H)-\pi(x_j) \right) \le \left( \frac{2}{1-\tau}+o(1) \right) \frac{JH}{\log X}. }

Since

X^{-\nu}\to0,

this cannot contradict the near-mean excess

\boxed{ \sum_j \left( \pi(x_j+H)-\pi(x_j) \right) \ge \left( 1+cX^{-\nu} \right) \frac{JH}{\log X}. }

The standard dimension-one linear sieve gives the same structural diagnosis.

If one uses only the deterministic divisor remainder above, the available level is

D\lesssim H.

When the upper linear sieve is in the range

1\le s\le3,

its function is

\boxed{ F(s)=\frac{2e^\gamma}{s}. }

At a prime-scale sifting point

z\asymp X^{1/2},

the parameter is

s=\frac{\log D}{\log z} \lesssim 2(1-\tau).

Thus the classical sieve constant remains bounded away from $1$ .

The family size does not change $s$ .

The third result concerns entropy.

The inertia scale compresses the continuum of possible starts to a grid of size

\boxed{ B_{\rm grid} \asymp \frac{X}{L} = X^{\tau+\nu+o(1)}. }

But the target exceptional estimate

|\mathcal E_\nu^+| \ll X^{1-c}

corresponds to an exceptional fraction

\boxed{ \frac{J}{B_{\rm grid}} \ll X^{-c+o(1)}. }

Thus inertia reduces the number of candidate positions, but not the power of $X$ required in the tail probability.

A representative-wise Chernoff or factorial-moment argument still needs logarithmic cost

c\log X.

At relative deviation

\delta=X^{-\nu},

the order remains

\boxed{ r \asymp \delta^{-1}\log B_{\rm grid} = X^\nu(\tau+\nu+o(1))\log X. }

Hence the polynomial complexity class $X^\nu$ survives inertia compression.

The analysis identifies the only place where a separated family can still help:

Type-II or other genuinely prime-specific cancellation across the selected family.

To make this precise, define the comparison density

\boxed{ \rho_{\mathcal C} = \frac{JH}{X}. }

On the dyadic ambient interval define

\boxed{ w_{\mathcal C}(n) = a_{\mathcal C}(n) - \rho_{\mathcal C} \mathbf1_{[X,2X]}(n). }

At threshold precision

\delta A_{\mathcal C} = JHX^{-\nu},

the aggregate Type-I error condition

JD\ll\delta JH

only reaches

\boxed{ D\ll HX^{-\nu}. }

Thus the threshold-accurate Type-I exponent is

\boxed{ \gamma_{\rm th} = 1-\tau-\nu. }

A power-preserving Vaughan / prime-producing-sieve extraction would therefore require a complementary Type-II range strong enough to bridge the missing width

\boxed{ 1-\gamma_{\rm th} = \tau+\nu. }

Open:

F-RH-028+
EXCEPTIONAL_FAMILY_THRESHOLD_TYPE_II

Given a maximal separated excess family $\mathcal C$ , prove a Type-II estimate for $w_{\mathcal C}$ at relative precision $X^{-\nu}$ over a complementary log-range sufficient to combine with

\gamma_{\rm th}=1-\tau-\nu

in a Vaughan / Heath–Brown / Ford–Maynard prime-producing identity.

A schematic dyadic form is

\boxed{ \left| \sum_{\substack{m\sim M\\n\sim N\\mn\asymp X}} \alpha_m\beta_n w_{\mathcal C}(mn) \right| \ll JH X^{-\nu-\eta} }

for bounded coefficients in the required balanced ranges.

If such a theorem held with the necessary range geometry, then one could obtain

\sum_n\Lambda(n)a_{\mathcal C}(n) = JH + O\left( JHX^{-\nu-\eta'} \right),

contradicting the defining prime excess of every interval in the family.

No such theorem is currently certified.

The key point is that it is now the only family-specific place where $J$ could provide new cancellation.

Current extremal interval-sieve work provides useful calibration.

Banks, Ford and Tao study extremal interval sieves and show that sieve survivor geometry can be highly nontrivial.

Jha's 2026 Poisson-tail work successfully combines extremal interval sieve estimates with concentration inequalities, but under a strong Hardy–Littlewood hypothesis and in a regime where the interval mean grows slower than every fixed power of $\log X$ .

The F-RH regime has

\lambda=X^{1-\tau}/\log X,

and seeks unconditional polynomial relative precision.

Thus those results validate the architecture but do not supply F-RH-028+.

No RH theorem is claimed.

1. Elementary polynomial-threshold inertia

Let

T=HX^{-\nu}.

For integers $x,y\in[X,2X]$ and fixed $H$ ,

\begin{aligned} \Delta_H(y)-\Delta_H(x) &= \psi(y+H)-\psi(x+H) \\ &\quad -\left(\psi(y)-\psi(x)\right). \end{aligned}

Each $\psi$ increment over an interval containing at most $|y-x|+1$ integers is bounded by

(|y-x|+1)\log(4X).

Hence:

Theorem 1.1 — Polynomial-threshold inertia

\boxed{ |\Delta_H(y)-\Delta_H(x)| \le 2(|y-x|+1)\log(4X). }

\Delta_H(x)>T

and

|y-x| \le \frac{T}{8\log(4X)}-1,

then

\boxed{ \Delta_H(y)>\frac{T}{2}. }

The same holds with both signs reversed.

Create:

B-RH-169
POLYNOMIAL_THRESHOLD_PRIME_INTERVAL_EXCEPTIONS_HAVE_INERTIA_LENGTH_H_X_MINUS_NU_UP_TO_LOGARITHMS
CERTIFIED

2. Relation to classical inertia

Bazzanella and Perelli prove that for fixed relative thresholds

0<\delta'<\delta

with

\delta-\delta' \ge e^{-\sqrt{\log X}},

an exceptional point generates an interval of exceptions of length comparable to

(\delta-\delta')H.

For

\delta=X^{-\nu},

the published hypothesis eventually fails.

Theorem 1.1 replaces that input in the present polynomial regime, with only a logarithmic loss.

3. Maximal separated family

Let

L=HX^{-\nu}(\log X)^{-1}.

Choose a maximal $L$ -separated subset

\mathcal C = \{x_1,\ldots,x_J\} \subset \mathcal E_\nu^+.

Maximality implies a covering by $O(L)$ neighborhoods.

Therefore:

Theorem 3.1 — Exceptional mass to center count

\boxed{ |\mathcal E_\nu^+| \ll JL. }

|\mathcal E_\nu^+| \ge X^{1-c},

then

\boxed{ J\gg X^{\tau+\nu-c-o(1)}. }

Create:

B-RH-170
A_FAILURE_OF_THE_ONE_SIDED_EXCEPTIONAL_EXPONENT_FORCES_X_TO_TAU_PLUS_NU_MINUS_C_SEPARATED_EXCESS_CENTERS
CERTIFIED

4. Extraction of disjoint H-scale intervals

Because the centers are $L$ -separated, any interval of length $O(H)$ contains at most

O(H/L) = X^{\nu+o(1)}

centers.

A greedy algorithm therefore selects an $H$ -separated subfamily of size

\boxed{ K\gg J\frac{L}{H}. }

Hence if the exceptional measure is at least $X^{1-c}$ ,

\boxed{ K\gg X^{\tau-c-o(1)}. }

Create:

B-RH-171
EXCEPTIONAL_MASS_X_TO_ONE_MINUS_C_FORCES_X_TO_TAU_MINUS_C_DISJOINT_H_SCALE_EXCESS_INTERVALS_WHEN_C_IS_BELOW_TAU
CERTIFIED

5. Strip-dominated versus lag-dominated family geometry

Recall the root forcing exponent

\min(d,\tau).

If $d<\tau$ , choose $d<c<\tau$ . A failure at exponent $c$ forces polynomially many disjoint intervals.

If $\tau\le d$ , every admissible root exponent satisfies $c>\tau$ and the disjoint-family lower bound becomes subconstant.

Create:

O-RH-192
THE_EXTREMAL_DISJOINT_INTERVAL_FAMILY_MECHANISM_IS_INTRINSICALLY_A_STRIP_DOMINATED_D_LESS_THAN_TAU_TOOL
CERTIFIED

6. Compatibility with the Poisson window

The upper-tail concentration window is

d<\nu<\frac{1-\tau}{2}.

Together with $d<\tau$ , such parameters exist if and only if

d<\frac13.

Equivalently,

\kappa<\frac23.

7. The interval-family multiset

Define

a_{\mathcal C}(n) = \sum_{j=1}^J \mathbf1_{(x_j,x_j+H]}(n).

Then

A_{\mathcal C} = JH+O(J).

For every modulus $q$ ,

A_{\mathcal C,q} = \sum_j \left( \frac{H}{q}+O(1) \right).

Thus:

Theorem 7.1 — Family divisor law

\boxed{ A_{\mathcal C,q} = \frac{JH}{q}+O(J). }

Create:

B-RH-172
A_MULTISET_OF_J_LENGTH_H_INTERVALS_HAS_DIVISOR_REMAINDER_O_J_PER_MODULUS_INDEPENDENT_OF_THE_FAMILY_SIZE
CERTIFIED

8. Aggregate Type-I level

For fixed $B$ ,

\sum_{q\le D} \tau(q)^B \left| A_{\mathcal C,q} - \frac{A_{\mathcal C}}q \right| \ll JD(\log D)^{O_B(1)}.

Since

A_{\mathcal C}\asymp JH,

the relative error is

\boxed{ \frac{D}{H}X^{o(1)}. }

Create:

B-RH-173
THE_DETERMINISTIC_TYPE_I_LEVEL_OF_AN_EXTREMAL_INTERVAL_MULTISET_IS_CONTROLLED_BY_H_NOT_BY_JH
CERTIFIED

9. Brun–Titchmarsh family bound

For

H=X^{1-\tau},

Brun–Titchmarsh gives

\pi(x_j+H)-\pi(x_j) \le \frac{2H}{\log H}(1+o(1)).

Therefore

\boxed{ \sum_j \left( \pi(x_j+H)-\pi(x_j) \right) \le \left( \frac{2}{1-\tau}+o(1) \right) \frac{JH}{\log X}. }

The defining excess is only $1+X^{-\nu}$ , so the classical family upper sieve leaves a constant gap much larger than the target deviation.

Create:

O-RH-193
SUMMING_BRUN_TITCHMARSH_OVER_AN_EXTREMAL_EXCESS_FAMILY_DOES_NOT_APPROACH_THE_ONE_PLUS_X_MINUS_NU_RESOLUTION
CERTIFIED

10. Linear-sieve interpretation

The dimension-one upper linear sieve has

F(s)=\frac{2e^\gamma}{s}

in the initial range.

With deterministic level $D\lesssim H$ and prime sifting scale $z\asymp X^{1/2}$ ,

s\lesssim2(1-\tau).

The family size $J$ does not appear.

This is a method-scope result, not an impossibility theorem for added Type-II information.

11. Inertia grid entropy

The inertia scale is

L=X^{1-\tau-\nu+o(1)}.

Hence

\boxed{ B_{\rm grid} = X/L = X^{\tau+\nu+o(1)}. }

The target center count is

J\ll X^{\tau+\nu-c+o(1)}.

Thus the desired exceptional fraction remains

X^{-c+o(1)}.

12. Resolution order after inertia compression

A near-Poisson representative event with relative deviation $\delta=X^{-\nu}$ still requires order

\delta^{-1} \log(B_{\rm grid}/J).

At the target fraction,

\log(B_{\rm grid}/J)=c\log X+o(\log X).

Therefore:

Theorem 12.1 — Inertia does not change the resolution complexity class

\boxed{ r_{\rm grid}\asymp X^\nu\log X. }

Create:

O-RH-194
INERTIA_COMPRESSES_THE_NUMBER_OF_CANDIDATE_INTERVALS_BUT_DOES_NOT_REMOVE_THE_POLYNOMIAL_X_TO_NU_CONCENTRATION_COMPLEXITY
CERTIFIED

13. Threshold-accurate Type-I level

Prime detection must be accurate to

\delta A_{\mathcal C} = JHX^{-\nu}.

The aggregate Type-I error to level $D$ is

JDX^{o(1)}.

Therefore threshold-small Type I only reaches

\boxed{ D\ll HX^{-\nu-o(1)}. }

Thus

\boxed{ \gamma_{\rm th}=1-\tau-\nu. }

Create:

B-RH-174
AT_RELATIVE_PRECISION_X_MINUS_NU_THE_EXTREMAL_INTERVAL_SEQUENCE_HAS_DETERMINISTIC_TYPE_I_EXPONENT_ONE_MINUS_TAU_MINUS_NU
CERTIFIED

14. Why a Type-II theorem is the only surviving family-specific leverage

A Vaughan / Heath–Brown identity can preserve fixed-power Type-I / Type-II errors.

But threshold-accurate Type I now reaches only

X^{1-\tau-\nu}.

The complementary logarithmic gap is

\boxed{ \tau+\nu. }

Thus any threshold-native prime asymptotic for the selected family must obtain genuinely prime-specific balanced information across a substantial complementary product range.

15. F-RH-028+ — exceptional-family threshold Type II

Let

\rho_{\mathcal C} = \frac{JH}{X}

and

w_{\mathcal C}(n) = a_{\mathcal C}(n) - \rho_{\mathcal C}\mathbf1_{[X,2X]}(n).

Open:

F-RH-028+
EXCEPTIONAL_FAMILY_THRESHOLD_TYPE_II

A schematic target is:

for a sufficiently broad balanced range with $MN\asymp X$ ,

\boxed{ \left| \sum_{m\sim M} \sum_{n\sim N} \alpha_m\beta_n w_{\mathcal C}(mn) \right| \ll JHX^{-\nu-\eta} }

for bounded coefficients, uniformly over every maximal separated excess family $\mathcal C$ .

If this can be combined with the Type-I exponent $1-\tau-\nu$ in a quantitative prime-producing identity, then

\sum_n\Lambda(n)a_{\mathcal C}(n) = JH+O(JHX^{-\nu-\eta'}),

contradicting the defining excess.

No proof is given.

16. Current extremal-interval literature calibration

Bazzanella and Perelli introduced the inertia and decrease properties of the short-interval PNT exceptional set.

Banks, Ford and Tao show that extremal interval-sieve survivor geometry is nontrivial and cannot be replaced by naive independent Bernoulli heuristics.

Jha's 2026 Poisson-tail work combines extremal interval sieve estimates with concentration inequalities under a strong Hardy–Littlewood hypothesis; its interval mean grows subpolynomially in $\log X$ , far below the polynomial mean here.

These works validate the architecture but do not supply F-RH-028+.

17. Strategic verdict for F-RH-027+

The inertia / maximal-family attack does achieve two things:

it turns exceptional measure into the size of a separated structured family;
in the strip-dominated regime it produces polynomially many disjoint prime-rich intervals.

But by itself it does not solve the resolution problem.

The two naive routes retain the old complexity:

LOCAL SIEVE:
family size J cancels from the divisor level.

REPRESENTATIVE-WISE CONCENTRATION:
inertia grid still needs X^nu-order resolution.

Thus F-RH-027+ survives only through a genuinely new balanced / Type-II theorem for the exceptional family.

18. State transition

Advance candidate state

v1.78\to v1.79.

Add:

B-RH-169
POLYNOMIAL_THRESHOLD_PRIME_INTERVAL_EXCEPTIONS_HAVE_INERTIA_LENGTH_H_X_MINUS_NU_UP_TO_LOGARITHMS

B-RH-170
A_FAILURE_OF_THE_ONE_SIDED_EXCEPTIONAL_EXPONENT_FORCES_X_TO_TAU_PLUS_NU_MINUS_C_SEPARATED_EXCESS_CENTERS

B-RH-171
EXCEPTIONAL_MASS_X_TO_ONE_MINUS_C_FORCES_X_TO_TAU_MINUS_C_DISJOINT_H_SCALE_EXCESS_INTERVALS_WHEN_C_IS_BELOW_TAU

B-RH-172
A_MULTISET_OF_J_LENGTH_H_INTERVALS_HAS_DIVISOR_REMAINDER_O_J_PER_MODULUS_INDEPENDENT_OF_THE_FAMILY_SIZE

B-RH-173
THE_DETERMINISTIC_TYPE_I_LEVEL_OF_AN_EXTREMAL_INTERVAL_MULTISET_IS_CONTROLLED_BY_H_NOT_BY_JH

B-RH-174
AT_RELATIVE_PRECISION_X_MINUS_NU_THE_EXTREMAL_INTERVAL_SEQUENCE_HAS_DETERMINISTIC_TYPE_I_EXPONENT_ONE_MINUS_TAU_MINUS_NU

O-RH-192
THE_EXTREMAL_DISJOINT_INTERVAL_FAMILY_MECHANISM_IS_INTRINSICALLY_A_STRIP_DOMINATED_D_LESS_THAN_TAU_TOOL

O-RH-193
SUMMING_BRUN_TITCHMARSH_OVER_AN_EXTREMAL_EXCESS_FAMILY_DOES_NOT_APPROACH_THE_ONE_PLUS_X_MINUS_NU_RESOLUTION

O-RH-194
INERTIA_COMPRESSES_THE_NUMBER_OF_CANDIDATE_INTERVALS_BUT_DOES_NOT_REMOVE_THE_POLYNOMIAL_X_TO_NU_CONCENTRATION_COMPLEXITY

Open:

F-RH-028+
EXCEPTIONAL_FAMILY_THRESHOLD_TYPE_II
OPEN

F-RH-027+ remains open.

No RH certificate is created.

19. Recommended next action

Attack F-RH-028+.

The next paper should derive an unconditional baseline Type-II estimate for arbitrary separated interval families:

a_{\mathcal C}(n) = \sum_j \mathbf1_{x_j<n\le x_j+H}.

The exact question is:

How small can
sum_{m~M,n~N} alpha_m beta_n
(a_C(mn)-JH/X)
be proved using only:

- separation of x_j;
- interval geometry;
- large-sieve / dispersion methods?

Does any saving improve with J?

If the best general bound has no useful $J$ gain, the upper-tail family route reaches a balanced-complexity wall.

If a fixed power of $J$ is gained, there may finally be a genuinely threshold-native upper-tail mechanism.

20. Conclusion

Polynomial-threshold inertia is available.

It creates clusters of the expected power length.

But cluster multiplicity does not improve the deterministic sieve level.

The family-size factor cancels from local divisor distribution, and the inertia grid does not change the $X^\nu$ concentration complexity class.

The only remaining possible gain from a large exceptional family is balanced multiplicative cancellation.

That is now isolated as F-RH-028+.

CSM_RH Paper 88 — Polynomial-Threshold Inertia, Extremal-Family Sieve-Level Conservation, and a Threshold-Type-II Gate f

CSM_RH Paper 88

Polynomial-Threshold Inertia, Extremal-Family Sieve-Level Conservation, and a Threshold-Type-II Gate for the Prime-Excess Tail

Abstract

Strip-dominated exceptional mass

Lag-dominated exceptional mass

Early-bootstrap family window

1. Elementary polynomial-threshold inertia

Theorem 1.1 — Polynomial-threshold inertia

2. Relation to classical inertia

3. Maximal separated family

Theorem 3.1 — Exceptional mass to center count

4. Extraction of disjoint H-scale intervals

5. Strip-dominated versus lag-dominated family geometry

6. Compatibility with the Poisson window

7. The interval-family multiset

Theorem 7.1 — Family divisor law

8. Aggregate Type-I level

9. Brun–Titchmarsh family bound

10. Linear-sieve interpretation

11. Inertia grid entropy

12. Resolution order after inertia compression

Theorem 12.1 — Inertia does not change the resolution complexity class

13. Threshold-accurate Type-I level

14. Why a Type-II theorem is the only surviving family-specific leverage

15. F-RH-028+ — exceptional-family threshold Type II

16. Current extremal-interval literature calibration

17. Strategic verdict for F-RH-027+

18. State transition

19. Recommended next action

20. Conclusion